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EXCHANGE RATE MODELS tr'OR INDIAAII APPRAISAL OF FORECASTING PERTORMANCE

Samrat Bhattacharya

Dissertatlon subfiitted to the University of Dethi in partial fullillmentof the requirements for the award sf the degree of

MASTEN OT PHILOSOTTTV

Department of EconomicsDelhi School of Economics

University of DelhiDelhi - I'10 (m7

India

JuIy, 2000

Dr. Paut llut(Supenristx)

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Delhi School of Eoonomics ' .

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This is to ce,rtiff that this otudy *Exchangs Rate Modets for India : An appaisal offorecasting'Se ance" is based on my original'resemb {ront. My indeb,ted e to

other works/publicatiirns has been duly ackriowldgd lffiin. This study has not bffisuUmiited in part or fuil for ey omrreipU*ra ordegree of,anlr other tmiversity.

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Ackuowledgement

I am grateful to my supervisor Dr.Pami Dua for the enormous patie,lrce aria irrtsest

that she has shown towtrds my dissertatioa. It is only with her hClpfirl guidauce *rat I

could successfully complete thr prese,lrt study.

I arn also &tailkful to my co-sup€rvi$or Dr.Par&a Ssn who helpod rus to bdtei

undsrstand the thesretical foundations of my resesroh wor,k

rout whose help it was not

possible to- conslete the present st$dy.

I axn t}rankful to'Ivk.Amit, Mr.Vinayu,r, ard bfr.Sqis€y for their assisffiroe in ttre

computer lab.

SAMRAT BHATTACHARYA

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TABLE OF CONTENTS

CHAPTER I: Introduction .....1

CHAPTERII: Survey of Literature ..... .........6

Section A. Review of the Theoretical Foundations of the Models of the

Exchange Rate Determination . .. . .:. . .. . ....6

Section B Empirical Survey of Asset Markets Models. ...,:.......:.......16

Section B (I). EstimationResults ..:..... .........16

Section B(II) Out-of-Sample Forecasting Performance.. .....29

CHAPTER III: Econometric Methodology ......42

Section A. Unit Root Tests . ... .. . . ..........42

Section B. Box-Jenkins Methodology.......::.. ......48

Section C. Cointegration Methodology ......50

Section D. Vector Autoregression Methodglogy .....52

CHAPTER IV: Measures of Forecast Evaluation ... .... .. ..61

CHAPTER V : Data Source & Definitions. .. ....73

CHAPTERYI: EmpiricalResults ........:.::..::... ......74

Section A. UnitRootTests. ....:... :..:....:.....:.... ..........75

Section B. Cointegrationlong-Run equilibrium & Vector error Correction

Model ......... ......:...86

Section C. BayesianvAR... ......,......... ........:.....,95

Section. D. ARIMA . .. . :... .....9:9

CHAPTERVII: Conclusion Result. ......124

Bibliography

Appendix

Chapter I

INTRODACTION

The notion of exchange is central to economics. The analysis of

exchange and exchange ratios suggests that there are broadly three kinds of prices .

relative prices, which reflect the exchange of goods for other goods and which exist in

both barter and monetary economies ; money prices, which reflect the exchange of

money for other goods, and exists only in a monetary economy; and thirdly, the

general level of prices, which reflects the average price of all commodities and exists

only in an economy with money. However, there is a fourth kind of price which is of

our interest. This is the price of one money (or medium of exchange) in terms of

another money (or medium of exchange). Hence, rather than exchanging money for

goods or services, money can be exchanged for another money. This price is called

the exchange rate. The exchange rate may be defined as the domestic price of foreign

currency, or as its reciprocal, the foreign price of domestic curency. In this paper, we

employ the former definition.

India followed a pegged exchange rate regime till 1991. The onset of

the economic crisis in l99l brought about a change in the government policy with the

aim of bringing India in line with the world economy. There was a seachange in the

Indian foreign exchange market after the economic liberalisation of 1991. In the pre-

liberalisation era, the basketJinked exchange rate policy regime, with the RBI

performing a market clearing role, provided very limited freedom to the market. The

post-1991 period, however, saw a movement towards a market-determined exchange

rate regime following the recommendations of the High Level Committee on Balance

of Payments (Chairman : Dr.C.Rangarajan,1991). The Report of the Expert Group on

Foreign Exchange Markets in India (Chairman : O.P.Sodhani, 1995) aimed at

integration of domestic foreign exchange market with foreign exchange markets,

more operational freedom to dealing banks and widening and deepening of the

markets. The principle underlying the conduct of the exchange rate policy under the

market based regime is to allow the market forces determine the exchange rate with

the monetary authority ensuring that the exchange rate reflects the fundamentals of

the economy.

In the background of the above discussion, it becomes evident that spot

exchange rate in India is fast becoming a very important market-determined variable.

One needs to better understand the behaviour of the spot exchange rate in the new

open economic environment in India. There is now more need to produce forecasts of

the exchange rate as it affects the economic agents in a far greater way than it used to

do a decade ago. Business houses, in this new environment, need to have 'good'

forecast of the exchange rate so that they can take adequate measures to minimise the

exchange rate related risks. Government also needs a 'good' forecast of exchange rate.

This is more so for an underdeveloped country like India where imports are a major

component of trade. Any major fluctuation in the exchange rate could affect import

(as well as exports) adversely leading to a deteriorating trade balance. This is more so

for the commodities like crude oil. So it will be advantageous for the government to

have 'reliable' and 'good' forecast of the exchange rate so that they can hedge to avoid

any adverse implications. Moreover, given the poor performance of the exchange rate

models, it becomes challenging to an academician to model the exchange rate

dynamics such a way so as to generate 'useful' forecasts.

The most challenging question that a forecaster faces is that " Is

Random Walk the best forecasting model ?". This question has haunted forecasters

since the seminal work of Meese and Rogoff(1983). Lot of research work has got into

this but without much sucsess. The present study is another attempt in this direction.

Here an attempt has been made to model the exchange rate dynamics iir a way so as to

generate accurate, rational and efficient forecast. The present study is the culmination

of the two earlier research work by the same author. Bhattacharya (199S) attempted to

model exchange rate by Box-Jenkins methodology, while another study (1999) tested

various competing models of exchange rate determination. In both these papers,

random walk turned out to be a better performer over other competing models, barring

the univariate ARIMA models, in terms of out-of-sample forecast performance. The

present study attempts to extend the earlier work and attempt to model the spot

exchange rate to generate reliable forecasts.

OBJECTIVES :

We lay out some specific objectives of the proposed study.

Most important and the root of all the heated discussion lies the argument that the

exchange rate follows a random walk. So it becomes utmost important to check

whether exchange rate follows a random walk or not. We work with the US-India

spot exchange rate. We propose to use the traditional unit root tests like

(Augmented Dickey-Fuller) ADF and Phillips-Perron. However, given the

drawback of these traditional and most popular test we intend to employ other

tests for unit root, like KPSS (Kwiatkowski, Phillips, Schmidtand Shin (Tgg2))

test, Bayesian unit root test. For a detailed discussion of these tests the readers are

requested to refer to chapter IfD.

To test the various competing models of exchange rate determination based on

economic fundamental namely moneta"ry models of exchange rate determination

which includes flexible-price monetary model, Dornbusch's sticky price monetary

model, Frankel's real interest differential model and Hooper-Morton model. These

models are the most frequently tested models of exchange rate determination

based on fundamentals of the economy. (For a detailed discussion on these

competing models refer chapter II).

The structural models impose ad-hoc restrictions on the coefficients of the

estimation models. To avoid any arbitrary restriction on the data generating

process, one moves into the realm of atheoretical modelling like Vector

Autoregressive Regression (VAR). The concept of Cointegration helps us to test

the monetary models in a cointegrating framework and leads to the estimation of

vector Error Correction Models (VECM) based on the long-run cointegrating

relationship for generating out-of- sample forecasts.

Economic forecasting involves not only data and statistical/econometric models,

but also the forecaster's personal belief about how the economy behaves and

where it is heading at any moment. So a task of forecaster, in practice, is to blend

data and personal belief according to a subjective procedure. Bayesian Vector

Autoregressions (BVAR) model attempts to blend forecaster's subjective belief

and data in a scientific way (refer chapter III).

All the competing models have been ranked on the basis of their forecasting

performance. A battery of forecast evaluation measures has been carried out to

assess the quality of the forecasts generated from the competing models. We use

the time series properties of the actual and predicted series to evaluate the

forecasting performance of the competing models.(All these tests are discussed in

Chapter IV).

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The present study is organised as follorvs : Chapter II reviews the

literatufe regading the theoretical foundations of the monetary models, their

ernpirical validation and out-of-sample forecasting performance ; Chapter III outlines

the econometric methodology used in the present work; Chapter IV describes the

various forecast Evaluation measures employed in tho pr6$ent study; Chapter V grves

the data source and definition of the variables used in this study; Chapter VI provides

a detailed aualysis of the empirical recults obtained and finally Chaptor YII coastudes

the study, outlining few limitations of the present study.

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Chapter II

SURWY OF LITERATURE

This chapter is organised as follows. Section I gives a brief theoretical

exposition to the various theories of the exchange rate determination. We discuss the

flexible-price monetary model, Dornbusch's sticky price formulation, Frankel's real

interest differential model, and Hooper-Morton model. We have also briefly described

the basics of the portfolio balance model of exchange rate determination. In Section II

the empirical results pertaining to the asset market models of exchange rate

determination have been outlined with Section II.A. giving an account of the

estimation results of the asset market models of exchange rate determination with a

focus on the monetary models. Section II.B briefly outlines the out-of-sample

forecasting performance of the asset market models of exchange rate determination.

A. Asset Market Models : A Theoreticul Exposition

Prior to 1970s the models of exchange rate determination were based

on relative price levels and trade flows. Trade elasticities were thought to underlie the

supply and demand curves in the foreign exchange market. Since the exchange rate

began to float in the 70s, their fluctuations have resembled those of asset market

prices. Rather than following the movement of relative price levels, exchange rate

movements seem to be dominated by monetary conditions. The theoretical literature

has correspondingly turned to the asset market models of exchange rate

determination. The theoretical assumption that all asset-market models share is the

absence of substantial transaction costs, capital control, or other impediments to the

flow of capital between countries. However, beyond this common point, the asset

market models diverge in a number of different and complex routes.

In one class of asset-market models, domestic and foreign bonds are

perfect substitutes. Portfolio shares are infinitely sensitive to expected rates of return.

The uncovered interest parity must hold. However, then bond supplies become

irrelevant, and the exchange rate is determined in the money market. Such models

belong to the 'monetary approach' of exchange rate determination, which focuses on

the demand and supply of money. Within the monetary approach, however, there are

two different models.

In the first type of models I Frenkel (1976),Bilson (1978) ] it is

assumed that prices are perfectly flexible. Consequently, changes in the nominal

interest rate reflect changes in the expected inflation rate. The relative increase in the

domestic interest rate compared to the foreign interest rate implies that the domestic

currency is expected to loose value through inflation and depreciation. Demand for

the domestic currency falls relative to the foreign currency, which causes it to

depreciate instantly. So, we get a positive relationship between exchange rate and

nominal interest differential.

The second strand of models I Dornbusch (1976) ] assumes that prices

are sticky, at least in the short-run. Consequently, changes in the nominal interest rate

reflect changes in the tightness of monetary policy. When the domestic interest rate

rises relative to the foreign rate it is because there has been a contraction in the

domestic money supply relative to the domestic money demand without a matching

fall in prices. The higher interest rate at home attracts a capital inflow, which causes

the domestic currency to appreciate instantly. So, we gd a negative relationship

between the exchange rate and the nominal interest differential. This model is also

coined as the 'overshooting model' as the domestic currency appreciate

instantaneously more than it will in the long-run.

In the other class of the asset-market models, domestic and foreign

bonds are imperfect substitutes. This is the 'Portfolio-Balance approach' of exchange

rate determination, in which asset holders wish to allocate their portfolios in shares

that are well-defined functions of expected rates of return. According to the portfolio-

balance model, the relative quantities of the various assets and of the rate of

accumulation ofthese assets exert profound first-order effects on the exchange rate.

(A) THE FLDilBLE-PNCE MONETARY APPROACH :

Apart from the assumption that uncovered interest parity holds

continuously, the flexible-price monetary model relies on the twin assumption of

continuous purchasing power parity (PPP) and the existence of stable money demand

functions for the domestic and foreign economies.

Monetary equilibrium in the domestic and foreign country, respectively, are given by

and

7n= p+Q y-)"i

m* =p*+0y*-)"i*

(1)

(2)

where m: log of the money supply

p : log of the price level.

y: log of the domestic real income

/ : money demand elasticity with respect to income.

)" : money demand semi-elasticity with respect to interest rate.

( x denotes the foreign variable)

explicitly as being fundamentally relevant for understanding the evolution of the

exchange rate. Rather, the relevant concepts relate to three groups ofvariables : first

are those which are determined by the monetary authorities, second are those which

affect demands for domestic and foreign monies, and third are those which affect the

relative price structures.

The flexible price model has been criticised for its assumptions of

continuous PPP. Under continuous PPP, the real exchange rate,i.e., the exchange rate

adjusted for differences in price levels cannot vary. But, one of the prime

characteristics of the floating exchange rate regime has been the wide gyrations in the

real exchange rates between many of the major currencies. This led to the second

generation of monetary models pioneered by Dornbusch (1976)'

(B) Dornbusch's' Overshoofins' Monetarv Model :

In this model PPP does hold in the long-run, so that a given increase tn

the money supply raises the exchange rate proportionately as in the monetarist model,

but only in the long-run. However, in the short run, because of sticky prices, a

monetary expansion leads to a fall in the interest rate. This leads to capital outflow,

causing the exchange rate to depreciate instantaneously to give rise to the anticipation

of appreciation at just suflicient a rate to offset the reduced domestic interest rate, so

that the uncovered interest rate parity hold. This model thus explains the paradox that

countries with relative high interest rates tend to have currencies whose exchange rate

is expected to depreciate. However, the above analysis is done under the assumption

of full employment so that the real output is fixed. If output, on the contrary, responds

to aggregate demand, the exchange rate and interest rate changes will be dampened.

11

From (l) and (2),we get

The PPP condition is

(m -m *) = (p - p *)+0 0 - y x1 * l(i -i*) (3)

s=p-p*

Using the PPP condition we get the formulation of the monetary model

which has been extensively used in the empirical literature :

s = (m - m*) - Q0-yx) +)"(i-i*) (s)

This says that an increase in the domestic money stock, relative to the foreign money

stock, will lead to a rise in s, that is, in a fall in the value of the domestic currency in

terms of foreign currency (depreciation). An increase in domestic output appreciates

the domestic currency, i.e. a fall in s (/>0). This is because an increase in domestic

real income creates an excess demand for the domestic money stock. As agents try to

increase their (real) money balances, they reduce expenditure and prices fall until

money market equilibrium is achieved. As prices fall, PPP ensures an appreciation of

the domestic currency in terms of foreign culrency. Similarly, a rise in the domestic

interest rate reduces the demand for money and prices increase to maintain the money

market equilibrium and via PPP leads to a depreciation of the domestic culrency

(2>0).

There is another alternative but equivalent way of formulating the flexible

price model by imposing the uncovered interest parity condition on equation (5).

Uncovered interest rate parity implies that

(4)

(6)i-ix = E(As)

Z(As) : the expected depreciation of domestic currency'

s: log ofthe spot exchange rate.

where,

Combining (5) and (6), we get

s - (m - m*)- 0 0 - y*)+zE(As)

If the expectations are assumed to be rational, then by iterating forward, we can obtain

the following' forward looking' solution

s, = (t+2)'Zr|}7'l(*-*.)" ,*,td!",*i*{* y,**" I (8)

The superscript " e " stands for the expectations which are conditioned on information

set at time t. From equation (8) it gets clear that the monetary model, with rational

expectations, involves solving for the expected future path of the " driving variables "

- that is, relative money supply and income.

The assumption that the prices relevant for money market equilibrium

are the same as those relevant for the PPP can be relaxed by allowing for the price

level to be a weighted average of the prices of non-tradable goods and internationally

traded goods I Frenkel and Mussa (1985)]:

p=o px+(1-o) pr

p*=on p** +(1-o*)p*,

where , p, and pt denote the logarithm of the prices of non-tradable and tradable

goods, and o denotes the weight of non-tradable goods in the price index.

PPP holds only for tradable goods, so that

pr=S*p*r

This gives us the monetary model equation of the form

s=(m-m*)+00 - y*)+ )"(i-i *)+of(pr - pr>(P* , * p* *))

[assuming o=o*)

The monetary approach differs from the elasticities approach to

exchange rate determination in that concepts like exports, imports do not appear

(7)

10

Although the exchange rate will still depreciate, it may no longer overshoot, and the

interest rate may actuallY rise.

The overshooting model retains the money demand function and

uncovered interest pality condition of the flexible-price monetary models' Thus,

m= p+g y- )"i (l)

m* =p*+0 y*-Ai* Q)

i-i4= E(As) (3)

However, it replaces the instantaneous PPP condition with a'long run version .

s=p-P

In the short ruq when the exchange rate deviates from its equilibrium path, it is

expected to close the gap with a speed of adjustment 0 :

E(As) = -0(s-s)

The equation of exchange rate determination is given by

s - (m *m *)- Q 0 - Y*)*(1/AX i -i *)

s-(m-m*)-\0-y*)+y (i-i*) lr= -]la

The sign of the coefiicient of the relative income term is same as that of the flexible

price monetary model, Since the prices are perfectly flexible in the long run,

proportionality between money supply and prices holds in the long run which by PPP

(which also holds in the long run) imply a coefftcient of one of the relative money

supply term. However, the difference with the flexible price monetary model arises

when one considers the sign of the interest differential. In case of the flexible price

monetary model the sign of the coeflicient of the interest differential is positive,

whereas in case of the Dornbusch model it is negative (f <0)'

t2

Frankel (1979) argued that a drawback of the Dornbusch (1976)

formulation of the sticky-price monetary model was that it did not allow a role for

differences in secular rates of inflation. He develops a model in that he emphasizes

the role of expectation and rapid adjustment in capital markets. The innovation is that

it combines the Keynesian assumption of sticky prices with the 'flexible-price'

assumption that there are secular rates of inflation.

Equation (4) of the Dornbusch model is replaced by

E(As) = -d(s-F)+(n-r*)

where, fr ,fr * are the current rates of expected long run inflation at home and abroad.

This says that the expected rate of depreciation is a function of the gap between the

current spot rate and an equilibrium rate, and of the expected long run inflation

differential between the domestic and foreign countries. The theory yields an equation

of exchange rate determination in which the spot rate is expressed as a function of the

relative money supply, relative income level, the nominal interest differential (with

sign hypothesized negative), and the expected long run inflation differential (with sign

hypothesized positive).

It will be very fruitful to consider an equation with alternative testable hypotheses:

s - (m - m *) + 0 Q - y *) +a (r - r *)+ B (n - r*)

The alternative testable hypotheses are as follows :

Flexible-price Model

Sticky-price Model

Real Interest Differential Model

'. d>0,0<0,F=0

: a<0,5<0,F=0

: a<0,Q<0,p> 0

13

(c) THE PORTFOLTO BAL-ANCE MOpEL :

The literature on the monetary models of the exchange rate

determination focuses the role of the exchange rate in maintaining continuous

portfolio balance among existing stocks of financial assets. It assigned the exchange

rate no role in balancing the flow of demands and supplies of foreign exchange arising

from trade in goods and capital.

As in the flexible-price and sticky-price monetary models, the level of

the exchange rate in the Portfolio Balance model (PBM) is determined, at least in the

short run, by supply and demand in the markets for financial assets. The exchange

rate, however, is a principal determinant of the current account of the balance of

payments. The PBM is inherently dynamic model of exchange rate adjustment, which

combines the asset market, the current account, the price level, and the rate of asset

accumulation. One of the main feature of the PBM is that it assumes imperfect

substitution between domestic and non-money assets. In addition, the PBM is stock-

flow consistent, in that it allows for current account imbalances to have a feedback

effect on wealth and hence, on long run equilibrium.

We consider a small open economy model due to Branson et.al (1977),

where domestic residents hold domestic money stock, M, which are dominated in

home currency; domestically issued non money assets B ( i.e., domestic bonds ) ; and

foreign-issued non money asset F, which are denominated in foreign exchange. The

current account in the BoP gives the rate of accumulation of F overtime. The total

supplies of the three assets M, B, and F, to domestic holders are given at each point of

time. The rate of return on F is given by ,*, fixed in world capital market, plus the

t4

expected rate of increase in the exchange rate, s . The rate of return on B is the

domestic interest rate r, which is to be determined in the domestic financial market.

The asset market equilibrium conditions are given by

(l) M = m(r,r* + s)W [Money market] (m, <0,m, <0)

(2) B = b(r ,r* + s)W [Home asset market] (0, > 0,b 2 < 0)

(3) sF * -f (, ,r* + s)W [Foreignasset market] (.f , < O, .f , > O)

G) W = M +B+sF I Wealth constraint ]

The assets are assumed to be gross substitutes, so that I r,l ,

I tl *a I frlrln ,l

Thg case where the assets are perfect substitutes is given bV "f ,=b r-)o, in which

case equations (2) and (3) collapses to the uncovered interest rate parity condition :

r=r"+i

and the financial sector of the model collapses to the money market equilibrium

condition. The main implication of the above equations is that the exchange rate is

determined not just by money market conditions, as in the monetary model, but also

by conditions in bond markets.

The novel feature of the portfolio balance model is that it allows for

the dynamic stock-flow interaction between the exchang e rate, the current account

and the level of wealth. For instance, an increase in money supply would be expected

to lead eventually to a rise in domestic prices, but a change in prices will affect the net

exports and hence will have implications for the .current

acmunt of the balance of

payments. This in turn affects the level of wealth which, in adjustment to long-run

equilibrium, feeds back into asset market and hence exchange rate behaviour.

Therefore, the reduced form equation used for estimation purpose is of the form

s = g(m,m" ,b,b* ,CA,CA.)

15

where, b denotes domestic (non-traded) bonds, m denotes the domestic money supply,

and CA denotes the cumulated domestic current account balance ( * stands for the

foreign variables).

However, it is worth to point out that the introduction of the flow

component in the determination of the exchange rate is not unique to the portfolio

balance model. Hooper and Morton (1982), for instance, attempted to incorporate the

flow dynamics in the Dornbusch-Frankel sticky price formulation by allowing

changes in the long-run real exchange rate. These long-run real exchange rate changes

are assumed to be correlated with unanticipated shocks to the trade balances. This

enabled them to introduce the trade balance in the exchange rate determination

equation.

B. Empirical Survev of Asset Market Modek

B.L Estimation Results

Frenkel (1976) tested the flexible price version of the monetary model

for the Deutsche mark - U.S. Dollar exchange rate over the period 1920-23. This

period corresponds to the German hyper-inflation. Frankel argued that during the

period of hyperinflation, domestic monetary impulses will overwhelmingly dominate

the monetary equation, and thus the domestic income and foreign variables can be

dropped. Frankel reported results supportive of the flexible-price model during this

period. His estimated regression equation is

logs = - 5.135 + 0.9751og M + 0.59llogn

(0.731) (0.050) (0.073)

*: 0.994, D.W. : 1.91. (standard effors are given in the brackets)

l6

The elasticity of the exchange rate with respect to the money stock does not differ

significantly from unity (at 95Yo confidence level) while the elasticity of the spot

exchange rate with respect to forward premium(capturing the expected inflation and

hence, expected depreciation) is positive and significant.

Bilson (1978) tested for the Deutsche Mark - Pound Sterling exchange

rate (with forward premium substituted for the expected change in the exchange rate

and without any restrictions on the coefficients on domestic and foreign money) over

the period January 1972 throttgh April 1979. His results were in accordance with the

monetary approach. Putnam and Woodbttry (1979) estimated the monetary model for

the Sterling - Dollar exchange rate over the period 1972-74, and reported that most of

the estimated coeflicients were significantly different from zero at 5Yo significance

level.

Frankel (1979) considered the real interest differential model for the

mark - dollar exchange rate over the period luly 1974 - Feb.1978. He combined both

the features of the flexible price monetary model and sticky price monetary model to

obtain a real interest differential (RD) model. Frankel used long-term bond interest

differential as an instrument for the expected inflation term, on the assumption that

long-term real rates of interest are equalized. He was able to reject both the flexible -

and sticky - price versions of the monetary models in favour of the real interest

differential model. Frankel also tested the possibility that the adjustment in capital

markets to changes in the interest differential is not instantaneous by including lagged

interest differential as a regressor. However, the coeffrcient of the lagged interest

differential is insignificantly less than zero suggesting of the idea that capital is

perfectly mobile.

t7

Driskill (1981) presented an estimate of an equation representative of

the Dornbusch overshooting model for the Swiss franc - U.S. dollar rate for the period

1973-77 (quarterly data) and reported results largely favourable to the sticky-price

model. The novel feature of this p4per is the incorporation of trade balance responses

to relative price changes in the exchange rate equation. The major findings are as

follows : (a) the exchange rate overshoots in the quarter in which a monetary change

takes place by a factor of 2; (b) the exchange rate adjustment path to full equilibrium

is not monotonic but rather exhibits periods of appreciation and depreciation.

However, price adjusts monotonically; and (c) PPP holds in the long-run.

Although monetary models performed reasonably well up to 1978, the

euphoria was short-lived once the sample period is extended. Dornbusch (1980),

HaSmes and Stone (1981), Frankel (1982) and Backus (198a) cast serious doubts

about the ability of monetary models to track the exchange rate in-sample : few

coefficients were correctly signed; the equations had poor explanatory power as

measured by the coefficient of determination; and residual autocorrelation was a

problem.

Dornbusch (1980) estimated the flexible price monetary model for the

dollar-mark exchange rate using a quarterly data for the period 1973:2 to 1979:4. The

explanatory variables are relative nominal money supplies (logarithm of Ml,

seasonally adjusted), relative real income (logarithm of gross national product at 1975

prices, seasonally adjusted), nominal short-term interest differential (yield on money

market instruments) and nominal long-term interest differential (yield on domestic

government bonds). The estimated regression equation for the 1973:2 - 1979:4 period

is

18

s = - 0.03 (m - m. ) -1.05 (y- y. )+0.01 (i-i )+0.04 (i-i )

(1.e0) (2.07)

R2: 0.33 , D.W. : 1.83. (t-statistics are given in the brackets)

The above equation offers little support for the monetary model with

most of the coefficients being insignificant, and the overall explanatory power of the

equation being very low. The model also suffers from a very high serial correlation

problem. Dornbusch points out that this poor performance of the flexible-price

monetary model can be attributed to the breakdown of the PPP in the short run.

Haynes and Stone (1981) estimated it" Frankel's real interest

differential (RID) model for the period luly 1974 to February 1978 (Frankel's original

sarnple) and from July 1974 to April 1980 based on Cochrane-Orcutt procedure.

While for the shorter period, all the coefficients have signs supporting RID model,

the situation changes dramatically for the longer sample period. Not only the

coefficient of determination falls drastically from 0.61 to 0.38, the signs of relative

money supply and relative income are inconsistent with all monetary models.

Especially disturbing to the monetary approach is the fact that the sign on the relative

money is significantly negative rather than positive. This evidence from the longer

sample is similar to estimates of Dornbusch (1980) and Frankel (1982). Frankel called

this phenomenon - the price of mark rising as its supply is increased - the "mystery of

the multiplying mark ".

In response to this apparent collapse of monetary models, Dornbusch

and Frankel each offers modifications. Dornbusch (1980) specifies the current

account as a significant determinant of the exchange rate within both rational

expectations and portfolio balance models. Frankel (1982) extends the money demand

equation underlying the real interest differential model to include a wealth proxy.

(-0.07) (-0.e7)

t9

Empirical evidence based upon data extended beyond February 1978 supports the

modifications. Haynes and Stone (1981) have an alternative explanation for the poor

performance of the monetary models. They point out that all the above equation

specifies each explanatory variable in relative form which restricts that a given

increase in each of the domestic variables to have the same effect on the spot

exchange rate as an equivalent decrease in the foreign variable. Such subtractive

linear constraints are dangerous because linear restrictions, in general, not only yield

biased parameter estimates, but also it can lead to sign reversals when the variables

are positively correlated. Unconstrained estimates show that the model explains the

mark-dollar exchange rate equally well before and after February 1978. Furthermore,

evidence in their study tends to support the Chicago variant, which stresses the

significance of secular rates of inflation, over the RD and the Keynesian special case.

Driskill and Shefkin (1981) argued that the poor performance of the

monetary model could be attributed to the simultaneity bias introduced by having the

expected change in the exchange rate(implicitly) on the right hand side of the

monetary equations. One potential way of correcting this problem is to use the

rational expectations solution of the monetary model. Hoffrnan and Schlagenhauf

(1983), Woo (1984), Finn (1986) implemented a version of the "forward solution"

flexible-price model formulation, and found support for the rational expectations

model.

Hoffrnan and Schlagenhauf (1983) considered the flexible price

monetary model where the exchange rate is considered as the relative price of two

monies, implying that the exchange rate is determined by the relative demands and

supplies of those monies. Assuming a Cagan relative money demand function, the

spot exchange rate equation (in logarithm form) can be written as

20

s t =k -fm, -fr*, *T /*, -q y,- € (i", -i,)

By assuming that uncovered interest parity and rational expectations hypothesis holds,

the spot exchange equation can be written as

,, = +;k * fi*, - fi*., - fi ,, *fi f ,*, * rlr, r,.,The above equation has one unobservable variable E, s ,*, .

By applying a mathematical expectation, they arrive at the following form

,, = k+]-it=l-l't E,lm *i- nt*,*j-!,*j+y*,*;)l+e fi'l+e'

This equation illustrates the point that exchange rates depend upon

current and expected future values of exogenous variables specified by the monetary

model. Thus, changes in the expected value of these variables can result in abrupt

changes in the spot exchange rates. However, the appearance of expected future

values of exogenous variables which are unobservable requires the specification of

the process generating the exogenous variables. Hoffman and Schlagenhauf assumed

a differenced AR(l) specification for all the exogeneous variables :

Lm, = p *Lm ,*t* Fi, Qa)

L** , = p* ^L,m* ,_r+pr, (2b)

Ly , -- p ,Ly,-t*/4, Qc)

Ly"t = p* ,Ly* ,_r*Fo, Qd)

The j-period forecasts from the above AR(l) model was used to replace the

unobservable expected values in the exchange rate determination equation. The

appropriate way of estimating the rational expectations monetary model is to estimate

equations (1) and (2) as a system to account for the implicit cross-equation parameter

21

restrictions. A likelihood ratio test is performed to check the validity of the

restrictions. Hoffinan and Schlagenhauf applied the model to dollar/deutschemark,

dollar/franc and dollar/pound exchange rate. Thre results are for the monthly data

covering the period 1974'.06 to 1979'.12. The likelihood ratio test indicated that the

restrictions implied by REH could not be rejected for any of the currencies

considered.

Woo (1984) reformulates the monetary approach by ascertaining that a

money demand function with a partial adjustment mechanism had more empirical

support than a money demand function mechanism had morye empirical support than a

money demand function which assumed instantaneous stock adjustment. His study

covered the time period 1974.3 to 1981:10 for the dollar mark exchange rate. A

rational expectation hypothesis model was estimated and the restrictions implied by it

could not be rejected. Finn (1986) also considered the simple flexible-price monetary

model and its rational expectations extensions. The US-UK exchange rate over the

period 1974:5-1982.12. l{rrs result confirms to the rational expectations version of the

flexible price monetary model. The test of coeffrcient restrictions 'could not be

rejected (at 5Yo significance level) for the REH version, but was strongly rejected for

the simple version for the monetary model.

Backus (1984), on the other hand, didn't find many statistically

significant coeffrcients for the Dornbusch model. Papell (19S8) argued that the price

and exchange rate dynamics underlying the Dornbusch sticky-price model cannot be

captured by single-equation estimation methods. He reduced the structural model to a

reduced form, vector-autoregressive, moving-average model with nonJinear

constraints. He found support for the Dornbusch model for the period 1973:l to

19844. Barr (1989) empirically implemented a version of the sticky-price model

22

formulated by Buiter and Miller (1981). The model performed satisfactorily, in-

sample.

With the advent of the cointegration methodology there has been a new

fillip to the research in the asset market models of exchange rate determination. The

asset market models are considered as theories of long-run equilibrium. McNown and

Wallace (1989) tested the monetary model of exchange rate determination as a theory

of long run equilibrium. They used the Engle-Granger (1987) two-step cointegration

methodology to test for the presence of co-integration among the variables of the

monetary model. This hypothesis of cointegration has been tested for five

industrialised countries - U.k., Japan, Germany, France, and Canada - the U.S. Test

and estimations employed monthly data covering the period of floating exchange rates

- beginning in April 1973 for all countries except Canada and U.K., which

commenced floating in July 1970, and June T972, respectively, and upto the end of

1986. Their results were generally unfavourable to the monetary approach both in the

case of restricted model (which assumes equality of foreign and base country) and the

unrestricted model. Only the restricted model for France with U.S. as base country

supports the hypothesis of cointegration.

Given the problems with the Engle-Granger test, the above result was

not surprising. MacDonald and Taylor (1991) used the multivariate cointegration

technique proposed by Johansen and Juselius (1990) to test for the long run

relationship between monetary variables and the exchange rate. They also considered

four of the five countries in the McNown and Wallace study, namely, Germany, U.S.,

Japan, and U.K. Their study covers the time period January 1976 to December 1990.

They took Ml as the money supply, income is measured by IIP and interest is long

term rate given in IMF's International Financial Statistics. The money supply and

23

industrial production series are seasonally adjusted. Two interesting results stemmed

from their work. First, the presence of cointegrating vectors provided a valid

explanation of the long-run nominal exchange rate. Two, for the German Mark - US

Dollar rate, a number of popular monetary restrictions cannot be rejected.

MacDonald and Taylor (1993) used the data for the deutsche mark -

U.S. dollar exchange rate over the period January 1976 to December 1990. Their

major empirical findings are as follows. First, the static monetary approach to the

exchange rate determination has got some validity when considered as a long-run

equilibriurh condition. Secondly, when the exchange rate fundamentals suggested by

the monetary model are assumed, the speculative bubble hypothesis is rejected and

thirdly, the full set of rational expectations restrictions imposed by forward looking

monetary model are rejected. However, their testing procedure of the rational

expectation version of the monetary model is different from the earlier tests of the

forward looking models of exchange rate determination (Hoffman and Schlagenhauf

(1980) Woo (1985) and Finn (1986)). MacDonald and Taylor's method of obtaining a

forward looking solution relies on the multivariate cointegration methodology and its

application to present value models. The forward looking solution of the flexible-price

monetary model can be written as

s, = (t +A)'Zrhr' E(*,*,t1,)

where, x,=lm' *f y'f,.

This is the basic equation of the forwardJooking monetary approach to the exchange

rate @MAER). An implication of the present value model of the exchange rate is that

the exchange rate should be cointegrated with forcing variables contained i, 4 .

24

This implies that

Lt= s, -ffi,+m* r+r7y,-0!', u /(0)

Previous researchers (Hoffman and Schlagenhauf (1980) Woo (1985) and

Finn (1986)) implemented the present value exchange rte model in first-difference

form. However, as Engle and Granger (1987) pointed out that if a vector of variables

are cointegrated, then an empirical formulation in first difference misspecifies the data

generating the process. Thus, if the variables are cointegrated, one should follow

Campbell and Shiller (1987) to test the forward restrictions.

We first need to estimate a VAR of lag length p for the vector

g = lA^xr,....,...,A x t_ p+t, L r,.....,L,_ o*rl

Define g' and h' as selection vectors with unity in the (p+l)th and first elements

respectively, so that,

Lr=g/2,

and Lx,=h/r,

(2)

(3)

The multiperiod forecasting formula is given by

Elr,*rlH ,)= Ak z, (4)

where, .F/ , is the restricted information set consisting of current and lagged values of

L, and Ar,.

st - xt = ,i t*l,E(Lx,,tlIt)

Projecting both sides onto .F/, and using (2), (3), and (a) we obtain,

8t z, - ,!, t#l' htAiz,

- ht VIAQ - WA)-' ,,

25

FMAER imposes on the VAR for (L, , M,)' the following 2p linear restrictions :

Ho : gl (I - WA) - h' ryA - A

We can also define the " theoretical spread " as

Li = h' VA(I * ryA)-t zt

Thus, testing the restrictions is tantamount to testing H , . L,=t ,. Manifest

differences in the behaviour of the time series of I, and lr would be indicative of

economically important deviations from the null hypothesis.

Hendrik and Jayanetti (1993) used a black market exchange rate to

allow them to consider a longer time period which may enable cointegration to better

capture the true long-run relationship between exchange and the explanatory variables

as suggested by the monetary approach. They limit their analysis to India, Pakistan

and Sri Lanka and worked with the annual data. They estimated an equation of the

form

so = ao * or,so + az(m - m*) + at(y - y-) + aa(r - r*)where, s, is the black market exchange rate, s o is the offrcial spot exchange rate.

The oflicial exchange rate is included because the black market exchange rate is

dependent on the official exchange rate and the administration of the official market.

If the set of policies and institutions that govern the legal exchange market is stable,

statistical analysis should find that the black market is related to both official

exchange rate (the spillover effect) and the monetary variables (the underlying shifts

in supply and demand) in a stable manner. They found strong support for the

monetary model using Johansen cointegration methodology. For each country the

26

model with no trend was rejected. For India the cointegrating vector, when

normalised on s, is obtained as [ 1, -0.596, -0.967,0.528, -0.054].

Benjamin and Mo (1995) undertook a study based on monthly data for

US-German exchange rate. They tested for the cointegration for all the three versions

of the monetary model : Frenkel-Bilson, Dornbusch-Frankel, and Hooper-Morton. In

all the three models only one cointegrating vector is detected which suggest that the

long-run relationship exists for the US dollar / Deutschmark exchange rate and the

economic fundamentals. Engsted (1996) reexamines the performance of the monetary

models of the exchange rate (MMER) during the German hyperinflation period of

1920s. The purpose of this paper is to derive and test the cointegration implications of

Frenkel's (1976) model. Also, based on the cointegration result, he derives and test

the exact restrictions that the rational expectations imposes on a bivariate monetary

model for exchange rate and money supply. They used the concept of multi-

cointegration to test for the restrictions imposed by the rational expectations version

of the monetary model. Despite the very strong assumptions inherent in the model,

viz, cagan-type money demand function, instantaneous Purchasing Power Parity,

rational expectations, the exact version of the MMER gives a very accurate

description of the deustchemark-sterling exchange rate during the German

hyperinflation period.

Choudhry and Lawler (1997) applies the Johansen cointegration

technique to examine the validity of the monetary model of exchange rate

determination as an explanation of the Canadian dollar - US dollar relationship over

the period of the Canadian float (October 1950 - May 1962). All data are monthly and

seasonally unadjusted. The money stock variable used for both countries is Ml, the

long term interest rate is represented by the long-term government bond rate, and the

27

industrial production is used as a proxy for income. The exchange rate is expressed as

Canadian dollar per US dollar. The ADF tests are applied with monthly seasonal

dummies turning out to be I(1). A single cointegrating vector is identified whose

coefficients conform to the restrictions imposed by the monetary model (only the

proportionality relationship between money supply and exchange rate is getting

rejected) which lends support to the interpretation of the model as describing a long-

run equilibrium relationship. This support is reinforced by the results derived from the

associated error-colrection model, which identify a short-run tendency for the

exchange rate to revert to the equilibrium value defined by the estimated long-run

model.

Diamandis, Georgoutsos and Kouretas (1998) re-examines the

monetary model of exchange rate determination from a long-run perspective using the

monthly data from January 1976 toMay 1994 for the Deutschemark - Dollar, Dollar-

Pound and Yen-Dollar exchange rates. A novel feature of the analysis is the

implementation of the testing procedure suggested by Paruolo (1996) to examine for

the presence of I(2) and I(1) components in a multivariate context. Two cointegrating

vectors are identified for all cases by using the maximum eignvalue test statistic. To

identify the two cointegrating vectors, independent linear restrictions on each vector

was imposed. Some commonly imposed restrictions on the monetary model were used

for one vector while the other was restricted to provide the Uncovered Interest Parity

relationship. However, the Likelihood Ratio test rejected all the jointly imposed

restriction and they were unable to associate the forward-looking version of monetary

model with either vector, conditional upon the other describing the Uncovered Interest

Parrty condition. This outcome may be attributed to the failure of the UIP condition to

hold in the long run. However, the unconditional version of the monetary model to the

28

exchange rate may still be a valid framework for interpreting the long-run movements

of the Deutschmark-dollar, pound-dollar and yen-dollar exchange rate.

B.II. Out - of - Ssmole Forecasting Performance

A model is often judged by its out-of-sample forecasting performance.

A model may have very good in-sample properties like high adjusted squared

correlation coeffrcient, good in-sample forecasts, no serial correlation problem, but it

may perform very poorly in terms of out-of-sample forecasting performance. A model

is finally used to generate out-of -sample forecasts and if it performs poorly in this

respect, then one may have to reduce his/her confidence on the model.

The monetary models - flexible price monetary model, sticky price

monetary model, real interest differential model - performed well in-sample till 1978-

79. These models, when tested on the extended time period covering upto early 80s,

also performed well in-sample, albeit after some modifications (see Dornbusch

(1980), Frankel (1982). However, the picture is totally different when one considers

the out-of-sample forecasting performance of these models.

Meese and Rogoff (1983) compares time series and structural models

of exchange rates on the basis of their out-of-sample forecasting accuracy. Each

competing model is used to generate forecasts at one to twelve month horizons for the

dollar/mark, dollar/yen, and trade-weighted dollar exchange rates. A rolling

regression methodology is adopted to generate dynamic out-of-sample forecasts. In

this methodology, the parameters of each model are estimaed on the basis of the most

up-to-date information available at the time of a given forecasts. The competing

structural monetary models are flexible-price monetary (Frenkel-Bilson) model, the

29

sticky-price monetary (Dornbusch-Frankel) model, and the Hooper-Morton model. A

variety of univariate time series techniques are also applied to the data. They also

considered an unconstrained vector autoregressive (WAR), composed of variables in

equation (I). A convenient normalization for the estimation of the VAR is one in

which the contemporaneous values of each variable is regressed against lagged values

of itself and all other variables. For example, the exchange rate equation is given by

s t : ai1 s .-t+ ........... + a in s t-n * B' t X - r * .......... + B' i, X - n * u'tit

where, Xit = {* rm* rlr! * rfr rT* , rfr' ,fr" ,TBTTB *) .

To reduce overparameteizatron of the VAR, they constrain the domestic and

cumulated trade balances to have same coefficients. The VAR model is important

because it does not restrict any vaiables to be exogenous a priori, and is therefore

robust to estimation problems like simultaneous equation bias, which plagues the

above discussed structural models.

Each model is initially estimated for each exchange rate using

data up through the first forecasting period, November 1976. As mentioned already,

forecasts are generated at one, three, six and twelve month horizons. The purpose of

considering multiple forecast horizons is to see whether the structural models do

better than time series models in the long run, when adjustment due to lags and / or

serially correlated error term has taken place. It is expected that when lags and serial

correlation are fully incorporated into the structural models, a consistently estimated

true structural model will outpredict a time series model at all horizons in large

sample. Out-of-sample forecast accuracy is measured by three statistics - Mean Error

(ME), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE). Table

below gives the RMSE for the US dollar / pound sterling exchange rate at one, six and

30

twelve month horizons over the November 1976 through June 1981 forecasting period

for exchange rates for representative versions of each model.

Root Mean Square Forecast Errors

(in percentage terms)

(Source : Meese and Rogoff (1983))

The above table is a representative of the general results obtained by Meese and

Rogoff. None of the models achieves lower, much less significantly lower, RMSE

than the random walk model at any forecasting horizon. The structural models, in

particular, failed to improve over the random walk model in spite of the fact their

forecasts are based on realised values of the explanatory variables. Allowing for

separate coefficients on domestic and foreign real incomes and money supplies

yielded no gain in out-of-sample forecasting accuracy. They cited several possible

reasons to explain the poor performance of the structural models. They concluded that

there could be problems with respect to the building blocks of the structural exchange

rate models : uncovered interest parity, proxies for inflationary expectations, goods

market specifications, and the common money demand specification.

A lot of research has gone into to refute the conclusions of Meese and

Rogoff. As discussed earlier, Woo (1985) and Finn (1986) estimated the rational

expectations version of the flexible-price model. Woo (1985) argued that a money

demand function with a partial adjustment mechanism had a more empirical support

than a money demand function which assumed instantaneous stock adjustment.

Model Random

Walk

Forward

Rate

Univariate

Autoregression

VAR Frenkel

-Bilson

Dornbusch-

Frankel

Hooper-

Morton

$/pound I Month 2.56 2.67 2.79 s.56 2.82 2.90 3.03

2 Month 6.4s 7.23 7.27 t2.97 8.90 8.88 9.08

3 Month 9.66 11.62 13.35 21.28 t4.62 t3.66 L4.J I

3l

Following Goldfeld (1973), they included a lagged money term in the money demand

specification to capture the partial adjustment in money holdings. The study focused

on mark-dollar monthly exchange rate covering the period T974:3 to 1981:10. He

used the last twenty observations in the sample, that is, 1980:3 to 1981:10, for the out-

of-sample forecast comparison with the random walk model. The result showed that

the rational expectation version of the monetary model outperform the random walk

model at every forecasting horizon under the mean-absolute-error criteria. In terms of

root- mean-square-effor also, the structural model outperform the random walk

model.

Finn (1986) evaluates the forecasting performance of monetary and

random walk models of the exchange rate. Instrumental-variable estimates of the

simple monetary model are not supported by the data, while the full-information-

maximum-likelihood estimates of its rational expectations counte rpart are. Monthly

data is used over the period 1974:5 to 1982:12 for explaining the dollar-sterling

exchange rate. The forecasting period embraces 1980:1 through 1982:12 and the

forecasting performance is evaluated for the rational expectations version of the

monetary and the random walk models. In terms of root-mean-square-effor and mean-

absolute-error, the two models are very closely ranked for the one- and six-month

forecasts - the random walk model performing marginally better at one month

horizon and rational expectations monetary model marginally better at the six-month

horizon. For the twelve month horizon forecasts, both models are closely ranked - the

random walk model performing marginally better on the mean-absolute-error criterion

and better by approximately 1.1 percentage point on the root-mean-square-elror

criterion. In light of the above results, Finn concluded that the rational expectation

monetary models forecasts as well as the random walk model.

32

Schinasi and Swamy (1989) reexamines the forecasting performance of

models reported by the Meese and Rogoff (MR) without imposing the restriction that

the regression slopes are fixed over time. Major result of their study is that when all

coefficients are allowed to vary, the conventional models of exchange rates employed

by MR yielded more accurate forecasts than their fixed coefficient counterparts and

more accurate than the random walk models. The study, however, supported most of

the MR conclusions regarding fixed coefficient models, but contrary to the MR study,

Schinasi and Swamy found significant improvements in the forecasts of fixed

coefficient models that include lagged adjustment. The structural models estimated by

MR and used for forecasting nominal spot exchange rates (do11ar-mark, dollar-yen,

dollar-pound) is given below:

s, = Bo+ Br(mt- m.t)+ Bz(yt-y.)* Bz(r,-r.,)+ p+(tTt" -v,". )+ Bs(TBt-T?.,)+u,

Schinasi and Swamy argued that the sequential estimation of fixed

coefficient regressions (that is,'rolling regression') is not the appropriate technique for

capturing the variations in coefficients overtime. At the high level of aggregation of

exchange markets, there is little reason to believe that behavioural parameters are

fixed. There is a wide diversity of participants in foreign exchange rate markets with

relatively small and highly variable market shares. Even if each participant reacted to

macroeconomic developments according to a stable fixed coefficient reaction

function, it is difficult to argue that macroeconomic variables would be related to

exchange rates by a simple fixed coefficient relationship, without also assuming that

individual reaction functions are identical.

We briefly present the stochastic coefficient representation of the

exchange rate models :

yt=X'tB

p,= B+a

&=Q tt*vt

E(vt) = Q

E(vtv") = Ao if Fs and 0, otherwise.

where, xt,Bt,B,tt,vt are all (kxl) vectors, (D and A,a a.rg (kxk)mafrcos, .rr

represents the vector of the explanatory variables in (1), and yt is the natural

logarithm of the spot exchange rate. In (3), each coefficient in each period, B i, , has

two components : a time-independent fixed coefficient, B, , and a time-dependent

stochastic component, 6;r.

Combining (2), (3) and (4a), we can view the stochastic coefficient representation as a

fixed coefficient model with effors that are both serially correlated and

heteroscedastic, where the form of serial correlation and heteroscedasticity is very

general :

yt = X't*I,lt

l,lt=X'ttt

tt=@tt-t*lt

They estimated all the competing models from March 1973 through March 1980 and

generated the multistep-ahead forecasts for the period April 1980 through June 1981.

The competing models include, apart from the stochastic coefficient model, fixed

coefficient model, and random walk model. The stochastic coefficient model turn out

34

to produce superior forecasts than the fixed coefficient model and also outperformed

the random walk model. They also generated multistep-ahead forecasts of the Box-

Jenkins (1970) type time series models, ARIMA(I,1,0), ARIMA(O,1,1) and

ARIMA(1,1,1) and found them to be inferior to the multi-step ahead forecasts of the

random walk model with or without drift.

As already discussed, with the advent of the technique of cointegration

and VAR there has been a new lease of life to this topic. Many of these papers

exploited the long-run and short-run properties of the monetary models to generate

out-of-sample forecast that outperformed the random walk in terms of RMSE and

MAE. MacDonald and Taylor (1993) used a vector effor correction model to generate

out-of-sample forecasts that are superior to those generated by a random walk

forecasting model. They found for the deutschemark-dollar exchange rate existence of

a cointegration relation that corresponds to the static monetary approach exchange

rate equation. Thus, the monetary model can be interpreted as having at least long-run

validity. According to the Granger representation theorem, if a cointegration

relationship exists among a set of I(1) series then a dynamic error-conection of the

data also exists. So they estimated the error-correction model for the initial period

1976:l to 1988:12 and reserved the last 24 datapoints, corresponding to the period

1989:01 through 1990:12 for post-sample forecasting performance. They performed a

d5mamic forecasting exercise for a number of forecasting horizons. The dynamic

elror-correction model outperforms the random walk model at every forecast horizon

as shown in the table below :

35

Dynamic Forecast Statistics

approx y equa

percentage differences divided by 100.

(Source : MacDonald and Taylor (1993)).

This shows that imposing the monetary model as a long-run equilibrium condition on

a dynamic, error-colrection model led to dynamic exchange rate forecasts" at every

forecast horizon considered.

Hoque and Latif (1993) compared the forecasting performance of

unrestricted VAR model, a Bayesian VAR model, a structural model and an error

correction version of the structural model. The purpose was to obtain the best

forecasts for the Australian dollar vis-d-vis the US dollar in terms of root mean square

error (RMSE) of forecasts. For estimation quarterly data from 1976Q1 to 1990Q1 has

been used. The period 1990Q2 to 1991Q1 has been used for ex post prediction. Five

variables chosen for VAR / BVAR system included exchange rate, current account

balance, three month forward rate, relative long-term interest rate, and relative price

level. The BVAR model was estimated with several degrees of tightness (2), decay

(d) andweights (w) as follows: )":0.I,0.25,0.3; d:1,2;w:0.01,0.15. A

structural model (due to Wallis(1989)) was also constructed to compare its forecasting

Horizon(Months)

RMSE fromError- Correction Model

RMSE from RandomWalkModel

t2 0.131 0.148

9 0.103 O.TI2

6 0.081 0.088

J 0.043 0.0s3

2 0.032 0.040

1 0.028 0.030

Note : Figures are logarithmic differences and are therefo re approximatelv equal to

36

perfoflnance with the multivariate time series model. It was found that the structural

model performs best in terms of RMSE than either of the two time-series models.

BVAR model performs better than the unreskicted VAR, but not as well as the

structural model. An attempt was also made to improve the forecasting performance

of the structural model by considering the time-series properties of the variables

involved in the exchange rate equation. The cointegration property among the

variables have been exploited to generate forecasts from an eror-coffection model.

The error-correction version of the structural model displayed a better forecasting

performance compared to the simple structural model.

Liu, Gerlow and Irwin (1994) analyses the forecasting accuracy of fuIl

vector autoregressive (FVAR), mixed vector autoregressive (MVAR) and Bayesian

vector autoregressive (BVAR) models of the US dollar lYen, US dollar I Canadiart

dollar, and US dollar / Deutsche mark exchange rates. The VAR models are based on

the theoretical model of monetary / asset exchange rate determination developed by

Driskill et.al (1992). The models are estimated over the in-sample period L973:3 -

1982:12. For the out-of-sample period covering 1983:1 through 1989:12, l-, 3-, 6-,

and l2-month forecasts are generated. Performance criteria include bias tests,

informational content tests, and market timing ability tests. The variables included in

the model are logarithm of the exchange rate, logarithm of the relative real income,

the logarithm of relative price levels, the interest rate differential and the trade balance

between the two countries. They employed the Litterman's prior to estimate the

Bayesian VAR.

In terms of bias tests, BVAR model's performance is better than FVAR

and MVAR. To determine if the forecasts generated from the alternative VAR models

contain additional information beyond a random walk process informational content

37

test developed by Fair and Shiller (1989, 1990) was employed. In the case of the US

dollar I Yen exchange rate, at forecast horizons of 1- through 6-months, forecasts

generated by the FVAR, MVAR, and BVAR models did not contain additional

information beyond that produced by the random walk forecasts. However, the BVAR

model dominated the FVAR and MVAR at the l2-month forecast horizon, and it

contained additional information not generally not found in a random walk model. In

terms of market timing ability test, the forecasts generated from the FVAR models,

for all three exchange rates, have no significant market timing value. In other words,

the FVAR model is not capable of significant predictions of the directional movement

of the exchange rates. MVAR forecasts exhibit significant market timing ability

across all forecast horizons for the US dollar I Canadian dollar exchange rate.

Forecasts generated from a BVAR model also have significant market timing value

for the US dollar I Catadian dollar rates and US dollar lYen across 3-to 12-months

forecasts horizons. Thus, out-of-sample forecasting performance indicate that the

forecasting performance of restricted VARs (MVAR and BVAR) is substantially

better than that of the unrestricted VARs (FVAR).

Benjamin and Mo (1995) used the multivariate cointegrating

methodology to generate long-run forecasts of the US dollar / Deutschmark exchange

rate.' They used three competing structural models - Frenkel-Bilson, Dombusch-

Frankel and Hooper-Morton - and in all the three models, only one cointegrating

vector was detected, suggesting that the long-run cointegrating relationship exists

between the exchange rate and economic fundamentals. They initially estimated the

models for the period 1973:04 - 1988:07 and out-of-sample forecasts were obtained

by using the rolling regression for the period 1988:8 - I993:l (60 months). The

forecasts were evaluated in terms of RMSE and MAE statistics. The forecasts were

38

generated using the error-correction versions of the three models. These forecasts

from the structural model clearly outperformed the random walk model at every step.

This finding was especially significant in that the multistep-ahead forecasts of the

structural models outperformed even the one-step-ahead forecasts of the random walk

model.

Chinn and Meese (1995) examined the predictive performance of the

standard structural exchange rate models using both parametric and nonparametric

techniques. They examined four bilateral rates (Canada, Germany, Japan, and the

U.K.) relative to the US dollar, using monthly data for the period 1973:03 - 1990:11.

They argued that the post-Bretton Woods era was too short to extract reliable

estimates of he long-run elasticities by direct estimation (either by Engle-Granger

methodology or by Johansen and Juselius). So they impose a set of coefficient

restrictions for each of the candidate models and used them to generate the error-

correction term.

Chinn and Meese estimated the monetary models by OLS and

instrumental variables (IV) procedures in unconstrained first differences, and error

correction models. At one month forecast horizon, the structural models procedure

discouraging result compared to the naiVe random walk model (with or without drift)

in terms of RMSE statistic. The predictive performance of the structural models

shows some improvements at horizons beyond one-month period. Error correction

models with an elror correction term lagged once do not, in general, produce the best

RMSE at the yearly predictive horizons.

Bhawani and Kadiyala (1997) employed the black market data for

exchange rates in developing countries to investigate the forecast performance of

several exchange rate models. Unlike other studies which used the ' actual ' realized

39

values of the exogenous variables, this study employed expected future values of the

exogenous variables (predicted outside the model). As representatives of the structural

monetary models they estimated the reduced form equation for Bilson-Frenkel

flexible price model and sticky price Dornbusch model. They also estimated an effor-

correction version of the structural models.

The models have been used to generate forecasts at one, three, six and

twelve month forecast horizon for the Indian rupee / US dollar, Mexican peso / US

dollar, and Pakistan rupee / US dollar bilateral spot exchange rates. Forecasts for all

models are based on rolling regressions. Initial sample period for India covers the

period 1913-89, for Mexico it covers 1982-89 and for Pakistan it covers 1978-88. The

numberof 1-, 3-,6-,12-monthaheadforecastsequal 36, 34,31 and25respectively.

Three criterion are used for forecast evaluation - ME, MAE, RMSE. The error-

correction version of the Bilson-Frenkel model outperformed the simple random walk

model at all forecast horizon for the Indian rupee / US dollar exchange rate. While in

case of Pakistan the effor-coffection model outperforms the random walk model at all

forecast horizon, except the one-month forecast horizon, for Mexico peso / US dollar

the simple random walk model exhibits the least forecast error at all horizons,

followed by the effor-coffection model.

Choudhry and Lawler (1997) estimated an eror-correction version of

the monetary model for the Canadian dollar - US dollar exchange rate over the period

of the Canadian float 1950-62. To test the adequacy of the monetary error-correction

model a forecasting exercise was carried out. Forecasts are generated for three, six,

nine, and twelve month forecasting horizon over the period June 1961 to May 1962.

For the comparison purpose forecasts were also made with two alternative models - a

simple random walk model and a random walk model with drift. RMSE statistic has

40

been used for evaluating the forecasting performance. The error-corection model

outperforms both the random walk models across the range of forecast horizons.

Reyiew of the Survey of Literature

soon after the breakdown of the Bretton Woods Agreement. More specifically, the

monetary models performed well in the years 1975-T980. (Frankel (1976), Bilson

(1 978), Frankel (197 9).

coefficients, persistent serial correlation problem in the 1980s, following which

few modifications to the original structure were put forward. (Dornbusch (1980),

Frenkel (1982), Haynes and Stone (1981), Driskill (1981).

Meese and Rogoff (1983). Given the general poor performance of the monetary

models vis-a-vis the random walk forecasts, a lot of research has gone into

refuting the negative conclusions reached by Meese and Rogoff (1983).

a long-run equilibrium phenomenon. (MacDonald & Taylor (1991, 1993, 1994),

Choudhry and Lawler (1997), Diamandis, Geougoutsos & Kouretas (1993)).

autoregression formulation of the monetary models of the exchange rate

determination produced forecasts which beats the random walk forecasts, mostly

in the developed country context. (MacDonald & Taylor (1993, 1994), Liu,

Gerlow and Irwin (1994), Bhawani & Kadiyala (1997), chaudhry & Lawler

(1ee7)).

41

Chapter III

E CONOME TRIC METHOD OLOGY

The present chapter discusses the econometric methodology used to

generate forecasts of the exchange rate, while the next chapter takes up the discussion

of various forecast evaluation measures. Starting point of any time series analysis

involves checking of the stationarity properties of the series under study. This is

achieved by the traditional unit root tests of (Augmented) Dickey-Fuller test (ADF),

Phillips-Perron (PP) test as well as two other test of unit root - KPSS Test, which is

essentially a non-parametric tests and Bayesian unit root test. Various unit root tests

are performed to get a clear picture about the presence of stochastic trend in the data.

These tests are described in more detail in section A. Section B discusses the

univariate modeling motivated by the Box-Jenkins three step methodology. Section C

takes up the discussion of the concept of cointegration among the variables which

helps us to impose economic structure on the variables under consideration. Section D

discusses the atheoretical modeling involving vector autoregression (VAR) models as

proposed by Sims (1980). This includes full VAR (FVAR), vector error correction

model (VECM) and Bayesian VAR (BVAR).

Section A : Unit Root Tests

Unit root hypothesis has received much attention in the economic and

econometric literature since the seminal work of Nelson and Plossar (1982). A non-

stationary series has the following properties :

42

(a) There is no long-run mean to which the series returns.

(b) The variance is time-dependent and goes to infinity as time approaches infinity.

Unit root becomes important in the context of spurious regression involving several

non-stationary varalbles as proposed by Granger and Newbold (1974). A spurious

regression has high R', t-statistics that appear to be signifrcarfi, but the results are

without any economic meaning. So it is very important to find out the stochastic

properties of . the variables under study so that we do not ran into such spurious

regression problems.

In the first modern attempt to test for the unit roots, Nelson and Plossar

(1982) tested 14 historical macroeconomic time series for the US by the Augmented

Dickey-Fuller (ADF) test. They analyzed the logarithms of all of these series (except

for the interest rates, which was treated in levels) and found empirical evidence to

support a unit root for 13 of them (exception being the unemployment rate). Meese

and Singleton (1982) studied various exchange rate time series and could not reject

the null hypothesis of a unit root. In the present thesis, we work out the sequential

testing procedure of the (augmented) Dickey-Fuller test as opposed to the simple

Dickey-Fuller test performed by the above mentioned studies.

(i) (Auementedt Dickey-Fuller Test :

Dickey-Fuller consider three different equations that can be used to test

the null of unit root :

Ly,

Ly, *Et

L)tri+t I Et

(1)

p

= aU*r + \fiLyni+t * tt

p

= ao * atlt-t + l_, BiLyn+r

= ao + arlrt + azt + f,^ fri

(2)

(3)Ly,

43

Here the nullof the unit root is given by

i}t :0.

The first equation is a pure random walk model, the second add an intercept or drift

term, and the third includes both a drift and linear time trend.

Enders (1995) suggests a sequential testing procedure to test for the

presence of a unit root when the form of the true data generating process (DGP) is

unknown. The motivation of doing this sequential procedure can be traced to Cambell

and Perron (1991). They pointed out that the major problem with the Dickey-Fuller

(DF) test is that tests for a unit root is conditional on the presence of the deterministic

regressors and tests for the presence of the deterministic regressors are conditional on

the presence of a unit root. This follows from the fact that if the estimated regression

includes deterministic regressors that are not in the actual DGP, the power of the unit

root test against a stationary alternative decreases as additional deterministic

regressors are included. Furthermore, if the estimated regression omits a deterministic

trending variable present in the true DGP, such as azt, the power of the t-test goes to

zero as the sample size increases. This necessitates the following sequential testing

procedure ofunit root.

Step 1. Estimate the most general model (eq.(3)) and test the null of unit root

(ar : 0) by € s statistic. If the null is rejected, conclude that unit root is not present in

the series and stop the sequential procedure. If the null hypothesis is not rejected we

proceed to step 2.

Step 2. At this stage we determine whether trend term is needed to be included

in the eq.(3). This is achieved by testing the significance of the trend term under the

null of unit root by using e B, statistic or Q. statistic. If the trend is not significant

we proceed to step 3. On the other hand, if the trend is significant we go back to step

44

1 and retest for the presence of unit root using the standardized normal distribution. If

the null of unit root is rejected stop the sequential procedure and conclude that the

series is stationary. If not, then conclude that unit root is present and ptoceed to

step 3.

Step 3. Estimate equation (2), that is, one without a trend but a drift (constant)

term and test the presence of unit root by using the g ustatistic. If the null of unit root

is rejected conclude that the series does not contain a unit root and stop the sequential

procedure. Ifnot, then proceed to step 4.

Step 4. Here we determine whether a drift term is needed to be included in our

regression model. This is done by using 6 ,, statisti c or Q , statistic. If the drift term

is not significant proceed to step 5. If, on the other hand, it is significant, we return to

step 3 and retest the null of unit root by using the standard normal distribution.

Rejection of null of unit root will lead us to abandon the sequential procedure and

conclude that the series does not contain a unit root. Non-rejection of the null will

lead us to step 5.

Step 5. Finally, we estimate the simple model, that is, one without a drift or

trend term (eq.(l)).We use the g statistic to test for the presence of unit root. If the

null is rejected, we conclude that the series does not contain a unit root. Otherwise, we

conclude that it contains a unit root.

(ii) Phillips-Peruon Test :

The distribution theory supporting the DF test assumes that the errors

are statistically independent and have a constant variance. Phillips-Perron developed a

test that allows that disturbances to be weakly dependent and heterogeneously

distributed.

45

They considered the following regression :

!r=a o*a r!,q*ll,

lt = do * drla + d2(t - Tl2) + 1t1

where T : No. of observations.

The most useful test-statistics are as follows :

Z (t a. ),: Use to test the null hypothesis a r =l

Z(td): Use to test the null hypothesis dr = 1.

Between the Dickey-Fuller and Phillips-Perron test, the later is

preferred because it has better power. This implies that if the null of unit root is not

rejected by the DF test but rejected by the PP test, then we rely on the PP test and

conclude that the series does not contain a unit root.

(iii) KPSS Test:

A feature of the Dickey-Fuller and Phillips-Penon tests is that they

make the unit root the null hypothesis and given the low power of the former test it is

very difficult to reject the null of unit root. KPSS, therefore, argue that in trying to

distinguish between stochastic and deterministic trends, it is natural to consider both

the null of trend stationary and difference stationary. They developed a test of unit

root where null hypothesis is taken to be the absence of unit root. This is essentially a

non-parametric test.

Let Y1 be a sample of T observations. KPSS assumes that the series

can be decomposed into the sum of a deterministic trend, a random walk, and a

stationary elTor :

Yt = (.t *rt*tt

Here 11 follows a random walk.

(1)

(2)

46

Let e1 , t:1,2,....T, be the residuals from regression of !, on an intercept and time

trend. Let 62 " be the estimate of the error variance from this regression (the sum of

squared residuals divided by T). Define the partial sum process of the residuals :

ts' = ,E,

u' ,r: lrzr.....rT.

Then, the LM statistic is given by

LM =\5, ,I 6, ,i=1

KPSS uses an estimator of sample error variance (6', ) of the form

TIT

^s21/1 = T-'Zu' ,+27 -1!w(s,/) +\e,e,-,

Y" w(s,l) is an optional lag window that corresponds to the choice of a spectral

window. KPSS uses the Bartlett window, w(s,l)=t-t/*1 , which guarantees the

non-negativity of the estimated sample variance. The lag parameter / is set to correct

for residual serial correlation.

In event of testing the null hypothesis of level stationarity instead of

trend stationarity, we define et as the residual from the regression of y on intercept

only (i.e. € t = !,-y), the rest of the test statistic is unaltered. The test is an upper tail

test. The critical values are given in KPSS (1992).

(iv) The Sims-Bayesian Unit Root Test :

Sims (1988) argues that conventional tests for the presence of unit

roots, such as DF tests, are fundamentally flawed. The relevant question should be

how probable is null of unit root relative to the other competing hypotheses. The

classical econometricians cannot give the probability that a hypothesis holds. What

47

they can tell us is whether a hypothesis is rejected or not rejected (Koops, 1992).

Further, while the classical inference is sharply affected by the presence of a trend and

drift term, the Bayesian flat prior theory is not.

Consider the following autoregression model :

lt=Pit-t*8t

The test statistic is the square of the conventional rstatistics for P = t. This is

compared to the Schwarz criterion which has an asymptotic Bayesian justification.

This is approximately given by

r = 2log (#) - bg (oil + 2tog(t - 2 tts1

where, o' o= , o' is the variance of a , and for monthly data s: 12.Z v,G *r)'

" Alpha" gives the prior probability on the stationary part of the prior; the remaining

probability is concentrated on p = 1 If t2 > t we reject the null hypothesis of unit

root.

Section B : Box-lenkins Methodology

Box and Jenkins (1976) popularized a three-stage method aimed at

selecting an appropriate model for the purpose of estimating and forecasting a

univariate time series. This is found to generate forecast which is as good as, if not

better than the multivariate models. The advantage of autoregressive integrated

moving average (ARIMA) models is the relatively little information set used in

estimation and forecasting - only the .lagged values of the dependent variable and

emor term are required. The ARIMA (p,d,q), where p is the order of the

autoregressive process, q is the order of the moving average process and d is the

degree of integration

o'

48

o@)0-B)" , = O(B)u ,+6 ,

where, the polSmomials in the backward operator (B) are given as follows :

Q(B) = r-Q, B -.......-Q o B'

e@) =l-0 tB--.-...-o ,B'

In the identification stage, we visually examine the time plot of the

series, autocorrelation function, and partial autocorrelation function. We use the tests

of stationarity - (Augmented) Dickey-Fuller test and Phillips-Perron test - to check

formally whether the series contain a unit root or not. If, from the above tests, we

conclude that the level is non-stationary we then work with the difference of the

series. The number of differencing to be done depends on whether the differenced

series is stationary or not.

In the estimation stage, we fit various plausible models and

significance of their coefficients are examined. Our aim is to choose the most

parsimonious model with no serial correlation in the eror term. The two most

commonly used model selection criteria are the Akaike Information Criterion (AIC)

and Schwartz Bayesian Criterion (SBC) :

AIC = T ln(residual sum of squares) + 2n

SBC : Tln(residual sum of squares)+nln(T)

where, n: number of parameters estimated (p+q+possible constant term)

T: number of usable observations.

Ideally, the AIC and SBC should be as small possible. Of the two criteria, the SBC

has superior large sample properties. The serial correlation test is performed by using

the Lung-Box-Pierce Q statistic

LQ=n(n+rZ*

49

This follows a chi-square distribution under the null hypothesis of no serial

correlation in the error term.

' The third state in the Box-Jenkins methodology involves diagnostic

checking. Here, we primarily see whether the residuals from an estimated model are

serially uncorrelated. Any evidence of serial correlation implies a systematic

movement in the series that is not accounted for by the ARMA coefficients included

in the model. The Ljung-Box Q statistics of the residuals are used to test the presence

ofserial autocorrelation in the residuals.

Finally, we can use the selected model to obtain the forecast of the

series.

Section C : Cointesrstion Methodolow

Granger and Newbold (1974) warned of a serious empirical

consequence of estimating models with non-stationary variables. When both the

dependent variable and the explanatory variables in a time-series regression are non-

stationary, spurious correlations are likely to occur: variables appear to be significant

when in fact they are not. The symptoms of this spurious correlation include a high R2

combined with a low Durbin-Watson statistic. These synrptoms are familiar features

of estimated exchange rate models [Boothe and Glassman (1987)]. This led to the

concept of co-integration as proposed by Engle-Granger (1987) to test for the

presence of spurious regressions. The test of co-integration allows us to find out

whether the non-stationary variables have any meaningful relation among them.

50

(i) Engle-Graneer Two Steo Methodolosy (EG) :

Suppose there are two variables Y1 and21, both being I(r). The long-

run equilibrium relationship is estimated by

Yt=00+8121+e1

The estimated residuals are saved. If the deviations from long-run equilibrium are

found to be stationary, the Yl and 21 series are cointegrated of the order (1,1). So in the

second stage, we perform the unit root tests of DF/ADF and PP to check the

stationarity of the estimated residuals.

There are quite a few problems with Engle-Granger procedure of

testing the presence of Cointegration among the variable. One major problem with the

two-step procedure is that the estimation of the long-run equilibrium regression

requires that the researcher place one variable on the left-hand side and use the others

as regressors. So, in practice, it is possible to find that one regression indicated the

variables are cointegrated, whereas reversing the order indicates no cointegration.

Moreover, it is not possible to find the presence of multiple cointegrating vector

among the variables. For all these drawbacks, it is suggested to test for the presence of

cointegration among the variables in a multi-variate framework by using Johansen-

Juselius Maximum likelihood method (JD (1990).

(ii) Johanseru-luselius Cointesration Test :

Consider the p-dimensional vector autoregressive model with

Gaussian errors

! , = At! ,_t*......+A *! rtc+Y.D+e ,

where .y, is a pxl vector of stochastic variables, D is a vector of nonstochastic

variables, such as seasonal dummies or time trend. Johansen test assumes that the

51

variables in y , are I(l). For testing the hypothesis of cointegration the model is

reformulated in the error-corection form

Ly , = f , A.y,-, *.......tf o-, L! ,-o*r+lIy ,-r+ p+y .D*, ,

The hypothesis of cointegration is formulated as a reduced rank of the lI -matrix

H ,:fI - o0'

where a and p are (pxr) matrices of full rank. The null hypothesis implies that the

process Ay, is stationary, "y, is nonstationary, but B'y, is stationary relations

among nonstationary variables. This is basically an eror-correction formulation

which allows for the inclusion of both differences and levels in the same model

thereby allowing one to investigate both short-run and long-run effects in the data.

The number of distinct cointegrating vectors can be obtained by

checking the significance of the characteristic roots of II matrix. The tests for the

number of characteristic roots that are significantly different from unity can be

conducted by using the following two test statistics:

ltrnr" = -, ,Jr ln (1 - ;*l)

).max - -T In (1-,tr,+l)

where, 7 ,', ar" the estimated values of the characteristics roots (also called

eignvalues) obtained from the estimated II matrix; T is the number of usable

observations.

Section D : Vector Autoreeression (VAR) Methodology

Vector autoregression (VAR)

critique to the structural macroeconometric

approach has been developed as a

modeling where arbitrary coefficient

52

restrictions and lag structures are imposed on the data-generating process. VAR

allows the data to speak for itself by allowing data to determine the lag structure and

doing away with the arbitrary exogeneity assumption that the structural econometric

models often make. Moreover, estimation of VAR becomes very important in the

context of multivariate cointegration tests proposed by Johansen-Juselius.

Consider a pth order VAR of n variables

! t = Ao+At!,_tl .......+A p!,_p+e t

where y , is an (nxl)vector containing each of the n variables included in the VAR,

A o is (nxl) vector of intercept terms, A ris (nxn) matrices of coefficients and

e ,is an (nx1) vector of error terms. VAR modeling has been popularrzed due to

Sims (1980). The variables included in the VAR are selected according to the relevant

economic model. VAR model scores over the traditional structural modeling by

abstaining from assuming exogeneity / endogeneity of the variables under

consideration. Rather, it treats all the variables symmetrically. When all the variables

included in the VAR have the same lag length in each of the equation of the VAR

system then it is known as a FUIMR (FYAR). Each separate equation can then

been efficiently estimated by simple OLS. However, it is possible to employ different

lag lengths for each variable in each equation. Such a system is called Mixed VAR

(MVAR). If some of the VAR equations have regressors not included in the others,

seemingly unrelated regressions (SUR) provide efficient estimates of the VAR

coefficients.

The choice of the lag length in VAR is an important issue, as the

inclusion of unnecessary lags will lead to severe overparameteization problem as

inclusion of lags quickly consumes the degrees of freedom. . The two commonly used

measures are the multivariate generahzation of the AIC and SBC :

53

AIC=Tlogl2l+2

SBC = rlog lll + Nlog (7)

where, l>l is tfre determinant of the variance / covariance matrix of the residuals; N is

the total number of parameters estimated in all equations. The model based on the

lowest AIC and SBC is chosen.

Another test that is often employed for selecting the lag length is the

likelihood ratio test. Let X, and X, be the variance / covariance matrices of the

unrestricted and restricted system respectively. Asymptotically, the test statistic is

given as

LR = e-c)( tog lX,l _ log l>,1)

has a chi-square distribution with degrees of freedom equal to the number of

restrictions in the system. Here, logll,l is the natural logarithm of the determinant

of I, ; T is the number of usable observation; and c is the number of parameters

estimated in each equation of unrestricted system. If the calculated value of the test

statistics exceeds the tabulated critical value, we reject null hypothesis, i.e., we reject

the restriction.

A block exogeneity test is useful for detecting whether to incorporate

a variable into a VAR. Consider a three variable system x, y and z. To test whether

variable z should be there in the system is similar to testing whether lags of z in x and

y equations are zero. This cross equation restriction is tested by performing a

likelihood ratio test. For this we need to estimate the x and y equations using p lagged

values of x, y and z and calculate the (unrestricted) variance I covaiance matrix X , .

Reestimate the two equations excluding the lagged values of z and calculate the

54

(restricted) variance / covariance matrix X, . Next, we calculate the likelihood ratio

statistic :

LR = Q-c)( log lX.l - log l>,1)

This statistic has a X' distnbution with degrees of freedom equal to the p lagged

values of z.

(i) Vector.Enor Coruection Model (VECM) :

Given that the variables are integrated of order 1, that is, I(1), and that

the variables are cointegrated I as concluded from the Engle-Granger and Johansen-

Juselius test for cointegration ], we can construct an error-correction model captures

both the short-run and long-run dynamics which may significantly improve the

model's forecasting performance. Consider the two variable system with y and zberng

the two variables. The two variables are integrated of order I and furthermore they are

cointegrated, that is, there exists a linear combination among them which is an I(0)

variable. Given this one can estimate the error-correction model of the following form

Ly , = f ,+ F ,a,_r+28,,(i)Ay ,_,*ZF ,r(i)Lz t_i*t yti=l

Lz , = F ,+ 0 ,a,_r+\F ,, Q)Ly ,_,*ZF ,r(i)a,z t-i* € ,ti=1 i=1

where O,-r= !,-r-fr 12,-, is the error-correction term, p, is the parameter of the

cointegrating vector, and e y,,€ ,, are the white noise disturbances.

Multivariate geteralization of the VECM is given as below

Ly,= f ,A.y,_r*.......*Io_, L/,_o*r+I-I.y Fr+p+Y.D+e ,

OLS is an efficient estimation strategy if each equation contains the same set of

regressors.

55

However, it will be appropriate to mention that the above formulation

of the eror-coffection model is only one of the approach to the problem and is

certainly does not exhaust the literature. The early development of the error-correction

model was very much the work of the London School of Economics by Phillips,

Sargan and Hendry. Phillips (1954,1957) introduced. the terminology of error

correction to economics in his analysis of feedback control mechanisms for

stabilization policy. The current popularity of error-correction model could largely be

attributed to the David Hendry, whose influential article with Davidson, Srba and Yeo

(1978) on aggregate time series relationship between consumer's expenditure and

income , proved to be a very important cornerstone. Work by Engle and Granger

(1987) on the effor-corection model is different because they take into consideration

the cointegration property of the time series data.

(ii) Bavesian Vector Autoreeression (BVAR) :

It is often rightly pointed out that economic forecasting is an art,

perhaps because it involves not only data and groups of equations, or statistical

models, but also the forecaster's personal beliefs about how the economy behaves.

The Bayesian approach to statistics, a general method for combining beliefs and data

in economic forecasting models. Bayesian Vector Autoregression (BVAR) modei has

been developed explicitly along Bayesian lines, provide modelers more flexibility in

expressing their beliefs as well as an objective way to combine those beliefs with

historical data.

BVAR models strikes a balance between Unrestricted Vector

Autoregression (UVAR) and structural econometric modeling. UVAR model relates

56

future values of a vector of variables to past values of that vector. From Bayesian

point of view, UVAR models allows the data speak for themselves. This implies

complete ignorance on the part of the modeler regarding the value of the coefficients

included in the UBVAR. The forecasting problems of large UVAR models stem from

the fact that economists often have too little data to isolate in a model's coefficients

only the stable and dependable relationships among its variables which leads to poor

forecasting performance.

On the other hand, in the structural econometric models which are

widely used for economic forecasting, overfitting problem of the UVAR model is

tackled by including in each equation of the model only a few variables (or lags of

variables) that economic theory suggests are most directly related to the variable that

the equation forecasts. Such exclusion of variables from an equation amounts to

certainty that their coefficients are zero. Certainty is an absolute belief, not subject to

revision by any amount of historical evidence. Although these restrictions can prevent

overfitting in a structural model, they are often too rigid to accurately express the

modeler's true beliefs and tend to cause useful information in the historical data to be

ignored.

A BVAR model have striking similarity with a UVAR as in both types

of models each variable is allowed to depend on the current and past values of all the

variables included in the model. But at the same time it also differs from a WAR

model and resemble a structural econometric model by using prior beliefs to reduce

overfitting. However, the sources of the prior beliefs and the ways they are used are

generally different in a BVAR model than in a structural model. Whereas economic

theory is the main source of prior beliefs, a BVAR to guess which values of all the

57

coefficients will lead to the best forecasts and to specifr an extensive system of

confidences in each coefficient.

Steps in Building s BVAR:

The first step is to construct an unrestricted VAR model. A n-variable

unrestricted VAR can be written as

Y (t) = A(L)Y (t)+ x(t)+u(t), t : 1,2,........, T.

Y(t) is an (nxl) vector of variables observed at time t, A(L) is an (nxn) matrix of

polynomials in the lag operator L, X(t) is an (nxnk) block diagonal matrix of

observations on k deterministic variables,p is the (nkxl) vector of coefficients on

the deterministic variables, u(t) is the (n xl) vector of stochastic disturbances, andX

is an (nxn) contemporaneous covariance matrix. The coefficient on Z o is zero fot

all the elements of A(L), i.e., only lagged values of elements of y appear on the

right-hand side.

The i-th equation of the model is given by

y(i,t) =\laU, j,c) y(j,t - j) + x' 1t1B1i) + u(i,t)j=t r=t

where, y(i,t) and u(i,t) are the ith elements of y(t) and u(t) respectively, BQ) is

the (ftx1) subvector of B corresponding to the ith equation and a(i,j,r) is the

coefficient on the c thlagof the jth variable of the ith equation.

The second step in the BVAR modeling is to speciff

the priors for the coefficients of the variables. BVAR model assumes an independent

normal prior distributions for each of the nz m YARcoefficients, such that a(i, j,r)

is assumed to have mean 6(i, j,c) and variance s'(i, j,r). The value chosen for

58

any particular 6 (i,7, r) would represent the ' best guess ' for the value of a (i, j,r) ,

and the value chosen for the corresponding ^s 2

Q, j,c) would reflect the degree of

confidence in that guess (smaller values reflecting greater confidence). The Minnesota

prior is the most often used prior which takes into account the fact that many

economic time series follows a random walk.

The Minnesota prior means are given by

5(i,i,r1=1 ,f i= j and r=l-0 otherwise

The Minnesota prior simplifies the choice of values for the s(i, j,r)

by specifying the following standard deviation function :

s(i, j,r) =ly sG)f(i, j)l(.{i) (4)sj

where s,is the standard error of a univariate autoregression for y(i,t), y is the

'overall tightness'parameter, g(r) is a function which describes the tightness on the

rthlag relative to first lag, and f (i,j) is a function giving the tightness on the jth

variable in the ith equation relative to the ith variable. Since the variables in the model

are likely to be of different magnitudes, the ratio of the standard errors, j, tt

included as a rescaling factor to make units comparable. Equation (4) reduces the

number of prior variandes fuom nzm to n2+2 parameters. Each f (i,j)can be

thought of as the i, jth element of an (nxn) matrix which must be specified. We

assume a symmetric prior for f (i, j)

f (i,j)=t if i=i=w otherwise

59

where w is a constant. The syrnmetric prior reduces the problem of choosing n2

parameters to the problem of choosing a single hyperparameter, w. The value of w

gives the relative tightness on the coefficients of the ' other ' variables in the ith

equation.

The choice of the lag tightness function, g(r), should be such that it

reflects the increasing confidence that coefficients are close to zero as the length of

the lag increases. Two possible functions are possible

Harmonic Function : SG) = r-o

Geometric Function : SG) = dt-l

In both cases, the single decay parameter, d, must be chosen. For the harmonic

function, the choice of a larger d implies more rapidly increasing tightness and thus a

more rapidly decreasing s(i, j ,c) as lag length increases. For the geometric function,

the choice of a smaller value of d implies more rapidly increasing tightness. The

overall parameter, y, gives an overall measure of confidence in the prior, with smaller

values coresponding to greater confidence.

60

Meusures of Forecast Evaluution

The present chapter discusses alternative forecast evaluation measures

for judging the forecast performance of the competing models. This is essential to

determine the qualitative performance of the forecasts. Moreover, it has been noticed

from the various empirical works that the ranking of the forecasts change quite

substantially under altemative forecast evaluation criterion. Given this fact it may be

upto the forecaster to give subjective weights to the alternative forecast evaluation

criterion to na:row down to one particular forecasting model. Again, the alternative

criterion may give very rankings to the forecasts from the various models at different

forecast horizons. So one may, in practice, find different models producing the best

forecast at different forecasts horizon.

I. Measures of Forecast Accuracy :

The crucial object in measuring forecast accuracy is the loss fturction

L(Y ,*tr,t ,*,,,,), often restricted to L(e,*0,,), which charts the "loss", "cost" or

"disutility" associated with various pairs of forecasts and realizations. Here f ,*0., is

the k-period ahead forecast errors and e *k,t = Y ,*o -t,**,, is the k-period ahead

forecast effors. In addition to the shape of the loss function, the forecast horizon (k) is

also crucial importance. Rankings of forecast accuracy may be very different across

different loss functions and / or different horizons. This result has led some to argue

the virtues of various " universally applicable " accuracy measures. Clements and

6t

Hendry (1993), for example, argue for an accuracy measure under which forecast

rankings are invariant to certain transformations.

Nevertheless, let us discuss a few stylized statistical'loss functions,

because they are widely used and serve as popular benchmarks. Accuracy measures

are defined on the forecast errors, € r*k,t=Y ,**-Y ,**,, , or percent errors,

p t+t.t = (Y ,*r, -t ,*0,,) lY ,*r,. The most common overall accuracy measures are the

Root Mean Squared Error (RMSE) and Root Mean Squared Percent Error (RMSPE)

defined below :

RMSPE =

Two other forecast accuracy statistics that have appeared in the literature are

RMSE=ff*,,

MAE = +*l ,.r)(1) Mean Absolute Error :

(2) Mean Absotute Percentage Error : MA\E = lLl, ,.r,,1

Anyway, the above measures do not provide much direct information

about whether something better might be achieved in terms of forecasting. It is

common to compare the performance with a set of naive forecasts, given by

\. _t/t t+k,t - I t'

Theil's inequality statistics helps in this comparison :

I r r ,*r -t ,**,,)'t=1

T

Itr ,*o-Y ,)'t=1

**o

U2=

62

U2:0 implies Perfect Forecast.

U2 : 1 implies forecast is as accurate as the naTve forecast

model, that is, the model should not be used for forecasting.

IJ2 > I implies forecast inaccurate relative to naive forecast.

U2<I implies forecast accurate relative to naTve forecast.

II. Forecast Rationalilv :

Forecast accuracy measures discussed above gives us an indication of

the relative performance of alternative forecasting models. However, it is essential to

check the properties of .the forecasts generated from the competing models

individually. There are a number of notions of rationality of the forecast which

includes those of unbiasedness and efficiency. One of the properties of the optimal

forecast is that forecast "rro..

have a zero meafl.In other words, optimal forecasts are

unbiased estimates of the actual series. In the present context it means that bias test

determine if the model forecasts are systematically higher or lower than actual

exchange rate. This is.also often referred as a requirement of weak rationality which

implies that forecasts are consistent in the sense that forecasters are not systematically

mistaken their forecast.

Tests ofunbiasedness are based on a regression ofthe form :

A,*n:d+BP *h*€ t+h (1)

The joint hypothesis a=0 and B =1 entails unbiasedness. An F-test is a valid test

statistic for the joint hypotheses if the error term, p, is i.i.d. Howeyer, a problem

associated with the above regression is that serial correlation is introduced into the

error term for equations corresponding to 3 -, 6-, 9-, and 12-month ahead forecasts.

For h > 1 (where h is the forecast horizon), the forecast horizon will exceed the

63

sampling frequency (assumed to be 1), so that forecasts overlap in the sense that they

are made before knowing the error in the previous forecast. Thus this test of

rationality does not rule out serial correlation in the error process of a moving average

of order (h-1). It also seems likely that conditional heteroscedasticity exists in the

error term. To correct for these problems, a heteroscedastic, autocorrelation consistent

covariance matrix (Newey and West (1987) is used to estimate the standard errors of

coefficients in the above equation. With the use of the Newey-West estimator, the

joint test statistic is distributed as a Chi-squared.

Although the joint hypothesis a = 0 and B= 1 is popularly described

as a test of unbiasedness, it can also be viewed as a test of efficiency, in the sense of

checking that forecasts and their errors are uncorrelated. If there is a systematic

relationship between the two, then this could be exploited to help predict future errors,

and could be used to adjust the forecast-generating mechanism accordingly. Mincer

and Zarnowitz (1969) used the concept of efficiency in this sense. They define

forecast efficiency as the condition that p =l in equation (1), so that the residual

variance in the regression is equal to the variance of the forecast eror.

When the data are non-stationary, integrated series, then a natural

further requirement of the relationship between the actual and forecasts is that they

are cointegrated. Infact, Cheung and Chinn (1998) proposed a test of rationality

based on the time series properties of the actual and the predicted series : the forecast

and the actual series (a) have the same order of integration, (b) are cointegrated and

(3) have a cointegrating vector consistent with long run unitary elasticity of

expectations. This means that cointegrating vector involving actual and the predicted

series should be (1 -1). We extend the unbiasedness test in this cointegrating

64

framework by testing the restriction (1 -1 0) on the cointegrating relationship

involving Actual, forecasts and a constant.

Hendry and Clements (1998) argued that the above tests of unbiasednsess or

rationality.may be too slack in that they are satisfied by more than one predictor, or

conversely, too stringent given the typical non-optimality of most forecasts, stemming

from the complexity of economic relationships, and the open-ended number of

variables that could conceivably affect the variable(s) of interest. Such criterion may,

therefore, be of limited value as means of forecast selection leading to a plethora of

forecasts, or non at all, ifonly these criteria are used.

It has been pointed out that the regression based test of unbiasedness

rest on strong statistical assumptions. Unsystematic forecast errors need not have

fixed variance through the sample period, nor need they be normally distributed. Such

deviations from the classical assumptions may compromise the efficiency of the

regression statistic. Moreover, standard testing procedures associated in (1) are only

valid as5rmptotically when the disturbances are correlated with future values of the

regressors. To deal with these problems, Campbell and Ghysets (1995) introduced a

nonparametric testing methodology to assess the unbiasedness of forecasts. The focus

of this nonparametric approach is on the median of the forecast effors rather than the

mean. However, for symmetric distributions with finite mean, median-unbiasedness

and mean-unbiasedness are equivalent. Nonparametric tests may be more reliable than

the regression-based procedures, particularly in small samples.

We consider the tests of the unbiasedness of one-step ahead forecast

effors. Let the one-period ahead forecast effors be written as

E ir=Sr-s" r-,

65

Define a function

u(z)=1if z>0

= 0 , otherwise

The role of the u (z) function is simply to indicate whether the forecast error is

positive or negative.

Consider the signed test statistic :

st =

where t:|,2,......,T is the number of forecast errors.

Under the null hypothesis that the forecast errors are independent with zero median,

the sign statistic is distributed with Bi(T,llz),that is, as the binomial distribution

with the number of trials T and probability of success 0.5. In large samples, the

studentized version of the statistic is standard normal,

s ,-T l2 - N(0,1)

lr /4Thus, significance may be assessed using standard tables of binomial or normal

distributions. It should be noted that the sign test does not require distributional

symmetry.

Consider another statistic given by

T

w , =2"@,, )'R* ,,t=l

where R* ,, is the rank of lE ,,1 with lE ,,1,. lz' ,.1 teing placed in ascending

order.

Under the null hypothesis that forecast errors are independent and symmetric about

mean zero (and hence about azero median), W , is distributed as Wilcoxon signed

rank test. The intuition of the test is that if the underlying distribution is symmetric

T

Z"@ u)t=l

66

about zero, a "very large" (or "very small") sum of the ranks of the absolute values of

the positive observations is "very unlikely" to be high. The exact finite-sample

distribution of the signed-rank statistic is free from nuisance parameterS and invariant

to the true underlying distribution. Moreover, in large samples, the studentized

version of the statistic is standard normal,

*,-ryP- N(0,1)

The results from the bias test will enable us to tell whether forecasts are unbiased

predeictors of the actual exchange rate. A forecast may be beating the random walk

forecasts, one of the main focus of the present study, but it may be actually biased.

Then it may be difficult to tell unambiguously whether we should use those forecasts

even when they are beating the random walk.

III. Testing the equalilv of orediction mean squared erroys :

The comparisons of mean squared error (MSE) or root mean square

effors (RMSE) are merely descriptive indicating one set of forecasts has made

relatively small errors than another. This does not any way tell us whether the

difference in the MSE or RMSE between the competing the forecasting models is

significant or not. So one may get a lower MSE or RMSE compared to the simple

random walk model but that difference may not be statistically significant. So it is

very essential to find out whether the difference in the MSE or RMSE arising are

statistically significant or not.

Diebald and Mariano (1995) proposed a test of the null hypothesis of

no difference in the accuracy of two competing forecasts which allow for forecast

r (T +DQr +D

67

effors that are potentially non-Gaussian, non-zero mean, serially correlated, and

contemporaneously correlated. Suppose that a pair of h-steps ahead forecasts have

produced errors (e ,, ,e 2,) , t=1,2,......n. The quality of a forecast is to be judged on

some specified function g(e) of the forecast error, e. Then, the null hypothesis of

equality of expected forecast performance is

Els@u)-s(er,)l=o

Define d , = g(e u) - g(e r,)

Then, the test statistic based on observed sample mean is given by

7 =n-'la ,t=1

One problem is that the series d , is likely to be autocorrelated. It can be shown that

the variance of d is, asymptotically,

v G) * n-t ll ,*zfr o J

k=l

where f o is the kth autocovariance of d , . This autocovariance can be estimated by

f r=n-'\{a,-a)@,-o-d)t=k+l

The Diebold-Mariano test statistic is then

s, = [ tr @)]''' .a

Under the null hypothesis, this statistic has an asymptotic normal diskibution.

Diebold and Mariano considered mean squared eror as the standard of forecast

quality, that is, g(e) = e' .

68

V. Information Content Test :

Forecast encompassing tests enable one to determine whether a certain

forecast incorporates (or encompasses) all the relevant informatioh in competing

forecasts. The idea of forecast encompassing was formalized and extended by Chong

and Hendry (1986). Suppose we have two forecasts, f' ,*0,, artd t' ,*r,,,. Consider the

regression

Y ,*k = F o+ F ,t' ,*0,,+ F ,t' ,*k,,*t ,*k,,

If (B * F, F, ) = (0,1,0), one says that model 1 forecast encompasses model 2, and

if (P o,B,F r):(0,0,1), then model 2 forecast encompasses model 1. For any

other (B o ,0 , , P , ) values, neither model encompasses the other, and both forecasts

contain useful information.

Fair and Shiller (1989, 1990) take a different but related approach

based on methodology. Their test is popularly known as the Information Content

Test. They argued that many econometric models are used to forecast economic

activity which differ in structure and in the data used. So their forecasts are not

perfectly correlated with each other. This necessitates the developing of some test

which enables to find out whether each model have a strength of its own, so that each

forecast represents useful information unique to it, or does one model dominate in the

sense of incorporating all the information in the other models plus some. They

developed the following regression based test

(Y,*o-Y,) = d+ P(Yt,*k,,-Y,)+y(Y' t+r,z-Y,)*€,*r.,

If neither model I nor model 2 corrtun any information useful for k-period-ahead

forecasting of Y ,, then estimates of B an d 7 should be zero.In this case the estimate

of the constant term a would be the average of the k-period-change in Y. If both

69

models contain independent information of the k-period-ahead forecasting, then p

and y should both be non-zero. If both models contain information, but the

information in, say, model 2 is completely contained in model I and model 1 contains

further relevant information as well, then B but not 7 should be non-zero. So one

estimates the above equation for different model's forecasts and test the hypothesis

H ri? =0 and the hypothesis ,E/ z:T=0. lf , is the hypothesis that model I's

forecasts contain no information relevant to forecasting k period ahead not in the

constant term and in model 2, and H , is the hypothesis that model 2's forecasts

contain no information not in the constant term and in model 1.

Fair and Shiller's test bears some relation to encompassing tests but is

not exactly identical to it. For instance, Fair and Shiller does not constrain B arf y to

sum to one, as usually the case for encompassing tests. However, it is not difficult to

perform the forecast encompassing test in the Fair-Shiller framework. We can test the

null hypothesis (a, f ,y): (0,1,0) or (0,0,1). Under the null of forecast

encompassing, the Chong-Hendry and Fair-Shiller regressions are identical. When the

variable being forecasted is integrated, however, the Fair-Shiller framework may

prove more convenient, because the specification in terms of changes facilitates the

use of Gaussian asymptotic distribution theory.

YI. Market Timing Test:

Studies by Gerlow and Irwin (1991) show that statistical evaluation

measures may not yield results consistent with the actual trading profits generated by

exchange rate models. Hence, it is useful to consider a measure of economic value of

a model. Henriksson and Merton (1981) developed a test which is essentially a test of

70

the directional forecasting accuracy of a model. The directional accuracy has been

shown to be highly correlated with actual trading profits.

From a sample of N actuals and forecasts and their probabilities, form

the following contingency table, and test the independence of actuals and forecasts

Forecasts I

Actuals

<0 >0

<0

>0

P ,, (O ,,) P ,, (O ri) P,.(O,)

P ,, (O .i,) P ,, (O ii) I-p t (O i)

P,(o.,) l-p r.(o i) 1(o)

Where p ,i is the joint probability that an observation will belong to the ith row and

jth column, p.i end pi. tre the marginal probabilities, with . denoting summation

over either columns or rows. In parentheses we denote the number of observations in

each cell.

Thus,

Pr=b and Pr=P.t

Pzzl- p,

The null hypothesis that a direction-of-change forecast has no value is that the

forecasts and actuals are independent, in which case p i = p r. p .i, for all ij. (for a

brief discussion on directional analysis of forecasts refer Ash, Smyth and Heravi

(1ee8)).

The formal H-M test relies on a theorem in Merton (1981) that shows,

without recourse to a model of equilibrium returns,that a necessary and sufficient

condition for a rational investor to modiff his prior beliefs is that the sum of the two

7t

conditional probabilities of a correct

implies that the forecast has no value

forecasts, pttpz, exceeds one. This also

when actual and forecasts of a series

distributed independently and pttpz=\. Consequently, the test proposed

Henriksson and Merton tests whether actual and predicted series are independent.

The uniformly most powerful unbiased test for independence is

R.A.Fisher's Exact Test (Fisher (19a1)) which is identical to H-M's test for predictive

values. Thus the nonparametric test proposed by H-M is asymptotically equal to the

simple X2 testof independence in a 2x 2 contingency table. The true cell probabilities

are

by

are unknown, so one uses a consistent estimates i, , =! ^ao

one consistently estimates the expected cell counts under the null,

^ o,o,E ,, = +. One, therefore, can construct the statistict,o

Under the null, C - Z' ,.

b -o ''. Theno

E r=p i. P .i ,bY

-E u)'(o,ic=zi,i=l E,j

72

Chupter V

Data Soarce and Deftnitions

We use monthly data in the present study. The time period covered in

our study is from January 1993 to JuJy 1999. The exchange rate used in this study is

Indian Rupee-US Dollar bilateral spot exchange rate defined as the number of Rupees

per unit of Dollar. Money supply is measured by Ml for both India and US. Output

is proxied by the General Index of Industrial Production (IIP). For India, IIP figures

for the period January 1993 to March 1998 are given for the base year 1980-81:100,

while for the period April 1998 to July 1999, IIP figures correspond to the base year

1993-94:100. The two series are spliced to obtain the whole series at 1980-81 prices.

For US, the IIP figures correspond to the base year 1992:100.Interest rate figures for

both countries coffespond to 3-month Treasury Bill rates. Prices used in this study are

Consumer Price Index (CPf numbers. For India, we use the CPI figures for the

Industrial Workers (base : 1982:100); while those for US corresponds to the CPI for

all commodities (base : 1982:100). All the above mentioned data for India are

obtained from Handbook of Statistics on Indian Economy, Reserve Bank of India,

1999 ; while for US they are obtained from the Federal Reserve Bank, Minneopolis

website hnp://www.stls.frb.org/fred/. Monthly data for imports and exports are

obtained from the Monthly Abstracts of Statistics, Government of India, and given in

terms of rupees. Monthly output, prices, money supply and trade balance data are

seasonally adjusted.

73

Chapter VI

Empirical Results

This chapter discusses the empirical findings of the present study.

First, we present the unit root test results on various time series variables. We carried

out a host of unit root tests - (Augmented) Dickey-Fuller (DF), Phillips-Perron (PP),

KPSS and Bayesian unit root test. Results of the unit root tests are reported in

Section A. A11 the variables under the present study tums out to be an integrated

process of order 1. Given that the series are I(1) it is natural to test for the presnece of

cointegration among the economic variables of interest. We employ the Engle-

Granger (EG) and Johansen-Juselius (JJ) tests of cointegration to estimate the long-

run equilibrium relationships. Section B describes the cointegration test results and

the existence of the monetary model as a long-run equilibrium relationship. We also

test for the presence any particular version of the monetary model in the Indian

economy. This section also reports the estimation result of the vector error correction

model for the purpose of generating forecasts of the exchange rate. Section C

describes the estimation results of the various formulations of Bayesian vector

autoregression based on the Minnesota prior and thereafter, forecasts are generated

from these models. We also estimate an univariate ARIMA model (Section D) by

employing Box-Jenkins methodology for generating forecasts of the exchange rate.

This often serves as the benchmark model for forecast comparison. A forecasting

exercise has been carried out whereby forecasts are generated for one to twelve

month forecast horizon. Finally, we run abattery of forecast evaluation tests to assess

the quality of the forecasts, reported in Section E.

74

A11 variables, except three-month treasury bill rates for both countries,

are given in logarithm. Our initial estimation period is from January 1993 to

December 1996. A rolling regression technique has been adopted whereby one

sample point is added and estimation is carried out. This is continued till the last

sample point, viz. July 1999, is reached. Unit root tests have been carried out by this

rolling regression technique. This will enable us to find out whether the stochastic

trend is present in the variables under consideration for whole of the forecasting

period, that is, from January 1997 to July 1999. Unit root tests have been performed

on both the individual variables, that is on the spot exchange rate, IIP of both

countries, money supply (M1) of both countries, and on the three-month treasury bill

rates, as well as on the relative value of the variables, that is, on the relative money

supply, relative interest rate and relative IIP.

Section A : Unit Root Tests

We first carry out the traditional Dickey-Fuller (DF) and Phillips-

Perron (PP) test to find out the presence of stochastic trend in the series. Logarithm

of the bilateral spot exchange rate is a natural candidate to begin the analysis as this

is the central variable of the present study. Tablesl.l (A) and (B) gives the DF and

PP test results for the levels of the variables. We have presented here the full sample

result for all the variables, that is, covering the period January 1993 to July 1997.

The conclusion remains unchanged for any of the sub-periods starting from the

period I 993 :01 -1996:12.

DF test requires a sequential testing procedure, starting from a general

model which allows for a deterministic trend in the data to a simple model which

75

tests for the presence of a pure random walk. In the general model the statistic that

tests for the null of the unit root.is given by e ,. The calculated value of the test

statistic is -2.3064 whereas the critical value is -3.41 (at 5% significance level). Since

it is a left-tail test, this leads to the inference that a unit root is present in the spot

exchange rate. Cambell and Perron (1991) argued that absence / presence of a

deterministic trend affects the stochastic trend inference. So it is natural to test for a

joint hypothesis of the presence of a stochastic trend but absence of a deterministic

trend. This is given by the / , statistic and it is a F-test. The calculated value of the

/ rstatistic is3.2314 which is less than the critical value of 6.25 (5% significance

level) implying the presence of a stochastic trend but no deterministic trend. We then

move to the next model - one with a constant and no trend. Here the null of unit root

is evaluated by the f , statistic. As indicated in table 1.1.(A), the null hypothesis gets

accepted at 5oh significance level. We use this regression to carry out another F-test

to find out whether the DGP (Data Generating Process) is a random walk with drift.

This is carried out by using / , statistic. The constant term is turning out to be

insignificant thereby rejecting the possiblility of a random walk with drift. We

therefore proceeded to find out whether the data generating process (DGP) is a pure

random walk. This has been caried out by estimating the simplest specification - one

with no constant and no trend. Here the relevant.statistic is q . The calculated value

of this statistic is 2.2078. Given this is a left-tail test, the null hypothesis of unit root

does not get rejected atany of the 1, 5 and 10% significance level. So from the DF

test one can conclude that the series is integrated of order 1, i.e.,I(1).

Given the low power of the DF test which has the implication that it

tends to accept the null hypothesis of unit root 'too' often one needs to substantiate

76

the result of the DF tests with some other tests. Phillips-Perron (PP) test is a natural

candidate since it also has the same null of the presence of a unit root and has a better

power than the DF test. Result of the PP test for the levels is given in Table 1.1(B).

We are concerned with the statistics Z(td, )and Z(ta- , ) whose critical values

corresponds to those of e ,and 6 , statistics. The calculated value of the

Z (t d , ) statistic is -2.2456 which is less than the critical value at lYo, 5o/o, and l}Yo

significance level, indicating the presence of a unit root. This conclusion is

reinforced by the Z (t a. , ) statistics. Thus both the DF and PP tests leads to the

same conclusion of the presence of a stochastic trend in the logarithm of the spot

exchange rate series, that is, it is I(1).

To find out whether the exchange rate series is integrated of higher

order, unit root tests have been carried out for the first difference of the series.

Results of the DF tests are given in Table 1.2 (A). The calculated e s statistic

(-4.01 86) falls in the critical region at arry of the 106, 50 and, l0o/o significance level.

Given the sequential nature of the DF test, we stop where the null of unit root gets

rejected. The results of the PP test, given in Table 1.2 (B) also supports the DF

results. So, from the DF and PP tests we can finally concluded that the spot exchange

rate is integrated of order 1, i.e., it is I(1).

We now look into the time series properties of other variables which

may help to explain the movements of the exchange rate - output (proxied by the

index of industrial production (IIP)), money supply (as measured by Ml) and interest

rates (measured by the three-month treasury bill rates). This choice of the variables is

motivated by the monetary model. The results of the Dickey-Fuller and Phillips-

Perron tests are given in Table 1. We refrain from giving a detailed description of the

test procedure here. However, the table below gives the final conclusion from these

tests.

Summary of the Dickey-Fuller and Phillips-Perron test

Time Span : January 1993 - July 1999

presence

is I(1).

Except for the Indian money supply, as given by Ml, there seems to be

unanimous conclusion by both the tests that the series under consideration are I(1).

Only in Indian Ml there seems to be different conclusions produced by the two

different tests. So we may need to look into some other unit root tests to come into

any clear cut conclusion regarding this variable. In fact, latter on we perform both the

KPSS and Bayesian unit root tests to cross-check our conclusion from the DF and PP

tests.

We are, however, more interested to cany out a detailed unit root

analysis of the relative value of the variables - relative IIP, relative money supply and

relative interest rates - as these variables will be later used to explain the exchange

IS,

Exchange Rate Y Y

Indian IIP Y Y

Indian Treasury Bill Rate Y Y

Indian M1 N Y

US IIP Y Y

US Treasury Bill Rate Y Y

US M1 Y Y

Y o Unit root in the series under eratlon

18

rate behaviour. The results are given in Table 2. Consider the results for the

logarithm of the relative index of industrial production. Dickey-Fuller test results for

the levels are given in Table 2.I(A). The calculated values of the C E ,e , and E

statistics are -1.6628, -2.117 and 0.1319 respectively. All of these falls in the non-

rejection region at l, 5 and l0o/o significance level. The PP test results (given in

Table 2.1(B) also supports the conclusion of the DF test. So a stochastic kend is

present in the relative IIP. To find out whether the series is integrated of higher order,

we undertake DF and PP test on the first differences of the relative IIP. Results are

reported in Table 2.2 (A) and (B). The DF test on first differences fails to reject the

null of unit root at l%o and 5%o significance level. However, the PP test strongly

reject the null of unit root in the relative IIP series. So we can conclude that the

relative IIP series is I(1).

We now look at the logarithm of the relative money supply as given by

M1. DF tests reported in Table 2.1(A) indicate that the series is nonstationary in

level. The calculated values of the e , ,C , and g statistics are -1.1841, -1.2796 and

-2.4556 respectively. The series is tuming out to be nonstationary at loh significance

level. However, at 5%o and LlY, significance level it is coming out to be stationary.

The PP test on the levels (Table 2.1(B)) strongly supports the null of unit root at any

of the three significance level under consideration here. The DF and PP tests on first

difference of the series strongly rejects the null of unit root. In fact, the calculated

value of the f , statistic is -5.0838 (Table 2.2 (A)) which leads to the rejection of the

null of unit root in the first step only. Thus, we can conclude that the logarithm of the

relative money supply is I(1).

Finally, we consider the relative interest rates measured by the

difference between the two country's three-month treasury bill rates. This variable is

79

not in logarithm term. The calculated values of the e e ,e o and g statistics are

2.007, -2.1418 and -1.770 respectively. This clearly indicates the presence of unit

root in the relative interest rate series at lo/o and 5o/o significance level. The PP test

results on the levels strongly supports the DF conclusion. The calculated value of

Z (t d , ) and Z (t a. , ) are -1.5205 and -1.5685 respectively (Table 2.1 (B)), which

falls in the acceptance region. Thus both the tests clearly points out towards the

presence of a stochastic trend in the level of the series. To infer on the degree of

integration we carry out the DF and PP tests on the first differences of the series.

Results are given in Table 2.2 (B). Both DF and PP test concludes that the first

difference ofthe relative interest rate is stationary. So the relative interest rate series

is I(t).

Summary of the Dickey-Fuller and Phillips-Perron test

Time Span : January 1993 - July 1999

presence

is I(1).

It is a well established fact that standard unit root tests fail to reject the

null hypothesis of a unit root for many economic time series. In these standard unit

root tests (DF and PP) the unit root is the null hypothesis to be tested. An explanation

for the common failure to reject a unit root is simply that most economic time series

are not very informative about whether or not there is a unit root.

Relative IIP Y Y

Relative Ml Y Y

Relative 3-Months TB Rate Y Y

the o Unit root in the series on ls

80

DeJong et. al. (1991) provide evidence that the DF tests have low power against

stable autoregressive alternatives with roots near unity, and Diebold and Rudebusch

(1990) show that they also have low power against fractionally integrated

alternatives. Kwiakowshi et. al. (1992) developed a unit root test, popularly know as

theIKP,SS test, which tests the null hypothesis of stationarity against the alternative of

a unit root. We intend to apply this test to crosscheck our inferences from the

standard DF and PP tests.

KPSS test consists of two test statistics - ry r, where the null of level

stationarity is tested, and rl , , where the null of trend stationarity is tested, against

the alternative of a unit root. In the calculation of the KPSS test statistic, the choice

of lag truncation parameter / is very important. In the presence of large and

persistent positive serial correlation, the long-run variance S'1f typically increases

monotonically in l, so that KPSS statistic decreases as / increases. KPSS

recommended calculation of ,S2 (/) out to a value of / such that the long-run

variance estimate and hence the KPSS test statistic have " settled " down. However,

setting / too high can result in significant loss of power. Keeping this trade-off in

mind we have chosen the truncation lag parameter / to be equal to five.

Table 3 and 4 reports the KPSS test for the levels of the variables

under this study. At the outset it should be remembered that all the variables except

the interest rates are in logarithm. Table 3.1. presents the result of the KpSS test

when the null hypothesis is that the series is level stationary. For the exchange rate

series, the null of level stationarity is getting rejected even at 1% significance level at

all values of / as the calculated value of the test statistic is greater than the critical

value. Table 3.2 gives the KPSS test result where the null hypothesis is trend

81

stationarity. In the case of exchange rate, the calculated value of rl , statistic exceeds

the critical value at lo/o significance level at all /. Thus the KPSS test supports the

conclusion of the DF and PP tests that the bilateral India-US spot exchange rate

indeed contain a unit root.

KPSS test has been applied individually to the levels of all the series

under the present study. As far as the null of level stationary is considered, except for

the Indian and US three-month treasury bill rate and the US Ml, the null of level

stationary is getting rejected atlYo significance level at all I for all the variables.

The US TB rate is getting rejected at 5% significance level atl>3, while that for US

Ml the rejection level is taking place at 10% significance level for all /. In case of

the Indian treasury bill rate, for l>3,the calculated value of the ? rstatistic is less

than the critical value (at any of the 1, 5 and 10% significance level), implying that

the null of level stationary is not getting rejected. However, given that the power of

the KPSS test declines as / increases, we conclude broadly that the null of level

stationary is getting rejected for the Indian three-month treasury bill rate. In the case

of null of trend stationary, the calculated value of the ry "

exceeds the critical value at

10% significance level at all 1, except for the US IIP. For US IIP, at l>3 the null of

trend stationary is not getting rejected even at 10% significance level. Invoking the

same argument as in the case of the Indian treasury bill rate we can broadly conclude

that US IIP is indeed trend stationary. Thus, KPSS test on the level of the variables

under the present study reinforces the DF and PP tests result that all the series indeed

contain a unit root. Table below summarizes the result of the KPSS test.

82

(Level)

implies that unit root is present in the relevant series.

We also test for the level and trend stationarity for the relative value of

the variables - relative money supply, relative IIP and relative treasury bill rate. A11

the variables, except the relative interest rate, are given in logarithm term. The results

of the KPSS test are given in Tables 4.1 and 4.2.Table 4.1 give the results for the

KPSS test when the null hypothesis of level stationary. For all I , the null hypothesis

gets rejected at 5o/o significance level as the calculated value of the ? , statistic

exceeds the critical value. Table 4.2 reports the calculated value of the 7 , statistic

for testing the null hypothesis of trend stationary. For relative IIP and relative money

supply series, the calculated value of the test statistic exceeds the critical value at lo/o

significance level for all /. For the relative interest rate series, the calculated value

Exchange Rate N N

Indian IIP N N

Indian Treasury Bill Rate N N

Indian Ml N N

US IIP N N

US Treasury Bill Rate N N

US Ml N N

N stands rejection of the respective null In other words, it

83

of the r7 , statistic exceeds the critical at l0%o significance level for all / except at

/:5. So we can conclude that all the series of relative terms indeed contain a unit

root, reaffirming the resulting obtained from the DF and PP tests.

Summary of the KPSS Test

Time Span : January 1993 - JuIy 1999 \

(Levels)

rr N rr stands for the rejection o respectlve nu is. In other words, it

implies that unit root is present in the relevant series.

As mentioned earlier, most economic time series are not very

informative about whether or not there is a unit root, which attributes towards the

poor performance of the standard unit root tests. Bayesian unit root analysis offers an

alternative means of evaluating how informative the data are regarding the presence

of a unit root, by providing direct posterior evidence in support of stationarity and

nonstationarity. In Bayesian analysis the choice of the prior distribution occupies a

very important position. When there is no a priori belief regarding the distribution of

the parameter, a diffuse or non-informative prior is used. Often, a uniform prior is

used to represent the ignorance over the parameter space. This is known as the flat

prior. We employ this flat prior in our analysis.

Sims (1988) notes that it would be appropriate to put some probability

a uniformly on the interval (0,1) and some probability (1- o ) on p=1, where p is

ative Interest Rate

Relative Money Supply

84

the autoregressive parameter. A lower limit for the stationary part of the prior is also

specified such that prior for p is flat on the interval (lower limit, 1). Following Sims

we take a : 0.8 since for this level the odds between stationarity and the presence of a

unit root are approximately even. The results of the Bayesian unit root test are

reported in Table 5. As usual, all the variables except the interest rates are taken in

logarithm term.

Table 5.1 reports the Bayesian unit root result for the level of the

exchange rate series is presented. Here the squared t (0.046) is less than the Schwarz

limit (8.195) thereby strongly supporting the presence of a unit root in the the

exchange rate series. The'marginal alpha'is also above 0.90. 'Marginal alpha' is the

value for alpha at which the posterior odds for and against unit roots are even. A

higher value of ' marginal alpha ' favours the presence of unit root. A high ' marginal

alpha ' of 0.9215 supports the presence of unit root in the exchange rate series. For all

the series reported in the Table 5.1. the squared t is less than Schwartzlimit indicating

the presence of a unit root in the level of these series. However, it should be noted that

the ' marginal value ' is very low for two series - Indian Ml and US Ml. Also,

although for these two series the squared t is less than Schwarz Limit, they are not

very less than the latter value. So one need to cautiously interpret the result of unit

root for these two series and need to be substantiated with other unit root tests.

Table 5.2. reports the Bayesian unit root test for the relative variables -

relative IIIP, relative money supply and relative interest rate. For all the series the

squared t is less than the Schwarz limit indicating the presence of unit root in all

these series. However, the ' marginal alpha' for all the three variables are not very

high. So it would be appropriate to substantiate the Bayesian test with other unit root

test to conclude about the nature of these time series.

85

Finally, we present a summary of the results obtained from the various

unit root test regarding the presence of a unit root in the series under consideration.

Summary of the Unit Root Tests

Time Span : January 1993 - July 1999

(Levet)

presence

Section B : Cointegration, Long-ran Equilibrium Relationship and

Vector Ewor Coruection Model

(i) Cointegration:

From the above discussion on unit root tests we can conclude that all

the series under consideration are non-stationary. Granger and Newbold (1974)

pointed out the possibility of spurious regression in the event of running an OLS

Exchange Rate Y Y Y Y

Indian IIP Y Y Y Y

Indian Treasury Bill Rate Y Y Y Y

Indian Ml N Y Y Y

US IIP Y Y Y Y

US Treasury Bill Rate Y Y Y Y

US M1 Y Y Y Y

Relative IIP Y Y Y Y

Relative Treasury Bill Rate Y Y Y Y

Relative Money Supply Y Y Y Y

" y " stands for the presence o unit root in the series ton.

86

involving non-stationary variables. A direct fallout of such an event is the presence

of very high R2 (orR') among the " economically unrelated " variables. A common

indicator of spurious regression is R2 > d, where d is the Durbin-Watson statistic.

However, two I(1) series could infact have some economic relationship between

them and hence are " co*integrated ". Engle and Granger (1987) developed the

concept of cointegration based on the time series properties of the variables where

they talked about the possibility of a linear combination the I(1) variables which is

(0). This is a rather special condition, because it msans that all the series

individually have extremely important long-run components but that in forming a

linear combination these long-run components cancel out and vanish.

We present the result of the Engle-Granger (EG) cointegration test in

Table 6 for the full sample period January 1993 to July 1999. We work with the

relative value of the variables - relative IIP, relative three-month treasury bill rates,

relative money supply as given by Ml, and exchange rate. A11 variables except the

relative three-month treasury bill rate are in logarithm. Our choice of the variables is

motivated by the monetary models of exchange rate determination (discussed in

Chapter II). To recapitulate, monetary models of exchange rate determination

postulates existence of an equilibrium relationship among exchange rate, money

supply, interest rate,output, prices and expected inflation, depending on the variant of

the monetary model we are estimating. We first test for the Flexible-price and

Dombusch's sticky price version of the monetary model which postulates an

equilibrium relationship between the exchange rate, money supply, interest rate and

output.

The calculated value of the DF statistic (constant, no trend regression)

is -3.3001 which is greater than critical value of -3.81 (10% significance level) and

87

hence lies on the non-rejection zone. This implies that the residuals from the

regression are non-stationary and hence the variables are not co-integrated. Even

when we allow for a trend in the regression equation the residuals are turning out to

be non-stationary. The PP statistics also reinforces the DF results.

It is well established, both theoretically and empirically, that EG test

has several problems. First, it is very sensitive to the normalisation in the sense that

with one variable as dependent variable , it may not reject the null of no-cointegration

but with some other variable as dependent variable it may show cointegration.

Furthermore, it presumes that only one co-integrating vector exists among the

variables, thus ruling out the possibility of multiple cointegrating relationships

among the variables, which may be justifred by the economic theory.

Johansen (1988) and Johansen and Juselius (JJ) (1990) developed a

cointegration test among the variables which takes into account the drawbacks of the

EG test. It uses the maximum likelihood estimates to find out the number of co-

integrating vectors among the variables. JJ test treats all the variables symmetrically

and thus it prerequires an estimation of vector autoregression (VAR) where apriori

all I(1) variables are assumed to be endogenous. Optimal lag length selection is an

important problem for the estimation of VAR. There is apparantly a trade-off

involved in the lag length selection problem. A smaller lag length will preserve

degrees of freedom but could induce serial correlation problem in the system,

whereas a longer lag length would exhaust the degrees of freedom very quickly but

remove the serial correlation problem.

We are interested in choosing a lag length which gives us a VAR with

no serial correlation problem in the individual equation of the system since presence

of serial correlation will lead to forecasts which consistently overpredicts or

88

underpredicts the variables concerned. Also, it Cheung and Lai (1993) pointed out

that serial correlation is a serious problem for the Johansen approach and that the

usual lag length selection criteria (Akaike Information Criterion and Schwarz

Bayesian Criterion) may be inadequate, particularly in the presence of moving arerage

errors. In the present study we employ four tests to select the optimal lag length of

VAR on which we would be performing the JJ test - Akaike Information Criterion

(AIC), Schwarz Bayesian Criterion (SBC), Likelihood ratio test (LR), and LM tests

for serial correlation. The result of these tests are given in Table 7. The criterion for

choosing lag length on the basis of AIC or SBC is to choose that lag length that gives

us the minimum of these two statistics. Both these statistics point towards a lag length

of 1. The LR test, which under null hypothesis, follows a chi-square test, also gives us

a lag length of 1. We therefore re-estimated the VAR system with one lag of all the

variables and tested for the presence of serial correlation in the system. We found

serial correlation problem in the system. This leads us to move to a higher lag length

at which there are no serial correlation problem. We choose a lag length of three.

It is worth to point out that we are working with the relative value of

the variables. One prime reason for restricting the coefficients to be equal for both

countries is to preserve degrees of freedom given relatively small post-liberalisation

period in India. Ideally we would like to allow the coefficients to be unequal for both

countries. Whether this affects the forecasting performance is not very clear from the

literature, for instance, Meese and Rogoff (1983) found that even after allowing the

coefficients to vary across the countries there was no significant improvement in the

forecasting performance of the monetary models.

We employed the block-exogeniety test to determine whether to

incorporate a variable into a VAR. This will enable us to determine whether lags of

89

one variable granger cause any of the variables in the system. This is again a

likelihood ratio test and follows a chi-square distribution under the null hypothesis.

The results of the block exogeneity test is given in Table 8. The results clearly

indicate that null hypothesis of non-inclusion of a particular variable is getting

rejected for all the variables. For instance, if we test the null hypothesis that the lags

of the exchange rate is not present in any of the equation of the system, the LR test

statistic value is given by 25.3496 with a p-value of 0.003 indicating that we can

reject the null hypothesis even at 1% significance level. Similar results hold for other

variables indicating that none of the variables under consideration can be excluded

from our system of VAR.

With the lag length of three in VAR, we proceed to calculate the two

statistics - I ^u* and )" oo, statistics - used in the JJ test. 2 *u* has got a much

sharper alternative than )" ,,," since the later tests the null hypothesis that the number

of distinct cointegrating vectors is less than equal to r against a general alternative

whereas the former tests the null hypothesis that number of cointegrating vectors is r

against the alternative of (r+1) cointegrating vectors. So we rely on the )" ,u* statistic

to pin down the number of cointegrating vectors. The results of the cointegration test

is given in Table 9. Trace statistic rejects the null of no-cointegration at 5oh and lo/o

significance leve. 2 nu* statistic indicates that there are two cointegrating vectors as

the calculated values of the statistic exceeds the critical values at 5Yo and ljoh

significance level.

variables motivated by the monetary models of exchang e rutedetermination.

90

among the variables suggested by the monetary models. While 2 oo" statistrcs

gives us one cointegrating vector, the ), -u* statistic indicates the presence of two

cointegrating vectors.

vectors between the exchangerate, money supplies, ou@uts and interest rates.

(ii) Long-run Equilibrium Relationship :

We present the estimation result of the cointegrating vectors in Table

10. As noted earlier, we have two cointegrating vectors. The signs of the second co-

integrating vector confirms to those postulated by the monetary models of exchange

rate determination. More specifically, it confirms to the flexible-price version of the

monetary models of the exchange rate determination.

For convenience we report below the long-run economic relationship

that makes economic sense

s = 7.8110 + 1.3669 (m-m.)-2.5361(y-y-)+0.3902(r-r.)(2t.57)** (21.07)x* (31.00)*x (9.98)**

where, s is the bilateral spot exchange rate (Rupee-Dollar exchange rate), y is output

as proxied by the Index of Industrial Production (IIP), r is the interest rate as proxied

by the three month treasury bill rate, and m is the money supply as measured by the

M1. (* indicates a corresponding variable for the foreign country, here it is USA). All

the variables are in logarithm, except the interest-rate. The figures given in the

brackets are the calculated value of the likelihood ratio of testing the null of the

significant presence of the relevant variable in the cointegration relati.onships. Under

91

the null of no significant presence, the statistic follows a chi-square distribution with

one degree of freedom. Given that the critical value of chi-square with one degree of

freedom is 5.84 at l0o/o significance level and 3.24 at 5o/o significance level, we can

conclude that the variables are significantly present in the cointegrating vector.

The long-run relationship confirms to the flexible-price version of the

monetary model. As postulated by the flexible-price version of the monetary models,

exchange rate is negatively related to the relative income differential. The mechanism

is that as domestic income goes up relative to the foreign income, there is an excess

demand for money in the domestic economy. To clear the money market, prices

decrease which in tum, leads to an appreciation of the domestic currency via the

Purchasing Power Parity (assuming to be holding continuously in this class of

models). This then ensures the negative relationship between the exchange rate and

the output. Moving on to the interest differential term, our results confirm to the

theoretical signs of the flexible-price monetary model. Here, the exchange rate is

positively related to interest differential. As domestic interest goes up relative to the

foreign interest rate, which basically reflects higher domestic expected inflation,

people hold less domestic money, thereby creating an excess supply in the domestic

money market. To clear the money market, prices increase and as Purchasing Power

Parity condition is assumed to hold continuously, this leads to a depreciation of the

domestic curency (an increase in s). Finally, relative money supply and the spot

exchange rate are positively related, again confirming to the theoretical conclusions of

the flexible-price version of the monetary models. An increase in the domestic money

supply vis-a-vis the foreign money supply creates an excess supply in the domestic

money market. Domestic prices increase to clear the money market which, via the

purchasing power parity condition, leads to a depreciation of the domestic currency.

92

LOne of the restrictions, implied by the monetary models, that is

frequently tested is the long-run proportionality betrveen money supply and exchange

rate. We have tested this restriction in our cointegrating framework. This test imposes

a value of one to the money supply coefficient and carry out a likelihood ratio test.

Under the null hypothesis of the restriction being valid, the test statistic follows a chi-

square statistic with one degrees of freedom. The calculated value of the statistic is

given as X2 (1)=19.848 withap-valueof 0.000. Thisimpliesthattherestrictionof

long-run proportionality between the exchange rate and money supply gets rejected at

the conventional l' , 5o/o and l0% significance level. However, as noted by

MacDonld and Taylor (1993, 1994) and Choudhry and Lawler (1997), the rejection of

this proportionality restriction does not invalidate the monetary model as a long-run

equilibrium condition given that signs of the cointegrating vector corresponds to those

postulated by the monetary model. We, therefore, conclude that the flexible-price

monetary model is a valid framework for analysing exchange rate behaviour in the

Indian Economy.)

We now focus on the second cointegrating vector which does not

confirm to the economic theory of the monetary models of exchange rute

determination. We present the second long-run equilibrium relationship given as

s =3.7770 + -0.1158(m-m.)+2.0145 (y-y" )-0.5409 (r-r.)(3.ee43) (0.0877)(0.046) (0.767)

(1.3018)(0.2s4)

(1.3851)(0.23e)

First parenthesis gives us the likelihood ratio value of testing the null

hypothesis of no significance of the relevant variable in the cointegrating

relationships. The second parenthesis contains the corresponding p-values. Only the

constant term was turning out to be significant at the SYosignificance level. All other

93

variables are not only incorrectly signed but also statistically not significant as

indicated by a very high p-value. Plots of the two cointegrating vectors clearly bring

out the mean -reverting tendency of the economically meaningful cointegrating

vector. The other co-integrating vector, one not confirming to the monetary model,

does not show the same mean-reverting tendency.

ERROR CORRECTION TERMS

Fotu

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

T

\.aA r^ /\nn.A-^ Ar, ,- rrv vv

^\A^ .J\\^^\- J"w

YEAR

ECMl is the plot of the error correction term which confirm to the Monetary model

ECM2 is the plot of the error correction term which does not confirm to the Monetary model

(iii) Vector Error Correction Model :

We use the economically meaningful cointegrating vector to develop a

vector effor colrection model (VECM) to generate out-of-sample forecasts. Engle and

Yoo (1987) pointed out that forecasts taken from cointegrated systems are 'tied

together'because the cointegrating relations must 'hold exactly in the long-run'. They

demonstrate in a series of Monte Carlo experiments that incorporating cointegration

into the forecasting model, can reduce mearl squared forecast erors by up to 40%o at

94

medium to long forecasts. Lin and Tsay (1996) found that for simulated data imposing

the'correct'unit-root constraints implied by cointegration does improve the accuracy

of the forecasts. However, they obtained a mixed results for the real data sets.

Result of the full sample estimation of the vector error correction

model (VECM) is given in the Table 11. The estimation results are not very

encouraging, except for the error-correction term which is negative (-0.08032) and

significant (t-statistic is -3.1284). This shows that the any deviations from the long-

run relationship gets corrected by 8 percentage points each period. The individual

t-statistics of other regressors are mostly not significant, except the first lag of the

relative interest term. But this is not totally unexpected because of the possible

presence of the multicollinearity problem among the lagged regressors. But what is

really important for the effor-corection model is that the error correction term should

be negatively signed and statistically significant, thereby justifying the estimation of

the error-correction model, depicting the short-run adjustments to the long-run

equilibrium.

Section C : Bayesian Vector Autoregressions

Continuing with our multivariate analysis, we turn to the bayesian

estimation of the monetary models. Selection of priors is the most important task

involved in the bayesian estimation of the vector autoregressive models. Bayesian

model assumes an independent normal prior distribution for each of the coefficients.

"""iV" use the most commonly used Minnesota prior which involves specification of

three "hyper-parameters" - )", overall tightness parameter ; w, symmetric weights

given to the lags of other variables in the equation; and d, decay parameter controlling

95

for the decreasing importance of the lags of the variables. Since the ultimate motive of

developing a bayesian VAR is to generate forecasts which beats the random walk, we

choose that prior which minimises the average of the one to twelve-step mean squared

errors and Theil's U statistics for the out-of-sample forecasts. That is, we estimate the

bayesian model initially with each prior over the period Jan. 1993 - Dec.1996, and

then used a rolling regression method to generate a sequence of one to twelve-step

ahead forecasts. This gives us, for example, 31 one-step ahead forecasts, 29 tfuee-

steps ahead forecasts and so on. We compute the root mean square erors and Theil's

U statistics for each of the forecast horizons and take an average of these statistics.

We choose that prior which gives us the minimum of the ayerage root mean square

erors and Theil's inequality. However, this is also not a very easy proposition as

there could be an infinite number of such combinations. We relied on the past

empirical studies on the bayesian analysis to restrict the prior choice space among the

combinations of )" :0.2,0.1, w :0.4,0.5, 0.6, and d: 1.0, 2.0 (refer Doan (1990),

spencer (1993), Todd (198a)). We use a harmonic decay function to tighten up the

prior with the increasing lags.

The result of this analysis is given in Table 12. We start with the

parameters of the prior recommended by Doan (1990). The overall tightness, I , and

the harmonic lag decay, d, are set at 0.2 and 1 respectively. A symmetric interaction

function - f(ij) - is assumed with w:0.5. The average of the root mean square errors

(RMSEs) and Theil's U statistic is given by 0.05417 and 0.9097 respectively. We then

did a grid search for the optimal values of the prior. We kept the overall tightness

prior equal to 0.2, but reduced the value of w to 0.4. It should be noted that reducing

the value of w, that is, decreasing the interaction, tightens the prior. We have marginal

improvement in terms of average RMSE (0.05376) and Theil's U (0.9035). Keeping

96

the harmonic lag decay fixed at 1.0 and varying the values of )"and w, we find that

), : 0.1 and w:0.4 produces the minimum of the average RMSE and Theil's U. We

then varied the harmonic lag decay, d, parameter to 2. So searching over all the

parameter values we conclude that the combination of ),:0.1, w : 0.5, and d:2.0 is

optimal as this gives us the minimum average RMSEs and Theil's U. We use this

prior to generate the forecasts from the bayesian VAR.

We attempted to estimate the Frankel real-interest differential and

Hooper-Morton model of exchange rate determination. As akeady discussed in the

chapter II, Frankel's model incotporates an expected inflation term in the exchange

rate determination equation, while the Hooper-Morton model additionally introduces

a cumulative trade balance to capture the role of stock-flow interaction in exchange

rate determination. Various proxies have been tried out in capturing the expected

inflation differential between countries, most popular being the long-run government

bond rate differential and past twelve-month inflation differential. However, because

of the lack of time series data on long term government bond rate for India and

inherent backward nature of the other proxies, we decided to fit an ARIMA model to

the inflation series of both the countries and generate twelve-month ahead forecasts as

the proxy for the expected inflation differential. For the cumulative trade balance, we

restricted ourselves to the bilateral cumulative trade balance between the two

countries to remove third counky effect that often get introduced if we take overall

trade balance for both the countries.

Dickey-Fuller and Phillips-Perron test results for the inflation series for

both countries, generated from the consumer price index numbers (CPD, and the log

of the cumulative bilateral trade balance (in logarithm term) are given in the Table 13.

The results indicate that we can reject the null of no-unit root for both the series. So

97

we went on to fit an ARMA model for both the inflation series. Our total sample

period spans 1980:01 to 1999:07, but we adopt a rolling regression method to

generate twelve-month ahead forecasts where our initial estimation period is from

1980:01 to 1993:01. Forecasted inflation series for both the countries turned out to be

stationary.

We face a stumbling block due to the fact that both the expected

inflation differential and the bilateral cumulative trade balance series tumed out to be

stationary. This prevented us from testing the theory in the cointegration framework

in the sense that we cannot include them directly in the cointegration relationship and

check whether their signs confirm to the theory. But still to find out whether the

presence of these variables does play any role in forecasting of exchange rate, we use

these variables in developing a bayesian vector autoregression. We use a non-

informative or flat prior on these I(0) variables in our Minnesota prior selection

framework. We employ the same prior as we did earlier in the bayesian estimation of

the monetary model. We call this model Bayesian VAR with deterministic variable

(BVARD).

As discussed earlier, there is a debate whether it is appropriate to

restrict the coefficients to be equal for the home and foreign countries in the monetary

model. We argued that because of relatively small post-liberalisation period we

refrain from allowing different coefficeints for the two countries. However, the

bayesian approach helps us to avoid the degrees of freedom problem. We therefore

estimate a bayeseian version of the monetary model where we allow the coefficients

of the regressors to vary between the two countries. This we term as an

o' LJnrestricted Bayesian Vector Autoregression (UBVAR) ". One clarification

should be done at this stage. Our usage of the word ounrestricted' have got a special

98

meaning of allowing the coefficients of the regressors to vary across the countries in

the exchange rute determination equation. One should not be counfused by

considering bayesian VAR as unrestricted model as in the bayesian VAR we always

impose restrictions in the form of the prior values of the parameters. We employ the

same grid search method as in the restricted bayesian model selection. The results are

reported in Table 14. The combination of )":0.1, w - 0.4, and d:1.0 is turning out to

be the optimal prior based on the average of the RMSE and Theil's U statistics. We

use this prior specification to generate the forecasts from the unrestricted bayesain

vector autoregression (UBVAR).

Section D : Autoregressive Integrated Moving Average Models (ARIMA)

So far we have discussed multivariate models which require a

relatively large information set for estimation and updating the forecasts from these

models may be a costly procedure. There is one class of univariate models,

popularized by Box-Jenkins, which uses very little information set for estimation and

forecasting and updating forecasts from these models are very easy compared to the

multivariate models. These are autoregressive moving average (ARIMA) models.

These models are popularly used for generating short-term forecasts and they performI

quite well in terms of out-of-sample forecasting performance compared to the

multivariate forecasting models.

First step in building an ARIMA model is to look at the autocorelation

function (ACF) and partial autocorrelation (PACF) functions to have an idea about the

nature of the data generating process. The plots of ACF and PACF are given in the

appendix. Plots of ACF and PACF on the levels of the log of the exchange rate shows

99

that ACF dies down very slowly while PACF has one spike dt lag 1 after which it dies

down very rapidly. Problem with this visual inspection is that this kind of ACF and

PACF behaviour is generated by both an AR(l) process and a simple random walk

process. So it is not possible to conclude from the naked eye whether the series is

stationary or non-stationary. This information is very crucial as ARIMA modeling

presupposes the series to be stationary. ACF and PACF plots of the first difference of

the series shows that both of them dies down very rapidly. This increases the

suspicision that the log of the spot exchange rate series could be non-stationary.

Infact, when we perform the formal tests of unit root on the spot of the exchange rate,

discussed at the beginning of the chapter, it indeed turned out that the log of the

exchange rate series is first-difference stationary.

We proceeded to fit an ARIMA model on the first difference of the log

of the exchange rate series. We give the estimation results for various specification of

the ARIMA models spanning the time period 1993:01 - 1996:12 (our first estimation

period) and 1993:01 - 1999:07 (full sample period) in Table 15. We choose that

ARIMA specification that passes through a battery of model adequacy measures -absence of serial correlation, significant coefficient estimates and satisff the principle

of parsimony. Apart from the above mentioned battery of tests one also needs to keep

in mind that the chosen model satisff the two important requirements of inevitability

and stationarity. Based on all these criterion, we choose the ARIMA (2,1)

specification. This model has been used to generate the forecasts of the exchange rate

and compared with the multivariate forecasts from the VAR models.

100

Section E : FORECAST EVALUATION

Monthly exchange rate forecasts were generated for the simple random

walk (RW), ARIMA, vector error corection model (VECM), bayesian vector

autoregression (BVAR), bayesian vector autoregression with deterministic variables

(BVARD), and unrestricted bayesian vector autoregression (UBVAR) models across

I-,3-, 6-,9-,and l2-month forecast horizons. It should be noted that we work with

the logarithm value of the spot exchange rate and forecasts are also generated for the

logarithm value of the spot exchange rate. For compariosn for forecast performance

we have also developed two multivariate models - a level VAR (LVAR), which is

justifiable given that the variables are cointegrated, and a bayesian vector error

correction model (BVECM), which is nothing but the usual bayesian BVAR

augmented by the effor coffection term. A flat prior is used on the coefficient of the

elror colTection term.

Evaluation tests were carried out over the out-of-sample period

1997:01-1999:07- We adopt a rolling regression estimation methodology for

generating the out-of-sample forecasts. In this method we initially estimate all the

models over the period 1993:01-1996:12 and forecasts are generated for the period

1997 01-1999:07. Next, we increase our estimation period by adding one sample

point and reestimate all the models for the period 1993:01-tr997:01and based on this

we generate our out-of-sample forecasts. This forecasting strategy gives us 31 one-

month ahead ,29 three-month ahead, 26 six-month ahead, 23 nine-month ahead, and

20 twelve-month ahead forecasts. For the error-coffection models, however, the

cointegrating vector was obtained from the full sample estimation and was fixed at

their long-run values while estimating the models for the various sub-samples.

101

(i) Descriptive Statistics :

Descriptive statistics on forecast accuracy are reported in Table 16.

Forecast error is calculated as the spot rate minus the forecast rate. Table reports two

most frequently used descriptive statistics - root mean square error (RMSE) and

Theil's U statistic (U). It also reports three other statistics that also sometime appear in

the forecasting literature - mean absolute error (MAE), root mean square percentage

error (RMSPE) and mean absolute percentage error (MAPE).

One Month Forecast Horizon :

We first look at the one-month ahead forecast performance of the alternative models.

One of the most challenging task that a forecaster faces is to generate forecasts that

beats the naiVe forecasts charucteized by simple random walk forecast, especially at

the short forecast horizon. The most popular statistic to find out these is Theil's

inequality statistic. The model which beats the random walkforecast has a value of U

less than.l. In case of one-month ahead forecasts, only BVECM model has U>1.

VECM produces the best forecasting performance with U:0.86629, followed by

UBVARI, BVARD and LVAR. In terms of RMSE, again, VECM produces the best

forecast as it possesses the minimum RMSE (0.01287). This is less than the RMSE of

the RW forecast (0.01486). ARIMA occupies the second position with a RMSE value

of 0.013943 followed by UBVAR with RMSE of 0.013971. LVAR, BVAR, and

BVARD very marginally outperforms the RW model in terms of RMSE. In terms of

other three reported statistics, VECM and ARIMA consistently outperforms the RW

model in the one-month horizon. For the other models, the picture is not that clear.

For instance, in terms of RMSPE, LVAR, BVAR, BVARD, BVECM and UBVAR

102

BVECM and UBVAR outperforms the random walk, while all of them show a poor

performance in terms of MAE and MAPE. We, however, rely on the Theil's U and

RMSE to conclude that most of the models beat the random walk forecasts at the one-

month horizon with VECM model posting an all-round impressive performance

followed by ARIMA, UBVAR, BVAR, BVARD and LVAR.

Summary of One Month Ahead Descriptive Statistics

Noie : The numbers in the cells are the rankings of the various forecasting models

across various forecast evaluation measures at this forecast horizon.

Three Month Forecast Horizon :

We now move on to the three-month ahead forecasting performance

analysis of the competing models. Here the rankings of the various models changes

substantially compared to the one-month ahead forecasts. VECM, which was tuming

out to be the best model in terms of one-month ahead forecasting performance, posted

a very poor perfofinance in the three-month horizon. Indeed, it failed to beat the

random walk forecasts in terms of RMSE and also had a Theil's U value greater than

one. ARIIMA model continued with its impressive performance by beating the

I I t is worth to point out again that the term Unrestricted Bayesian VAR has been used in this paper toindicate that the coefficients of the regressors in the exchange rate determination model are allowed to

Descriptive

StatisticsRW ARIMA VECM LVAR BVAR BVECM BVARD UBVAR

RMSE

THEIL'S U

MAE

RMSPE

MAPE

r03

random walk model in terms of all the descriptive statistics. But it was the IIBVAR

which produced the best forecast in this horizon by posting the lowest RMSE

(0.029712) and Theil's U (0.90191) among the competing models. BVAR, BVARD

and LVAR also outperformed the RW model in terms of RMSE and Theil's U. For

other statistics, the situation is the same as the one-month ahead forecasts with the

models out-performing the random walk model in terms of RMSPE, while

performing worse in terms of MAE and MAPE. Thus we can conclude that in the

three-month horizon, it is the UBVAR which occupies the top rank followed by the

ARIMA, BVAR, BVARD, and LVAR. BVECM continued to show a poor

performance in the three-month ahead forecast horizon also while VECM joined this

group.

Summary of Three Month Ahead Descriptive Statistics

Note : The numbers in the cells are the rankings of the various forecasting

across various forecast evaluation measures at this forecast horizon.

Descriptive

Statistics

ffiTHEIL'S U

MAE

RMSPE

MAPE

RW

)

5

2

5

2

ARIMA VECM LVAR BVAR BVECM BVARD UBVAR

2

2

1

I

I

8

8

7

8

6

6

6

6

7

7

aJ

J

4

4

5

7

7

8

6

8

4

4

5

J

4

1

1

J

2

J

vary across the countiries.vary across the countrres.

104

Six Month Forecast Horizon :

At the six-month ahead forecast horizon, it is the UBVAR model

which gave the best forecasting perfonnance in terms of the descriptive statistics.

VECM, which posted a very poor perfofinance at the three-month horizon, bounced

back to register an impressive performance at the six-month horizon by beating the

random walk forecast both in terms of RMSE and Theil's U statistics. ARIMA

continued with its impressive performance by occupying a close second position after

the UBVAR model. BVAR also outperformed the random walk forecasts at this

horizon. BVECM continued with its dismal performance at the six-month horizon

while the LVAR forecasts were unable to beat the random walk forecasts. In terms of

RMSPE, all the models which have U<1, beats the random walk forecasts. Contrary

to earlier forecast horizon, ARIMA and UBVAR beats the random walk forecasts in

terms of MAE and MAPE criterion also. So we can conclude that at the six-month

forecast horizon AR[MA, UBVAR, BVAR, and BVARD outperformed the random

walk forecasts at the six-month ahead forecasts.

Summary of Six Month Ahead Descriptive Statistics

Note : The numbers in the cells are the rankings vanous

across various forecast evaluation measures at this forecast horizon.

DescriptiveRW ARIMA VECM LVAR BVAR BVECM BVARD UBVAR

Statistics

RMSE

THEIL'S U

MAE

RMSPE

MAPE

6

6

aJ

6

J

2

2

1

1

1

5

5

6

J

4

7

7

7

7

7

J

aJ

4

5

6

8

8

8

8

8

4

4

5

4

5

1

I

2

2

2

105

asting models

At the nine-month ahead forecast horizon, ARIMA, VECM,

BVAR, BVARD and UBVAR models beats the random walk forecasts by the virtue

of having a Theil's U statistic value of less than one and RMSE less than the

corresponding figure for the random walk model. UBVAR model again outperformed

all other models in terms of RMSE (0.06398) and Theils U statistics (0.785805).

VECM comes a close second with a RMSE value of 0.06615 and U:0.812410.

BVAR model occupies the third position followed by the ARIMA and BVARD.

LVAR and BVECM model performed very poorly and was unable to beat the random

walk model as they have a Theil's U value of greater than one. In terms of RMSPE

and MAE the models with U<l beats the random walk model, while in terms of

MAPE, ARIMA, VECM and UBVAR model beats the random walk forecast. So at

the nine-month ahead forecast horizon I-IBVAR model outperforms all other models

in terms of all the statistics followed by the VECM.

Summary of Nine Month Ahead Descriptive Statistics

Note : The numbers in the cells are the rankings of the various forecasting models

across various forecast evaluation measures at this forecast horizon.

Descriptive

Statistics

mTHEIL'S U

MAE

RMSPE

MAPE

RW ARIMA VECM LVAR BVAR BVECM BVARD UBVAR

6

6

6

6

4

4

4

2

J

1

2

2

J

1

2

7

7

7

7

7

J

aJ

5

5

5

8

8

8

8

8

5

5

4

4

6

1

1

1

2

J

106

Twevle Month Forecast Horizon :

At the twelve-month forecast horizon, the rankings of the models

remains unchanged compared to the nine-month ahead forecasts.'VECM model

outperformed all other models including the random walk model, followed by the

UBVAR, BVAR, BVARD, ARIMA and LVAR. A11 of these models also beats the

random walk forecasts.

Summary of Twelve Month Ahead Descriptive Statistics

Note : The numbers in the cells are of the varibus forecasting models

across various forecast evaluation measures at this forecast horizon.

Major ftndings From Descriptive Statistics :

considered.

better than the univariate benchmark model across the forecasting horizon.

However, no one multivariate model consistently out-predicts the forecasts from

the ARIMA model. At one month forecast horizon it is the VECM, at three and

six month it is UBVAR, at nine month they are IIBVAR, VECM and BVAR,

DescriptiveRV/ ARIMA VECM LVAR BVAR BVECM BVARD TIBVAR

Statistics

RMSE

THEIL'S U

MAE

RMSPE

MAPE

7

7

7

6

5

5

5

4

6

6

6

7

7

1

3

2

aJ

aJ

8

8

8

8

8

4

4

J

5

6

2

2

5

2

2

4

4

107

while in the twelve month they are VECM, IIBVAR, BVAR and BVARD. A11

these rankings are based on RMSE and Theil's U statistic.

The performance of the ARIMA model deteriorates over the longer forecast

horizon. This is expected as ARIMA model is specifically used to generate

forecasts over the short and medium forecast horizon. Infact, it consistently

occupies the second position at one, three, and six month ahead forecast horizon.

VECM vs Bayesian Vector Autoregression : No clear cut conclusion emerges

from the present analysis - it depends on the forecast horizon. At one month and

twelve month forecast horizon, it is the VECM which outperforms all the bayesian

models. However, at other forecast horizon bayesian models, specifically,

UBVAR, outperforms the VECM. Again, this conclusion is based on the RMSE

and Theil's U statistics.

Among the bayesian models, it is the UBVAR which is turning out to be the best

performer across the forecast horizon followed by the BVAR.

) Forecast performance of all the models deteriorate as we move to the future which

is evident from the increasing RMSE, RMSPE, MAE and MAPE. This confirms

to the theory that forecast errors increases as we try to generate forecast in the

distant future.

From the above discussion on the descriptive statistics, we conclude

that BVECM and LVAR are not performing as well compared to the other competing

models. So we drop these two models from our further analysis. We concentrate on

the following 5 models - VECM, BVAR, BVARD, UBVAR and ARIMA - in our

further analysis of the quality of the forecasts.

108

getting accepted at 10 , 5o/o and 10% signifrcance level. For the other models, the null

of unbiasedness is getting accepted at only 1% significance level and not at 5o/o or

t0% significance level. The efficiency test results arc far more encouraging compared

to the one-month forecast horizon. For all the models null of efficient forecasts are

getting overwhelmingly accepted. Thus, we can conclude that at three-month forecast

horizon forecasts from all the competing models are both unbiased and efficient.

We move on to the nine-month forecast horizon (Table 19.3). The

results are not very encouraging at this forecast horizon. Except the forecasts from

the ARIMA model, none of the model produces unbiased forecasts. Even for the

ARIMA model, the null of unbiasedness of forecasts get accepted at 1o/o significance

level. Moving on to the efficiency test, the results are agun turning out to be negative.

None of the forecasts are tuming out to be efficient.

Summary of Unbiasedness and Efficiency Tests :

unit root. This holds across all the forecast horizon.

Actual and Predicted series are cointegrated, at one, three and nine month forecast

horizon. While they are cointegrated by both the Engle-Granger and Johansen-

Juselius methodology at one and nine month ahead forecast horizons, they are

cointegrated only by the Johansen-Juselius method at six month forecast horizon.

At one month ahead forecast horizon, none of the forecasts are unbiased. Only

ARIMA and VECM turned out to be efficient (Mincer-Zarnowitz) at this forecast

horizon.

111

A t*h:d+BP,*o*€,*r,

where A ,*o is the actual series, P ,*o is the predicted series and h is the forecast

horizon.

The null of unbiasedness is given by the joint hypothesis of

a,=0, and F =l

while, the null of efficiency is given by

f=lGiven the existence of cointegration between the actual and predicted

series, it is natural to test for the above hypothesis as restrictions on the co-integrating

vector. The restrictions on the co-integrating vector could be thought of as the

following restrictions : (1 -1 0) as the unbiasedness restrictions and (1 -1) as the

efficiency restrictions, where the elements in the bracket corresponds to the actual

series, predicted series and a constant respectively. Results are given in Table 19.

Table 19.1 reports the result for the one-month ahead forecast horizon. As far as the

bias test is considered, none of the model's forecasts are turning out to be unbiased as

the calculated value of the chi-square statistic exceeds the critical value at the

conventional significance levels. In case of the efficiency test, ARIMA and the

VECM's forecasts are turning out to be efficient as the null hypothesis cannot be

rejected at lo/o,5o/o and 10% significance level for ARIMA and lo/o for the VECM.

All other tests fails to pass this test. Thus, at one-month forecast horizon, only

AzuMA and VECM qualiff among the competing models as far as the efficiency

criterion is considered. All other models fails to pass both the tests.

At the three-month forecast horizon, the unbiasedness and efficiency

criterion is satisfied by all the competing forecasts (Table 19.2). We first focus on the

unbiasedness test results. For the ARIMA and VECM the null of unbiasedness is

110

(ii) Forecast Rationality : Unbiasedness and Efficiency Tests

(a) Parametric Tests :

It is important to consider the time series properties of the actual and

the predicted series with the advent of the unit root and cointegration literature

(Cheung and Chinn, 1998 ; Hendry and Clements, 1993). Two of the prime

requirements for ' consistent ' forecasts are (1) actual and predicted series should be

integrated of the same order, and (2) they should be co-integrated. Otherwise the

forecast effors will have unbounded variance. The summary results of the unit root

tests, across the forecast horizon, are reported in Table 17. We employ the traditional

Dickey-Fuller and Phillips-Perron test of unit root to assess the order of integration of

the actual and predicted series. The results indicate that the actual and predicted series

are integrated of order 1, that is, they contain a stochastic trend. Thus, the first

requirement of consistent forecast, viz., the actual and the predicted series are

integrated of the same order gets satisfied. We employ the Engle-Granger and

Johansen-Juselius cointegration test to find out the existence of cointegration between

the actual and predicted series. The summary results are given in Tables 18. Results

indicate that the actual and the predicted series are co-integrated by both EG and JJ

methodology at one- and nine-month forecast horizon, while it is co-integrated by the

JJ methodology at the three-month forecast horizon but not by the EG method.

However, the null of no-cointegration could not be rejected by EG and JJ method at

six- and twelve-month forecast horizon. For all the cases where co-integaratiotn

exists, there is only one co-integrating vector.

Two of the desirable properties underlying a rational forecasts are

unbiasedness and efficiency properties of the predicted series (refer chapter IV). To

test for the unbiasedness and efficiency we consider the following regression

109

unbiased or efficient.

efficiency tests could be attributed to the relatively small out-of-sample forecast

horizon. Given alarge out-of-sample forecast period, one may able to overturn the

negative results obtained above.

(b) Non-Parumetric Tests :

So far we have discussed the parametric test of unbiasedness

and effrciency and the results have been, in general, very discouraging. However, as

discussed in the chapter IV, parametric unbiasedness test does not perform well in a

small sample size. Our relatively small sample period of 3l forecasted values is a

good justification for carrying out non-parametric tests of unbiasedness which has

better small sample properties. We employ two non-parametric tests - sign-test and

wilcoxon rank-sum test. While the former tests for median unbiasedness rather than

mean unbiasedness, the later one, under the assumption of symmetric distribution of

forecast elrors, tests for mean unbiasedness. The results of these tests are reported in

Table 20. Sign-test is a two tail test where we reject the null hypothesis if S <r or if

S>n-f, where n is the number sf *'ve and -'ve errors and t is the critical value. In

case of one-month ahead forecast errors, t:9.993. So reject the null of median

unbiasedness at 5oZ significance level if S is less than equal to 9.993 or if S is greater

than equal to 21.007 .In case of ARIMA forecasts at one-month forecast horizon the

calculated value of S is 15 which falls in the no-rejection zone. We, therefore,

conclude that ARIMA forecasts are unbiased based on the sign test.

As mentioned earlier, we also calculate the Wilcoxon rank sum test to

check for median unbiasedness which under the assumption of s5rmmetric distribution

tt2

also implies mean unbiasedness. We reject the null of mean (: median) : 0 at the

significance level a if thecalculated uul'rr" of the statistic $fRS) exceeds W ,-o,, or

if it is less than W orz. If the calculated value of WRS is between W orz arrd W ,_o,,

or equal to either quantile, accept the null hypothesis. In case of one-month ahead

forecasts, we reject the null hypothesis if WRS > 348 or if WRS < 148, otherwise we

accept the null hypothesis. For ARIMA forecasts, the calculated value of WRS is 273

which does not fall in the critical region. Thus we can conclude from the Wilcoxon

rank-sum test that the ARIMA forecasts are mean unbiased. So both the non-

parametic test of biasedness gives us the same result that ARIMA forecasts are

unbiased at one-month ahead forecasts. This is in contrary to the parametric test of

unbiasedness where the nullof unbiasedness was strongly getting rejected. Given the

fact that these tests have good finite-sample power and are insensitive to deviations

from the standard assumptions of normality and homoscedasticity that are very

critical for carrying out the parametric tests, we conclude that the ARIMA forecasts

are indeed unbiased.

We carried out non-parametric unbiasedness test, both sign and

Wilcoxon rank-sum tests, for the forecasts generated from the VECM, BVAR,

BVARD and UBVAR. The results are reported in the Table 20. The result clearly

indicates that for none of these models the forecasts were biased at the one-month

forecast horizon, except the VECM forecasts. In case of the forecasts from the

VECM, the null of unbiasedness is getting rejected by the Wilcoxon rank-sum test.

However, the sign test accepts the null of median unbiasedness. So, in general, the

nonparametric test overturns the conclusions of the parametric tests of unbiasedness

discussed earlier where none of the forecasts are turning out to be unbiased. We

refrain from performing the non-parametric bias test as either the test procedure could

ll3

be very conservative, even asymptotically, or it could have avery low power (Diebold

andLopez (1996)).

Summary of the Rationality Tests :

for all the models at one month ahead forecast horizon. At three month forecasts

horizon all the models produce unbiased forecasts. Again, at the nine month

ahead horizon, none of the model's forecasts are tuming out to be unbiased.

of forecasts from all the models, except the VECM. For forecasts from the

VECM, while the sign test does not reject the null of unbiasedness, the Wilcoxon-

rank sum test does reject the null. Given that the Wilcoxon-rank sum test assumes

that forecasts errors are symmetrically distributed, one may fall back on the sign

test to conclude that forecasts are unbiased.

form ARIMA and VECM are efficient. None of the other model's forecasts are

efficient. In case of three month ahead forecast, all the forecasts are turning out

to be efficient. However, at the nine month ahead forecast horizon, the results are

again tuming out to be negative.

(tii) Equality of Forecast Eruors Test :

Diebold and Mariano (1995) pointed out that mere looking at the

various forecast accuracy statistics and concluding that one model is outperforming

the other is not correct. We employ a test developed by them which tests for the null

hl.pothesis of no difference in the accuracy of the two competing models. We use the

tt4

loss-function of the form g (e) = e' , which allows one to test whether there is any

significant difference between the root-mean-square effors of the competing models.

This is important because it may so happen that in terms of root-mean-square emors

some model may be outperforming the random walk model but there may not be any

statistically significant difference between the two root-mean-square elrors.

We present the results of the Diebold-Mariano (DMS) test in Table2l.

Let us first consider the one-month ahead forecasts. The calculated value of the DMS

for the pair of simple random walk (SRW) model and the ARIMA model is 1.2535,

which is insignificarrt at 10% significance level. (This test statistic, under the null

hypothesis, asymptotically follows a standard normal distribution, and it is a two-tail

test). This implies that there is no significant difference between the forecast error of

the SRW and ARIMA models. The DMS between the SRW and VECM is -3.7199

which is statistically significant at 10% significance level, implying that there is

significant difference between the forecast errors of these two models. This in turn

implies that the RMSE for the VECM is different from that of the SRW model. Given

that the RMSE of the VECM is lower than that of the SRW model, we can conclude

from the Diebold-Mariano test that the RMSE of the VECM is significantly lower

than that of the SRW model at one-month ahead forecast horizon. However, for other

models - LVAR, BVAR, BVARD and tiBVAR- the DMS is not statistically

significant implying that the forecast effors of these models are not significantly

different from that of the SRW model.

We proceed to test for the equality of the forecast errors at other

forecast horizon between simple random walk forecasts and other competing models.

The results are again not very encouraging. At the three-month horizon, forecast

erors of none of the competing model is significantly different from that of the

115

simple random walk forecast. At the six-month ahead forecast horizon, agun, none of

the model's forecast error is different from that of the simple random walk model. At

the nine-month ahead forecast hoizon, only the ARIMA model's forecast errors is

statistically different from that of the simple random walk model. Given that it has a

lower RMSE (infact, lowest among the competing models) we can conclude that

ARIMA model is significantly out-performing the simple random walk model. At the

twelve-month ahead forecast horizon, it is again the ARIMA model that has got

statistically significant different RMSE from the simple random walk model.

Summary of the Diebold-Mariano Test :

significantly different from the SRV/ model. Forecast effors are also significantly

different (as a pairwise) between the ARIMA and VECM, VECM and BVAR,

VECM and BVAR, and between VECM and UBVAR. Difference of forecast

effors also imply that there is statistically significant difference between the root

mean square elTors from these models.

At three month ahead forecast horizon, test fails to provide evidence of

significantly different forecast effors among the competing models.

At six month ahead forecast horizon, none of the model's forecast errors are

statistically different from that of the random walk model. Only cases where

forecast effors were significantly different from each other are those of BVAR and

UBVAR, and BVARD and LIBVAR.

At the nine month ahead forecast horizon, it is only the ARIMA model which

produces forecast erors that are significantly different from that of the SRW.

ll6

At the twelve month ahead forecast hoizon, again, none of the model's forecasts

are significantly different from those of the random walk model.

(iv) Information Conterut Test :

Fair and Shiller (1989, 1990) noted that the superiority of a particular

model in terms of forecast accuracy does not necessarily imply that the forecasts from

other models contain no additional information. Moreover, when the RMSEs are close

for two forecasts (this is particularly true in our study), little can be concluded about

the relative merits of the two. This led Fair and Shiller develop an Information

Content Test to find out whether one set of forecasts has more 'information' than the

competing models. This will help us to conclude whether the forecasts from the

competing model has more information relative to the simple random walk model.

The results of the information content test is given inTable22.

At the one month forecast horizon, only the forecasts from the VECM

have more information than the simple random walk forecasts. Considering the

ARIMA and VECM as a pair, we found that the coefficient of the ARIMA term is

insignificant whereas the VECM term is statistically significant at 5Yo significance

level. This implies two things. One, VECM has information beyond that provided by

the simple random walk model. Second, information of ARIMA model is completely

contained in VECM and VECM contains further relevant information than the

ARIMA model. Other pairs with the ARIMA model do not have coefficients

significant implying that none of the model contains any information useful for the

one-month ahead forecast of the spot exchangerate. Considering other combinations

with the VECM, viz., (VECM,BVAR), (VECM,BVARD) and (VECM,UBVAR), we

tt7

find that in all these cases the coefficient of the VECM term not only to be positively

signed but also statistically significant at 5%o significance level. However, the

coefficient of the other variable with the VECM is always turning out to be

statistically insignificant. For the BVARD and UBVAR pair, the coefficient of the

BVARD term was significant but negatively signed which " is a perverse result in

economic terms as this implies that the information in the BVARD models is

negatively correlated with the actual exchange rate changes " (Liu, Gerlow and

Irwin (1994)).

Beyond one-month ahead, the results of the information content test is

not very encouraging. At all other forecast horizon, viz. tltree to twelve months, either

the coefficients are statistically insignificant or they are significant but negatively

signed implying a perverse relation in economic terms. This implies that none of the

models at more than one-month ahead forecast horizon contains information which is

useful to forecasting the spot exchange rate beyond that provided by the simple

random walk model. It should be noted that forecast errors are autocorrelated of order

MA(k-l), where k is the forecast horizon, for which we have used the Newy-West

heteroscedastic-autocorrelation consistent estimator for carrying out information

content test beyond one month horizon.

Summary of the Information Content Test :

(VECM) forecasts which contains 'information' beyond that contained by that of

a random walk model. 'Information' from all other model are contained in the

VECM forecasts.

ll8

encouraging. None of the model contained any 'information' beyond that of the

simple random walk model.

(v) Direction-of-change Forecast Analysis :

Direction-of-change forecasts are often used in financial and economic

decision making (e.g. Leitch and Tanner, 1991). The question as to whether a

direction-of-change forecast has value involves comparison to a naive benchmark -

the direction of change forecast is compared to a ' naive ' coin flip. Cumby and

Modest (1987) developed a test based on Merton's (1981) work for evaluating the

direction-of-change forecasts. They tested the null hypothesis that a direction-of-

change forecast has no value by testing the null of independence between the actual

changes and forecasted changes. The test statistic follows a chi-square distribution,

under the null of independence, with one degree of freedom.

Results are reported in Table 23. At one-month forecast horizon, none

of the models predicted changes in the spot exchange rate accurately. This is because

none of the calculated value of the chi-square statistic are statistically significant. The

picture improves somewhat at the three month ahead forecast horizon. Here forecasts

from the VECM and UBVAR models predicted changes in the exchange rate that has

some value to the consumers as both of them has calculated chi-square statistic which

is very marginally significant at l0o/o significance level. It should be mentioned again

that predicted changes has 'value' if the actual and predicted changes are not

independent. At the six-month ahead forecast horizon, it is only the VECM which

predicted changes in the exchange rate that has got some value in the sense that the

119

actual changes and predicted changes of the exchange rate are statistically not

independent. At nine- and twelve-month forecast horizons, the direction-of-change

forecasts results are again not positive since all of the models produces forecasts that

is statistically independent from the actual changes.

Summary of the Direction-of-Change Analysis :

competing models are statistically independent from the actual changes in the

exchange rate.

depreciation and appreciation of the spot exchange rate that is not statistically

correlated with the actual changes.

has some 'value' in the sense that the predicted changes in the exchange rate is not

statistically independent from the actual changes.

Major Findings of the Forecusting Exercise :

Across the forecast horizon, that is, from one to twelve month ahead forecasts,

simple random walk forecasts are beaten by most of the models as shown by the

descriptive statistics such as root mean square effors (RMSE), Theil's U

statistic (U).

The benchmark univariate ARIMA(2,1,1) model produced better short and

medium period forecasts, viz. one, three and six month ahead forecasts, as

indicated by RMSE and U ; however, its performance deteriorates in the long-run

forecast horizon of nine and twelve months.

120

The multivaiate models able to outperform the ARIMA model across the forecast

horizon. At one month horizon, it is the vector effor correction model (VECM), at

three and six month horizon it is the Unrestricted Bayesian VAR (IJBVAR)2, at

nine month horizon they are VECM, UBVAR, and BVAR, while at the twelve

month honzon they are VECM, BVAR, BVARD and UBVAR.

Among the multivanate models, the performance is mixed in terms of out-

performing the random walk model. While BVAR, BVARD and TIBVAR

consisitently outperforms the random walk across all the forecast horizon in terms

of RMSE and U, BVECM always performs worse than the random walk forecasts

in terms of these descriptive statistics.

VECM and LVAR have shown mixed performance. VECM ou@erforms the

random walk model at all forecast horizon, except at the three month horizon.

LVAR outperforms the random walk forecasts only at one and twelve month

forecast horizon.

From the analysis of the descriptive statistics, it can be concluded that none of the

model performs best across all the forecast horizon. While VECM forecasts

occupies the first position in one and twelve month forecast horizon, while

UBVAR occupies the top slot in three, six and nine month forecast horizon.

VECM forecasts are turning out to be efficient at one month ahead forecast

horizon, although there is no clearcut evidence on the unbiasedness of these

lorecasts at this horizon.

At one month ahead forecast horizon, other models which includes ARIMA,

BVAR, BVARD and UBVAR, turning out to be unbiased in terms of non-

parametric tests of unbiasedness.

2 It is worth to mention again that we use " Unrestricted Bayesian VAR " to point out the fact that in

121

All the forecasts satisfy the condition of unbiasedness and efficiency at three

month ahead forecast horizon. Although VECM performs very poorly in terms of

the descriptive statistics at this horrzon, it is also turning out to be unbiased and

efficient.

At one month ahead forecast horizon, VECM forecast erors are statistically

different from all other model's forecast elrors including that of the random walk.

This evidence, combined with the fact that VECM has the minimum RMSE,

implying that it is significantly lower than the competing models.

In terms of the information content test, also, it is the VECM which gives the best

performance at the one month forecast horizon. VECM forecasts contained

'information' beyond those contained in other competing models including the

random walk.

a As regards the performance of the models in correctly predicting the appreciation

and depreciation of the exchange rute, it is the VECM and ARIMA models which

significantly predicts the same at the three month forecast horizon. This

corroborates our earlier hypothesis that VECM forecasts being unbiased and

efficient, inspite of having very poor descriptive statistics, could only imply that it

is correctly predicting the increase/decrease of the exchange rate.

From the above discussion of the various tests of forecast evaluation,

one can conclude that we can show some confidence in the forecasts at one and three

month forecast horizon. At one month ahead forecast horizon, VECM forecasts has an

edge over its competitors in terms of RMSE, Theil's U, Efficiency and Unbiased test,

Diebold-Mariano statistics and Information Content Test. At three month forecast

this Bayesian VAR formulation we allow for the coefficients to vary across the country.

122

horizon, has an edge over its competitors in terms of RMSE, Thsil's u,

Efficiency and Unbiased testg and direction-of-change analysis. Beyond three month

forecast horizon, one has to check the forecast output very carefully as there is no

clearcut evidence on the quality of the forecasts. This is not wholly rmexpected.

Forecasts of a furancial variable l1ke the exchange rate over the long frrecast horizonI

is, in all probability, could go haywire

123

Chupter VII

Conclusion

The forecasting perfornance of exchange rate models has received

considerable attention since the breakdown of the Bretton Woods, when exchange rate

began to float. The debate opened up with the work of Meese and Rogoff (1983),

whose work on exchange rate forecasting of various theoretical and atheoretical

models of exchange rate determination brought out the poor out-of-sample forecasting

performance of the asset market models. In fact, the most serious result was that the

naiVe forecasts of the simple random walk model outperformed the various asset

market models in out-of-sample forecasting performance. Lot of research input has

gone into overtuming the conclusions reached by the Meese and Rogoff. The present

study is an attempt in that direction and can be thought as a culmination of the earlier

works by the author (Bhattacharya1998,1999).

In this dissertation we develop forecasting models involving both

univariate and multivariate time series techniques with the aim of outperforming the

random walk model in terms of out-of-sample forecasting performance. We consider

the monetary models of exchange rate of determination - flexible price monetary

model, sticky price monetary model, real interest differential model and Hooper-

Morton model. A11 these models are essentially'monetary' in nature - they assume

that exchangerate is determined by the relative demand and supply of the two monies

as exchange rate is thought to be as nothing but the relative price of two monies.

With the advent of the cointegration technique there is a new fillip to

the on-going empirical analysis of the asset market models. Exchange rate is now

considered to be determined , in the long-run, by the economic fundamentals. It is

hypothesized that if the asset market models hold, then there must be a cointegrating

t24

relationship between the exchange rate and the variables determining it. We

,therefore, expected to find a long-run equilibrium relationship between exchange

rate, money supplies, interest rates and outputs as postulated by the monetary models

of exchange rate determinhtion. The presence of cointegration among these variables

confirmed the monetary models as a valid framework for evaluating exchange rate

beltaviour.

The next question that needs to be addressed is which version of the

monetary model is supported by the empirical study. This could be determined by

signs of the variables in the cointegrating vector. Our empirical study supports the

flexible-price version of the monetary model of exchange rate determination. Given

the existence of an economically meaningful cointegrating vector, we proceed to

develop an effor correction model which would capture the short-run adjustments

towards the long-run equilibrium. Finally, forecasts are generated from this vector

error coffection model (VECM).

Alternative time series models are also developed for generating

competing forecasts. A commonly used univariate time series model is developed by

using the Box-Jenkins methodology. These class of models have empirically shown to

produce good short and medium-run forecasts. Other multivariate models developed

in the present study are bayesian in nature where a modeler's prior beliefs are

incorporated in the model estimation. In the present study we developed various

bayesian vector autoregressive models by using the Minnesota prior developed by

Litterman (1979).

Four different bayesian VAR model have been developed. First is the

bayesian VAR (BVAR) model which is simply the bayesian counterpart of the

monetary model used in the cointegration analysis. Second one incorporates the

t2s

expected inflation and cumulative bilateral trade balance between India and USA as

deterministic variables by using a flat prior on them in an otherwise BVAR model.

This we name as bayesian VAR with deterministic variable (BVARD). This is

basically a Hooper-Morton model developed in the bayesian framework. Third model

in the bayesian class is what we call a bayesian vector elTor correction model

(BVECM) which incorporates the error correction term (with a flat prior on it) in the

BVAR model, where the error correction term is obtained from the cointegrating

relationship. Finally, we develop the unrestricted bayesian vector error correction

(UBVAR) model. The term ' unrestricted ' has been used to denote the fact that we

have allowed the coefficients to vary across the countries in the bayesian framework.

That is, instead of assuming Minnesota prior on the coefficient of the, say, relative

money supply, we now assume the same prior on separate coefficients of the money

supply term of each country.

We employ a rolling regression method of estimation and generated

the out-of-sample forecasts from the competing models. Forecasts are generated for

one, three, six, nine and twelve month ahead forecasts. We carried out a battery of

tests to assess the quality of the forecasts from the competing models. Among the

tests we used to track the performance of forecasts includes various descriptive

statistics like root mean square errors, Theil's U statistic, unbiasedness and efficiency

tests, Diebold-Mariano equality of forecasts eror test, information content test and

direction-of-change analysis. We use the time-series properties of the actual and

predicted series to evaluate the forecasting performance of the competing models.

On the basis of various forecast evaluation criterion we conclude that

at one month ahead forecast horizon, it is the vector effor corection model (VECM)

which outperforms all the models including the simple random walk model, while at

126

the three month horizon, it is the unrestricted bayesian VAR (UBVAR) which

occupies the first position. Although beyond three month horizon, multivariate models

based on economic fundamentals as well as the univariate ARIMA model

outperforms the random walk forecasts in terms of the various descriptive statistics,

these forecasts do not seem to satisff the desirable properties ofa good forecast. This

lead us to conclude that beyond the three month horizon one should interpret the

forecast results very carefully.

We conclude here by noting down few limitations of the present study.

implicitly imposed the restriction of equal coefficienls between the domestic and

foreign country by working with relative money supplies, relative outputs and

relative interest rates. This may be a very restrictive assumption. Because of our

relatively small sample size, given by the post-liberalisation period, we are forced

to take this route to avoid degrees of freedom problem.

dependent on the definition of the money supply employed. With M3 as the

measure of the money supply, we are unable to find any long-run equilibrium

relationship that confirms to the monetary models of the exchange rate

determination.

because of the absence of time series data on the three month treasury bill rate for

India. Regular auction of the Indian three month treasury bill rate takes place from

January 1993 only.

t27

We also tried out call money rate in the case of India as the measure of the short-

term interest rate which gave us alarger sample size, that is, from August 1991 to

July 1999. However, we are unable to find any long-run equilibrium relationship

by using either M3 or Ml as the measure of the rironey supply.

Thus, it seems that the existence of the monetary model as a long-run equilibrium

is not very robust to the choice of the variables included in the determinants of the

fundamentals.

exchange rate determination. One would like to test for the portfolio balance

model of exchange rate determination which allows for imperfect substitutability

of assets. However, we refrain from doing this analysis as data on the breakdown

of asset holding by economic agents were not available.

To sum up, in this dissertation we had undertaken an exercise to find

out the forecasting performance of the monetary models of the exchange rate

determination. Our prime motive was to obtain forecasts from the monetary models

of the exchange rate determination which beats the simple random walk forecasts.

We have been successful in generating forecasts from the flexible-price monetary

model which beats the simple random walk forecasts. However, our analysis is

restricted by the relatively small post-liberalisation period in India. In future, we

would like to evaluate the forecasting performance of the portfolio balance models of

the exchange rute determination. Given the on-going research on the market

microstructure approach, we would like to model the agent's behaviour in the foreign

exchange markets which would help us to better forecast the exchange rate

movements than the traditional macroeconometric approach.

r28

Table: 1.1@)

PHILLPS-PERRON TEST

Models:

(l)!t = d.o * d.rta * d.z(t - Tl2) + p1

(2) y, = a*o *a*, !,-t*lt,

Time Period : Janua.ry 1993 - July 1999

Levels

ModelNull HypothesisTest Statistic

Exchange Rate

Indian IIP

Indian CPI

I,ndian Treasury Bill Rate

Indian Ml

US IIP

US CPI

US Treasury Bill Rate

US M1

(1)

H o:d, --lZ (tdr)

-2.2456

-1.7275

-2.1986

-t.4379

-2.0032

-2.t257

-1.4693

-t.0721

-3.6085

(2)

H o:a't =lZ(t a.)

0.2886

-1.7648

-0.9084

-1.3888

-2.6941

-0.4101

-1.5310

-r.8529

-2.3792

Table : 1.2(A)

DICKEY-FULLER TEST

Time Period : February 1993 - Juty 1999

First DifferencesModelNull HypothesisTest Statistic

Exchange Rate

Indian IIP

Indian CPI

Indian Treasury Bill Rate

Indian Ml

US IIP

US CPI

US Treasury Bill Rate

US M1

(3)

Ho:a, =at =00,

4.6993

2.6307

2.84s0

s.3971

3.0426

3.7011

(3)

H o:a, =0tr1

-4.0186

-3.0609

-2.0997

-2.3831

-5.9088

-4.0596

-3.2828

-2.4647

-2.7023

a)H o:a, =0

tr$

-2.5604

-2.1545

-2.3972

-2.t721

-2.6333

Q)

4,q*r=0,

3.2826

2.4504

2.8761

2.3637

3.5581

(1)

Ho:ar=0T

-t.6432

-1.1070

-2.4155

-2.1105

-2.6835

fable: 1.2(B)

PHILLPS.PERRON TEST

Time Period : February 1993 - Juty 1999

First Differences

Critical Values

Perron Test

ModelNull HypothesisTest Statistic

Exchange Rate

Indian IIP

Indian CPI

Indian Treasury Bill Rate

Indian Ml

US IIP

US CPI

US Treasury Bill Rate

US Ml

(1)

H o:d, =1

Z (td,)

-t0.3200

-12.2440

-6.8433

-7.7047

-10.6940

-9.8343

-7.4110

-7.7271

-4.3868

(2)

H o'.a* t =l

4o.r)-10.0100

-12.1010

-6.8330

-7.7566

-9.9658

-9.8943

-7.3344

-7.3726

-4.0959

Dickey-Fuller Test

SignificanceLevel

aa

-3.96-3.41

-3.13

0,

ffi6.255.34

xu 0'

3.654.59

3.78

x

T%

5%t0%

-3.43

-2.86-2.s7

-2.58-1.95-1.62

Significance Z (t dr) Z(t a. )Level

t%5%10%

-3.96 -3.43

-3.41 -2.86-3.13 -2.57

TABLE :2.1(A)

DICKEY.FULLER TEST

Time Period : January 1993 - July 1999

Levels

TABLE :2.1(B)

PHILLPS - PERRON TEST

ModelNull HypothesisTest Statistic

(3)

H o:a, =0xr

(3)

H o:a, =a,

0,

(2)

Ho"a,=0xu

(2)

4:q-4-0,

(l)H o:a, =0

T

Relative IIP

Relative CPI

Relative InterestRateRelative Money

-1.6628

-2.7484

-2.0067

-1.1841

2.2043

3.9054

2.3198

t.t29t

-2.ttt7

-0.7423

-2.1418

-1.2796

2.6783

4.6t31

2.3766

3.6534

0.1319

2.6064

-1.1770

-2.4556

Levels

ModelNull HypothesisTest Statistic

R"lrt"- IP

Relative CPI

(1)

H o:d, =lZ (td,)

(2)

H o:a* t =l

fg,)-2.2603

-0.7545

-1.568s

-r.7250

Relative Interest Rate

Relative Money Supply

-1.9434

-2.2526

-1.5205

-1.4638

TABLE .2.2(A)

DICKEY.FULLER TEST

Time Period : January 1993 - July 1999First Differences

TABLE | 2.2(B\

PHILLPS. PERRON TEST

Notes : All the variables are in logarithm term, except the relative interest rate.Exchange rate is the bilateral Indian Rupees / US Dollar exchange rate.Relative IIP is given by the logarithm difference of two country's Industrial

Production Index.Relative CPI is given by the logarithm difference of two country's Consumer Price

Index for industrial workers.Relative interest is given by the 3-Month Treasury Bill rate differential.Relative money supply is given by the logarithm difference of two country's money

supply where Ml has been used as the measure of the money supply.

ModelNull HypothesisTest Statistic

Relative IIP

Relative CPI

Relative InterestRateRelative MoneySupply

(3)

H o:a, =0

_::_-3.0418

-2.0403

-3.3488

-5.0838

(3)

H o:a, = a,

_0,4.6279

2.s354

5.6113

(2)

H o:a, =01u

-2.7660

-2.1861

-3.3130

(2)

4:q-'q=0,

3.82s8

2.4977

s.4948

(t)H r:a, =0

__:_2.6678

-1.2858

-0

First Differences

ModelNull HypothesisTest Statistic

R.trt"- m

Relative CPI

(1)

H o:d, =l

3!:!--12.3050

-6.6488

-7.9321

-11.0020

(2)

H o'.a* t =lZ(t a.)

Relative Interest Rate

Relative Money Supply

-12.1940

-6.666r

-7.93t8

-10.6130

APPENDIX

Table : 1.1(A)

DICKEY.F'ULLER TEST

Models:p

(1) A !t = at !trtZO,t!y_l+ii=2

P

(2) L ! t = ao * at !,-t +ZbiL y y_1*ii=2

p

(3) A ! t = ao + azt + at !rt +Z4L ! y_t*i,'- t

Time Period : January 1993 - July 1999

Levels

Notes : IIP stands for the Index of Industrial ProductionCPI stands for the Consumer Price Index NumberTreasury Bill rate is of the maturity period of 3-Months

All the variables, except the treasury bill rate, are in logarithms.

ModelNuliHypothesisTest Statistic

Exchange Rate

Indian IIP

Indian CPI

Indian Treasury Bill Rate

Indian Ml

US IIP

US CPI

US Treasury Bill Rate

US Ml

(3)

Ho"a,=g

_::_-2.3064

-t.448t

-2.8842

-t.8278

-2.0472

-2.1925

-t.3499

-1.0191

-2.3681

(3)

Hr"a, =s,

_d,3.2314

1.9739

4.5t26

t.7200

3.8109

2.436t

2.0188

2.3850

2.8041

-0(2)

H o:a, =g

__t,0.2139

-1.7890

-0.9665

-1.8553

-2.5350

-0.4081

-t.6176

-1.8638

-1.8328

(2)

4:q:4=

--!r-2.4t66

6.2192

6.0261

1.7284

8.5396

33.t250

65.5110

2.3370

1.72s7

(1)

H o'.a, =Q

c

2.2078

2.9411

3.2585

-0.4789

3.0340

8.1562

1 1.1750

"0.73t6

-0.3056

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Bayesian Unit Root Test(limit : 0.5, alpha : 0.8)

Time Period : January 1993 - July 1999(Levels)

Table: 5.2

Bayesian Unit Root Test(limit:0.5, alpha = 0.8)

Time Period : January 1993 - JuIy 1999

Variables

Exchange Rate

Indian IIP

Indian CPI

Indian Treasury BillRate

Indian Ml

US IIP

US CPI

US Treasury BillRate

US Ml

Squared t Schwarz Limit Marginal Alpha

8.195 0.9215

8.038 0.7316

10.333 0.9568

6.787 0.7170

8.494 0.3593

10.s41 0.9727

11.109 0.9330

7.263 0.6219

8.049 0.2761

0.046

2.808

0.914

1.703

6.426

0.t67

2.617

3.747

6.753

Variables

Relative IIP

Relative CPI

Relative Interest Rate

Relative Monev S ly

Squared t Schwarz Limit

6.327

9.7t4

7.014

8.514

Marginal

5.644

0.591

3.t47

2.737

0.2t91

0.9s02

0.5797

0.7818

, Tahle r 6

. EnglrGrmgorTe*t'of Cofutegrafioni'..

'::. Sarnple Psrtod : Jirriurry 1993 - July 1999

Test Type Constant, No Trend eonstant, Tremd

Dickey-Fuller

Phillips-Perron

-3.3001

-3.2383

-2.3381.

-3.1413

No Coirrtegration

No Cointqgration

Table : 7

Lag-length Selection for Unrestricted Vector Autoregression

Sample Period : January 1993 - JuIy 1999

Notes: (1) AIC stands for Akaike Information Criterion(2) SBC stands for Schwarz Bayesian Criterion(3) LR stands for the Likelihood Ratio Test Statistic(4) LM stands for the Langrange Multiplier test for Serial Correlation(5) In column (a) the value corresponding to 3 lags is the LR test statistic valuefor testing the null of 3 lags versus 4 lags. (p-values are given in the bracketsbelow)

No. of Lags

(1)

AIC

(2)

SBC

(3)

LR

(4)

LM

(s)

4lags -19.4383 -17.3371

3 lags -19.5818 -t7.974721.2656(0.168s)

18.557(0.2e23)

t9.3664(0.2s01)

No serialCorrelation

SerialCorrelation

SerialCorrelation

2 lags

1 lag

-19.7607 -18.6483

-19.9292 -19.3112

Table 8

Block-Exogeneity Test

Sample Period : January 1993 - JuIy 1999

Number of Lags in the VAR: 3

Notes : (1) p-values are given in the parenthesis.

(2) under the null hypothesis, LR statistic follows a chi-square distribution with 9degrees of freedom.

Null Hypothesis : Variable is not present in Likelihood Ratio (LR) Statistic

Spot Exchange Rate

Relative IIP

Relative Interest Rate

Relative Money Supply

25.3496(0.003)

20.$a6(0.01s)

18.7968(0.027\

30.3714(0.0000)

Johansen-Juseliustl*;iation Test Resuu

Sample Period : 1993:01 - 1999:07

Number of Lags in VAR: 3

Cointegration test based on the Trace Statistic

(Constant in the cointegrating Vector)

Cointegration test based on the Maximum Eignvalue Statistic

(Constant in the cointegrating Vector)

Eignvalues

0.37339

0.25635

0.06584

0.027t0

Nul1Hypothesis

AlthemativeHvpgqE,t

r)1

::-

6s.299s

29.7746

7.2642

2.0883

s%c.Y. t0% c.Y.

49.9500

31.9300

17.8800

7.s300

r-0

r<l

r<2

r<3

r>2

r>3

r=4

s3.4800

24.8700

20.1800

9.1600

Y""".0.37339

0.25635

0.06584

0.02710

Nul1 Althemative

35.5249

22.5t04

5.1759

2.0883

5% C.V. t0% c.Y.

25.8000

19.8600

13.8100

7.5300

Hvoothesis

r=0

r<l

r<2

r<3

r=7 28.2700

22.0400

15.8700

9.1600

r=2

r=3

r=4

Table : 10

. Cointegrating Vectsrs

Sample Period : January 1993 - July 1999

Number of lags in VAR: 3

Note:Sisthebilateralspotexchangeratg (y-y' ) istherelativellP, (i-i*)istherelative interest differential, (*-*' ) is the relative money supplies and C is theconstant. * indicates a foreign variable.

S (v- v. ) (i-i ) (m-m') C

1.0000

t.0000

-2.4145 0.5409 0.0058 -r.2363

2.5361 -0.3902 -t.3669 -7.8110

o!+$nrr)

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Table: 12

Bayesian Vector Autoregression Parameter Selection

Simple Flexible Price Monetary Model with Log of exchanle rate,log of Relative' UP, Relative 3-Months Treasury Bill Rates, and Log of Relative Money Supplygiven by Ml.

THEIL'S INEQUALITY STATISTICS

Harmonic Lag : d: 1.0

Ilarmonic Lag : d:2.0

ROOT MEAN SQUARE ERRORS

Harmonic Lag : d: 1.0

Priors l" = 0.1 )'"=0.2

ffi0.9097920.916s19

w:0.4w:0.5w:0.6

0.8756060.8792030.882678

Priors 7" = 0.1 )u=0.2ffi0.8916140.896867

w:0.4w:0.5w:0.6

0.8756060.8651120.867559

Priors l" = 0.1

ffi0.0520290.052225

)"=0.20.0537680.0s41740.054s89

w:0.4w:0.5w:0.6

Harmonic Lag: d:2.0

Priors l. = 0.1

ffiffi?v=0.2

ffi0.0s27470.053062

w:0.4w:0.5w:0.6

0.0509910.051 130

TABLE : 13

DICKEY.F'ULLER TEST

Models:p

(1) A !t = ar !t_rtZO,tly_1+ii=2

p

(2) A !t = ao * at !,-r +ZbiL y y_t*ii=2

p

(3) A !t =ao+azt*at !,_t+Ib,A !y_r+i!-a

Time Period : January 1993 - July 1999

Levels

ModelNull HypothesisTest Statistic

Expected Inflationfor IndiaExpected Inflationfor USABilateral CumulativeTrade Balance

(3)

Ho:a, =a,

0,

12834

2.5423

(2)

4,%n=

---ir-3.23s1

1.6114

(1)

Ho:ar=0a

ffi-1.0499

(3)

H ,:a, =0T1

(2)

H o:a, =0xu

-0

-2.5606

-2.1992

-3.6565

-2.5436

-t.7923

PHILLPS. PERRON TEST

LevelsModelNull HypothesisTest Statistic

Expected Inflation for India

Expected Inflation for USA

Bilateral Cumulative TradeBalance

(1)

H o:d, =lz (tdl)

(2)

Ho"a*

Z(t a,^0069

-5.5164

-3.5262

r=1-r)

-6.8640

-5.9s24

-3.9806

Table: 14

BAYESIAN VAR PARAMETER SELECTION

Simple Flexible Price Monetary Model with Log of exchange rate, log of rrps, 3-Months Treasury Bill Rates, and Log of Money Supplies given by Ml.

THEIL'S INEQUALITY STATISTICSHarmonic Lag z d: 1.0

IlarmonicLag:d:2.0

ROOT MEAN SQUARE ERRORS

Harmonic Lag : d: 1.0

llarmonic Lag : d:2.0

Priors t!!_0.8324120.8436270.856376

]:!2-0.8452380.8661760.885023

w:0.4w:0.5w:0.6

Priors l" :0.1 ]:!2-0.842t040.8587040.873395

w:0.4w:0.5w:0.6

0.8375280.8473570.858362

Priors l. = 0.1 _?u:0.2

0.049t490.0504780.051632

w:0.4w:0.5w:0.6

0.0488350.0495810.050431

Priors l, = 0.1 )v=0.2

w:0.4w:0.5w:0.6

0.0492980.0499530.050687

0.049t020.0s01590.051063

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\t OO .+ C.l tf\O O\ C.l \A [n#ooootaoOOOH\OtI\N\O*OIooootr-OOOOO

\O$oo-t\O O\ ri (n COqn*O\-c\lOtoOrioOO\ (\l t-- C.l cOOOOOOOOOOOO

Or\OoOOOC-lca-Or.) o\ t-- r- oo \o t+ (\.1 oOC\ooC{OtsOOOOOOOO-.;

(\t u)

It H Er Er 7Hg a s t IF-A

oQ\o(l)

!Er LoLL

frl ljo)9a0Ftrl _-+r IEI{E sEE ;

-9l-l a)r;q)

H E.Ei5 EIl{ o.rdU) or E0 a|-IFS;EiaE<€1={ -

E*5stuEr4 q)5-oE!?Da-U)PU)'- frl rl '-trJ Ar '- fr)!d (, rr] FrEE3-<*x>>;q)

oz

acg(Jq)Lotu15cl(L)

oEIeI

q)

(1)

H

Trble : 17

Summary Results of Unit Root Tests on Actuals and Predicted Series

i.li,!'

Notes : "Y' stands for the non-rejeetion of the null of unit root.We employed Dickey-Fuller and Phillips-Perron unit root tests.

Models/Forecast ACTUALS ARIMA VECM BVAR BVARD UBVARHorizon

lStep Y Y Y Y Y Y

3 Steps

6 Steps

9 Steps

12 Steps

Y

Y

Y

Y Y Y Y

Table: 18

Summary Results of Cointegration between Actual and Forecasted Series

Notes : "EG' stands for the presence of cointegration by the Engle-Granger procedure."JJ" stands for the presence ofcointegration by the Johansen-Juselius procedure.'\IOC" stands for no cointegration by either Engle-Granger or Johansen-Juseliusprocedure.

Models/ForecastHorizon

ACTUALS BVAR BVARD UBVAR

I Step

3 Steps

6 Steps

9 Steps

12 Steps

EG, JJ

JJ

NOC

EG, JJ

NOC

EG, JJ

JJ

NOC

EG, JJ

NOC

EG, JJ

JJ

NOC

EG, JJ

NOC

EG, JJ

JJ

NOC

EG, JJ

NOC

EG, JJ

JJ

NOC

EG, JJ

NOC

EG, JJ

JJ

NOC

EG, JJ

NOC

Table:19

Bias and Efficiency Tests of the Forecasts

Table 19.1

One-Month Ahead Forecasts

Models Bias Test

l1i1-E10.0831(0.006)

25.7335(0.0000)

84.8132(0.0000)

86.0046(0.0000)

88.1885(0.0000)

Efficiency Test(1 -t

ARIMA r.7984(0.180)

3.8779(0.04e)

40.1989(0.0000)

37.7419(0.0000)

27.507t(0.0000)

VECM

BVAR

BVARD

UBVAR

Table 19.2

Three-Month Ahead Forecasts ,?

Models Bias Test

l1-rg_3.562s(0.168)

3.0312(0.220)

8.5920(0.014)

8.1199(0.017)

8.5609(0.013)

Efficiency Test(1 -1)

ARIMA 0.0603(0.806)

0.1002(0.7s2)

0.3904(0.s32)

0.3104(0.s77)

0.1599(0.68e)

VECM

BVAR

BVARD

UBVAR

- _1.3Fq!!:]iw:tr

Teble 19.3

Niue-Month Ahead tr'orecasts

Bias Test Test(1-l 0 -r)

ARIMA

VECM

8.7265(0.013)

14.4646(0,001)

12.1932(0.002)

12.2484(0.002)

12.4788(0.002)

7.0984(0.00s)

17.!,47g(0.000)

15.6964(0.000)

r6.0102(0.mm)

16.0?48(CI.0m)

BVAR

BVARD

UBVAR

Table:20

Nonfarametric Test of Unbfusodness

Models

-ARIMA

VECM

BVAR

BVARD

TJBV'AR

:'i1"':t2

20

11

11

11

Wilcoxon Rank-Sum Test

202

413

209

197

213

Notes : (1) Reject the null of median urfiiasednese,at 5yo significsnce level if th€calculated value of the sign statistic is less than equal to 9.993 or if S is greater thar equalto 21.007.

(2) Reject the null of median (= mean) unbiase&ess if the calculated valuo of theWilcoxon rank sum test statistic exceds 348 or if it is less than 148.

Table :21

Diebold-Mariano Test for Equality of Mean Square Errror

One-Month Ahead Forecasts

( * indicates significant at 5% and 10% significance level)

Three-Month Ahead Forecasts

( * indicates significant at 5% and 10% significance level)

Six-Month Ahead Forecasts

Models ARIMA

mVECM BVAR

0.9002-0.32313.9532.

BVARD

@,0.20933.9681 .0.7808

UBVAR

@0.1433

4.0129 *

1.87561.5758

SRWARIMAVECMBVARBVARD

-3.7199.-3.8118 .

Models ARIMA

@VECM

-a8642-1.0256

BVAR

m-0.76810.4857

BVARD UBVAR

ffi-0.28291.04681.71291.8107

SRWARIMAVECMBVARBVARD

0.5057-0.84840.9790-0.2936

Models

ffi:ARIMAVECMBVARBVARD

ARIMA

mVECM

ffi-0.3045

BVAR BVARD

03884-1.0748-0.19080.1338

UBVAR

0.3612-1.0364-0.2207

0.8069-0.10060.4903

4.1578 *

3.5814 *

( * indicates significant at 5% and 10% significance level)

Table 21 (Gontd..)

Nine-Month Ahead Forecasts

( * indicates significant at 5% and 10% significance level)

Twelve-Month Ahead Forecasts

Models

SRW:ARIMAVECMBVARBVARD

ARIMA VECM

037690.1815

BVAR

ffi-1.0851-1.6305

BVARD

ffi-0.9493-1.4588-0.6234

UBV

3.6993. 1.35670.3083-0.0132-0.1485-0.7423

Models ARIMA

Tssz

VECM

ffi0.5192

BVAR BVARD

ffi-0.2598-1.62670.5474

UBVAR

0.4234-0.3079-1.4459

1 .0159ARIMAVECMBVARBVARD

0.5495-0.43121.49711.7156

( * indicates significant at 5% and 10% significance level)

Table:22

lnformation Content Test

One-Month Forecast HorizonDependent Variable : Actual

Independent Variables

Constant

0.0014(0.6888)(0.4e66)

0.0089(1.4e43)(0.1463)0.0094

(1.s200)(0.13e7)0.0093

(1.3406)(0.1908)0.0069

(r.66e0)(0.1083)0.0077

(1.71e8)(0.0e65)0.0086

(1.s482)(0.1328)0.0102

(1.ss21)(0.131e)0.0087

(1.2248)(0.230e)0.0082

(1.1646)(0.2s40)

ARIMA VECM

0.7783(2.0128)(0.0538)

BVAR BVARD UBVAR

0.1495(0.33s8)(0.73es)0.4873

(1.zete)(0.2069)0.5006

(1.3468)(0.1 888)0.5076

(1.24s4)(0.2233)

-0.97s6(-1.1414)(0.2634)

-1.1792(-t.ts73)(o.2s6e)

-t.1273(-0.e106)(0.3702)

0.9232(2.7s42)(0.0102)

0.9499(2.8183)(0.0088)0.9s94

(2_e020)

(0.0071)

-1.2573(-1.se13)(0.1,228)

-1.5857(-1.50e6)(0.t424)

-1.7207(-r.4003)(0.1724)

-0.463r(-0.662e)(0.s128)-1.1323

(-1.ss76)(0.1306)

-0.486s(-0.4813)(0.6340)

-1.9372(-1.e002)(0.067s)

0.5661(0.53ee)(0.5e35)t.3637

(1.00s7)(0.3 r31)

Table 22 (Contd..)

Three-Month Forecast Horizon

Independent Variables

Constant ARIMA

ffi(-0.5018)(0.6200)0.5857

(0.ee82)(0.3273)0.5990

(0.e4t4)(o.3ss2)0.5183

(0.es8s)(0.3467)

VECM BVAR BVARD UBVAR

0.0244(1.8706)(0.0727)0.0368

(2.0108)(0.0s48)0.0382

(2.r448)(0.041s)0.0455

(2.0628)(0.04e3)

0.0392(2.1 880)(0.0378)0.0408

(2.3s08)(0.0266)0.0464

(2.137T)(0.0422)0.0424

(3.37es)(0.0023)0.040s

(2.1064)(0.04s0)

0.0361(1.8258)(0.07e5)

-0.1061(-t.6643)(0.1081)

-1.7209(-2.3706)(0.0255)

-1.9t07(-2.6737)(0.0128)

-2.2970(-2.32t7)(0.02383)

-0.0385(-0.60e3)(0.s476)-0.0388

(-0.0388)(0.s644)-0.02s1

(-0.410e)(0.684s)

-1.4653(-1.8s06)(0.07s6)

-1.6509(-2.11,66)(0.0440)

-1.9920(-1.8828)(0.0705)

21.7534(1.e724)(0.0se3)-t.3042

(-0.8462)(0.4s01)

-24.4491(-2.ts78)(0,0404)

-2.4150(-2.33e8)(0.0272)

-0.3179(-0.1837)(0.40s1)

1.0069(0.7022)(0.4888)

Table 22 (Contd..)

Six-Month Forecast Horizon

Independent Yariables

Constant

0^0654(2.6se)

(0.0140)0.0638

(2.e64s)(0.006e)0.0658

(3.0e8s)(0.00s1)0.0905

(4.26t0)(0.0003)0.0726

(4.0468)(0.000s)0.074t

(4.2e83)(0.0003)

0.0970(s. l 63s)(0.0000)0.0798

(s.6582)(0.0ooo)0.1078

(3.68s3)(0.0012)0.0888

(3.47ss)(0.0020)

ARIMA VECM BVAR BVARD UBVAR

-1.1424(-1.312s)(0.2023)r.6287

(1.5s80)(0.132e)1.5493

(1.440s)(0.1632)1.8545

(1.7726)(0.08es)

-0.1914(-0.4604)(0.8481)

-1.8766(-6.e4s4)(0.0000)

-1.9740(-6.710s)(0.0000)

-2.8901(-6.s024)(0.0000)

0.3421(0.8661)(0.3e80)0.34t7

(0.8736)(0.3e13)0.3716

(0.8e36)(0.3808)

-1.6658(-4.515)(0.0002)

-1.787s(-4.3368)(0.0002)

-2.5138(-4.tt78)(0.ooo4)

10.311l(1.se63)(0.r24r)0.9375

(0.4123)(0.683e)

-12.2369(-1.8702)(0.0742)

-0.58s2(-0.3es0)(0.6e6s)

-3.3427(-1.1028)(0.281s)-t.2186

(-0.6218)(0.5401)

Table 22 (Contd..)

Nine-Month Forecast Horizon

Constant

0.1205(5.53e1)(0.oooo)0.1199

(6.8388)(0.0000)0.1091

(7.721e)(0.0000)0.1456

(e.1 s6s)(0.0000)0.0989

(6.41s6)(0.0000)0.1031

(8.3285)(0.0000)0.1513

(12.1683)(0.0000)0.1162

(1 1.8887)(0.0000)0.1496

(e.22e2)(0.0000)0.1398

(s.6826)(0.0000)

ARIMA

+8477(-2.003)(0.0s8e)-1.t253

(-1.0721)(0.2e64)0.6362

(t.4s7L)(0.1606)1.0816

(2.t6tt)(0.0430)

Independent Variables

VECM BVAR BVARD UBVAR

-0.2358(-0.4777)(0.6380)

-0.8068(-2.07ee)(0.0506)

-1.5469(-6.st76)(0.0000)

-2.5035(-6.8618)(0.0000)

0.1 555(0.237e)(0.8144)0.7341

(1.67e2)(0.1087)0.7916

(1.8232)(0.0833)

-1.t476(-1.6074)(0.t236)

-1.9782(-4.7r37)(0.0001)

-2.9969(-s.0342)(0.0001)

0.0893(t.6672)(0.1111)0.0275

(0.367e)(0.7168)

-1.4346(-1r.5s36)(0.0000)

-0.3918(-0.2716)(0.7887)

-2.0333(-10.4634)(0.0000)-1.4262

(-0.73re)(0.4727)

Table 22 (Contd..)

Twelve-Month Forecast Horizon

Constant

0.173 t(s.e773)(0.0000)0.1469

(s.e083)(0.0000)0.ts29

(6.0033)(0.0000)0.1903

(8.51e2)(0.0000)0.1459

(6.6686)(0.0000)0.1517

(6.3306)(0.0000)0.1901

(8.2e84)(0.0000)0.1358

(11.8763)(0.0ooo)0.2884

(6.0e5e)(0.0000)

0.2595(6.8064)

ARIMA

-L7541(-2.se{e)(0.018e)-0.3219

(-0.4062)(0.68e7)-0.7243

(-0.8827)(0.38e7)0.4023

(0.s142)(0.6137)

lndependent Variables

VECM BVAR BVARD UBVAR

-0.4574(-1.337s)(0.1e87)

-0.9909(-4.43ss)(0.0004)

-0.9809(-e.0422)(0.0010)

-2.1099(-4.16s6)(0.0006)

-0.1015(-0.387e)(-0.3878)-0.2547

(-0.8378)(0.4138)0.0218

(0.105s)(0.10

-1.0301(-s.034s)(-s.034s)

-0.6921(-4.6921)(0.0002)

-1.9096(-s.304e)(0.0001)

-5.1222(-2.3273)(0.0326)2.4743

(2.s62t)(0.0202)

4.2983(r.e6o2)(0.0666)

t.9664(2.71e8)(0.0146)

-5.9470(-3.s180)(0.0026)-4.8945

(-3.8887)(0.0000 0.0012

Table : 23

Direction - of - Change Analysis

Notes : (1) Entries in the cell are calculated chi-square statistic with 1 degrees of freedom.(2) Critical value of the chi-square statistic with I degrees of freedom atl}Yo

significance level is 2.7 l.(3) * indicates that the statistic is significant at lloh significance level.(4) uindicates that the statistic is significant at 10.08o/o significance level, bindicates

that the statistic is significant at 10.06% significance level.

ARIMA

VECM

BVAR

BVARD

UBVAR

One-Month

0.6628

0.7484

0.0000

0.0000

1.6155

Three-Month

alg44

2.581t',

0.0000

0.0000

2.6966b

Six-Month Nine-Month0.1406

0.5312

0.1406

0.0000

0.0000

Twelve-Month1.2468

4.3681 *

0.1889

0.1639

0.6288

0.0553

0.1169

0.0000

0.0000

0.0000

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