a short-run monetary model of exchange rate determination: stability tests and forecasting

empec, Vol. 13, 1988, page 1-15

A Short-Run Monetary Model of Exchange Rate Determination: Stability Tests and Forecasting

By D. H. Richardson x and M. T. C. Wu 2

Abstract: The paper develops a short-run econometric monetary model of exchange rate determination. The model assumes a conventional money demand function, markets which are linked by interest arbitrage, adaptive expectations formation, and parameters which are stable over time. One-period-ahead forecasts of the mark/pound rate generated by the model compare favorably with naive model forecasts using monthly data. Stability tests provided evidence of parameter instability in 1976 but correction for it did not improve forecasting accuracy. The inability of monetary models to forecast accurately may be due to the underlying model assumptions rather than parameter instability.

Several recent studies of the forecasting performance of monetary exchange rate

models have found that forecasts generated by these models were not nearly as ac-

curate as the forecasts of naive models. For example, Meese and Rogoff (1983a,

1983b) found that forecasts based on monetary models produced one-month-ahead

forecasts which were ten percent to twenty percent less accurate (as measured by root-

mean square error) than the naive model forecasts. This inability of monetary models

to outperform naive models has called into question the desirability of using monetary

models for the purpose of forecasting exchange rates.

The monetary models used in these forecasting exercises assume that the exchange

rate function is stable, i.e., that the parameters of the function are invariant with

respect to time. The models were est imated and forecasts are generated under the as-

sumption that the relationship between the explanatory variables and the exchange

rate did not change over time. It has been suggested by Meese and Rogoff (1983a,

1989b), Frankel (1984) and others that the poor forecasting performance of the

monetary models might be due to the instability of the underlying structural relation-

ships.

1 David H. Richardson, Department of Economics, St. Lawrence University, Canton, New York 13617, USA. 2 Mickey T. C. Wu, Department of Economics, Coe College, Cedar Rapids, Iowa 52402, USA.

0377-7332/88/1/1-15 $2.50 �9 1988 Physica-Verlag, Heidelberg

2 D.H. Richardson and M. T. C. Wu

The purpose of this paper is to develop a short-run econometric monetary model

that may be used for forecasting purposes and subject it to stability tests. Our interest in performing these tests is to determine if parameter instability might affect the fore-

casting ability of the model. In Section 1 of the paper we set out the monetary model which provides the

framework for our empirical tests. The model follows Dornbusch (1978) in that the short-run exchange rate equation is derived through interest arbitrage conditions. The model does not assume that purchasing power parity holds in the short run.

The model is estimated in Section 2 using monthly data from West Germany and the United Kingdom over the 91/2 year period from January 1974 to June 1983.

Ex post forecasts are presented for the last 18 months of the sample period and the accuracy of the forecasts is compared to forecasts from naive models.

The tests of stability are given in Section 3. These tests include the cusum test and the cusum of squares test due to Brown, Durbin, and Evans (1975), the likelihood

ratio test of Quandt (1960) and the analysis of variance (Chow) test. The Chow test shows evidence of a parameter shift in late 1976. The model is reestimated using data since late 1976 and the results are aslo presented in this section.

1 T h e Mode l

In this section we derive an equation that is appropriate for forecasting exchange rates in the short run for a small open economy. The equation is derived from a model of exchange rate determination in which the exchange rate is defined as the relative price of national monies. Thus, disturbances affect the exchange rate through adjustments in

the asset markets and, more speicifically, through the interaction of the domestic and foreign supply and demand for money.

We begin with the money demand equations:

ma = ao + a l p + a2y - aai (i)

and

m~ = a~ + a~p* + a~ y * - a~i* (2)

where ma is the (natural) logarithm of money demand for the home country, p is the logarithm of the price index, y is the logarithm of real income, and i is the nominal interest rate. Equation (2) is the money demand equation for the foreign country

A Short-Run Monetary Model of Exchange Rate Determination 3

where asterisks denote variables and parameters for the foreign country. We assume that all coefficients are positive and that the money market is in equilibrium so that m a and m~ are equal to the observed money supplies.

In this model we assume that the asset market adjusts more rapidly to shocks than the goods market. Thus in the short run, arbitrage in the goods market may not be complete and, thus, the purchasing power parity condition need not hold exactly. In this case the short-run equilibrium exchange rate is determined by asset market equilibrium conditions and given capital mobility, domestic and foreign markets are linked through interest rates rather than prices. The exchange rate equation is therefore derived through interest arbitrage rather than goods arbitrage conditions. Covered interest arbitrage requires that the difference between domestic and foreign interest

rates be approximately equal to the forward premium on foreign exchange, i.e.:

F - S i - i* = (3)

F

where F is the forward rate and S is the spot rate defined as the domestic currency price of a unit of foreign exchange.

We can rewrite equation (3) as:

i - i* = f - s (4)

where f = In F and s = In S. Rearranging equations (1) and (2), and assuming a a = ~ ' gives:

aa(i - i*) = a o - o~ + a t P - a t P * + a2Y - c~y* - rn + m* (5)

Substituting equation (5) in (4) yields:

1 f - s = - - [So - o ~ + = I P - =~'P* + =2Y - = ~ Y * - m + m * ] ( 6 )

&3

s=f_O~o +a~ o q p § a2y et~y* 1 - - - - § ( 7 )

o~ 3 o~ 3 a 3 ~x3 {x 3 Cx 3 c~3

In determining the forward rate in equation (7) we follow Dornbuseh (1978, p. 23) and assume that profit maximizing speculators will set the forward rate equal to their expectation of the future spot rate, i.e., f r = Et(st+ 1). Any deviation from this condi-


tion would imply that future profits could be earned through the purchase and sale of the relevant currencies. We assume that these transactions would continue until the equality between the future rate and the expected spot rate is realized. Finally, we

assume that expectations about the future spot rate are formed adaptively, i.e.,

Et(St+l)-Et_l($t)=)t[s t -Et_l(St)] O<<.X < 1, (8)

It follows that:

ft=~st+(1-X)ft_l 0 ~ < X < I . (9)

Substituting (9) into equation (7) yields:

st=ft-1 a3(1 - X) a3(1 - x)P + ots(l - X) p*

~2 1 0~a(1 _ X) y + t~* y * + - - (m - m * ) (10)

o~3(1 - X) aa(1 - X)

The lagged rate in equation (10) can be considered a reflection of the expectation of the future spot rate by speculators. An expected depreciation of the spot rate in the future would put pressure on the current rate to depreciate as traders try to discard their holdings of domestic assets, including currency today to avoid future losses.

The other variables in equation (10) indicate that the exchange rate is determined by the relative supply and demand of domestic and foreign currencies. Given the equilibrium conditions, an increase in the domestic price level or real income raises the demand for money and thus causes the spot rate to appreciate while an increase in the domestic money supply generates an excess supply of money and thus causes the spot rate to depreciate. Changes in the foreign price level, foreign income, and foreign money supply have the corresponding opposite effects on the spot rate.


2 Es t imat ion and Forecas ts

Equation (10) can be expressed as follows:

$ $ $ $ = + [32Yt S t (30 +/31Pt +/31Pt + [32Yt

+/33(mt -- m ~ ) +/34ft- 1 + u t (111

where:

(So /30

a3(1 -X)

/31 = ~s(1 - k )

al'

a3(1 - k )

/32- o~3(1 - X)

a3(1 - X)

1 /33 = aa(1 -X)

and u is a random error term with mean zero and constant variance. The postulates of the monetary model imply that/3~', ~ and/33 are positive,/31 and/~ are negative, and /34 is equal to one.

Equation (1) can be simplified considerably if the coefficients of p and y in the domestic demand for money equation (1) are equal to the corresponding coefficients in the foreign demand equation (2). In terms of the parameters of equation (11), this hypothesis is:

/qo:


and the alternative hypothesis is that H~ is not satisfied. The F-statistic for this test was 2.78 which is less than the critical value of 3.09. We thus reject the hypothesis of

different coefficients at a 5 percent level of significance. The estimation and forecasts

are carried out using ordinary least squares regression in which the explanatory vari-

ables arep - p * , y - y * , m - m.*, and f t _ 1 . The estimated equation is:

s= 0.6846 - 0 . 0 8 1 4 ( p - p * ) - 0 . 0 6 6 1 ( y - y * ) + 0.1345 ( m - m * )

(0.2176) (0.0421) (0.0707) (0.0498)

+ 0.9653 ( f t - 1) (0.0224) (12)

T= 113 R 2 =0.9757 SER = 0.0239 DW= 1.18

The standard errors of the estimated coefficients are in parentheses. Note that all of the estimated coefficients have the hypothesized signs and that the coefficient of the lagged forward rate is not significantly different from one.

Since the coefficient o f f r _ 1 was highly significant in the estimated equation, we

tested the hypothesis that all of the coefficients in equation (12) except 130 and/34

were equal to zero. The F-statistic for this test was 3.35. Using a 5 percent level of

significance, we jecet the hypothesis that 131,132 and 133 are zero. The Durbin-Watson test statistic (DW) indicates positive first-order serial correla-

tion in the error terms. The estimated correlation coefficient is 0.41. In the light of this result, the model was reestimated using the Cochrane-Orcutt method of estima-

tion. The coefficient estimates and forecasts using Cochrane-Orcutt did not differ substantially from those of equation (12). We therefore decided to employ ordinary least squares in the forecasting and testing for stability.

One-step-ahead ex post forecasts of the exchange rate were made for the last 18 months of the sample period, i.e., January 1982 to June 1983. Forecasts at the end

of each month were computed using only data available up to that month. Moreover, the actual values of the explanatory variables were used in the forecast period. Measures of predictive accuracy are presented in Table 1 along with measures of predictive accuracy of two naive models. The first naive model is the random walk (with drift) where the spot rate is regressed on the spot rate in the previous month. The second naive model assumes that the previous one-month forward rate is an efficient predic- tor, i.e., the current spot rate is regressed on the forward rate in the previous month.

The accuracy of the forecasts is measured by the mean absolute error, root mean square error, and Theil's inequality coefficient as defined by:

Table 1. Measures of Predictive Accuracy

Model

1 m MAE=-- E Ist-stl

m t = l

mean absolute error root mean squared error Theil's U

Monetary 0.022 0.029 0.021

Random walk 0.020 0.026 0.018

Forward rate 0.022 0.028 0.020

RMSE =/ml--- t~l ( s t -g t ) 2

(13)

RMSE U=

1 m

m t = l

(14)

F i

F

(15)

where gt is the one-period-ahead forecast of the spot rate st.


F H ^ H J . J A S 0 N 0 J F H ^ H J

1982 HONTH 1983 Fig. 1. Forecast errors

i Honetory Hodel -

--Rondom Wo Forword Rote


As indicated in Table 1 the predictions from the random walk model are slightly

better than those obtained from the monetary model. The predictions from the naive forward model were roughly the same as those of the monetary model. Figure 1 shows that the three models produce forecasts that are highly correlated. In every case, the largest forecast ~rrors are produced when there is a substantial change in the spot rate.

It is not apparent from the plots of the forecast errors which method provides the best

forecast during periods in which the spot rate fluctuates substantially. The stability tests discussed in the next section were conducted to determine if parameter instability affected the forecasting performance of the monetary model.

3 Tes t s o f S tab i l i ty

Equation (10) was estimated under the assumption that the parameters of the model are stable over time. A general specification o f equation (10) which allows for time-

varying parameters would be:

St=Xt[Jt+U t t = 1,2 . . . . . T (16)

where

xt=(1,ft-l,Pt-P*,Yt yt,mt mr),

and ut are independently normally distributed error terms with means zero and variance ~ , cr~ . . . . , o~v. The stability hypothesis in this model is:

~o:

The first two tests of H o which we consider are the cusum and cusum of squares tests

proposed by Brown, Durbin and Evans (1975). These tests are based on the recursive residuals:


y , - xrrbr_ l wr = r = 6, . . . , T

~/(1 + x ' r ( X ; _ l X r _ l ) - l x r

(17)

where br_ 1 is the OLS estimate o f t based on the first r - 1 observations, i.e.,

br_ x = ( X t r _ l X r _ l ) - l X r _ l Y r _ l

where

f !

S t _ 1 = [X1 ,X 2 . . . . . X r _ l ] and Y r - I = [ Y z , Y 2 . . . . , Y r - z ] .

The recursive residuals are convenient for testing the stability hypothesis because the are independently normally distributed under H o.

The cusum test is based on the cumulative sums of standardized residuals

1" u

W , = - Z w / r = 6 . . . . . T (18) 0 i = 6

where wj is given by (17) and ~2 = ( Y t - - X T b T ) ' O ' T - X T b r ) / ( T - 5 ) . The procedure is to plot I4/r against r along with a pair of straight lines which are symmetric about

Ir r = 0. The straight lines are constructed so that the probabili ty of the sample cusums

crossing one of the lines under H o is equal to the level of significance. I f the sample plot o f Ir r falls outside the straight lines, the stability hypothesis is rejected.

The cusum statistics IV r were computed for the sample data and are plotted in Fig. 2 for r = 6 to T = 113. The sample statistics lie above the upper 5 percent significance line for five observations between r -- 81 and r = 85. I f a 1 percent significance

line were drawn on Fig. 2 it would lie above the entire Ir r plot. We thus reject the stability hypothesis H o at a 5 percent level of significance but not a 1 percent level using the cusum test.

The cusum of squares test o f H o is based on the cumulative sums of squares o f the recursive residuals, i.e.,

S, = ~ w ~ / / ~ w~ r = 6 . . . . . T (19) 1=6

30-

25-

20-

15-

10-

5-

0

-5


/

I I

OBSERVATION NUMBER

20 40 GO BO 100

Fig. 2. Cumulative sums of recursive residulas

Under the stability hypothesis H o these statistics have a Beta distribution with mean (r - 5 ) / ( T - 5). Using tables provided by Durbin (1969) one can construct lines parallel to the mean line such that the probability of the sample plot falling outside one of these lines is equal to the level of significance.

Plots of the cusum of squares statistics for the sample data are given in Fig. 3 along with the 5 percent significance lines. The data lie below the lower 5 percent significance line for four observations between r = 59 and r = 62. The plots would lie

entirely within 1 percent significance lines. The conclusion for the cusum of squares test is thus similar to the cusum test. We reject H o at a 5 percent level but not at a 1 percent level.

The Quandt log-likelihood ratio technique is appropriate for detecting the time at

which the regression parameters change abruptly from one value to another. The ratio kr is defined as the logarithm of the ratio of the maximum of the likelihood function

under Ho to the maximum of the likelihood function under the alternative hypothesis that the parameters after time r differ from the parameters up to time r. The procedure is to plot kr against r and determine the value of r where ~,r attains a minimum.


Io

0.9.

0.8-

0.7.

0.6-

0.5.

0.4.

0.3.

0.2-

0.1

0 0

_ ~ [ / l I OBSERVIATION NUNBEIR 20 40 fig BO I O0

Fig. 3. Cumulative sums of squares of recursive residuales

I 120

This value o f r indicates the time at which there was an abrupt change in the regression relationship.

The Quandt log-likelihood ratio for the monetary model fitted to the sample data is plotted in Fig. 4. It is interesting to note that there is no unique minimum value of the ratio. There are four relative minima of roughly the same magnitude. These minima occur at r = 9, 25, 32, and 106 which correspond to October 1974, February 1976, September 1976, and November 1982, respectively. We exclude the minima which occur in October 1974 and November 1982 because they are so close the beginning and end of the time period that estimates of the parameters cannot be considered reliable. Of the two minima which occur in 1976 the value of Xr is smaller in September. We thus choose September 1976 as the time period at which the regression parameters changed abruptly from one value to another.

The analysis of variance (Chow) test was employed to determine if the coefficient vector before September 1976 was significantly different from the coefficient vector after September 1976. The computed F-ratio for this test was 7.21. Since the one

12 D.H. Richardson and M. T, C. Wu

-1

-3

-4

-5

-6

-7

-B

-9

20 40 60 BO 100 I~O I I I I I I

Fig. 4. Quandt's log-likelihood ratio

percent significance level of this statistic is 3.20 we clearly reject the hypothesis that the regression parameters did not change in late 1976.

This result appears to contradict the findings of Boughton (1981) who investigated the money demand functions of Germany and the United Kingdom using quarterly data over the period 1960 to 1977 and did not find any significant evidence of instability. His findings are not inconsistent with ours because he considered only money demand functions. It is possible that the source of instability in the exchange rate function lies in the assumption made about the foreign exchange market and not in the money demand functions. Moreover, the shift that we detect occurs at the end of Boughton's data set. His results may have been different if his data series had extended to 1983.

One possible explanation for the shift in the mark/pound exchange rate equation is the abrupt change in monetary and fiscal policy which occurred in the United King- dom in late 1976. Between 1974 and 1976 the United Kingdom economy experienced high inflation (24 percent in 1975), a reduction in output, a depreciating exchange rate, and an adverse trade balance. In an effort to reverse these trends the government made commitments to the IMF in December 1976 which included a sharp reduction in public expenditures and a limit on domestic credit expansion. It is possible that the financial markets anticipated these policy changes and this was reflected in a shift in the parameter ~, in equation (8). This line of reasoning is similar to the Lucas (1976) argument that actual or anticipated changes in government policy can affect the be-


havior of market participants which in turn may change the parameters of the model under consideration.

A second explanation lies in the instability in the German money demand function. Frankel (1982, 1984) argues that wealth effects caused the German money demand to shift upwards between 1974 and 1980. Since we have not included wealth as an argument in the money demand function, this factor could account for our find- ing of instability.

In an effort to determine if parameter instability affected the forecasting ability of the monetary model, we reestimated the model using data from October 1976 to June 1983. The estimated equation is:

s= 1.2166 + 0 . 7 7 2 4 ( f r _ t ) - 0 . 1 7 1 6 ( p - p * ) + 0 . 1 6 7 0 ( y - y * ) (0.2623) (0.0503) (0.0530) (0.1048)

+ 0.2009 (m-m*) (0.0579)

T = 8 2 R 2=0 .8970 SER=0.0299 DW=1.30 (2o)

It many respects the reestimated equation is inferior to the original equation (12). The coefficient of the lagged forward rate is significantly less than one, the coefficient of y - y * has the wrong sign (even though it is not significant), and the coefficient of determination R 2 is lower.

We used equation (20) to compute one-step-ahead forecasts for the last 18 months of the sample period (January 1982 to June 1983). We followed the same procedure as outlined in Section 2 so the forecasts would be comparable. The measures of predictive accuracy using equation (20) were: MAE = 0.024, RMSE = 0.028, and Theil's U = 0.020. Comparing these measures with those of Table 1 indicates that forecasting ability of the monetary model was not improved by breaking the sample at the end of 1976. Therefore, it does not appear that, in this case, parameter instability affected the forecasting accuracy of the model.

4 S u m m a r y and Conclus ions

In this paper we have presented a short-run monetary model of exchange rate determination. The model assumes a conventional money demand function, markets which are linked by covered interest arbitrage, expectations about future exchange rates which are formed adaptively, and parameters which are stable over time. The model generates


one-period-ahead forecasts which have roughly the same degree of accuracy as the naive model forecasts using monthly data.

We also performed various tests of stability to determine if parameter instability affected the forecasting ability of the model. The results of these tests were not entire-

ly conclusive. We find some evidence of instability at a 5 percent level of significance

using the cusum test and the cusum of squares test, but we do not reject the stability hypothesis using these tests at a one percent level of significance. The Quandt log-likelihood ratio procedure and the analysis of variance (Chow) test indicate a shift in the parameters of the monetary model in late 1976.

We reestimated the model excluding data prior to 1976, but the accuracy of the

forecasts was not improved. Our results are, of course, restricted to a single specifica-

tion of a monetary model and a single data set. Nevertheless, our study suggests that instability may not be a major cause of poor forecasting performance. Efforts to respecify models to incorporate time-varying or shifting parameters may not be as worthwhile as a reexamination of the building blocks of monetary models.

A p p e n d i x

Data Definitions and Data Sources

S __

F -

p -

y -

M -

mark/pound spot exchange rate, monthly average. Sources: DRI data tapes.

mark/pound one month forward exchange rate, monthly average. Source: DRI data tapes. consumer price index, 1980 = 100, seasonally adjusted. Source: IMF Interna-

tional Financial Statistics.

industrial production index, 1980 = 100, seasonally adjusted. Source: IMF Inter.

national Financial Statistics.

money supply (M1), billions of deutsch marks for Germany, millions of pounds for United Kingdom, seasonal adjuste. Source: IMF International Financial

Statistics.


References

Boughton JM (1981) Recent instability of the demand for money: an international perspective. Southern Economic Journal 47:579-597

Brown RL, Durbin J, Evans JM (1975) Techniques for testing the constancy of regression rehtion- ships over time. Journal of the Royal Statistical Society 37:149-163

Dornbuseh R (1978) The theory of flexible exchange rate regimes and macroeconomic policy. In: Frenkel JA, Johnson HG (eds) The economics of exchange rates: selected studies. Addison- Wesley Publishing Company, Reading, Mass., pp 27-46

Durbin J (1969) Tests for serial correlation in regression analysis based on the periodog/am of least-squares residuals. Biometrika 56:1-15

Frankel JA (1982) The mystery of the multiplying marks: a modification of the monetary model. Review of Economies and Statistics 64:515-519

Frankel JA (1984) Tests of monetary and portfolio balance models in exchange rate determination. In: Bilson J, Marston R (eds) The exchange rate theory and practice. The University of Chicago Press, Chicago, pp 239-260

Lucas RE Jr (1976) Econometric policy evaluation: a critique. In: Brunner K, Meltzer A (eds) The Phillips curve and labor markets. North-Holland, Amsterdam, pp 19-46

Meese RA, Rogoff KS (1983) Empirical excahnge rate models of the seventies: do they fit out of sample. Journal of International Economics 14:3-24

Meese RA, Rogoff KS (1983) The out-of-sample failure ofempiricalexehange rate models: sampling error or misspeeifieation. In: Frenkel JA (ed) Exchange rates and international maeroeconomies. The University of Chicago Press, Chicago, pp 67-105

Quandt RE (1958) The estimation of parameters of a linear regression system obeying two separate regimes. Journal of the American Statistical Association 53:873-880

Quandt RE (1960) Tests of the hypothesis that a regression system obeys two separate regimes. Journal of the American Statistical Association 55:324-330

First version received January 1987 Final version received December 1987

a short-run monetary model of exchange rate determination: stability tests and forecasting

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