structural, var and bvar models of exchange rate determination: a comparison of their forecasting...

Journal of Forecasting, Vol. 14,201 -215 (1995)

Structural, VAR and BVAR Models of Exchange Rate Determination: A Comparison of Their Forecasting Performance

NICHOLAS SARANTIS AND CHRIS STEWART Kingston University, UK

ABSTRACT This paper compares the out-of-sample forecasting accuracy of a wide class of structural, BVAR and VAR models for major sterling exchange rates over different forecast horizons. As representative structural models we employ a portfolio balance model and a modified uncovered interest parity model, with the latter producing the more accurate forecasts. Proper attention to the long-run properties and the short-run dynamics of structural models can improve on the forecasting performance of the random walk model. The structural model shows substantial improvement in medium- term forecasting accuracy, whereas the BVAR model is the more accurate in the short term. BVAR and VAR models in levels strongly outpmdict these models formulated in difference form at all forecast horizons.

KEY WORDS exchange rate forecasting; structural models: vector autoregression; Bayesian vector autoregression

INTRODUCTION

This study compares the out-of-sample forecasting accuracy of a wide class of structural and time-series models of exchange rates over different forecast horizons. The existing literature on this subject has concentrated on US dollar bilateral exchange rates and focused mostly on the 1970s and early 1980s. In the present study, on the other hand, we examine the major bilateral exchange rates against the pound sterling (i.e. dollarlpound, D-mark/pound, yen/pound, and F-franc/pound) and cover the entire floating period up the end of the 1980s (i.e.

The seminal work of Meese and Rogoff (1983a) revealed that the structural models failed to match the out-of-sample forecasting accuracy of the random walk model irrespective of the forecast horizon. Subsequent studies by Alexander and Thomas (1987), Wolf (1987), and Schinasi and Swamy (1989) have re-evaluated these models by adopting a methodology which allows for time-varying parameters, Meese and Rogoff (1983b) allowed for simultaneous equation bias, while Meese and Rose (1991) have assessed the importance of non-linearities in structural models using a variety of parametric and non-parametric techniques. Their findings were mixed, with Meese and Rogoff (1983b) and Alexander and Thomas (1987) confirming the original results by Meese and Rogoff (1987), Wolf (1987) showing that the ex-post 0277-6693/95/0302O 1 - 15 Received February 1994 Q 1995 by John Wiley & Sons, Ltd Revised September 1994

197341- 199OQ3).

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predictions of the structural models are not uniformly dominated by the random walk, Schinasi and Swamy (1989) reporting that the structural models outperform the random walk model, and Meese and Rose (1991) concluding that accounting for non-linearities does not improve the ability of structural models to outpredict a random walk alternative. These studies have two important weaknesses. First, they only consider variants of the monetary approach to exchange rate determination.' They normally estimate three versions: the flexible price, sticky price, and sticky price with the current account monetary models. However, the literature on in-sample studies of exchange rate determination casts serious doubts on the monetary approach,* especially for the 1980s. Some of these authors themselves (e.g. Alexander and Thomas, 1987, p. 61) suggest that the structural models used may be inadequate. The second weakness concerns the dynamic specification of the structural models. Typically these models are estimated either in static form with or without first-order serial correlation or with a lagged dependent variable (e.g. Somanath, 1986; Schinasi and Swamy, 1989). This is a rather ad hoc and simplistic approach to a complex problem.

An important objective of this study is to address both these issues. Hence in addition to the monetary models we examine the portfolio balance model and a modified version of the uncovered interest parity hypothesis, both of which represent more recent theoretical developments. With regard to the specification of these models, we employ the cointegration- error correction methodology which allows us first, to investigate the existence of long-run equilibrium relationships consistent with alternative theoretical models, and second, to build the short-term dynamics by applying the error correction model and adopting the general to specific methodology.

Another objective of the present study is to examine a wider class of time-series models. None of the previous studies has considered multivariate time-series models, except Meese and Rogoff (1983a). who applied a VAR (vector autoregressive) model. In this paper we examine both VAR and BVAR (Bayesian vector autoregressive) models composed of all the variables from the respective structural models.* An important issue that we wish to investigate is whether VAR and BVAR models should be estimated in level or difference form.

The paper is organised as follows. The next section describes the competing structural models and the methodology used in their estimation. The third and fourth sections present the VAR and BVAR models. The fifth section outlines our forecasting strategy for generating out-of-sample forecasts. In the sixth we compare the forecasting performance of the competing models. The final section draws the main conclusions.

STRUCTURAL MODELS

Our research plan was to investigate all major theories of exchange rate determination normally used in empirical studies: the flexible price monetary model, the sticky price monetary model, the Hooper-Morton model, the portfolio balance model, and the uncovered interest parity model. The first three models are different versions of the monetary approach, with the

' The one exception is the study by Meese and Rose (1991) which has also considered Hodrick's (1988) model. 'For an excellent survey of this literature, see MacDonald and Taylor (1992). 'The seminal papers on cointegration are those by Engle and Granger (1987) and Johansen (1988). An excellent analysis of the dynamic modelling methodology adopted in this paper can be found in Cuthbertson et al. (1992). A

'Notice that Diebold and Nason (1990) have investigated the forecasting performance of non-parametric autoregression exchange rate models, but they did not provide a comparison with predictions by structural models.

ood analysis of the general to specific methodology can also be found in Pagan (1990).

N . Sarantis and C . Stewart Structural, VAR and BVAR Models 203

Hooper-Morton model (used by Meese and Rogoff, 1983; Alexander and Thomas, 1987; Schinasi and Swamy, 1989; and Meese and Rose, 1991) nesting the other two.

The exchange rate equations derived from the alternative theoretical models are essentially long-run equilibrium relationships. Therefore modelling should begin with an investigation of these equilibrium relationships. If we fail to uncover the existence of equilibrium relationships consistent with the respective theoretical models, the development of short-run structural equations based on such models will be inappropriate for forecasting exchange rates (see Baillie and Selover (1987)). Such an investigation can be carried out by applying the cointegration method (see Engle and Granger, 1987; Engle and Yoo, 1987; and Johansen, 1988).

The cointegration tests, together with a detailed analysis, for all the structural models mentioned above and all four sterling exchange rates over the period 1973Q1-199043 are reported in Sarantis and Stewart (1993).' These tests rejected overwhelmingly the existence of a long-run equilibrium relationship consistent with the flexible price monetary model, or the sticky price monetary model, or the Hooper-Morton model. Our findings are similar to those reported by Baillie and Selover (1987) for five bilateral exchange rates against the US dollar. This dismal evidence seriously undermines the validity of monetary models used in exchange rate forecasting, and therefore the empirical results obtained by Meese and Rogoff (1983a,b) and others using monetary exchange rate models. The cointegration tests, on the other hand, uncovered long-run equilibrium relationships consistent with the portfolio balance model and a modified uncovered interest parity relationship with backward-looking expectations for three exchange rates: D-mark/pound, yenlpound, and F-franc/pound. Application of the Johansen (1988) maximum likelihood tests for cointegration confirmed the uniqueness of these cointegrating vectors.

Consequently, the structural models we have used in our forecasting study are as follows.

Modified uncovered interest parity (MUIP)

e,= ( ~ ~ - a ~ ( ~ * - r ) , + a ~ ( n ' - n ) , + as(P'-P),+a,(CA/NY),-a,(CA/NY):+u, (1)

where e is the (natural) logarithm of the exchange rate, defined as the foreign currency price of domestic currency; r , n, p and (CAINY) represent the logarithm of nominal short-term interest rate, expected price inflation rate, the logarithm of the price level, and the ratio of current account to nominal GDP for the domestic economy (i.e. UK), respectively; asterisks denote the corresponding foreign variables; u is the error term. The variables (CAINY) and (CA/NY)' are proxies for the risk premium (see Fisher et al., 1990, for a similar treatment).

Equation (1) is a modified version of the real UIP relationship (adjusted for risk) with backward-looking expectations, and is based on a similar model used by the Bank of England for modelling sterling's effective exchange rate. A detailed discussion and investigation of this model is given in Sarantis and Stewart (1993).

Portfolio balance (PB)

'Sarantis and Stewart (1993) provide a brief description of the main theoretical exchange rate models and a detailed econometric evaluation of these models using the cointegration-error correction methodology. Further evidence on variants of the monetary exchange rate model is also provided in Sarantis (1994).


where (vnso) is the value of North Sea oil; m, k, b and f a r e the nominal stocks (in natural logarithm) of UK money, equities, bonds, and foreign assets (held by the private sector), respectively; and asterisks denote the corresponding foreign variables.6

Equation (2) is derived from Sarantis (1987). That study developed a portfolio balance model which includes money, bonds, equity and foreign asset markets, as well as the influence of North Sea oil, and was applied to five sterling exchange rates with satisfactory results.

In applying the structural models (1) and (2) to each sterling exchange rate, we have made use of the two-stage modelling methodology proposed by Engle and Granger (1987). In the first stage we estimate the equilibrium or cointegrating equations (1) and (2) with the method of Ordinary Least Squares. The uniqueness of these cointegrating vectors is examined with the Johansen (1988) multivariate cointegration tests. These tests support the existence of unique long-run equilibrium relationships consistent with the respective structural models (see Sarantis and Stewart, 1993; and Sarantis, 1994). In the second stage we employ the error correction formulation to model the short-run dynamics of each sterling exchange rate, utilizing our prior estimates of the long-run parameters:

Ae, = Po + $lAYt-i + P2Ae,-l-i - P3RESt-I + v, (3) where Y is the vector of explanatory variables in the respective structural model: is the lagged residuals from the corresponding cointegrating relationship; and v is the error term. In specifying the dynamic model (3) we have used the general to specific methodology, starting with lags of up to four quarters and nested down to a parsimonious representation of data, so that the final regression model passes the various diagnostic tests of its error term, functional form and parameter stability. These econometric results are reported in Sarantis and Stewart (1993).

VECTOR AUTOREGRESSIVE (VAR) MODELS

In the VAR approach each variable is a linear function of lagged values of all variables in the system. Hence a VAR representation is given by

X, = C + B(L)X, + U, (4) where X is an (n x 1) vector of all variables in the system, u is an (n x 1) vector of error terms, and B(L) is an ( n x n ) polynomial matrix in the lag operator L.

Given that we are employing two structural models in our forecasting investigation, we have also estimated two VAR models; one includes the variables of the MUIP model, referred to as VARl , and the other includes the variables of the portfolio balance model, referred to as VARZ An important issue in VAR modelling is whether to transform the data to render all variables stationary or to conduct the analysis in levels. Pagan (1990) points out that ‘the idea that z,(X,) can be expressed as a VAR has its origin in the theory of stationary process’ and hence suggests that ‘the sensible response ... would be to focus upon growth rates rather than levels of variables’. On the other hand, Engle and Yo0 (1987) argue that vector autoregressions estimated with cointegrated data will be misspecified if the data are differenced, and demonstrate that error correction and VAR in levels are competing specifications. To address the forecasting implications of this issue, we have used both level and stationary approaches. The VAR models in levels are denoted as LVARl and LVAR2 respectively, while VARs in difference (stationary) form are denoted by DVARl and DVAR2 respectively.

Due to the unavailability of data on k’ and b’, these variables were not included in the estimation of model (2).


Table I. LR tests for VAR lag lengths (levels)"

US dollar/ D-mark/ Yen/ F-franc/ Modelllags pound pound Pound pound

LVAR 1 1+2 48.24 (51.0) 55.16 (51.0) 73.18 (51.0) 62.13 (51.0) 243 55.55 (51.0) 43.92 (51.0) 34.04 (51 .O) 29.59 (51.0)

52.57 (51.0) 33.36 (51.0)

77.23 (92.8) 1=2

-

3+4 36.63 (51.0) 53.98 (51.0) 44.08 (51.0) - - - 1=2

4+5 - 30.97 (51.0) 1+3 96.90 (92.8) - 2+4 - 90.58 (92.8) Chosen lag I = 3 1=2

LVAR2 1+2 110.51 (66.3) 131.94 (83.7) 1-0.87 (83.7) 09.52 (83.7) 2+3 64.58 (66.3) 60.26 (83.7) 94.35 (83.7) 82.34 (83.7) 3+4 56.00 (66.3) 90.33 (83.7) 116.45 (83.7) 88.95 (83.7) 4+5 - 89.05 (83.7) 104.21 (83.7) 80.59 (83.7)

244 - - - 148.58 (155.4) 5+6 - 81.37 (83.7) 376.69 (83.7) - 2+ 5 - 171.99 (225.3) 218.59 (225.3) - Chosen lag 1=2 1=2 1=2 1=2

5% critical values in parentheses.

Table II. LR tests for VAR lag lengths (differences)"

US dollar/ D-mark/ Yen/ F- fmc / Modelllags pound pound Pound pound

DVARl 1+2 47.77 (51.0) 50.90 (51.0) 36.08 (51.0) 47.63 (51.0) 2+3 46.40 (51.0) 42.19 (51.0) 42.93 (51.0) 38.00 (51.0) 3+4 41.02 (51.0) 33.29 (51.0) 37.85 (51.0) 36.88 (51.0) Chosen lag I= 1 I = 1 I= 1 I = 1

DVAR2 1+2 55.22 (66.3) 71.34 (83.7) 80.59 (83.7) 87.63 (83.7) 2+3 56.25 (66.3) 58.93 (83.7) 79.05 (83.7) 127.33 (83.7) 3+4 64.73 (66.3) 57.67 (83.7) 108.13 (83.7) 24.75 (83.7) 4+5 - - 83.21 (83.7) - 1+4 - - 211.11 (225.3) - Chosen lag I = 1 I = 1 I = 1 I= 3

5% critical values in parentheses.


To determine the optimal lag length of the VAR system we have employed the likelihood ratio test statistic, LR, suggested by Sims (1980).’ The estimates of LRs for the two VAR models estimated in level and stationary form are shown in Tables I and II respectively. Given the number of observations, we have considered a maximum lag of four and then tested downwards to one lag. Where the test supports the four lag, we have estimated longer lag structures in order to confirm the significance of lag four. In cases where the LR statistics indicate two significant lags, we have tested one versus the other and the shorter lag is acceptable in all cases. The chosen lags for all exchange rates are indicated in the tables.

BAYESIAN VECTOR AUTOREGRESSIVE (BVAR) MODELS

To overcome the serious problem of over-parameterization associated with VAR models, Litterman (1986) and others have proposed the BVAR representation which provides a flexible method for incorporating prior information for all the coefficients.* Following Litterman (1986), we assume that each coefficient has an independent and normal distribution with zero mean, except for the coefficient on the first lag of the own variable which has a mean of unity.

The standard deviations of the prior distributions are given by

S ( i , 0 = yf(i, j )g (Os j /s i (5) The function g describes the lag pattern and is assumed to be harmonic: g( I) = 1 -d . The function f gives the tightness of the prior for variable j in equation i relative to the tightness on the own lags. This function is assumed to be symmetric and has the form, f(i, j ) = w. The parameter y is the overall tightness of the prior; it is the standard deviation of the prior on the first own lag in each equation.

The most important hyperparameters in the construction of BVAR models are w and y. The parameter w measures the weight attached to the coefficients of other variables relative to that of the own lagged value. This parameter takes values between zero and one. When w = 1, all variables in each equation have equal weight when forecasting the own variable. As w approaches zero the model approaches a set of univariate autoregressions. In our experimentation we have considered three values for w: 0.3, 0.5, 0.8. For the parameter y we have assumed a relatively ‘loose’ value of 0.2 and a ‘tight’ value of 0.1. This has given us six alternative specifications. Using lags up to the fourth order for all variables, each formulation was estimated both in levels and stationary form and used to produce out-of-sample forecasts.

We found that forecasting performance is sensitive to the choice of hyper parameter^.^ The

If AR(m) is the unrestricted VAR and AR(f) the restricted one, where rn and I are the respective lags, then the LR statistic for testing AR(1) against AR(m) is given by

LR = (T- c)(lnlnrl -lnlQml) where T is the number of observations, c is the correction factor which is equal to the number of regressors in each equation in AR(m), and i2 is the covariance matrix of residuals of AR(f) and AR(m) respectively. The statistic LR is asymptotically distributed as x2 with kz(m - I ) degrees of freedom, where k is the number of regressions. ‘As far as we are aware, this is the first application of BVAR models to exchange rates. It is worth noticing that Meese and Rogoff (1983a) suggest that application of this methodology may improve on the random walk model, but the authors do not pursue this investigation. ’ A r t i s and Zhang (1990) have found a similar result in applications of the BVAR model to macroeconomic forecasting.

N . Sarantis and C. Stewart Structural, VAR and BVAR Models 207

Table III. BVAR hyperparameters (levels)

Exchange rate LBVARl LBVARIL

Y W Y W

US dollar/pound 0.2 0.8 0.2 0.8 D-mark/pound 0.1 0.3 0.2 0.8 Yen/pound 0.2 0.3 0.2 0.8 F-franc/pound 0.2 0.8 0.2 0.5

Table IV. BVAR hyperparmeters (differences)

Exchange rate DBVARl DBVAR2

Y W Y W

US dollar/pound 0.2 0.8 0.2 0.8 D-mark/pound 0.2 0.3 0.2 0.3 Yenlpound 0.1 0.8 0.1 0.3 F-franclpound 0.2 0.3 0.2 0.3

combinations of values for w and y producing the best forecast for each exchange rate are reported in Tables El (for BVARs in levels) and IV (BVARs in differences). A point worth making is that relatively ‘loose’ values for both y and w produce better forecasts than ‘tight’ values for the overwhelming majority of models and exchange rates.

FORECASTING STRATEGY

Following Meese and Rogoff (1983a) and other researchers, we adopted the strategy of sequential estimation for generating out-of-sample forecasts for one to ten horizons. The parameters of the models are based on the most recent information available at the time the forecast is made. More precisely, the models were initially estimated using data from 197342 (the beginning of the recent floating period) to 198604. The choice as to where to begin forecasting was predicated on the desire to produce medium-term forecasts and have sufficient number of observations for each forecast step. This has enabled us to produce ten-steps-ahead (quarters) forecasts, which is normally the longest forecast horizon considered by professional forecasters in this area. Forecasts are then generated at horizons of one to ten quarters. Next, data for 1987Q1 are added to the sample and the parameters of each model are re-estimated. New forecasts are then generated for one- to ten-quarters-ahead. This process continues through to the last forecast period, 199042.

The generation of forecasts with cointegration-error correction models requires the setting up of a simultaneous equation system to capture both the long-run equilibrium and short-run relationships. Hence the forecasting system for the MUIP (modified uncovered interest parity)


and PB (portfolio balance) structural models consists of the following equations:

A e , = B o + B ' , A Y , - , +B&-1-i -B3RESI-l (6a)

e, = e+, + Ae,

RES, = e, - a'Y,

Note that equation (6a) is the error-correction equation (identical to equation (3)), equation (6b) is used to obtain the predicted level (in logs) of the exchange rate, and equation (6c) is the cointegrating equation (i.e. equations (1) and (2) respectively) rearranged so that the residual term becomes the dependent variable.

It should be stressed that in the case of the structural models, both the cointegrating equation and the error correctionmodels were re-estimated at each forecast point, as the data for the latest quarter were added to the sample. These parameter estimates (and the respective cointegrating residuals, RES) were used in the system (6a)-(6c) to generate one- to ten-quarters-ahead forecasts. These updated forecasts were stored in a separate file, which was subsequently used to construct the accuracy statistics described below. On the other hand, the updating of the VAR and BVAR models and forecasts was carried out with the Kalman filter which computes automatically all the accuracy statistics for different forecast steps. lo

Following previous researchers in this area, we have used actual realised values of all explanatory variables in the prediction period to generate out-of-sample forecasts with the structural models. This problem does not arise in the case of VAR and BVAR models where all variables are endogenous."

The accuracy of forecasts for each forecast horizon is measured by the root mean square error (RMSE) and the Theil U statistic, which are defined as follows:

[A(t + j + k) - F(t + j + (7)

U = RMSE(model)/RMSE(random walk) (8)

where k = 1, 2, . . . 10 denotes the forecast step; N is the total number of forecasts in the prediction period; A ( t ) and F ( t ) are actual and forecast values respectively.

The U statistic is the ratio of the RMSE for the estimated model to the RMSE of the simple random walk model which predicts that the forecast simply equals the most recent information. Hence if U c 1, the model performs better than the random walk model without drift; if U > 1, the random walk outperforms the model. The forecasted value used in the computation of the RMSE and U statistics is the level (in logarithm) of the exchange rate, so these statistics can be compared across the different models. In the case of the VAR and BVAR models formulated in stationary form, the generated forecasts were the changes in the log of the exchange rate. The latter were subsequently converted into forecasted levels using past levels and forecasted exchange rate changes.

"The estimation and simulation of the structural models were done on Microfit and PC-TSP. while the RMSE and U statistics for the forecasts produced by these models were computed on Quattro. The VAR and BVAR forecasts were yroduced on PC-RATS.

I The only exception is North Sea oil (vnso), which has been treated as an exogenous variable and, therefore. we have used its actual values during the prediction period.

N. Sarantis and C. Stewart Structural, VXR and BVXR Models 209

EMPIRICAL, RESULTS

The forecasting results for the competing models and each exchange rate over varied horizons are summarized in Table V-VII." Also shown are RMSE and U averages for forecast steps one to four and one to ten. The most striking results from these tables concerns the relative forecasting performance of the two structural models. The modified UIP strongly outperforms the portfolio balance model for all exchange rates and forecast horizons. The average RMSE (over steps one to ten) for the MUIP is reduced by roughly 80% for the D-mark/pound, 84% for the F-franc/pound and by 3 1 % for the yen/pound. Over the shorter horizon of one to four steps, the average RMSE for the Muzp is reduced by 72% for the D-mark/pound, 75% for the F- franc/pound and 28% for the yen/pound. This suggests that the gap between the two models becomes wider over longer forecasting horizons. We therefore choose the MUIP as our representative structural m0de1.l~. Similar evidence on the dominance of the MUIP model is also provided by the BVAR and VAR formulations of the variables included in the respective

Table V. Statistics on forecasting accuracy: deutschmarklpound

Forecast step One-step lbo-step Three-step Four-step Five-step Six-step Observations 15 14 13 12 11 10

RMSE U RMSE U RMSE U RMSE U RMSE U RMSE U

ModifiedUIP 0.027 0.900 0.045 0.939 0.056 0.900 0.063 0.847 0.065 0.776 0.066 0.736 Portfoliobalance 0.076 2.511 0.139 2.924 0.201 3.209 0.263 3.551 0.325 3.871 0.362 4.056 LVAR 1 0.037 1.223 0.065 1.364 0.087 1.392 0.108 1.458 0.128 1.521 0.144 1.611 LBVARl 0.026 0.862 0.043 0.907 0.055 0.884 0.065 0.883 0.077 0.912 0.089 0.991 DVARl 0.028 0.933 0.053 1.108 0.072 1.152 0.090 1.217 0.112 1.336 0.132 1.478 DBVARl 0.031 1.032 0.055 1.163 0.074 1.183 0.087 1.180 0.108 1.283 0.128 1.427

Forecast step Seven-step Eight-step Nine-step Ten-step Average Average Observations 9 8 7 6 one to four one to ten


ModifiedUIP 0.064 0.735 0.061 0.729 0.057 0.758 0.052 0.835 0.048 0.897 0.055 0.816 Portfoliobalance 0.379 4.374 0.400 4.810 0.368 4.909 0.324 5.233 0.170 3.059 0.284 3.949 LVAR 1 0.149 1.715 0.153 1.845 0.153 2.042 0.137 2.215 0.074 1.360 0.116 1.639 LBVARl 0.094 1.083 0.098 1.175 0.099 1.325 0.088 1.419 0.047 0.884 0.073 1.044 DVARl 0.146 1.680 0.158 1.902 0.171 2.281 0.175 2.824 0.061 1.103 0.114 1.591 DBVARl 0.139 1.608 0.152 1.825 0.163 2.166 0.169 2.725 0.062 1.139 0.111 1.559

"Since we were unable to uncover any significant cointegrating vector, and consequently to develop a structural model, for the US dollarlpound rate, we decided to exclude this exchange rate from our forecasting comparison. Of course, one could still develop VAR and BVAR models for this exchange rate. This was done during our experimentation and the general finding was that the BVAR in levels tended to produce the most accurate forecasts. "This evidence about the superiority of the modified U P model over the portfolio balance model supports a similar result based on non-nested tests in Sarantis and Stewart (1993).


Table VI. Statistics on forecasting accuracy: yen/pound

Forecast step One-step lko-step Three-step Four-step Five-step Six-step Observations 15 14 13 12 11 10


ModifiedUIP 0.039 1.283 0.064 1.220 0.074 1.064 0.077 1.008 0.081 0.993 0.088 1.026 Portfoliobalance 0.052 1.714 0.082 1.570 0.104 1.486 0.112 1.477 0.117 1.428 0.130 1.510 LVARl 0.043 1.401 0.052 0.998 0.070 0.994 0.096 1.265 0.109 1.327 0.122 1.418 LBVARl 0.025 0.802 0.043 0.823 0.060 0.853 0.074 0.970 0.085 1.037 0.100 1.166 DVARl 0.034 1.110 0.062 1.187 0.090 1.286 0.106 1.392 0.120 1.469 0.127 1.474 DBVARl 0.030 0.978 0.053 1.007 0.079 1.123 0.085 1.118 0.099 1.206 0.111 1.297



ModifiedUIP 0.094 1.087 0.100 1.173 0.103 1.266 0.101 1.272 0.063 1.144 0.082 1.139 Portfoliobalance 0.141 1.620 0.147 1.723 0.151 1.843 0.150 1.887 0.088 1.562 0.119 1.626 LVARl 0.130 1.496 0.133 1.558 0.139 1.697 0.135 1.695 0.065 1.165 0.103 1.385 LBVARl 0.115 1.330 0.127 1.487 0.133 1.627 0.130 1.635 0.050 0.862 0.089 1.173 DVARl 0.132 1.521 0.135 1.580 0.162 1.987 0.192 2.407 0.073 1.244 0.116 1.541 DBVARl 0.122 1.403 0.144 1.696 0.177 2.170 0.193 2.429 0.062 1,057 0.109 1.443

structural models. Consequently, we do not report the forecasts produced by the BVAR and VAR formulations of the portfolio balance model (i.e. LVAR2, LBVAR2, DVAR2 and DBVAFt.2) in order to focus on the ‘key contenders’.

Another issue of interest is whether the VAR and BVAR models should be estimated in level or difference (stationary) form. Looking at the VAR and BVAR formulations of the variables included in the MUIP model (VARl and BVARl), the levels approach strongly outperforms the difference approach for all exchange rates and forecast horizons, except for the VAR model of the D-marklpound rate over the short term horizon. It should be noted, however, that even these short-tern VAR forecasts are considerably inferior to those produced by the BVAR model. The gap between the two modelling approaches is more striking in the case of the BVAR model. For steps one to ten, the average RMSE for the BVAR model in levels (LBVARl) is reduced by 34% for the D-marklpound, 18% for the yenlpound and by 25% for the F-franclpound rate.I4

Comparing the forecasting performance of the best structural, VAR and BVAR models, several observations are made. Looking at the forecasted levels of the D-mark/pound rate, the VAR model yields the largest RMSEs at all horizons, with its average value being 56% higher

“To our knowledge, the only study to have compared the out-of-sample forecasting performance of VAR models in level and difFercnce form is that by Shoesmith (1992) for regional and state non-agricultural employment in the USA. “he results of that study show that the stationary VAR produces more accurate fonecasts than its level counterpart under different lag structures. We should note, however, that Shoesmith applied the VAR model to a non-integrated system, whereas our VAR and BVAR models are applied to cointegrated systems.

N . Sarantis and C . Stewart Structural, VAR and BVXR Models 211

Table VII. Statistics on forecasting accuracy: french franc/pound

Forecast step One-step Two-step Three-step Four-step Five-step Six-step Observations 15 14 13 12 11 10

RMSE U RMSE U RMSE U RMSE LI RMSE U RMSE U

ModifiedUIF’ 0.026 0.877 0.043 0.873 0.051 0.765 0.053 0.647 0.051 0.540 0.047 0.465 Portfolio balance 0.096 3.215 0.152 3.080 0.200 2.980 0.268 3.251 0.341 3.614 0.399 3.912 LVAR 1 0.034 1.143 0.047 0.944 0.052 0.767 0.055 0.666 0.061 0.650 0.066 0.650 LBVARl 0.021 0.698 0.037 0.741 0.051 0.764 0.067 0.814 0.080 0.849 0.087 0.869 DVARl 0.029 0.956 0.050 1.016 0.070 1.034 0.086 1.046 0.101 1.074 0.112 1.095 DBVARl 0.028 0.948 0.049 0.991 0.069 1.023 0.086 1.037 0.102 1.082 0.116 1.140



ModifiedUIP 0.043 0.428 0.036 0.373 0.035 0.398 0.042 0.581 0.044 0.791 0.043 0.595 Portfolio balance 0.403 4.007 0.350 3.589 0.288 3.250 0.249 3.464 0.179 3.131 0.275 3.436 LVARl 0.067 0.666 0.070 0.719 0.076 0.857 0.078 1.084 0.047 0.880 0.061 0.815 LBVARl 0.089 0.886 0.085 0.874 0.083 0.937 0.079 1.101 0.044 0.754 0.068 0.853 DVARl 0.111 1.109 0.107 1.100 0.099 1.116 0.086 1.195 0.059 1.013 0.085 1.074 DBVARl 0.118 1.179 0.120 1.228 0.115 1.303 0.103 1.431 0.058 1.000 0.091 1.136

than that under the BVAR model. The BVAR model in levels produces the most accurate forecasts over steps one to three, while the structural MUlP model consistently outperforms the BVAR model for forecast steps one to ten. This is reflected in the average values of the RMSE. For steps one to four, the MUIP performs slightly worse than the BVAR, but for steps one to ten the MUIP improves on the BVAR by roughly 25%. A similar pattern is observed for the F- franc/pound rate. The BVAR model in levels is the most accurate over steps one and two, but the MUIP displays a considerable improvement over forecast steps three to ten, showing a reduction in the average RMSE (over steps one to ten) by 30% over the VAR and 37% over the BVAR model. In the case of the yen/pound rate, the BVAR model in levels is again producing the more accurate forecasts over steps one to four, whereas the structural MUIP model strongly outperforms all models over four to ten forecast steps. As a result of the widening gap over the longer horizons, the average RMSE (for steps one to ten) under the MUlP is reduced by 8% over the BVAR and 20% over the VAR. This contrasts with the short-term loss (over steps one to four) of 21% to the BVAR model. The VAR model is outperformed by the BVAR model at all forecast horizons, with its average RMSE value being 16% higher than that under the BVAR model for one to ten steps. The reported gains in medium-term forecasting accuracy under the structural model for all exchange rates, at the expense of loss of some short-term accuracy to the BVAR model, support 4 priori expectations about the relative performance of structural versus multivariate time-series models, and are also remarkably similar to the pattern obtained by Engle and Yo0 (1987) through simulations.

A controversial issue in the literature is whether the random walk model yields more


accurate forecasts than the structural models.” The U statistics indicate that the best structural model (MUIP) outperforms consistently the random walk model at all forecast horizons in the cases of the D-marklpound and F-franclpound rates. On the other hand, the random walk model produces more accurate forecasts for the level of the yenlpound rate at all horizons. This finding is rather surprising, given the strong evidence of cointegration and error correction for the yenlpound rate reported in Sarantis and Stewart (1993). A potential explanation is the relative stability of the yen/pound exchange rate during the forecasting period 198741-199043; this rate exhibited a standard deviation of 0.055 and a coefficient of variation of 0.010, while the maximum and minimum values (in logs) were 5.599 and 5.412 respectively.

CONCLUSIONS

In this paper we have investigated the out-of-sample forecasting performance of a wide class of structural, BVAR and VAR models for bilateral sterling exchange rates, There are a number of empirical findings worth mentioning. The lack of statistical evidence in support of long-run relationships consistent with the various versions of monetary models used by previous researchers suggests that these structural models are inappropriate for exchange rate forecasting. Instead one should focus on the portfolio balance (PB) and a modified version of the uncovered interest parity (MUIP) as representative structural models. Our results show that the MUIP model produces more accurate forecasts than the portfolio balance model at all forecast horizons and all exchange rates. Proper attention to the long-run properties and the dynamics of structural models of exchange rate determination can improve on the forecasting performance of the random walk model. The best structural model strongly outperforms the random walk model at all forecast horizons for two out of three bilateral exchange rates (i.e. D-marklpound, F-franclpound). The only exception is the yenlpound rate which would have been predicted more accurately by the random walk model than by any structural model. These findings suggest that the dispute over structural versus random walk models is far from over and that the jury is still out. Any claim that exchange rates are not longer driven by fundamentals (Williamson, 1993) would seem to be premature, particularly for the European currencies.

The overall ranking of the models varies over different forecast horizons. The structural model shows substantial improvement in medium-term forecasting accuracy of exchange rate levels, whereas the BVAR model is more accurate in the short term (normally up to three or four quarters), though the average forecast errors for all steps (one to ten) are lower under the best structural model for all exchange rates. An implication of these results is that optimizing short- and long-term forecasting with cointegrated systems will often require separate efforts. Our results show that the best VAR and BVAR models in levels improve considerably on the

out-of-sample forecasting performance of these models formulated in difference form, This evidence provides support in favour of the levels approach in VAR and BVAR modelling. Worth mentioning is also the finding that the unrestricted VAR is generally outperformed by the restricted BVAR model.

”In contrasting our results with those of previous studies cited in the first section, we should bear in mind not only differences in the exchange rates and sample period covered but also in the frequency of data used. In the present study we have used quarterly data, while most previous researchers used monthly data.

N. Sarantis and C. Stewart Structural, VAR and BVAR Models 213

APPENDIX: MEASUREMENT OF VARIABLES AND DATA SOURCES'6

(log) foreign currency/pound sterling spot exchange rate, average of daily telegraphic transfer rates in London- Financial Statistics. (log) UK money stock (Ml) in pounds-data supplied by the Economics Division of the Bank of England. (log) foreign money stock (Ml) in foreign currency (i.e. for the USA, West Germany, Japan and France)- OECD Main Economic Indicators. (log) UK interest rate on 3-month inter-bank loans, monthly averages-Financial Statistics. (log) foreign money market interest rate (Federal Funds interest rate for the USA), monthly averages- OECD Main Economic Indicators and International Financial Statistics. (log) UK long-term interest rate (yield of long-term government bonds), monthly averages- OECD Main Economic Indicators. (log) foreign long-term interest rate (USA: yield of government composite bonds > 10 years; Germany and Japan: yield of long-term government bonds; France: yield of long-term bonds guaranteed by government), monthly averages- OECD Main Economic Indicators. (log) UK producer price index- OECD Main Economic Indicators. (log) foreign producer price index (GDP implicit price deflator for France)-OECD Main Economic Indicators. UK gross domestic product at current prices (in pounds)-OECD Main Economic Indicators. foreign gross domestic product at current prices (in foreign currency)-OECD Main Economic Indicators. UK current account balance in pounds- OECD Main Economic Indicators. foreign current account balance in foreign currency (data for Japan were in UK dollars and were converted in yen using the yen/dollar exchange rate)-OECD Main Economic Indicators. (log) foreign asset stock held by the UK private sector, F, computed as follows:

F, = F,-l + [CA, - A(RESERV),]

where A(RESERV) is the change in non-gold foreign reserves (these were in US dollars and were converted into pounds using the dollar/pound rate). The benchmark was 1971(1). For an explanation and use of this formula see Branson et al., (1977) and Keating (1985). Data for RESERV were obtained from International Financial Statistics. (log) foreign asset stock held by the private sector of the foreign country, F', computed by the formula:

F*, = F*,-l+ [CA', - A(RESERV)',]

where the change in non-gold foreign reserves was converted from US dollars to that country's currency using the corresponding exchange rate between the US dollar and the country's domestic currency.

I6Notice that all variables were tested for stochastic seasonality and that these seasonal roots (wherever found) were removed by applying appropriate filters. See Sarantis and Stewart (1993) for details about the measurement of variables.

214

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(log) UK personal sector holdings of equity assets-data provided by the Centre of Economic Forecasting of the London Business School. (log) UK personal sector holdings of public sector debt-data provided by the Centre for Economic Forecasting of the London Business School. Value of North Sea oil production in pounds-data provided by the Centre for Economic Forecasting of the London Business School-expressed as a percentage of UK nominal GDP.

ACKNOWLEDGEMENTS

Financial support from the Kingston University Research Fund is gratefully acknowledged. We thank an anonymous referee and the editor, Professor K. Holden, for their useful comments. As usual, we are responsible for the organisation and contents of the paper.

REFERENCES

Alexander, D. and Thomas, L. R., ‘Monetary/asset models of exchange rate determination: how well have

A r t i s , M. J. and Zhang. W., ‘BVAR forecasts for the G-7’, International Journal of Forecasting, 6

Baillie, R. T. and Selover, P. D., ‘Cointegration and models of exchange rate determination’, International

Branson, W. H., Halhmen, H. and Masson, P., ‘Exchange rates in the short run: the dollar deutschmark

Cutherbertson, K., Hall, S. G. and Taylor, M. P., Applied Econometric Techniques, London: Harvester

Diebold, F. X. and Nason, J. A., ‘Nonparametric exchange rate prediction’, Journal of International

Engle, R. F. and Granger, C. W. J., ‘Co-integration and error corntion: npresentation, estimation and

they performed in the 1980’sY International Journal of Forecasting, 3 (1987), 53-64.

(1990), 349-62.

Journal of Forecasting, 3 (1987), 43-51.

rate’. European Economic Review, 10 (1977), 303-24.

Wheatsheaf, 1992.

Economics, 28 (1990). 315-32.

testing’, Econometrica, 55 (19871,251-76. Engle, R. F. and Yoo, B. S., ‘Forecasting and testing in co-integrated systems’, Journal of Econometrics,

35 (1987). 143-59. Fisher, P. G:, Tunna, S. K., Turner, D. S., Wallis, K. F. and Whitley, J. D., ‘Economentric evaluation of

Hodrick. R. J., ‘Risk, uncertainty and exchange rates’. Jounral of Monetary Economics, 23 (1988),

Johansen, S., ‘Statistical analysis of cointegrated systems’, Journal of Economic Dynamics and Control,

Keating, G., The Production and Use of Economic Forecusts, London: Methuen, 1985. Litteman, R. B., ‘Forecasting with Bayesian vector autoregressions-five years of experience’. Journal

MacDonald, R. and Taylor, M. P., ‘Exchange rate economics: a survey’, IMF Staff Papers, 39 (1992),

Meese, R. A. and Rogoff, K., ‘Empirical exchange rate models for the seventies: do they fit out of sample?’ Journal of International Economics, 14 (1983a), 3-24.

Meese, R. A. and Rogoff, K., ‘The out-of-sample failure of empirical exchange rate models: sampling error or misspecification?’, in Frankel, J. A. (ed.), Exchange Rates and International Macroeconomics, Chicago: University of Chicago Press, 1983b.

Meese, R. A. and Rose, A. K., ‘An empirical assessment of non-linearities in models of exchange rate determination’, Review of Economic Studies, 58 (1991), 603-19.

Pagan, A. R., ‘Three econometric methodologies: a critical appraisal’, in Granger, C. W. J. (ed.), Modelling Economic Series, Oxford Clarmdon Press, 1990.

the exchange rate in models of the U.K. Economy’. Economic Journal, 100 (1990), 1230-44.

433-59.

12 (1988), 231-54.

of Business and Economic Stutistics, 4 (1986). 25-38.

1-57.


Sarantis, N., ‘A dynamic asset market model for the exchange rate of the pound sterling’, Weltwirtschaftliches Archiv, (1987), 24-37.

Sarantis, N., ‘The monetary exchange rate model in the long-run: an empirical investigation’, Weltwirtschaftlishes Archiv, (1994), 698-71 1.

Sarantis, N. and Stewart, C., ‘Monetary and asset market models for sterling exchange rates: a cointegration approach’, Economics Discussion Paper 93/1, 1993, Kingston University (forthcoming in the Journal of Economic Integration, 1995).

Schinasi, G. J. and Swamy, P. A. V. B., ‘The out-of-sample forecasting performance of exchange rate models when coefficients are allowed to change’, Journal of International Money and Finance, 8

Shoesmith, G. L., “on-cointegration and causality: implications for VAR modelling’, International

Sims, C., ‘Macroeconomics and reality’, Econometrica, 48 (1980), 1-48. Somanath, V. S., ‘Efficient exchange rate forecasts: lagged models better than the random walk’, Journal

Williamson, J., ‘Exchange rate management’, Economic Journal, 103 (1993), 188-97. Wolf, C. C. P., ‘Time-varying parameters and the out-of-sample forecasting performance of structural

(1989), 373-90.

Journal OfForecasting, 8 (1992), 187-99.

of International Money and Finance, 5 (1986), 195-220.

exchange rate models’, Journal of Business & Economics Statistics. 5 (1987), 87-97.

Authors’ biographies: Nicholas Sarantis is Reader in Economics, and Course Director of the MSc in Business and Economic Forecasting, at the School of Economics, Kingston University. He has been a consultant on macroeconomic forecasting for Economic Models Ltd and DRI Europe Ltd. His research interests include macroeconomic modelling and forecasting, exchange rate modelling and forecasting, macroeconomics, international finance, and applied econometrics. He has published numerous articles in academic journals in the above areas. Chris Stewart is part-time Lecturer and Research Assistant at the School of economics, Kingston University. He holds an MSc degree in economics from Birkbeck College. His research interests include time-series forecasting methods and applied econometrics. He has co-authored an article published in Applied Economics.

Authors’ address: Nicholas Sarantis and Chris Stewart, School of Economics, Kingston University, Penrhp Road, Kingston-upon-Thames, Surrey KT1 2EE, UK.

structural, var and bvar models of exchange rate determination: a comparison of their forecasting...

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