The Out-of-Sample Forecasting Performance of Exchange Rate Models when Coefficients are Allowed to Change

International Finance Discussion Papers Number 301

January 1987

THE OUT-OF-SAMPLE FORECASTING PERFORMANCE OF EXCHANGE RATE MODELS WHEN COEFFICIENTS ARE ALLOWED TO CHANGE

by

Garry J. Schinasi P.A.V.B. Swamy

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in pub:.ications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

This study examines the out-of-sample forecasting performance of models of exchange rate determination without imposing the restriction that coefficients are fixed over time. Both fixed and variable coefficient versions of conventional structural models are considered, with and without a lagged dependent variable. While our results on fixed coefficient models support most of the Meese and Rogoff conclusions, we find that when coefficients are allowed to change, an important subset of conventional models of the dollar-pound, the dollar-deutsche mark, and the dollar-yen exchange rates can outperform forecasts of a random walk model. The structural models considered are the flexible-price (Frenkel-Bilson) and sticky-price (Dornbusch-Frankel) monetary models, and a sticky~price model which includes the current account (Hooper-Morton). We also find that the variable coefficient version of the Dornbusch-Frankel model with a lagged dependent variable generally predicts better than the other models

~' eonsidered including the random walk model.

The Out-of-Sample Forecasting Performance of Exchange Rate Models When Coefficients Are Allowed to Change

by Garry J. Schinasi P.A.V.B. Swamy! I. Introduction The seminal work of Meese and Rogoff (1983a,b), hereafter referred to as MR, casts serious doubt on the ability of international macroeconomic theory to predict exchange rate movements.2 These studies concluded that linear fixed-coefficient regressions of exchange rates on variables such as relative money supplies, indices of industrial production, short interest rates, and trade balances, failed to match the out-of-sample forecasting performance of a simple random-walk model. MR's main results were reported as "robust to a variety of [fixed coefficient ] estimation techniques, specifications of the underlying money demand functions, alternative serial correlation or lagged adjustment corrections, and measures of forecast accuracy."3 One might be tempted to conclude from these studies that economic

variables convey little or no useful information about exchange rate

\The authors would like to acknowledge the helpful comments of Don Alexander James Barth, Frank Diebold, Eric Fisher, David Howard, Karen Johnson, Jaime Marquez, Ralph Tryon, and the participants of seminars presented in the International Finance Division at the Federal Reserve Board and at the Bank of Japan in Tokyo. The authors take full responsibility for any remaining errors or misconceptions. Views expressed in this paper are those of the authors and do not reflect the views of the Board of Governors or the staff of the Federal Reserve System.

2Some other follow-up studies are the following: MR (1985), Backus (1984), Woo (1985), Finn (1986), Edison (1984), Hakkio (1986), and Samanath (1986).

3See MR (1985, p.5).

-2movements. We find this conclusion unsatisfying. We share the view suggested by MR that "...their disappointing performance is most likely to be attributable to simultaneous equation bias, sampling error, stochastic movements in the ... underlying parameters, or misspecification" (1983a, p.17).4

This study evaluates the forecasting performance of models reported by MR (1983a) without imposing the restriction that the regression slopes are fixed over time (misspecification of the dynamics of exchange rate models is also examined). Although there are a number of studies that have relaxed the fixed coefficient assumption, they have done so in rather restrictive ways, requiring extensive and generally unavailable prior information.? The present study avoids these problems by applying a general technique for estimating models with stochastic coefficients (see Swamy and Tinsley (1980)) that encompasses as special cases the Kalman filtering technique, the method of Hildreth and Houck (1968), ARCH models (Engle (1982)) and others. Since multi~step- ahead forecasts (to be defined rigorously later in the paper) are not necessarily inferior to one-step-ahead forecasts (Swamy and Schinasi (1936)), we compute the former, using models with fixed and stochastically varying coefficients, to

extend the work of MR without duplicating their efforts.

4see Isard (1986) for a complementary view.

5Wolff (1985) and Alexander and Thomas (1986) estimate models using the Kalman filtering technique with known covariance matrices and essentially substantiate the MR results. Makin and Sauer (1986) use the Hildreth and Houck (1968) technique for estimating restrictive variable coefficiert models to study policy regimes and do not perform the MR experiments.

-3-

The major result of our study is that when all coefficients are allowed to vary, the conventional models of exchange rates tested by MR can yield forecasts which are more accurate than their fixed-coefficient counterparts and more accurate than the random walk model. The present study does however support most of the MR conclusions regarding fixed coefficient models, although, contrary.to their study, we find major improvements in the forecasts of fixed coefficient models that include laggec. adjustment.

The paper is organized as follows. Section 2 presents the excharige rate models tested and describes the forecasting strategy adopted in the MR (1983a) study. It also presents the alternative fixed coefficient estimators used in the present study. Section 3 briefly discusses why variable coefficient models are appropriate for exchange rate modelling.. Section 4 presents the stocnastic coefficient representation of the exchange rate models tested by MR, and examines the assumptions required to estimate them. It also describes useful approximations to the minimum average mean square error linear predictor of an individual drawing of a dependent variable, given the independent variables, in a stochastic coeff:.cient model. Section 5 describes the forecasting strategy of the present study. Section 6 compares the out-of-sample forecasting performance of the random walk models and the variable and fixed coefficient representations of the exchange rate models. The final section

draws conclusions from the study.

2. The Models and the Method 2.1 Exchange Rate Models The structural models estimated by MR (1983a,b) and used for forecasting nominal exchange rates are presented below. One general

specification covering all these models as special cases is equation

(1):

e*

x ¥ ¥ e (1) i By + 8, (m, m,) + Ba(y, y,) + B3(ry- r,) + By Cm, 1

+ 8, (TB, - TB, ) + u,.

where, lower case letters indicate natural logs except for interest rates,

* indicates a foreign variable, and where,

Ss = the spot exchange rate ($/DM, $/¥, $/£)

m = money supply

y = industrial production

r = short-term interest rate

m@= expected inflation rate®

TB= cumulative trade balance

u = disturbance term which may be serially correlated.

The original MR exercises estimated the following fixed coefficient versions of equation (1): (i) Frenkel-Bilson (purchasing power parity) ‘ which assumes 8y=85=0; (ii) Dornbusch-Frankel (slow price adjustment) which assumes Br =0; and (iii) Hooper-Morton which is equation (1) with _ unequal coefficients for the trade balances.7

These models are all variants of the monetary model of exchange rates

6To try proxies for the unobservable variable (1f- nf*) different fron those of MR, we replace this variable by a distributed lag in the CPI inflation rate differential.

TThe most recent vintage of MR (1985) performs root mean square forecast error (RMSE) exercises on the nominal exchange rate models, with three more years of data, and real exchange-rate versions of these same structures.

~5and differ only in the way they treat price adjustment .8 In this Study we also explicitly report results for these models with a lagged dependent variable as an explanatory variable, a specification that explicitly allows for short run deviations from long run purchasing power parity,

MR(1983a) used monthly data over the period March 1973 through June 1981. The fixed coefficient versions of (1) were initially estimated for each exchange rate using data up through October 1976. Forecasts were generated at four different horizons using the actual realizations of all explanatory variables for a prediction period. Then the data for November 1976 were added to the sample, and the parameters of each model were reestimated. New forecasts were generated at the same horizons, etc. In the MR studies, this sequential estimation yielded fixed-step-ahead forecasts which were generally inferior to those given by the random walk model. The estimation procedures used in this sequential estimation were ordinary least squares, approximate generalized least squares (correcting for serial correlation in the error term), and Fair's instrumental variables. MR also considered six

univariate time series models involving a variety of prefiltering

8see Bilson (1978,1979), Frenkel (1976), Dornbusch (1976), Frankel (1979, 1981) and Hooper and Morton (1982).

-6-

techniques and lag length selection criteria’, a random walk with drift parameter, and an unconstrained vector autoregression; none could ‘outpredict the random walk model: st = St-1 + at, where {ay} is white noise with mean zero and constant variance. 2.2 Alternative Fixed Coefficient Estimates

This study also applies ordinary least squares and approximate generalized least squares with a correction for serial correlation in the error term to equation (1). To extend MR's work on fixed coefficient models, we also applied alternative estimators of fixed coefficients not applied by them to the structural models. These estimators were: (i) posterior mean for Shiller's smoothness prior on distributed lag coefficients; (ii) Almon's polynomial distributed lag; and (iii) ridge regression estimator with a smoothness restriction on lag coefficients.!9 These three estimators were considered with and without a first-order serial correlation correction for the error term. We experimented with several lag lengths and degrees of polynomials that

were required to apply these estimators.

9The six time series models used were the following: (1) an unconstrained autoregression (AR) in which the longest lag considered (M) is set equal to (n/log n), where n is the sample size; (2) AR in which lag lengths are determined by Schwartz's criterion; (3) AR in which lag lengths are determined by Akaike's criterion; (4) long AR estimated by using observations that are arbitrarily weighted by powers of 0.95; (5) the Wiener-Kolmogorov prediction formula; (6) AR estimated by minimizing the sum of the absolute values of errors. The prefiltering techniques involve differencing, deseasonalizing, and removing time trends.

10For a theoretical derivation of these alternative estimators, see Thurman, Swamy, and Mehta (1986) and Kashyap, Swamy, Mehta, and Porter (1936).

-7- 3. Why Are Variable Coefficient Models Appropriate?

There are several reasons why the coefficients of exchange rate models may change over time. First, even if the explanatory variables capture all information used by traders, there is no reason to. believe information is used in the same way over all policy regimes and over all time horizons; parameters can change over time. As argued in another paper (see Swamy and Schinasi (1986a)), sequential estimation of fixed coefficient regressions ("rolling") is not the appropriate technique for capturing variations in coefficients over time.

Secondly, at the high level of aggregation of exchange markets, there is little reason to believe that behavioral parameters are fixed. There is a wide diversity of participants in foreign exchange 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 were identical.

Thirdly, most if not all of the empirical literature has assumed that coefficients are fixed over the relevant sample period. Most of this literature decisively rejects economic theory as having any ability to produce accurate predictions. Yet, it would be unreasonable to reject theories that have been tested on only a very limited subset

of models, namely linear fixed coefficient models. This study takes one

- 8 small step in the direction of expanding the set of models for testing _ theories of exchange rate behavior.

One cannot relax the assumption of fixed coefficients wit.aout imposing some other structure on coefficients. It would be preferable to apply existing economic theory to explain variations in coefficients, but there is unfortunately a paucity of such exchange rate theory. Asa preliminary investigation, we have chosen a stochastic coefficient model of exchange rate behavior, one in which coefficients have a systematic component, fixed over time, and a stochastic component that can vary in each time period.

While these stochastic coefficient models cannot "explain" variations in coefficients over time -~ a phenomenon economists would like to explain for other relationships as well -- estimating such models can nevertheless uncover variations in coefficients.'! Furthermore, using the estimated variation in coefficients in an efficient statistical forecasting procedure might improve the

forecasting ability of the models suggested by economic theory, when compared to fixed coefficient models. Even allowing the coefficients to vary, however, may not result in improved forecasts if the underlying

exchange rate equation is not well specified. In such cases we further

11 For learning about such variations we cannot rely on the classical F-test (Rao (1973, pp.281-284)) of a one time change in the fixed coefficients within a samplé period, and its generalizations, because these tests have poor power against the most general alternative that all coefficients change in every period. This is so because under this most genera.. alternative hypothesis, time-varying coefficients are not consistently estimable (see the uncertainty principle formulated by Swamy and Tinsley (1980, p.117)).

-9experiment with dynamic versions of the static equations by adding a

lagged dependent variable as an explanatory variable.

4. The Stochastic Coefficient Representation of the Exchange Rate Mcdels and its Uses 4.1 The Stochastic Coefficient Models The stochastic coefficient representation of the models

estimated here is presented in equations (2) through (4),

(2) ye = xtBt for all t,

(3) BR = Bt &t

(Ya) e¢ = © €¢-1 + VE

(4p) E(v_)= 0

(4c) El(vevg) = dg if t=s and 0 otherwise,

where Xt, Bt» 8B, Et, Ve are all k x 1 vectors, ® and Ag are k x k matrices, Xt represents the vector of the explanatory variables in eq. (1), including a constant term, and Yt is the natural log of the spot exchange rate. Note that equations (2) - (4) represent a special case

of a more general variable coefficient specification which allows one to

describe variations in coefficients with explanatory variables, allows

- 10 for "simultaneous equations" complications, and allows for more general specifications of the error processes. !2

In equation (3), each coefficient in each period, 8j4, has t-wo components: a time-independent fixed component, Bis and a time-dependent stochastic component, ejt. e¢ is a vector stationary first-order autoregressive process (which can also be represented as a vector moving average process). Combining (2), (3), and (4a) reveals that the stochastic coefficient representation can be viewed as a fixed coefficient model with errors that are both serially correlated and heteroscedastic, where the form of serial correlation and

heteroscedasticity is very general:

12The general model developed by Swamy and Tinsley (1980) is as follows (1) ye = xEBe, for all t

(ii) Be = Bz + €t

; ;

1

q (iii) ef Oyer-4 + 654-5 * at, j=

i

where Bisakxm matrix, oj, i=1, 2,...p, and Oj» j=1, 2,...q are

k x k matrices of fixed but unknown parameters, zz, is a mx1 vector of explanatory variables, and ay is a white noise variable. Note that if some of the elements of x are correlated with 8, because of simultaneous equations complications, then those elements of Xt are also entered into z_ to account for such correlations. Note also that (iii) represents a wide class of time series specifications. Hence, the non~deterministic component of yz is a nonstationary process (an ARMA(p,q) with time-dependent coefficients); it is a complicated mixture of serially correlated and heteroscedastic error terms. The usual fixed and variable coefficient models can be shown to be special cases of this general model.

(5a) ye = xEB + ut (5b) up = Xt et

(Se) =¢ = © e¢-4 + VE

4,2 Estimation

Estimation of 8 can be viewed as an application of generalized least squares or Aitken estimation, assuming that © and Ag are known. Swamy and Tinsley (1980) have developed a minimum average risk linear estimator which for given a priori moments of 8, © and Aa» can be shown to be more efficient than the Aitken estimator. Because ® and Ag are not known and must be estimated, Swamy and Tinsley developed an iterative estimation procedure in which 6 and Ag are initially arbitrarily chosen but through iteration are consistently and efficiently estimated after initial consistent estimates of 8, % and

Ag are obtained. 13

13comparisons of the Swamy and Tinsley procedure with the Kalman and Bayesian procedures are given in Narasimham, Swamy and Reed (1986). It should be noted that as emphasized by Narasimham, Swamy and Reed (1986) the statistical notions of consistency and efficiency require the existence of the true values of parameters and hence do not apply to the estimators of B, ®, and Ag (or any other fixed parameter estimators either) if these parameters do not represent "real" physical quantities or if equation (3) does not have a natural interpretation in terms of physical quantities. Since this equation is about the 8 which are never observed, it is necessarily arbitrary. Therefore it is difficult to believe that there are model-free physical quantities associated with each of these arbitrary model parameters. It is difficult to take consistent estimation seriously as an essential procedure if one must always interpret the parameters as "real" physical quantities. Equations (3) - (4c) are used here as a way of obtaining good predictions of the observable exchange rates and are not estimated to interpret the 8,. The ®t are used only to index the set of distributions of yr.

- 12-

4.3 Prediction When 8, %, and A, are known, it can be shown that the minimum average mean square error linear predictor of yy.j; is 14

(6) ' . ' {tir m4 ~ Yruy = * B + xX, ,O 2 My Y XB),

T+i T+i T where Q = E(ee'), € = (€,,€n,---ep)', ' 9 ' Dy= diag(x1,X0,...,X7),

Ly= DyMDy= E(Dyee'D,), Mp consists of the last k columns of 2,

and

= (XqsX0,000,Xp)'-

Since 8, 6, and Ag are not known, and in fact estimated, equation

(6) with these estimated parameters may no longer represent a minimum average mean square error forecast. It can be shown that -8 is more

| precisely estimated than either @ or Ag, and so it might improve forecasts to drop the second term on the right-hand side of (6) based on ‘ estimated 8B, @ and Ag, since the second term is:- more heavily

influenced by the estimates of and Ag than the first term. In the results reported below, the best of the two alternative forecasts, (estimated) x74,8 and (estimated) equation (6), were chosen.

Furthermore, ® and Ag can take on various forms, Four or five

T4Note that equation (6) is equal to BCyT+4 | y,Xp+i2XBy%, Aq) if (Yretey" ) is jointly. normal.

-13combinations were tried: both @ and Ag were nondiagonal; both $ and Ag were diagonal; ® was diagonal and A, was nondiagonal; all elements of $6 except its leading diagonal element were zero and- Ag was nondiagonal; | certain columns and rows of $ were zero and Ag was nondiagonal. In the applications of the present paper, values of 8B, @ and Ag which

implied positive definite covariance matrices for the dependent variables and which produced the most satisfactory forecasts of st ina RMSE sense were selected by screening 25 iterations of the Swamy and

Tinslev procedure.

5. Forecasting Strategy and Procedures

So that the results of this study complement those of the original MR study, we use (the MR) monthly data. We use the MR data from March 1973 through March 1980 for estimation of variable and fixed coefficient versions of the structural models and then use these estimated models to generate the multi-step-ahead forecasts of the out-of-sample values of the exchange rates for the period April 1980 through June 1981. 5.1 Computable Formulas for the Structural Models

Let T denote the terminal period, March 1980, of the

estimation period. The operational versions of (6) are

a

‘ os ' R

(7a) Boag 7 Xpeg8

or

(Tb) 3 = x! B+ x! ota: pts! (y-xB) i= 1, 2, w+, 15 T+i T+i Tti “T xy , nd -

- 1W4-

where one of the elements of XT +i is s74j-; if a structural model includes a lagged dependent variable, and a "“" above a symbol indicates “that it is either a forecast of a variable or an estimate of an unknown constant. In any case XT +i represents an actual observation vector

on all the right-hand side variables in a structural nodel. Since the stochastic coefficient models (2) cover the fixed coefficient models as special cases, formula (7) with appropriate zero restrictions on % and Ag applies to the fixed coefficient models. It should be noted that the estimators of z, @ and Ag based on the Swamy-Tinsley procedure

involve only the observations for the estimation period and do not involve observations for the forecast period. For this reason, we call the vector (Sp41> eeey Sp+15) of forecasts generated by (7) a vector of multi-step-ahead forecasts. We would have called this vector a

one- step-ahead forecast had we reestimated the fixed parameters every _period for predictions using all past data prior to each of the forecast periods T+1, Tt+2,...,7T+15.

5.2 Computable Formulas for Random Walk

Define the following formulas for the random walk model,

(8a) Speq = St+i-t i=1, 2, ..., 15

(8b) Speq = St i= 1, 2, 00, 15

(8c) Spey = ifit+ sp i= 1, 2, ..., 15

where fi is the simple arithmetic mean of the changes in the values of

St in the estimation period.

- 15 - Equations (8a)-(8c) represent the formulas used to generate predictions from the random walk models. Formulas (8a) and (8b) apply to the random walk model without drift and formula (8c) applies to the mode]. with drift. To distinguish these predictions we call the predictions defined in (8a) the one-step-ahead prediction of the random walk model and those defined in (8b) and (8c) multi-step-ahead predictions of the random walk model without and with drift, respectively. Note that only the random walk model with drift requires

estimation of u for generating predictions.

5.3 Forecasting Strategy

As is clear from the definitions (7) and (8c), by multi-step-ahead we mean that the fixed parameters are estimated only once from the observations for the estimation period and are then used to produce forecasts over the entire forecast period, i.e., sequential estimation is not used. The strategy of sequential estimation and producing one-step-ahead predictions was not employed in this study for several reasons. First, we wanted to extend the interesting comparisons of MR to multi-step-ahead predictions without duplicating their effort. Second, we are not convinced that sequential estimation is the apprcpriate technique to generate forecasts. We have argued in another paper (see Swamy and Schinasi (1986)) that for fixed coefficient models, sequential estimation does not necessarily improve forecasting perfcrmance, and that sequential estimation of fixed coefficient

regressions is not the proper technique to use to capture parameter

- 16variation,!5 Furthermore, multi-step-ahead forecasts typify the econometric practice of private and public agencies who estimate ‘Large-scale econometric models only once in a period, in part because sequentially updating estimates tends to be prohibitively expensive. Multi-step-ahead forecasts more realistically represent the needs of the average user of exchange rate forecasting models. We therefore produced multi-step-ahead random walk forecasts for comparison with multi-step-ahead predictions of the stochastic coefficient models ancl view this comparison as more useful for evaluating forecast performarice. Following MR, we also use actual realizations of all explanatory variables in the forecast period ‘to generate the out-of-sample forecasts of the exchange rate variables. We then compare multi-step-ahead predictions of the variable coefficient models to multi-~step-ahead predictions of the fixed coefficient models and the _Yandom walk model with drift. We also compare all these mul ti-step-ahead forecasts to one-step-ahead forecasts of the random walk model. | We extended the MR computations for the fixed coefficient versions of the Dornbusch-Frankel and the Hooper-Morton models, which ‘involve differences in expectations of inflation, by modelling the unobservable expectations variable as a distributed lag in inflation differentials. We followed the following procedure for constructing the

proxies for the inflation expectations differential. First for each

15sequential estimation is not appropriate for stochastic coefficient. models if the variance-covariance matrix of coefficients is fixed over time, as is assumed here.

-17distrituted lag we chose a few combinations of the values of its lag length permitted by the sample size, degree of differencing appearing in Shiller's smoothness prior on its lag coefficients, and a zero or a nonzero value for its coefficient appearing in a first-order autoregressive process for the error term. We applied the four different estimators of fixed coefficients described in Section 2 for each distributed lag model. Multi-step-ahead forecasts were generated and a root mean square forecast error (RMSE) was computed for each equation and each coefficient estimator. Next we Sought whether the smallest of these RMSEs could be reduced further by changing either the lag length, the degree of differencing, or the zero or nonzero value of the autoregressive parameter. By this procedure we found the lag lengths and lag coefficients which yielded the "best" predictions and used these values to combine the current and lagged values of the inflation rate differential for each exchange rate. We used this linear combination in place of (1£- 1£*) in equation (2), the stochastic coefficient versions of the structural exchange rate models.

The approach described above is not designed, per se, to examine how wel]. a model is likely to perform in the future. It is instead designed to examine how well a model would have performed in a past period. Nor is it suggested here that the model that generates the most accurate forecasts also represents the "truth". The view taken in this paper is rather conservative: if a model cannot accurately predict certain observed events, then it is unlikely to be able to predict future events either.

The most. a probability forecast from a model can represent is a measure of

- 18 the confidence with which one expects that model to predict an event in the future, based on currently available evidence, and not based on information yet to be observed. The accuracy of multi~step- ahead forecasts can be judged by three Statistics: mean forecast error (ME), mean absolute forecast error (MAE)

and RMSE. They are defined as follows:

. 1 15 A 1 15 A (9b) MAE = TS z LYoeg Yoed . is 5 a 2. 1/2

(9c) RMSE = [i

where Yr4; is an i-step-ahead forecast of Yt+i and T is the terminal period of the fitting period. 6. Empirical Results

The major results of this study are summarized in Tables | and 2. Note that for each exchange rate, the one~step-ahead forecast of the random walk model is far superior to multi-step-ahead forecasts of the random walk model with or without a drift parameter. The results not reported here show that multi-step-ahead forecasts of the Box-Jenkins (1970) type time

series models, ARIMA (1,1,0), ARIMA (0,1,1,) and ARIMA (1,1,1,)16 are

l6The symbol ARIMA(p,d,q) denotes an autoregressive integrated moving average model, where d is the degree of differencing to achieve stationarity, p is

. the order of the autoregressive part, and q is the order of the moving average part.

-19inferior to multi-step-ahead forecasts of the random walk model with or

without drift.

6.1 How to Compare Predictions of Structural Models and Random Walk Models

Table 1 compares the forecasting performance of the random walk model to fixed and stochastic coefficient structural models without a lagged dependent variable, while Table 2 compares the forecasting perfornance of the random walk model to the structural models with a lagged dependent variable included. Because each of these two sets of structural models--with and without a lagged dependent variable--use different sets of information in generating multi-~step-ahead predictions, one has to be careful in comparing their predictions with those of the random walk.

In Table 1, multi-~step-ahead random walk forecasts are probably the proper benchmark to use in judging the relative forecast accuracy of the structural models without a lagged dependent variable. Comparing multi-step-ahead predictions of such structural models with one-step-ahead predictions of the random walk model gives the random walk model the unfair advantage of using possibly as yet "unobserved" spot rates to predict future values of the spot rate (for example, using Sp+;, to predict syp+i+1 for i21). The multi-step-ahead prediction of the random walk model is on more equal footing with the structural model's mul ti-step-ahead predictions, when the structural models do not include a lagged dependent variable.

In Table 2, multi-step-ahead predictions of the structural

models, including a lagged dependent variable, are compared with

-20one-step-ahead predictions of the random walk model. Although equation (7) represents a multi-step-ahead forecast, in the sense that it does nct use ‘data beyond period T to estimate fixed parameters used for predicticn, its vector XT +i does contain the observation on the lagged dependent variable for the time periods beyond T up to T+15..

| We now turn to the tables and report the major findings of this

study.

6.2 The Forecasting Performance of Conventional Models without Lagged__ Dependent Variables

With one exception (the Hooper-Morton model of the dollardeutsche-mark rate), when coefficients are allowed to change period by period, multi-step-ahead forecasts of all three structural models of all three currencies consistently outperform multi-step-ahead forecasts of the random walk model (see Table 1). The same cannot be said for the fixed coefficient versions of these conventional models. All three fixed coefficient models of the dollar-pound rate fail to outperform the multi- “ step-ahead forecast of the random walk model, while some fixed coefficient estimates of the models of the dollar-yen and dollar~deutsche-mark rates can outperform the mul ti- step- ahead forecast of the random walk model. With one exception (the Hooper-Morton model for the dollar-yen rate), multi-step- ahead forecasts of the time-varying parameter models are more accurate than those of their fixed coefficient counterparts.

Models of the dollar-deutsche-mark rate present the most contrast

in forecast accuracy. The Dornbusch-Frankel model, which includes a proxy

- 20a - Table 1

Root mean square forecast errors!»2

$e 8/8 $/DM Random walk: 1. multi-step-ahead: with drift 7.34 17.48 10.35 2. multi~step-ahead: without drift 7.24 17.51 10.30 Frenkel-Bilson>: 3. Leas’: squares 20.93 17.36 15.93 4, Serial correlation correction4 8.90 — 17 10.29 5. Stochastic coefficient 5.85 11.52 9.31 Dornbuseh-Frankel: 6. Least squares 18.74 17.81 4,81 7. Ser:al correlation correction4 10.53 17.60 14.52 8. Posterior mean 7.80 16.78 4.51 9. Almon 7.93 16.29 4,64 10. Ridge regression 10.68 16.77 4.55 11. Stochastie coefficient 5.75 9.10 3.00 Hooper-Morton: 12. Least squares 17.70 12.95 20.56 13. Serial correlation correction4 10.06 6.71 19.09 14. Posterior mean 9.55 12.62 18.65 15. Almon 9.38 12.01 ; 18.09 16. Ridge regression ‘ 10.12 6.48 18.72 17. Stochastic coefficient “7.18 7.08 13.49 Addendum: 18. One-step-ahead random walk 3.03 3.96 3.69

! Approximately in percentage terms.

Forecasts for all structural models were multi- step-ahead forecasts for 15 periods beyond the last sample point which was March 1980.

The estimators labelled Posterior Mean, Almon, and Ridge Regression are designec to estimate distributed lag models subject to smoothness restrictions and therefore do not apply to the Frankel-Bilson model.

First-order serial correlation correction.

-21for the differential in inflation expectations, has the lowest RMSEs. The RMSE of the stochastic coefficient version of the Dornbusch-Frankel model is low enough to outrank the one- step- ahead forecast of the random walk model, despite the extreme informational advantage (of using the actual previous value to predict the next observation of. the spot rate) of the one- step-ahead random walk model forecast presented in Table 2. In contrast, the Hooper-Morton model, which further adds the differential between cumulated trade balances of the United States and Germany as an additional explanatory variable, fails to outperform the multi-step-ahead forecast of the random walk model or any other model. These results indicate the importance of inflation expectations differentials (and therefore real interest rate differentials) and the unimportance of trade balances between Germany and the United States. Similar conclusions, though less extreme, can be drawn from the results for the dollar-pound

. rate, although the stochastic coefficient estimator does not outperform the one- step- ahead forecast of the random walk model.

Models of the dollar-yen rate are ranked quite differently. Only in the case of the dollar-yen rate does the inclusion of the differential between cumulated trade balances substantially lower RMSEs and only for the dollar-yen rate is the Hooper-Morton model superior to multi-step- ahead forecasts of the other models, probably owing to the relatively significant trade flows between Japan and the United States. For the dollar-yen rate, two of the fixed coefficient estimators for the Hooper-Morton model

outperform its stochastic coefficient version.

- 22 -

Note that when coefficients are fixed, the Dornbusch~Frankel model outperforms the Frankel-Bilson model (when the posterior mean estimator is used in our sample). Forecasts generated by the posterior mean can be improved by using the Almon or the ridge regression estimator in some cases,

To summarize this section, multi-step-ahead forecasts of the Stochastic coefficient versions of the structural models examined in this study are generally more accurate than multi-step-ahead forecasts of fixed coefficient models and multi-step-ahead forecasts of the random walk model. The Dornbusch-Frankel model is the most accurate multi-step-ahead forecasting model of the dollar-pound and the dollar-deutsche-mark rates, while the Hooper-Morton model is the best multi-step-ahead predictor of the dollar-yen rate.

6.3 Models with a Lagged Dependent Variable

Turning to Table 2, and comparing its values with those reported in Table 1, a striking observation one can make is that adding a lagged dependent variable makes a substantial difference in the forecasting ability of all three structural models. Both fixed and stochastic coefficient models for all three currencies produce multi-step-ahead forecastis that are more accurate than the mul ti-step-ahead forecast of the random walk model. Also note that stochastic coefficient representations of the ssructural models uniformly outperform the fixed coefficient versions.

Further inspection reveals that with one exception (the

Hooper-Morton model for the dollar-pound rate), multi-step-ahead forecasts

- 22a - Table 2

Models with lagged dependent variable: Root mean square forecast errors! »2

$/£ $/¥ $/DM __ Random- walk: 1. One-step-ahead Random Walk 3.03 - 3.96 3.69 Dornbusch-Frankel: 2. Least squares 3.66 5.29 2.69 3. Serial correlation correction3 3.63 5.10 2.87 4. Posterior mean 3.61 ; 4,98 2.79 5. Almon 3.54 4.03 2.56 6. Ridge regression 3.63 4,96 2.76 7. Stochastic coefficient 2.17 3.50 2.17 Hooper-Morton: 8. Least squares 449 5.79 6.58 9. Serial correlation correction3 4.49 5.66 7.91 10. Posterior mean 4.45 5.53 6.60 11. Almon 4,30 5.03 6.00 12. Ridge regression 4,46 5.51 6.59 13. Stochastic coefficient 3.85 3.27 3.17

Approximately in percentage terms.

All structural models were estimated with a lagged dependent variable as an explanatory variable. Forecasts for all structural models were multi-step-ahead forecasts for 15 periods beyond the last sample point which was Marcn 1980.

3 First-order serial correlation correction. :

- 23of the stochastic coefficient versions of all the structural models are more accurate than the one-step-ahead forecast of the random walk model. Except for the fact that the structural models are used to produce multi~step-ahead forecasts and the random walk model is used to produce one- step- ahead forecasts, comparison of their respective RMSEs is a "pure" measure of the contribution of economic variables (theory) in improving forecast accuracy.

Because one cannot derive the exact finite sample distributions of the RMSE statistics (for even much simpler models) one cannot make probability statements about how significant these differences are. But we do observe that including economic variables (i.e., using economic theory) along with a lagged dependent variable generally improves upon the one-step-ahead forecast of the random walk model when coefficients are allowed to vary, and generally does not improve upon the one- step- ahead forecast of the random walk model when coefficients are constrained to be fixed. Note however that the fixed coefficient estimators of the Dornbusch-Frankel model (with a lagged dependent variable) for the dollar~deutsche-mark rate gave more accurate forecasts of the (log) spot rate than the random walk model, regardless of whether multi-~step- or one-step-ahead forecasts are considered.

The relative ranking of the various structural models of the dollar-pound and the dollar-deutsche- mark rates match those of Table 1, while the ranking of models of the dollar-yen rate is reversed for the fixed coefficient case. Hence when one includes a lagged dependent

variable, the Dornbusch-Frankel model is generally superior for all three

- 24 -

currencies tested in this study. The major improvements in the models with stochastically varying coefficients and a lagged value of the depenclent

variable indicate that departures from purchasing power parity and

instability in the money demand functions, which underpin the structural

models of the exchange rate, may explain a significant part of the

instability in the exchange rates that apparently cannot be adequately

captured by the fixed coefficient models.

6.4 Further Results

The mean absolute forecast error statistics, which are gerierally smaller than RMSE, are not listed here since’ they yielded the same rankings as the RMSEs. The mean forecast errors, which are also not listed here, do not suggest systematic over- or underprediction for the models yielding the "best" predictions.

It is worth emphasizing that for each stochastic coefficient specification one of 25 different estimates of 8, @ and Ag, the one with the lowest RMSE, is selected by screening 25 iterations of the Swamy and Tinsley procedure. This is necessary because this iterative procedure does not converge unless sufficient a priori restrictions are imposed on 6. For example, when all the elements of $6 except the leading diagonal elenent are restricted to be zero, the iteration converged in several cases. But imposing constraints of this nature did not improve forecast accuracy. The forecasts generated by the initial estimates were better than those generated by the final iteration. One of the 25 different sets of

estimates of nondiagonal 6 and nondiagonal Ag yielded the lowest RMSE in

- 25 general. Ina few cases the diagonality restriction on 4 improved forecast accuracy.

The reason for following an iterative procedure which converges is to find maximum likelihood or least squares estimates of nonlinear models. If such estimates do not exist, however, then no iterative procedure converges to those estimates. In cases where 6 and Ag are both nondiagonal, the conditions for the existence of maximum likelihood estimates are not satisfied. In these cases our procedure of choosing the best predicting estimates of B, @ and Ag from among 25 different estimates is analogous to that of comparing inferences given by a class of prior distributions in a Bayesian analysis.

These computations show that arbitrary choices of the values of 8, ® and Ag do not guarantee improvements in forecast accuracy. Perhaps this explains why Alexander and Thomas (1986) could not reverse the MR conclusions using the Kalman filtering technique with the prior values ry = (89,0,..,0)' and @= 1. Wolff (1985) also evaluates the Kalman filter using the prior values 8 = (Bg,0,-.,0)' and 6 = I and finds that the forecasts generated by this filter are poor relative to the forecasts of the random walk model for the dollar-yen and dollar-pound rate. Our computations show that the use of these a priori values reduces the accuracy of forecasts.

7. Conclusion

The main result of this paper is that once one is willing to

relax the conventional assumption of fixed regression slopes, it is

possible to estimate structural models of exchange rate determination which

-~ 26 perform substantially better than the random walk model in predicting _ out-of-sample values of exchange rates. As was shown decisively by Meese and Rogoff, and others, fixed coefficient models of exchange rates -- with and without a lagged dependent variable, and with a variety of proxies for the differential of inflation expectations -- could not outperform the random walk model. Our findings are traceable, in varying degrees, to three sources, namely, relaxation of the assumption that coefficients are fixed, inclusion of a lagged dependent variable as an explanatory variable, and our detailed construction of a proxy on inflation expectations differentials. But the results reported here indicate that relaxing tre fixed coefficient assumption is the crucial source of the superior forecasting ability of the exchange rate models tested.

We would have preferred reporting results that explain the variability of coefficients with sound and rigorous economic principles. But until economic theory postulates empirically implementable hypotheses addressing why exchange markets are so volatile and why model coefficients |

vary over time, one can at least examine the type of stochastic coefficient models presented here before rejecting existing exchange rate models out of hand. This study demonstrates that stochastic coefficient models of exchange rate determination can be useful in improving the accuracy of

forecasts of exchange rates.

-27-

References

Alexander, D. and L.R. Thomas, 1986, Monetary/assét models of exchange rate determination: How well have they performed in the 1980's? Working Paper, Citicorp Investment Bank.

Backus, 1., 1984, Empirical models of the exchange rate: Separating the wheat from the chaff, Canadian Journal of Economics 17, 824-846.

Bilson, J.F.0., 1978, Rational expectation and the exchange rate, in: J. Frenkel and H. Johnson, eds., The Economics of Exchange Rates (Addison-Wesley Press, Reading).

Bilson, J.F.0., 1979, The deutsche mark/dollar rate -- A monetary analysis, in: Karl Brunner an Alan H. Meltzer, eds., Policies for employment, prices and exchange rates, Carnegie-Rochester Conference 4 (North-Holland Publishing Company, Amsterdam).

Box, G.E.P. and G.M. Jenkins, 1970, Time Series Analysis, forecasting and control (Holden-Day, San Francisco).

Dornbusch, R., 1976, Expectations and exchange rate dynamics, Journal of Political Economy 84, 1161-1074.

Edison, H. J., 1985, "The rise and fall of sterling: Testing alternative models of exchange rate determination, Applied Economics 17, 1003-21.

Engle, R.F., 1982, Autoregression conditional heteroscedesticity with estimates of the variance of U. K. inflation, Econometrica 50, 987-1008.

Finn, Mary G, 1986, "Forecasting the exchange rate: A monetary or random walk; phenomenon," Journal of International Money and Finance 5, 181-194,

Frankel, J.A., 1979, On the mark: A theory of floating exchange rates based on real interest differentials, American Economic Review 69, 610-622.

Frankel, J.A., 1981, On the mark: Reply, American Economic Review 71, 1075-1082.

Frenkel, J.A., 1976, A monetary approach to the exchange rate: Doctrinal aspects zand empirical evidence, Scandinavian Journal of Economics 78, 200-224,

Hakkio, C.S., 1986, Does the exchange rate follow a random walk? A Monte Carlo study of four tests for a random walk, Journal of International Money and Finance 5, 221-229.

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Hildreth, C. and J.P. Houck, 1986, Some estimators for a linear mociel with random coefficients, Journal of the American Statistical Association 63, 584-595.

Hooper, P. and J.E. Morton, 1982, Fluctuations in the dollar: A model of nominal and real exchange rate determination, Journal of International Money and Finance.

Isard, P., 1986, "The Empirical Modeling of Exchange Rates; an Assessment of Alternatives," Departmental Memorandum 86-36, International. Monetary Fund.

Kashyap, A.K., P.A.V.B. Swamy, J.S. Mehta and R.D. Porter, 1986, Further results on estimating linear regression models with partial prior information, Special Studies Paper, Federal Reserve Board, Washington, D.C.

Makin, J.H. and R.D. Sauer, 1986, Exchange rate determination with changes in the policy regime: The yen-dollar rate, The Review of Economics and Statistics XX, 164-169.

Meese, R.A. and K. Rogoff, 1983a, Empirical exchange rate models o° the seventies: Do they fit out of sample?, Journal of International Economics 14, 3-24.

Meese, R.A. and K. Rogoff, 1983b, The out-of-sample failure of empirical exchange rate models: Sampling error or misspecification, in: Jacob A. Frenkel, ed., Exchange rates and international macroeconomics (University of Chicago Press, Chicago).

Meese, R.A., and K. Rogoff, 1985, Was it real? The exchange rate interest differential relation, 1973-1984, International Finance Discussion Paper #268, Federal Reserve System, Washington, D.C.

Narasimham, G.V.L., P.A.V.B. Swamy and R.C. Reed, 1986, Productivity analysis of U.S. manufacturing using a stochastic coefficients production function, Special Studies Paper, Federal Reserve Board, Washington D.C.

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Swamy, P.A.V.B. and G.J. Schinasi, 1986 Should fixed coefficients be reestimated every period for extrapolation?, International Finance Discussion Paper #287, Federal Reserve Board, Washington, D.C.

Thurman, S.S., P.A.V.B. Swamy and J.S. Mehta, 1986, An examination of distributed lag model coefficients estimated with smoothness priors, Cummunications in Statistics-Theory and Methods, 15, 1723-1750.

Wolff, C.C.P., 1985, Time-varying parameters and the out-of-sample forecasting performance of structural exchange rate models, Mimeo, London Business School, London, England.

Woo, W.T., 1985, The monetary approach to exchange rate determination under rational expectation, Journal of International Economics 18, 1-16.

IF DP NUMBER

301

298

297

296 295

294

293

292

291

290

289

288

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International Finance Discussion Papers

TITLES 1987

The Out~of*Sample Forecasting Performance of Exchange Rate Models When Coefficients are Allowed to Change

1986 The International Debt Situation

The Cost Competitiveness of the Europaper Mar ket

The United States International Asset and Liability Position: A Comparison of Flow of Funds and Commerce Department Presentation

An International Arbitrage Pricing Model with PPP Deviations

The Structure and Properties of the FRB Multicountry Model

Short*term and LongSterm Expectations of the Yen/Dollar Exchange Rate: Evidence from Survey Data

Anticipated Fiscal Contraction: The Economic Consequences of the Announcement of Gramm*Rudman#Hollings

Tests of the Foreign Exchange Risk Premium Using the Expected Second Moments Implied by Option Pricing

Deposit Risk Pooling, Irreversible Investment, and Financial Intermediation

The Yen*Dollar Relationship: A Recent Historical Perspective

AUTHOR(s,)

Garry J. Schinasi P.A.V.B. Swamy

Edwin M. Truman

Rodney H. Mills

Guido E. van der Ven John E. Wilson

Ross Levine

Hali J. Edison Jaime R. Marquez Ralph W. Tryon

Jeffrey A. Frankel Kenneth A. Froot

Robert A. Johnson

Richard K. Lyons

Robert A. Johnson

Manuel H. Johnson Bonnie E. Loopesko

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal Reserve System, Washington, D.C. 20551.

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International Finance Discussion Papers

IFDP

NUMBER TITLES AUTHOR(s)

287 Should Fixed Coefficients be Reestimated P.A.V.B. Swamy Every Period for Extrapolation? Garry J. Schinasi

286 An Empirical Analysis of Policy Hali J. Edison Coordination in the U.S., Japan and Ralph Tryon Europe

285 Comovements in Aggregate and Relative B. Dianne Pauls Prices: Some Evidence on Neutrality

284 Labor Market Rigidities and Unemployment: Michael K. Gavin The Case of Severance Costs

283 A Framework for Analyzing the Process Allen B. Frankel of Financial Innovation Catherine L. Mann

282 Is the ECU an Optimal Currency Basket? Hali J. Edison

281 Are Foreign Exchange Forecasts Rational? . Kathryn M. Dominguez New Evidence from Survey Data

280 Taxation of Capital Gains on Foreign Garry J. Schinasi Exchange Transactions and the Non&neutrality of Changes in Anticipated Inflation

279 The Prospect of a Depreciating Dollar Jacques Melitz and Possible Tension Inside the EMS

278 The Stock Market and Exchange Rate Dynamics Michael K. Gavin

277 Can Debtor Countries Service Their Debts? Jaime Marquez Income and Price Elasticities for Exports Caryl McNeilly of Developing Countries

276 Posthsimulation Analysis of Monte Carlo Neil R. Ericsson Experiments: Interpreting Pesaran's (1974) Study of Non#nested Hypothesis Test Statistics

275 A Method for Solving Systems of First Order Robert A. Johnson Linear Homogeneous Differential Equations When the Elements of the Forcing Vector are Modelled as Step Functions

274 International Comparisons of Fiscal Policy: Garry Schinasi

The OECD and the IMF Measures of Fiscal Impulse