# The performance of alternative VAR models in forecasting exchange rates

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EISEVIER International Journal of Forecasting 10 (1994) 419-433 The performance of alternative VAR models in forecasting exchange rates Te-Ru Liu, Mary E. Gerlow, Scott H. Irwin* Department of Agricultural Economics and Rural Sociology, The Ohio State University, 2120 Fyffe Road, Columbus. Ohio 43210-1099, USA Abstract The purpose of this research is to analyze the forecasting accuracy of full vector autoregressive (FVAR), mixed vector autoregressive (MVAR), and Bayesian vector autoregressive (BVAR) models of the US dollar/yen, US dollar/Canadian dollar, and US dollar/Deutsche mark exchange rates. The VAR specifications are based on a monetary/asset model of exchange rate determination. Out-of-sample results (1983:1-1989:12) indicate that the forecasting performance of restricted VARs (MVARs and BVARs) is substantially better than that of unrestricted VARs (FVARs). Overall, the results show that a monetary/asset model in a VAR representation does have forecasting value for some exchange rates. Keywords: Vector autoregression, Forecasting. Exchange rates 1. Introduction The forecasting performance of exchange rate models has received considerable attention [e.g. Meese and Rogoff (1983); Woo (1985); Finn (1986); Alexander and Thomas (1987); Boothe and Glassman (1987); Canarella and Pollard (1987); Wolff (1988); Hoque and Latif (1993)]. Forecasting models involving the use of both univariate and multivariate time series tech- niques have been constructed and analyzed. Performance results have been mixed; however, multivariate vector autoregressive (VAR) models * Corresponding author. Tel. (614) 292-6399, Fax. (614) 292-4749. have exhibited some forecasting power [e.g. Canarella and Pollard (1987); Wolff (1988); Hoque and Latif (1993)]. The different conclusions reached in previous studies may depend on the economic theories used to construct models, forecasting methods, and evaluation criteria. First, the theoretical model used in most studies is a monetary/asset model based on assumptions of purchasing power parity and uncovered interest parity. Recently, Driskill et al. (1992) demonstrated that a different version of the monetary/asset model leads to improved forecasting perform- ance with respect to the Swiss franc/dollar ex- change rate. In this model, the assumption of rational expectations in the foreign exchange market is maintained, while incorporating im- 0169-2070/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0169-2070(94)00534-J 420 Te-Ru Liu et al. i International Journal of Forecasting 10 (1994) 419-433 perfect capital substitutability and current ac- count effects. The success of this model in predicting the Swiss franc/dollar exchange rate is the basis for applying it to other exchange rates. Second, the VAR system used in most previous research is a full VAR model, where all of the independent variables are included as lagged dependent variables in the system and the lag length of each dependent variable in each equa- tion is identical [Hoque and Latif (1993) ex- amined an unrestricted (full) VAR and a re- stricted (Bayesian) VAR model. They found the restricted model to be more accurate in forecast- ing the US/Australian dollar exchange rate]. The major problem associated with VAR models, even in small systems, is the number of in- significant parameters (over-parameterization), which can lead to poor out-of-sample forecasting performance. Two methods of restricting lag length in a VAR system have been proposed to overcome the problem of an over-parameterized model. One method is termed a mixed VAR model, where zero coefficients are allowed in the equa- tions. Therefore, the lag length of each variable in each equation may be different. Further, every variable need not be included in each equation. An alternative approach to address this problem is the Bayesian VAR model [Litter- man (1986a,b)]. The Bayesian VAR approach assumes that each coefficient has an independent and normal distribution and imposes prior in- formation regarding the mean and standard deviation of this distribution into the VAR model. Litterman argues that a Bayesian prior, which filters the useful signal from the accom- panying noise, may yield biased, but mean squared error superior forecasts. Finally, a wide array of forecast evaluation criteria have been used in previous studies. Conclusions regarding the performance of a given model tend to depend upon the criteria chosen [Boothe and Glassman (1987); Gerlow and Irwin (1991)]. Thus, the mixed results in earlier studies may be attributed partially to differences in evaluation criteria. The evaluation criteria used in previous studies can be grouped into three broad categories: bias tests, informational content tests, and profitability tests. Different evaluation criteria focus on different targets of forecasting accuracy. Bias tests determine if model forecasts are systematically higher or lower than actual exchange rates. Informational content tests ex- amine whether model forecasts contain useful information over and above that given by ran- dom walk forecasts. Profitability tests show the economic value of the forecasts. Given the apparent sensitivity of results to the particular test employed, it is important to evaluate fore- casting accuracy across all three types of tests. This will allow a fuller understanding of the forecasting performance of alternative exchange rate models. The purpose of this paper is to analyze the forecasting accuracy of full vector autoregressive (FVAR), mixed vector autoregressive (MVAR), and Bayesian vector autoregressive (BVAR) models of the US dollar/yen, US dollar/Cana- dian dollar, and US dollar/Deutsche mark ex- change rates. The VAR models are based on the theoretical model of monetary/asset exchange rate determination developed by Driskill et al. (1992). The models are estimated over the in- sample period 1973:3-1982:12. For the out-of- sample period 1983:1-1989:12, l-, 3-, 6-, and 12-month forecasts are generated Performance criteria include bias tests, informational content tests, and market timing tests. 2. Exchange rate modelling 2.1. VAR models A vector autoregressive (VAR) model is a general dynamic model with several endogenous variables in the system. In the VAR model, each variable is a linear function of lagged values of all variables in the system. Three widely applied methods of selecting the appropriate lag length are used in this study. First, a likelihood ratio test suggested by Sims (1980) and Tiao and Box (1981) is used to construct a full VAR model. Second, a final prediction error (FPE) criteria suggested by Hsiao (1979) is used to build a Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 421 mixed VAR system. Third, a Bayesian approach suggested by Litterman (1986a,b) is used to impose a prior distribution on the parameters of a full VAR. 21.1. Full VAR model A full VAR model is the standard form of a VAR model. In this approach, lag lengths for each variable are identical and every variable in the system is included in each equation. The full VAR model with a lag length of p (FVAR(p)) can be represented as: z, = i l#J(s)Z,_, + Et (1) s=l where 2, is an (m x 1) vector of variables mea- sured at time period t, (p(s) is an (m x m) matrix of the coefficients, p is the lag length of the variables, and er is an (m x 1) vector of random disturbances with the properties, (2) and E is the expected operator. A likelihood ratio test was suggested by Sims (1980) and Tiao and Box (1981) to determine the optimal lag length p in the full VAR model. They note that FVAR(p) can be viewed as an unrestricted model and FVAR(q) as a restricted model, where q zl-,> x (4 - wFL, - . . . - &d4-,> (5) where 4 is the estimate of 4. The statistic LR( p) is asymptotically distribut- ed as a chi-squared distribution with (p - q) x k2 degrees of freedom, where k is the number of equations. If the x2 value is greater than the critical value, then the null hypothesis, $(q + I>= . . . =$(p)=O, can be rejected and the model FVAR(p) will be chosen. On the other hand, if the x2 value is less than the critical value, then the null hypothesis, 4(q + 1) = 1 = 4(p) = 0, cannot be rejected and the model with shorter lags, FVAR(q), wiI1 be chosen. 2.1.2. Mixed VAR model In a mixed VAR model, different lag lengths are specified for each variable in each equation. Hsiao (1979) argues that imposing an identical lag length on all test variables may result in an over-parametrization problem (a large number of coefficients with insignificant t-statistics). Thus, Hsiao suggests a procedure using Akaikes (1970) minimum final prediction error (FPE) criterion to specify the order of parameters in a VAR model. The FPE criterion is derived assuming a quad- ratic loss function for each equation. The FPE is computed as &Z,, - Zii,) FFE== T T + N XT-N (6) where Zi is the ith variable, N is the total number of parameters in the equation, and T is the number of observations. The first term on 422 Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 the right hand side of the Eq. (6) is a measure of modelling error, while the second term, (T + N) /(T - N), is an adjustment for degrees of freedom. A modified version of Hsiaos procedure is used in this study to build a mixed VAR model [this modification of Hsiaos procedure was sug- gested by Kaylen (19SS)l. Assuming that k variables are considered as one group, the pro- cedure is as follows. First, consider the equation for variable Z,(i = 1, . . . , k). Then regress Zi on each variable, one at a time, with lags from 1 to m. The FPEs are computed by varying the variables from 1 to k with lags from 1 to m. All of the series are then searched to find the one series, Z,, and its associated lags, p, which give the minimum FPE. The best lags of this series are then introduced into the equation ,,, = 4ii,(O) + +ij,,zj,,-l + + 4ij,(P),,tF, +~,i, j=l,...k (7) where the coefficient &,(p) is the coefficient of variable j in equation i at the pth lag length. The next step is to add each variable from the remaining k - 1 variables into Eq. (7) in an effort to reduce the FPE value of the Z, equa- tion. The FPE criterion is then computed by varying the lags from 1 to m to examine whether the new FPE value is smaller than the old one. The procedure is then repeated for each of the k - 1 remaining variables. Thus, the FPE values are compared to select the second variable, Z,, with appropriate lag length, q. If FPE(p, q) is larger than FPE(p), then Z, does not cause Z, and Eq. (7) is used. If the converse is true, then Z, causes Z, and a new equation is specified: ,,, = +i,(O) + +ij(l>z,,c-l + + +ij(P>,,t-p + 4,(l>4,,F, + . + 4iir(&-, + Et (8) where Z, is the second variable into equation i and 4 is the lag length of Z,. To obtain the overall causality implications, the above proce- dure is repeated using the rest of the variables. Thus, a single equation for variable Zi will be specified. Finally, combining all the single-equa- tion specifications together, a mixed VAR system is identified. 2.1.3. Bayesian VAR model The Bayesian procedure is also used to cir- cumvent the problem of overfitting. The prior information imposed on the VAR model is the mean and standard deviation of the coefficients, c#+~(s), in Eq. (1). The coefficients apply to variable Z, with lag length of s in the ith equation. A random walk prior suggested by Litterman (1986a, b) may be consistent with an efficient market hypothesis and may be useful for exchange rate series. Assuming the 4s are jointly normally and independently distributed, the mean of c#+~(s) is ?ij('> = i 0 if i#j; i=j, s#l 1 if i=j s=l, where &;(s) is the mean of +ii(~) and s is the lag length. The assumption shows that the mean of the coefficient of an own-lagged dependent vari- able is 1 and that other coefficients are 0. The mean of the coefficient can also be expressed as a random walk process z, = Z,_, + CI (10) The variances of the distributions for coeffi- cients on lags vary as a function of the lag number; variance is tighter around lags further back in the distribution. The standard deviation of the prior distribution for the coefficient on lag e of variable j in equation i, 8Q, is given by the formula (11) where A, d, and wij are tightness information on the prior. A constant standard deviation on the coefficient, A, associated with the first own-lag of the dependent variable, sets an overall tightness for the entire prior. The rate of decay, d, controls the speed at which the standard devia- Te-Ru Liu et al. / International Journal of Forecasting 10 (1994) 419-433 423 tions of the +,js approach zero as the lag increases. The tightness weights, wi,, allow the standard deviations on lags of other series to be smaller than those on own lags. 2.2. Specification of the alternative VAR models 2.2.1. Specification of variables A significant decision in the construction of VAR models involves specifying the variables in the vector 2,. This study uses the monetary/asset model developed by Driskill et al. (1992) to specify the vector of variables. Driskill et al. derive a reduced form model of exchange rate determination from a structural model which includes money and foreign exchange markets. The exchange rate is a function of all exogenous variables. However, in order to incorporate all information in the money and foreign exchange markets, both exogenous variables and endogen- ous variables will be included in the VAR system. Thus, the vector of variables, Z,, can be repre- sented as: Z, = [c,, M,, P,, r,, 4, T,l (12) The variables included in this model are the logarithm of the exchange rate (e), the logarithm of relative money supply between domestic and foreign countries (M), the logarithm of relative price level between two countries (P), the logarithm of relative real income between two countries (Y), the interest rate differential be- tween two countries (I), and the trade balances between two countries (T). 2.2.2. Lag length Another important decision in building the alternative VAR models involves specifying lag lengths. This can be done either arbitrarily or by using a specified statistical testing procedure. The likelihood ratio test suggested by Tiao and Box (1981) is used to choose the lag length for the FVAR and BVAR models. A seven-lag structure is specified for the FVAR and BVAR specifications for dollar/yen exchange rate models, a six-lag structure for dollar/Canadian dollar exchange rate models, and an eight-lag structure for dollar/Deutsche mark exchange rate models. In a mixed VAR model, different lag lengths are specified for each variable in each equation. A modified version of Hsiaos procedure is used to specify the lag length of the mixed VAR in this study. The FPE statistic is used to select the lag lengths. The estimation is carried out on lags ranging from one through 18 for each variable in order to detect any trace of seasonality. The specific model structures are presented in the Appendix. 2.2.3. Determination of the Bayesian prior A crucial step in building a Bayesian VAR model is to determine the prior parameters (A, d, w). Numerical specification of parameters is based on suggestions from previous studies [Lit- terman (1986a, b); Bessler and Kling (1986); Kaylen (19SS)]. The setting on the overall tight- ness, A, for each coefficient will vary between 0.00 and 1.00 inclusively, directly controlling the importance of own lags. As A approaches zero, the estimated model will approach the random walk process, implying that it is not necessary to allow the data to have much impact on the resulting models. On the other hand, a tightness parameter setting close to 1.00 implies that the data has a strong effect upon the model. Doan and Litterman (1988) suggest that a reasonable procedure is to set the overall tightness parame- ter to 0.1 or 0.2. In this study, the overall tightness, A, is set to 0.2 for the three exchange rate models. The decay type (harmonic or geometric) and decay parameter (d) control the speed of decay within the lags. A harmonic lag with decay ranging from 0.00 to 2.00 has been used in previous studies. A decay tightness of 1.0 is used to specify the three BVAR exchange rate models. The prior on wLj determines the relative weight function giving the tightness on variable j in equation i, relative to variable i. As w,, ap- proaches zero, the model approaches a uni- variate AR process. A symmetric weight of 0.5 is a common choice and is used in this study. Thus, the prior information (A, d, w) of (0.2, 1.0, 0.5) 424 Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 has been chosen for each BVAR exchange rate model. 2.2.4. Estimation methods The full VAR models are estimated by or- dinary least squares (OLS) estimation. The mixed VAR models are estimated by seemingly unrelated regression (SUR). The Bayesian VAR models are estimated using a mixed estimation technique provided by Theil (1971), which uses the sample to modify prior judgements. The in-sample estimation period for all models is 1973:3-1982:12. Finally, a Kalman filter is used to update initial estimates during the out-of- sample forecasting period (1983: 1-1989: 12). 2.2.5. Data Most of the data series were collected from the OECD Main Economic Indicator Statistics. The exchange rates data series for the dollar/yen, dollar/Canadian dollar, and dollar / Deutsche mark are end-of-month rates. The money supply statistic is Ml in all countries. Interest rates are 3-month Treasury Bill or money market rates for the USA, Canada, and Germany. Because the Japanese 3-month rate data series is discontinu- ous, the call money rate (money market, day-to- day rate) is used to replace the 3-month Treasury Bill rate. Trade balances between the US and Canada and the US and Germany are from the Monthly Statistics of Foreign Trade. Trade bal- ances between the US and Japan are from the Economic Statistics Monthly published by the Bank of Japan. 3. Forecast evaluation 3.1. Descriptive statistics Monthly exchange rates forecasts were gener- ated for the FVAR, MVAR, and BVAR models across l-, 3-, 6-, and 1Zmonth forecast horizons. Evaluation tests were performed over the out-of- sample period 1983: 1-1989: 12. Descriptive statistics on forecast accuracy are reported in Table 1. Forecast error is calculated as the forecast rate minus the spot rate. The mean errors indicate that forecast rates tend to be lower than actual rates in most cases. The magnitude of the mean errors is generally in the range of - 18~ to 21 per hundred Japanese yen, - 3c to 0.02e per Canadian dollar, and - 6e to 0.25~ per Deutsche mark. Of the 12 RMSE statistics associated with exchange rate forecasts generated from the BVAR models ten are nominally lower than the RMSE associated with the forecasts generated from FVAR and MVAR models. The root mean squared error ranges from 2c to 23c per hundred Japanese yen, 0.8~ to 6c per Canadian dollar, and lc to 12c per Deutsche mark. As expected, RMSE tends to increase as the forecast horizon lengthens. 3.2. Bias test The purpose of a bias test is to determine if forecasts are unbiased estimates of the actual series [MacDonald and Taylor (1992)]. In other words, bias tests determine if model forecasts are systematically higher or lower than actual ex- change rates. The unbiasedness hypothesis is tested from the regression Z ,-tk - Z, = a + HZ:,,-, - Z,) + & (13) Here, Zr+k is the actual series at time t + k, Z, is the actual series at time t, and ZF,t+k is the k-month ahead forecast made at time t. The unbiasedness hypothesis of (a,@) = (0,l) is test- ed from the regression (13). An F-test is a valid test statistic for the joint hypotheses if the error term, Pi, is i.i.d. A problem associated with regression Eq. (13) is that serial correlation is introduced into the error term for equations corresponding to 3-, 6-, and 12-month ahead forecasts. This is due to the fact that forecast horizons overlap. The overlap- ping of time periods introduces a moving average process into the error term of order k - 1, where k is the forecast horizon [Granger and Newbold (1986)]. It also seems likely that conditional heteroscedasticity exists in the error term of Eq. (13). Consequently, test statistics based on OLS would be inappropriate. To correct for these Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 425 Table 1 Descriptive accuracy statistics for VAR exchange rate forecasting models, 1983:1-1989:12 Exchange Rate Mean error Root mean squared error FVAR MVAR BVAR FVAR MVAR BVAR model model model model model model One-month forecast horizon Dollar/Yen 0.00227 -0.00025 0.00044 0.0245 0.0211 0.0210 Dollar/Canadian $ 0.00017 -0.00231 -0.00071 0.0105 0.0088 0.0087 Dollar/D. mark -0.00593 0.00031 -0.00285 0.0383 0.0202 0.0184 Three-month forecast horizon Dollar/Yen -0.02337 -0.00150 0.00027 0.0499 0.0379 0.0384 Dollar/Canadian $ -0.00986 -0.00696 -0.00292 0.0183 0.0151 0.0147 Dollar/D. mark -0.01280 0.00116 -0.00778 0.0584 0.0365 0.0353 Six-month forecast horizon Dollar/Yen -0.08112 0.00470 -0.00261 0.1139 0.0578 0.0581 Dollar/Canadian $ -0.00483 -0.01375 -0.00749 0.0290 0.0232 0.0224 Dollar/D. mark -0.02001 0.00249 -0.01488 0.0808 0.0599 0.0572 Twelve-month forecast horizon Dollar/Yen -0.17639 -0.01711 -0.01502 0.2214 0.0985 0.0873 Dollar/Canadian $ -0.00845 -0.02619 -0.01953 0.0520 0.0408 0.0406 Dollar/D. mark -0.05327 0.00145 -0.03139 0.1151 0.0937 0.0921 Notes: The dollar/Yen exchange rate is expressed as dollar per hundred yen. The dollar/Canadian dollar exchange rate is expressed as dollar per Canadian dollar. The dollar/Deutsche mark exchange rate is expressed as dollar per Deutsche mark. FVAR is a full vector autoregressive model. MVAR is a mixed vector autoregressive model. BVAR is a Bayesian vector autoregressive model. problems, a heteroscedastic, autocorrelation consistent covariance matrix [Newey and West (1987)] is used to estimate the standard errors of coefficients in Eq. (13). With the use of the Newey-West estimator, the joint test statistic is distributed as a Chi-squared. Results from the exchange rate bias tests are shown in Table 2. In the case of exchange rate forecasts generated from the FVAR model, the hypothesis of unbiasedness was rejected for all cases across all forecast horizons. For the MVAR exchange rate forecasts, the hypothesis was re- jected for all cases with the exception of the dollar/yen exchange rate at the 3-month forecast horizon. For the BVAR model forecasts, the hypothesis of unbiasedness was not rejected for the dollar/yen exchange rate at l- and 3-month forecast horizons and for the dollar/Canadian dollar across all forecast horizons. BVAR fore- casts of the dollar/Deutsche mark were biased across all forecast horizons. In terms of bias, BVAR model performance was clearly superior. However, it should be noted that the superior performance of BVAR models was limited to forecasting dollar/yen and dollar/Canadian dollar exchange rates. Finally, it is interesting to examine the form of the bias in the cases where significant bias was present. In these cases, intercepts were in general insig- nificantly different from zero and slopes were in general significantly less than one. This indicates that the bias was not constant, but instead varied with the level of the forecast (a scaling effect). The slope coefficients also indicate that biased models had a tendency to over-predict the mag- nitude of exchange rate changes. 3.3. Informational content test To determine if the forecasts generated from the alternative VAR models contain additional information beyond a random walk process, an Table 2 Bias test results for VAR exchange rate forecasting models, 1983:1-1989:12 Exchange rate FVAR model MVAR model BVAR model a P R2 2 X - One-month forecast horizon Dollar/Yen 0.005 (1.68) Dollar/Canadian $ 0.001 (0.52) Dollar/D. mark 0.002 (1.03)Three-month forecast horizon Dollar/Yen 0.010 (1.46) Dollar/Canadian $ 0.002 (0.52) Dollar/D. mark 0.004 (0.74)Six-month forecast horizon Dollar/Yen 0.026 (1.20) Dollar/Canadian $ 0.003 (0.42)Dollar/D. mark 0.008 (0.70)Twelve-month forecast horizon Dollar/Yen 0.072 (1.41) Dollar/Canadian $ 0.004 (0.21) Dollar/D. mark 0.005 (0.20)~0.269** (-5.52) 0.106** (-6.37) 0.023** (~20.95) 0.003** (~2.90) 0.124** (-4.65) -0.174** (-16.80) 0.072** (-4.27) -0.058 (-3.42) -0.257** (-14.84) 0.176** (-3.61) -0.494** (~6.33) -0.621** (~7.02) 0.020 0.006 0.002 0.000 0.008 21.97** 0.057 320.16** 0.006 49.31** 0.001 12.72** 0.087 237.76* * 0.073 55.31** 0.074 41.95** 0.206 49.32** 36.87** 28.8X**46X.22** 13.04** 0.004 (1.65) 0.002* (2.04) 0.002 (1.16) 0.005 (0.65) 0.009% (3.00) 0.008 (1.46) 0.013 (0.69) 0.021 (4.35)0.020 (1.82)0.042 (0.83) 0.046** (4.05) 0.042 (1.52) ~0.165** (-6.44) 0.905 (-0.33) 0.054** (-10.27)0.418 (-1.13) 1.349 (0.91) -0.444** (-7.22) 0.330** (-2.44) 1.702 (1.75)-0.770**(-9.09) 0.116* (-2.51) 1.962 (3.51) -0.756**(-7.82) 0.001 s.43* 0.067 6.31* 0.001 105.62** 0.017 1.29 0.276 10.20** 0.047 52.37** 0.019 6.96* 0.445 21.48** 0.174 83.34** 0.006 11.23** 0.533 0.193 20.27** 61.51** 0.003 (1.09) 0.001 (0.71) 0.002 (1.05)-0.005 (-1.91) 0.668 (-0.89) 0.107** (-4.11) 0.007 0.309 (0.81) (~1.46) 0.003 1.053 (1.12) (0.12) 0.004 -0.509** (0.75) (-4.62) 0.015 (1.05) 0.008 (1.74) 0.335 (-3.43) 1.206 (0.59) 0.006 -1.006** (0.51) (~6.22) 0.034 0.448 (1.97) (-2.31) 0.027 1.513 (1.89) (1.23) 0.011 -1.536** (0.52) (-6.14) 0.000 3.730.031 I .33 0.002 l&37* 0.013 2.11 0.138 1.27 0.038 22.77*0.026 12.75** 0.205 3.53 0.164 42.58** 0.073 6.83 0.257 0.377 4.473s.94* Note: The bias regression is specified as follows:Z iik -Z, = o + P(Z:,+, - Z,) + CL,, where Z, + ~ is the actual exchange rate at time r + k and Z;, , ~ is the k-month ahead exchange rate forecast made at time t. FVAR is a full vector autoregressive model. MVAR is a mixed vector autoregressive model. BVAR is a Bayesian vector autoregressive model. *The f-values in parentheses are calculated for the null hypothesis that a = 0. h The f-values in parentheses are calculated for the null hypothesis that p = 1. xL statistics are calculated to test the null hypothesis of (a, /3) = (0, 1). *Significant at the 5% level. ** Significant at the 1% level. Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 427 informational content test developed by Fair and Shiller (1989, 1990) is used. Fair and Shiller argued that the procedure has two advantages over the comparison of root mean squared errors. First, if the RMSEs are close for two forecasts, the informational content test may be better able to discriminate between the forecasts. Second, even if the RMSE for one model is much higher than for another, the model with the higher RMSE may still contain more in- formation. The Fair-Shiller test involves running regres- sions of the actual change in the variable on paired forecasts of the change in the variable. The regression equation is Z f+k - Z, = a + P(Z;t,r+k - Z,) + Y(ZSr,r+k - Z,) + rur (14) where Zy, ,+k is the k-month ahead forecast of model 1 made at time t; and Z;, ,+k is the k- month ahead forecast of model 2 made at time t. The Fair-Shiller test is equivalent to testing a null hypothesis of no information in both fore- casts, H,: p = 0 and y = 0, against the alter- native hypothesis that both coefficients are non- zero or that at least one is non-zero. If both p and y are zero, then neither forecast 1 nor forecast 2 contains information significantly dif- ferent from that found in a random walk model. However, if the /3 (y) coefficient is non-zero, then forecast 1 (forecast 2) contains significant information above that found in a random walk model and there is no additional independent information found in forecast 2 (forecast 1). Finally, if both p and y are non-zero, significant economic information is revealed in both series and the information sets are independent of one another [Fair and Shiller (1990)]. Note that serial correlation is introduced into the error term in Eq. (14) owing to the overlap of 3-, 6-, and 12-month ahead forecasts. Thus, the Newey-West covariance estimator is used to calculate the estimated standard errors. Results from the Fair-Shiller information tests are presented in Table 3. In the case of dollar/ yen exchange rates, the coefficients associated with FVAR, MVAR, and BVAR forecasts were not significantly different from zero at forecast horizons of l- through 6-months. Hence, the dollar/yen forecasts generated by the FVAR, MVAR, and BVAR models at these horizons did not contain additional information beyond that produced by random walk forecasts. However, the coefficients associated with BVAR forecasts were significantly positive at the 12-month horizon. Hence, the BVAR model dominated the FVAR and MVAR models at the 12-month forecast horizon, and the forecasts generated from BVAR model contained additional infor- mation not found in a random walk model. In the case of dollar/Canadian dollar exchange rates at a l- month horizon, the coefficients associated with the MVAR model were signifi- cantly positive, whereas the coefficients for the FVAR and BVAR models were insignificant. In contrast, at forecast horizons of 3- to 12-months all coefficients were significant, except for the BVAR model in the paired comparisons with the MVAR model. However, these results need to be interpreted carefully, because the coefficients associated with FVAR forecasts were significant- ly negative across 3- to 12-month forecast horizons. The negative coefficients imply that the information in the FVAR models was negatively correlated with actual exchange rate changes, a perverse result in economic terms. Hence, it can be concluded that the MVAR and BVAR models dominated the FVAR model at these forecasting horizons. Finally, MVAR models had significant- ly positive coefficients in paired comparisons with BVAR models, indicating that the MVAR model dominated the BVAR model for the 3- to 12-month horizons. In the case of dollar/Deutsche mark exchange rate forecasts at short horizons, none of the models had significant coefficients, and hence did not have useful information beyond that found in a simple random walk model. At longer horizons of 6- and 12-months, the MVAR and BVAR models had significant coefficients. But, again, the coefficients were negative, indicating that the information in the MVAR and BVAR models was negatively correlated with actual exchange rate changes. Hence, it can be concluded that none of the models contained useful information Table 3 Information content test results for VAR exchange rate forecasting models, 1983:1-1989:12 $/Japanese yenConstant FVAR MVAR BVAR R= $/Canadian dollarConstant FVAR MVAR BVAR RZ$/D. markConstant FVAR MVAR BVAR X2One-month forecast horizon 0.005 -0.265 -0.102 (1.81) (-1.14) (-0.25) 0.004 -0.336 (1.23) (-1.35) -0.285 (-0.42) 0.068 0.032 0.074 0.002 (1.05) 0.002 (1.04) 0.002 (1.07)-0.019 (-0.34) -0.018 (-0.18) -0.027 (---0.23) 0.339 (0.62) 0.181 (0.23) 0.021 0.002 (1.77) 0.025 0.001 (0.69) 0.002 0.003 (1.99) -0.044 (-0.23) -0.043 (-0.21) 0.949** (2.82) 1.37.Y(2.15) 0.735 (1.65) -0.582 (-0.74) -0.020 (-0.13) -0.034 (-0.06) -0.083 (-0.22) Three-month forecast horizon 0.005 -0.012 0.419 (0.70) ((0.04) (0.82) 0.006 -0.060 (0.85) (-0.18) 0.005 0.373 (0.68) (0.32)0.337 (0.76) 0.044 (0.04) 0.017 0.011 (3.42) 0.014 0.004 (1.47) 0.017 0.011 (2.99) -0.372* (-2.21) -0.450* (-2.26) 1.710** (4.46) 0.327 0.007 (1.05) 0.004 (0.75) 0.007 (1.11) -0.301 (-1.37) 2.036* (3.57) 1.681$ (3.42) -0.882 (- 1.39) 0.192 -0.133 (-1.67) -0.174 (-1.25) 0.302 -0.324 (-1.14) -0.001 (0.00) .- 0.287 (-0.66) Six-month forecast horizon 0.017 0.100 0.382 (0.82) (0.45) (1.15) 0.011) 0.060 (0.94) (0.29) 0.005 -0.004 (0.71) (-0.01) 0.324 (1.71) 0.339 (0.47) 0.031 0.022 (4.52) 0.030 0.01 (1.86) 0.026 0.024 (4.52) -0.371** (-2.56) -0..590** (-2.30) 1.x74** (5.19) 2.2339** (4.33)1.74s** (3.62) -0.806 (-1.80) 0.499 0.312 0.474 0.017 (1.63) 0.006 (0.51) 0.014 (l.lY) -0.133 -0.657*(-1.47) (-3.90) -0.021 -0.963fl 0.164 (-0.22) (-2.59) --a..513 -0.625 0.0218 (-1.75) (-1.62) Twelve-month forecast horizon 0.058 0.223 0.307 (1.12) (0.96) (0.79) 0.057 0.179 (1.17) (0.91) 0.067 -0.9370.453** (2.22) 0.310 0.107 0.045 (4.36) 0.147 0.030 (2.51) 0.167 0.046 1 .Y80* * (8.02) -0.524** (-2.67) -0.x34** (-3.27) 2.340** 1.916* (3.75) -0.535 0.617 0.451 0.545 0.021 -0.388 -0.365 (0.79) (-1.02) (-1.18) 0.008 -0.174 (0.32) (-0.43) 0.022 -0.3.53 -1.315* 0.346 (-1.84) -1.271* 0.369 (1.49) (- 1.48) (2.59) (4.37) (6.54) (-1.14) (1.W (-0.97) (-2.64) Note: The regression is specified as follows: Z iik - 2, = 0 + P(G,+* -a + YG%+, --cl + CL,, where Z,,, is the actual exchange rate at time t + k, Z;,,r+k is the k-month ahead BVAR forecast of model 1 made at time r, and Zy,~,,, is the k-month ahead forecast of model 2 made at time I. FVAR is a full vector autoregressive model. MVAR is a mixed vector autoregressive model. BVAR is a Bayesian vector autoregressive model. *The r-values in parentheses are calculated for the null hypothesis that fi = 0 or y = 0. * Significant at the 5% level. ** Significant at the 1% level. Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 429 beyond that found in a random walk model in forecasting doliar/Deutsche mark exchange rates. Overall, the information content results were mixed. FVAR models did not exhibit significant informational content in any case. MVAR models exhibited significant informational con- tent in forecasting dollar/Canadian dollar ex- change rates at all horizons, but did not exhibit informational content when forecasting dollar/ Canadian dollar and dollar/Deutsche mark ex- change rates. BVAR models exhibited significant informational content in forecasting long-horizon dollar/yen exchange rates, but were bettered by MVAR models in forecasting dollar/Canadian dollar exchange rates and were of no significant value in forecasting dollar/Deutsche mark ex- change rates. Studies by Boothe and Glassman (1987) and Gerlow and Irwin (1991) show that statistical evaluation measures, such as those considered above, may not yield results consistent with the actual trading profits generated by exchange rate models. Hence, it is useful to consider a measure of the economic value of a model. In this study, the Henriksson-Merton test of market timing ability will be used to examine this dimension of forecast performance. The Henriksson-Merton test is essentially a test of the directional forecasting accuracy of a model. This test is employed for three reasons. First, the test is simple to implement. Second, the test does not require specification of an equilibrium asset pricing model. Third, direc- tional accuracy has been shown to be highly correlated with actual trading profits [Leitch and Tanner (1991)J. The Henriksson-Merton test requires specifi- cation of two binary variables. First, a forecast direction variable, F,, is defined, where F, = 1 if the exchange rate is forecast to rise over the period from time t - i to t, and F, = 0 if the exchange rate is forecast to remain constant or fall over the same period. Second, a market direction variable, A,, is defined, such that A, = I if the actual exchange rate increases between time t - i and t, and A, = 0 if the actual exchange rate remains constant or falls over the same period. Given the previous definitions, market timing ability depends upon the conditional probability of a correct forecast of the realized direction of exchange rate change. Conditional probabilities are defined by: pl = Prob[F, = O]A, = 0] (15) 1 - pl = Prob[F, = l/A, = 0] (16) p2 = Prob[F, = l/A, = 11 (17) 1 - p2 = Prob[F, = OIA, = l] (IS) Hence, pl is the conditional probability of correctly forecasting that the exchange rate will decrease or stay constant, and p2 is the con- ditional probability of correctly forecasting that the exchange rate will increase. Merton (1981) shows that the sum of the conditional prob- abilities of correctly forecasting the direction of the exchange rate change is a sufficient statistic for market timing value. More specifically, Mer- ton shows that the sum of conditional prob- abilities pl and p2 must exceed 1 for forecasts to exhibit market timing value. Henriksson and Merton derive a statistic to test the null hypothesis of no market timing value (H,: pl +p2 = 1). The test proposed by Henriksson and Merton is a Fisher exact test and, therefore, the most uniformly powerful unbiased test of this null hypothesis [Cumby and Modest (1987)j. The confidence level, c, associ- ated with the rejection of the null hypothesis is based on the hypergeometric distribution and is given by the following formula (19) where Nl is the number of times the exchange rate decreases or stays constant, N2 is the 430 Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 number of times the exchange rate increases, N tion, with marginal significance levels of 12% or equals the sum of Nl and N2, nl is the number less, MVAR forecasts exhibit reasonable evi- of times the exchange rate is correctly forecast to dence of market timing ability for the dollar/yen decrease or stay constant, n2 is the number of rate at 3- and 6-month horizons. Forecasts gener- times the exchange rate is incorrectly forecast to ated from a BVAR model also have significant decrease or stay constant, and n equals the sum market timing value for dollar/Canadian dollar of nl and n2. A one-tailed test is appropriate if rates across 3- to 12-month forecast horizons. the forecast series is not perverse (~1 + p2 < 1). Finally, with marginal significance levels of 15% Hence, the null hypothesis of no market timing or less, BVAR forecasts of the dollar/yen rate at is rejected for any level of significance greater horizons of 3- through 12-months also exhibit than 1 - c. some evidence of market timing value. The results of the Henriksson-Merton market timing test are given in Table 4. The forecasts generated from FVAR models, for all three exchange rates, have no significant market tim- ing value. In other words, the FVAR model is not capable of significant predictions of the directional movement of the exchange rates. MVAR forecasts exhibit significant market tim- ing ability across all forecast horizons for the dollar/Canadian dollar exchange rate. In addi- The market timing results indicate that the MVAR and BVAR models have an economically significant value in predicting the directional change in two of the three exchange rates. In addition, the market timing results show more evidence of forecasting value than the bias and informational content tests. This corroborates Boothe and Glassmans (1987) and Gerlow and Irwins (1991) finding that statistical and econ- omic evaluations may yield different results. Table 4 Market timing test results for VAR exchange rate forecasting models, 1983:1-1989:12 Exchange rate FVAR MVAR BVAR PI Pzb P, + P,= H-M p, a PIh P, + P2 H-M PI a Pzh P, f PzL H-M sigmlicance significance stgnificance level d level d level d Onr-month forecast horizon D0lhlriYen 0.2326 0.5854 0.8180 0.979 0.2558 0.7561 I.0119 0.550 0.2326 0.8049 1.0375 0.440 Dollar/Canadian $ 0.4474 0.6087 1.0561 0.383 0.8421 0.3696 1.2117* 0.026 0.6579 0.4565 1.1144 0.201 Dollar/D. mark 0.4000 0.5227 0.9227 0.824 0.1250 0.7955 0.9205 0.899 0.4000 0.4001 0.8091 0.976 Three-month forecast horizon Dollar/Yen 0.5714 0.2766 0.8480 O.Y52 0.2286 0.9149 1.1480 0.067 0.3429 0.7872 1.1301 0.144 Dollar/Canadian $ 0.4048 0.5000 0.9048 0.862 0.904X 0.3500 1.2548 0.005 0.73X1 0.5250 1.2631** 0.013 Dollar/D. mark 0.3611 0.5217 0.8828 0.902 0.0556 0.8913 0.9469 0.897 0.4722 0.4348 0.9070 0.855 Six-month forecast horizon Dollar/Yen 0.5862 0.1800 0.7662 0.994 0.1724 0.9400 1.1124 0.115 0.3793 0.7800 1.1593 0.104 Dollar/Canadian $ 0.6667 0.3256 0.9923 0.624 1.0000 0.2558 1.2558* 0.001 0.9444 0.4884 1.4326* 0.000 Dollar/D. mark 0.361 I 0.4884 0.8495 0.942 0.0278 0.8837 0.9115 0.978 0.4444 0.3721 0.8165 0.969 Twdve-month forecast horizon Dollar/Yen 0.8261 0.2600 1.0861 0.311 0.0000 0.9000 0.9000 1 .OOO 0.3913 0.8000 1.1913 0.076 Dollar/Canadian $ 0.5429 0.3421 0.8850 0.893 1.0000 0.2368 1.2368* 0.002 1 .OOOO 0.2895 1.2895* 0.000 Dollar/D. mark 0.5882 0.1795 0.7677 0.993 0.0000 0.8205 0.8205 1 .OOO 0.3824 0.2308 0.6132 1.000 FVAR is a full vector autoregressive model; MVAR is a mixed vector autoregressive model; BVAR is a Bayesian vector autoregressive model. P, is the conditional probability of correctly forecasting that the exchange rate will decrease or stay constant. Pz is the conditional probability of correctly forecasting that the exchange rate will increase. * P, + Pz is significantly greater than 1 at the 5% level of significance; ** P, + P7 is sigmficantly greater than 1 at the 1% level of significance. The reported significance level is equal to 1 minus the Henriksson-Merton confihence level, and is used to test the null hypothesis of oo market timing ability (H,: P, + P2 = 1). Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 431 4. Summary The purpose of this research was to analyze the forecasting accuracy of full vector autore- gressive (FVAR), mixed vector autoregressive (MVAR) > and Bayesian vector autoregressive (BVAR) models of the US dollar/yen, US dol- lar/Canadian dollar, and US dollar/Deutsche mark exchange rates. The VAR models were based on the theoretical model of monetary/ asset exchange rate determination developed by Driskill et al. (1992). The in-sample estimation period for the models was 1973:3-1982:12. For the out-of-sample period 1983: l-989: 12, l-, 3-, 6-, and 12-month forecasts were generated. Per- formance criteria included bias tests, informa- tional content tests, and market timing tests. The out-of-sample forecasting results indicated a sharp difference between the performance of unrestricted VARs (FVARs) and restricted VARs (MVARs and BVARs). Forecasts from FVAR models, in all cases, were biased and exhibited no significant informational content or market timing ability. In contrast, MVAR and BVAR model forecasts generally exhibited less bias, and in a number of cases had significant information content and/or market timing ability. Discriminating across the restricted models was more problematic, as forecast performance varied somewhat across exchange rate, forecast- ing horizon, and performance test. MVAR models exhibited the most significant informa- tional content, while BVAR models exhibited the least bias. The market timing abilities of the MVAR and BVAR models were roughly equal. The MVAR and BVAR models were most successful in predicting dollar/Canadian dollar exchange rates. These models were somewhat less successful in predicting dollar/yen exchange rates, and were of no value in predicting dollar/ Deutsche mark exchange rates. Overall, the results show that a monetary/ asset model in a VAR representation does have forecasting value for some exchange rates. Addi- tional research is needed to determine if further improvements in forecasting accuracy are pos- sible. One promising approach is suggested by MacDonald and Taylor (1993). They impose the monetary/asset model as a long-run equilibrium condition in a dynamic error-correction model, and allow the data to determine short-run dy- namics. MacDonald and Taylor report that ex- change rate forecasts for the error-correction model are more accurate than forecasts from a random walk model. An interesting question is whether error-correction models provide im- proved forecasting performance over the VAR models considered in this study. 5. Acknowledgment Funding support from the Farm Income En- hancement Program at the Ohio State University is gratefully acknowledged. 6. References Akaike, H., 1970, Statistical predictor identification. Annals of the Institute of Statistical Mathematics, 22, 203-217. Alexander, D. and L.R. Thomas, 1987, Monetary/asset models of exchange rate determination: How well have they performed in the 198Os?, International Journal of Forecasting, 3, 53-62. Bessler, D.A. and J.L. Kling, 1986, Forecasting vector autoregressions with Bayesian priors, American Journal of Agricultural Economics, 68, 144-151. Boothe P. and D. Glassman, 1987, Comparing exchange rate forecasting models: Accuracy versus profitability, Interna- tional Journal of Forecasting, 3, 65-79. Canarella, G. and S.K. Pollard, 1987, Efficiency in foreign exchange markets: A vector autoregression approach, Journal of International Money and Finance, 7, 331-346. Cumby, R.E. and D.M. Modest, 1987, Testing for market timing ability: A framework for forecast evaluation, Jour- nal of Financial Economics, 19, 169-189. Doan, T.A. and R.B. Litterman. 1988, Regression Analysis for Time Series Analysis (RATS) Users Manual, Version 3.00 (VAR Econometrics Co., Evanston, IL). Driskill, R.A., N.C. Mark and S.M. Sheffrin, 1992, Some evidence in favor of a monetary rational expectations exchange rate model with imperfect capital substitutability, International Economic Review. 33, 223-237. Fair, R.C. and R.J. 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Wolff, C.P., 1988, Exchange rates, innovation and forecast- ing, International Money and Finance, 7, 49-61. Woo, W.T., 1985, The monetary approach to exchange rate determination under rational expectations, Journal of International Economics, 18, 1-16. 7. Appendix. The structure of mixed VAR exchange rate forecasting models Dollar/Yen: M = F[M(-l), . , ) M(-13), Y(-l), . . ,Y(-16), I(-l), T(-1), T(-2)], T= F[T(-l), . , T(-13), I(-l), P(-1), . . . , P(-lo), Y(-l), Y(-2)], Y = F[Y(-l), . . . ) Y(-13), P(-1), . . . ) P(-3), M(-I), c(-l), . . . , q-3)], Z = F[Z(-1), e(-l), M(-1)], P=F[P(-l), . . .) P(-13), e(-l), e(-2) Y(-1), . ) Y(-lo)], e = F[e(-1), T(-1), P(-l)]. Dollar/Canadian dollar: M = F[M(-l), . . , M(-13), T(-l), P(-l), P(-2) q-1)1, T= F[T(-I), T(-2) Y(-1), . . . , Y(-4) M(-1), M(-2), P(-1), . . . , f(-3)1, Y = F[Y(-l), . . ) Y(-14), P(-1), . . ) P(-3), M(-l), d-1)1, I = F[Z(-1), T(-1), T(-2), T(-3)], P = F[P(-l), . . . , P(-9), T(-l), T(-2)]. e = F[e(-1), T(-1), Z(-l)]. Dollar/Deutsche mark: M = F[M(-l), . . , M(-13) Z(-l), P(-l)l, T= F[T(-l), . . , T(-3), Y(-1), P(-1)], Y = F(Y(-l), . . , Y(-14), P(-1), M(-I)], Z = F[Z(-1), e(-l)], P = F[P(-l), . . , P(-12), Y(-1), e(-l), T(-l), T(-2)1. e = F[e(-l), T(-1), T(-2)]. where M is the logarithm of relative money supply between domestic and foreign countries, Te-Ru Liu et al. I International Journal of Forecasting 10 (1994) 419-433 433 T is the trade balances between two countries, Y is the logarithm of relative real income between two countries, I is the interest rate differential countries, P is the logarithm of between two relative price the logarithm lag operator. level between two countries, e is of exchange rate, and (-8) is the Biographies: Te-Ru LIU is a Former Research Associate in the Department of Agricultural Economics and Rural Sociol- ogy at The Ohio State University. She earned a Ph.D. in Agricultural Economics from Ohio State University. Her research interests are in the areas of forecasting, exchange Mary E. GERLOW is a Former Assistant Professor in the Department of Agricultural Economics and Rural Sociology at The Ohio State University. She earned a B.A. degree from Texas A&M University and a Ph.D. in Agricultural Economics from Texas A&M Universitv. Her oublications are in the areas of futures and options markets, forecasting, and international trade in agricultural products. Scott H. IRWIN is an Associate Professor in the Department of Agricultural Economics and Rural Sociology at The Ohio State University. He earned a B.A. degree from Iowa State University and a Ph.D. in Agricultural Economics from Purdue University. His publications are in the areas of futures and options markets, forecasting, and investments. rate determination, and policy analysis

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