model specification and forecasting foreign exchange rates with vector autoregressions

Journal of ForecastingJ. Forecast. 20, 451–484 (2001)DOI: 10.1002/for.808

Model Specification and ForecastingForeign Exchange Rates with VectorAutoregressions

NATHAN LAEL JOSEPH*University of Manchester, UK

ABSTRACTThis study examines the forecasting accuracy of alternative vector autore-gressive models each in a seven-variable system that comprises in turn ofdaily, weekly and monthly foreign exchange (FX) spot rates. The vectorautoregressions (VARs) are in non-stationary, stationary and error-correctionforms and are estimated using OLS. The imposition of Bayesian priors inthe OLS estimations also allowed us to obtain another set of results. We findthat there is some tendency for the Bayesian estimation method to generatesuperior forecast measures relatively to the OLS method. This result holdswhether or not the data sets contain outliers. Also, the best forecasts underthe non-stationary specification outperformed those of the stationary anderror-correction specifications, particularly at long forecast horizons, whilethe best forecasts under the stationary and error-correction specificationsare generally similar. The findings for the OLS forecasts are consistent withrecent simulation results. The predictive ability of the VARs is very weak.Copyright 2001 John Wiley & Sons, Ltd.

KEY WORDS seasonality; forecasting; cointegration; vector autoregressions;error-correction models; Bayesian estimation, outliers

Multivariate vector autoregressive models have been extensively used to assess the forecastingperformance of monetary/asset models of foreign exchange (FX) rate determination. The empiricalresults from the unrestricted (full) vector autoregressions (VARs) generally show that models ofFX rate determination are unable to outperform the random walk in out-of-sample forecasts (Meeseand Rogoff, 1983; Liu et al., 1994). Unrestricted VARs tend to suffer from over-parameterization(over-fitting) and multicollinearity among the lagged variables, with the result that they producepoor out-of-sample forecasts. By imposing Bayesian prior restrictions in a VAR, the Bayesianvector autoregressive (BVAR) model of Litterman (1984, 1986) attempts to resolve both the over-parameterization and multicollinearity problems. Another approach that is often used to assessthe forecasting performance of models of FX rate determination relies on cointegration theory(see Granger, 1986) and error-correction methodology. Here, the use of standard error-correctionmodels (ECMs) results in some restrictions on the VARs thereby reducing the over-parameterizationproblem. A further advantage of the ECM is that since the time series in the model would be

* Correspondence to: Dr. Nathan Lael Joseph, Manchester School of Accounting and Finance, University of Manchester,Manchester M13 9PL, UK. E-mail: [email protected]

Received January 1997Copyright 2001 John Wiley & Sons, Ltd. Accepted October 2000

452 N. L. Joseph

cointegrated, the error-correction term would ensure estimation consistency. Chinn and Meese(1995) show that the monetary models of FX rate determination work well at long forecast horizons,when cast within an error-correction framework. It is also possible to derive a Bayesian error-correction model (BECM) which is related to the ECM but with the Bayesian priors imposed inthe model.

Researchers have also attempted to forecast FX rates using only the time series data of FXrates. Here, a relatively common approach to forecast daily and weekly FX rates is to employ non-parametric prediction techniques and non-linear models. Using standard forecasting criteria, mostempirical results show that the out-of-sample forecasts from those models are unable to outperformthe simple random walk (see e.g. Diebold and Nason, 1990). However, Satchell and Timmermann’s(1995) empirical results show that compared to standard measures of forecasting accuracy, a non-parametric sign test based on the predicted direction of the FX rate change is more suitable forevaluating the forecasts from non-linear models. Their results show that non-linear models performwell in predicting the sign of daily changes in FX rates.

This study differs from previous empirical studies of FX rate forecasting in many importantrespects. We focus on the comparative gain or loss in forecasting accuracy when different VARspecifications, for example, levels, differences, etc. are employed. Our focus on model specifi-cation is motivated by Engle and Yoo’s (1987) observation that I(1) variables will forecast withincreasingly wider confidence bands, while cointegrated linear combinations of such I(1) variableshave confidence bands which become finite as the forecast horizon increases. Engle and Yoo’s(1987) simulation results allowed them to favour an ECM based on Engle and Granger’s two-stepprocedure over an unrestricted VAR. More recently Clements and Hendry’s (1995a) simulationresults for a bivariate I(1) cointegrated system indicate that there is little gain in forecasting per-formance when reduced rank cointegration restriction is imposed, unless the sample size is small.Our study employs historical data within a larger system for a wider class of VARs. Our findingsalso have implications for other empirical work which shows that a class of technical trading rulescan generate economically significant returns in the FX spot market (Neely et al., 1997).

The VARs in our study are formulated in: (1) non-stationary (levels); (2) stationary (differences);and (3) error-correction forms. We identify those three (unrestricted) specifications as VARL, VARDand VECM, respectively, when they are estimated using standard OLS. We also imposed Bayesianpriors in the OLS estimations to obtain another set of forecast measures. The associated Bayesianmodels are identified as: (1) BVARL for the BVAR in non-stationary form; (2) BVARD for theBVAR in stationary form; and (3) BECM. The forecasting accuracy of the VARs is evaluated fordaily, weekly and monthly data. We are unaware of any previous study that considers the forecastingaccuracy of weekly FX rates across currencies, for sets of similar days-of-the week, i.e. Mondayto Monday, Tuesday to Tuesday, etc. Most empirical studies employ monthly FX rates along withother macroeconomic variables while others focus on daily data or data for a specific day of theweek. Our focus on several data intervals is motivated by the evidence concerned with variation inthe statistical distribution of FX rate returns (see McFarland et al., 1982).

This study contributes to the economic forecasting literature in the following ways. Our resultsfor daily, weekly and monthly FX rates indicate that the seven currencies are cointegrated at all dataintervals. So, a VAR in levels would fail to account explicitly for all the properties of the time seriesin the system, while a VAR in differences would ignore long-run information. To simplify matters,we consider both VARs to be mis-specified. The ECM however, is the correct specification for ourdata, but we find that this VAR does not generate superior forecasts. With predetermined lags inthe VARs, there was some tendency for important differences to arise in the forecasting accuracy

Copyright 2001 John Wiley & Sons, Ltd. J. Forecast. 20, 451–484 (2001)

Forecasting Foreign Exchange Rates with Vector Autoregressions 453

of the estimation methods. Joutz et al. (1995) had suggested that determining the lag length for theBVARs was not a crucial consideration. The best forecasts under the non-stationary specificationoutperformed the best forecast under both the stationary and error-correction specifications, whilethe best forecasts under the stationary and error-correction specifications are generally similar. Onekey finding is that although the VARs outperformed the random walk on the basis of the unit-freeTheil U (hereafter U) statistic, their forecasts are still very poor in terms of: (1) the magnitude ofthe forecast standard errors; and (2) the ability of the VARs to correctly predict the direction of theprice movement. Thus the apparent superior performance of our forecasts relative to the randomwalk is not too meaningful, although the random walk may not be a useful criterion for judgingforecasting performance (see also Dacco and Satchell, 1999).

As cointegration implies Granger causality in at least one direction, we stress that our finding thatthe series are cointegrated is not a sufficient condition for inferring market inefficiency. Here, wefollow LeRoy’s (1989) argument that the predictability of asset prices has no particular role to playin market efficiency considerations, in general. As such, we support the refined view that a marketis (informationally) efficient if agents are unable to generate risk-free returns above opportunitycost, given also, transaction costs and agents’ information sets.

The next section addresses the relative importance of model specification in forecasting exper-iments and the potential impacts of seasonal variation on forecasting accuracy. The third sectiondescribes both our research methodology and the data sets. Our empirical results are then presentedand we summarise our results and suggest some implications in the final section.

IMPORTANCE OF GOOD MODEL SPECIFICATION

The considerations associated with alternative specification and estimation methods in forecastingexperiments are interesting in many respects. The accuracy of the forecasts is likely to reflecthow well the specified model captures the underlying data generating process (DGP) of theseries, such that the relative gain or loss in forecasting accuracy would depend on the specifica-tion/transformation of the data. For example, using the invariant generalized forecast error secondmoment (GFESM), Clements and Hendry (1995a) show that the forecasts from a VAR in levelsoutperformed those of another VAR in which co-integration restriction was imposed in the form ofthe Engle–Granger two-step methodology. Also, BVARs generally appear to generate better out-of-sample forecasts than both unrestricted OLS VARs and univariate autoregressions (see LeSage,1989). Using the forecast error standard deviations, Joutz et al. (1995) report that the OLS ECMand BECM outperform the OLS VAR and BVAR (both in levels) only over short forecast hori-zons. As empirical evidence regarding the existence of cointegrating ranks among FX rates acrosscurrencies appears to depend on the time period of the study (Baillie and Bollerslev, 1989a; Vande Gucht et al., 1996), we do not know a priori which model specification would be appropriatefor our data sets.

The literature also suggests that failure to account for seasonal variation in time series datacan adversely affect the dynamic structure of econometric models (Osborn et al., 1988) and thatignoring seasonal variation is also likely to adversely affect forecasting accuracy (see Lee andSiklos, 1997). In general, forecasting accuracy tends to improve when the seasonal variation in thedata is properly accounted for (Reimers, 1997). Seasonal variation can be defined as ‘. . . just thatpart of the series which gives rise to the spectral lines at the seasonal frequencies’ (Nerlove, 1964,p. 260). Since the empirical evidence from the finance literature suggests that daily FX rates exhibit


454 N. L. Joseph

seasonal variation (see McFarland et al., 1982), this finding may have important implications forforecasting FX rates. Other empirical studies that incorporated daily seasonal and holiday dummyvariables in autoregressive-type ARCH and GARCH models (see Baillie and Bollerslev, 1989b;Copeland and Wang, 1994), also confirm that the volatility of daily FX rates varies at seasonalfrequencies. It is worth noting that when time series contain seasonal unit roots, the inclusionof deterministic seasonal components in the OLS model tends to generate inconsistent estimates(Franses et al., 1995). A preferred approach when using OLS would be to apply a difference filter�1 C BC B2 C B3 C B4�, where BkXt D Xt�k . The filter �1 � B5� removes both the unit root at zerofrequency and the seasonal unit roots in daily FX rates. A model with more seasonal cointegratingrelations than is required by the DGP tends to produce large forecast errors (see Reimers, 1997).

Finally, McFarland et al. (1982) show that the empirical distribution of FX rate changes dependson the interval at which the rates are generated. They found that the symmetric stable Paretiandistribution provides a more adequate fit for weekly changes compared with daily changes. Theapplication of ARCH and GARCH models to changes in daily, monthly and Wednesday FX spotrates also suggests that the statistical distribution of the FX rate changes depends on the interval ofthe data (see Baillie and Bollerslev, 1989b). Our use of weekly and monthly data sets is intendedto capture those effects and to aid our understanding of the behaviour of economic agents whomake decisions and form expectations in presence of changing patterns in FX rates. An alternativeargument can be put forward for analysing daily, rather than weekly or monthly series, sincethe FX market has been known to react quickly to central bank intervention and money-supplyannouncements (see Ito and Roley, 1987). In this case, the long-run effect of a sustained shockin FX rates can be observed more readily and the corresponding impact on forecasting accuracyanalysed. For those reasons, a set of daily FX rates is also analysed.

RESEARCH METHODOLOGY AND THE DATA SET

Specifying the OLS VARsThe OLS VAR (VARL) for the levels of daily FX rates can be written as

Yi,t D ˛i,0 Cp∑dD1

ˇi,dSd,t Cn∑jD1

m∑kD1

υi,j,kYj,t�k Cq∑rD1

�i,rXr,t C εi,t �1�

for i D 1, . . . , n. The subscript t is the time interval (daily) of the series and m is the order of thelag structure. The deterministic terms include a constant ˛i,0, q other variables Xr,t and possiblyseasonal dummy variables, Sd,t Ð υi,j,k is the coefficient of the jth variable at the kth lag in theequation of the ith variable Yi,t. εi,t is the disturbance (error) term.

The OLS VAR (VARD) for the differences of daily rates can be written as


ˇi,dSd,t Cn∑jD1

m∑kD1


�i,rXr,t C �i,t �2�

where denotes the (first) difference operator and �i,t is the disturbance term. Also, the OLSerror-correction model (VECM) can be written as


ˇi,dSdt C �iECTi,t�1 Cn∑jD1

m∑kD1


�i,jXr,t C �i,t �3�



where ECTi,t�1 is the error-correction term generated equation by equation from the level series inthe seven-variable system. To illustrate, assume instead a three-variable system where xt, yt and ztare the level series. If xt is the left-hand side variable in this system, the time series for the error-correction term would be �x � y � z�t�1, where t � 1 is the lag imposed in the system. Similarly,if yt and zt were in turn, left-hand-side variables, the series for the error-correction termswould be �y � x � z�t�1 and �z � x � y�t�1, respectively. This approach was used to estimate theerror-correction terms for our seven-variable system equation by equation.1 Each error-correctionterm was imposed (equation by equation) in each VAR in differences in order to generate theerror-correction specification.

The VARs for the weekly series are similar to those in equations (1), (2) and (3) but with theseasonal dummy variables omitted. The VARs for monthly series are similar to the VARs for weeklyseries but with pre- and post-Bank Holiday dummy variables included in each system.2

Specifying the Bayesian VARsA Bayesian approach to forecasting requires the imposition of fuzzy restrictions on the coefficients ofthe VAR. This means that the independent normal prior distribution for each of the n2m coefficientsneeds to be specified. Thus υ�i, j, k� in equation (1), for example, is assumed to have a mean ofϕ�i, j, k� and a variance of S2�i, j, k�, across indices i, j and k. To specify our prior distribution, weemployed the Minnesota prior which depicts: (1) an overall restriction or tightness; (2) cross-lagrestrictions; and (3) higher lag restrictions. This approach facilitates a substantial reduction in thedimensionality of the specification problem to a few hyperparameters (see Doan et al., 1984). Ourspecification of the Minnesota prior is such that it reflects the statistical observation that (say) Yi,tfollows a random walk around unknown deterministic components which can include a constant,˛i,0 and seasonal dummy variables, Sd,t, thus


ˇi,dSd,t C Yi,t�1 C #i,t �4�

Equation (4) suggests that Yi,t depends primarily on its own first lag and that although higher-orderlags may be important, their role is assumed to be less important compared to that of the first lag.

The Minnesota prior mean can be written as

ϕi�i, j, k� D{

1 if i D j and k D 10 otherwise

�5�

Here, the prior distribution is assumed to be Gaussian with a prior mean of unity on each coefficientof the first lag of the dependent variable, and a prior mean of zero on all other coefficients. The

1 Empirical work often employ a linear combination of the level series to generate the error-correction term (see Kremerset al. 1992), although typically for two- or three-variable systems. The error-correction series could also have been obtainedusing the residuals from the (static) level regression. This approach was not adopted since to do so would require muchcomputing time when re-estimating each observation in the error-correction term as it became available. By estimating theerror-correction series in the manner we have chosen, we can ensure that the Kalman filter in RATS consistently updatesthe forecasts of all the VARs in this study.2 It turned out that when restrictions were imposed on the cointegrating vectors, only pre- and post-bank holiday dummyvariables were required in the monthly VARs. Copeland and Wang (1994) had suggested that three dummy variables arerequired when working with daily data. All statistical results not fully presented can be obtained from the author.


456 N. L. Joseph

standard deviation function for the Minnesota prior for lag k of variable j in equation i for all i, jand k is represented by

Si�i, j, k� D f$igi�k�f�i, j�gsi/sjf�i, j� D

{1.0 if i = jw otherwise

�6�

The overall tightness is described by the parameter $i which is the standard deviation on the firstown lag in equation i. Small values of $i indicate increasing confidence in the prior. In equation (6),gi�k� is the function that depicts the tightness of the kth lag relative to the first lag. It reflects thedegree of confidence in the view that the coefficients at longer lags are closer to zero. We assumethat the function gi�k� has a harmonic shape such that gi�k� D k�*,i, where *i ½ 0 is the decay ratein equation i. A parameter of *i D 0 implies no shrinkage on the standard deviation at increasinglags while *i D 1 implies shrinkage at a rate of (1/k). We use the harmonic function rather than thegeometric function since the latter is likely to get too tight too quickly (see Doan, 1992). Also, inequation (6), si is the standard error of the univariate autoregression for variable j. The ratio si/siis a scaling factor that corrects for differences in the units of measurement of the variables acrossequations. The function f�i, j� depicts the tightness on variable j in equation i relative to thatof variable i. We employ the symmetric type of prior since its application reduces the problem tochoosing fewer parameter values. Thus, in equation (6), the same parameters for relative tightnessw are applied to all the off-diagonal variables in the system. The RATS software is used to generateall our forecasts. Here, a very large value for $i, such as 2.0 effectively eliminates the ‘Bayesianpart’ of the VAR while a very small value for w such as 0.001 eliminates the ‘vector part’ of theVAR and results in an autoregression.

Setting the hyperparametersTo estimate the BVARs, a harmonic decay of 1.00 was used in all cases. To impose the otherhyperparameters, we considered values of 0.30, 0.50 and 0.80 for w. Each value of w was matchedin turn with a value for $i that was relatively loose (0.20) to tight (0.10). This allowed us to choosebetween six different sets of forecasts. A flat prior was assigned to all deterministic terms. Theprocedure in RATS allowed us to impose only a flat prior on the error-correction term. However,Amisano and Serati’s (1999) results show that the use of a flat prior on the error-correction termtogether with an informative prior on the endogenous variables can adversely affect forecastingperformance. The six sets of forecasts that were obtained under those hyperparameters were notsubstantially different. We report the results where a common w value of 0.50 was applied to allthe Bayesian models. The associated value of $i D 0.20 was used for BVARL, while a value of$i D 0.10 was used for both the BVARD and BECM. Those hyperparameters appeared to workwell and were consistently applied to all the samples within the data sets.

Forecasting approach and comparability of the estimation methodsBefore generating the out-of-sample forecasts, we determined the optimal lag length over the estima-tion period for all the OLS VARs in levels and differences. The optimal lag length was determinedusing the likelihood ratio (LR) test statistic due to Sims (1980) but with a small sample correction(see Doan, 1992). To obtain the LR statistic, a constant was the only deterministic componentin the VARs containing daily and weekly data. In addition to a constant, the VARs for monthlydata also contained one pre- and one post-bank holiday dummy variable when the lag length was



determined (see footnote 2). Eight (nine) lags were required for weekly (daily) systems while fivelags were required for monthly systems. The Bayesian models employ the same lag lengths astheir OLS counterparts. We report on the ‘finished statistics’ of each forecast step from the simu-lated out-of-sample forecasts within the data range of the last 13 (48) observations of the monthly(daily/weekly) series for both estimation methods. To do this, a Kalman filter procedure was used toestimate the VARs using only the data up to the start period of each set of forecasts. As we movedthrough the forecast period, there was always one observation less on which to base the forecastat each step. The procedure in RATS automatically generated the forecasts, forecast measures andforecast standard errors for each estimation method. Our choice of the samples of 1, 2, . . . , 13(48)-step-ahead forecasts is arbitrary but allowed us to generate sufficient data points for evaluatingthe alternative systems.

Finally, it should be noted that the Bayesian estimation method could be considered to besubjective since in its usual application, the prior distribution is selected in an informative manner.In contrast, many frequentists argue that OLS estimates objectively, since it is an intuitively fairestimation method which is unbiased (see Efron, 1986). As such, it can be argued that our OLS andBayesian forecasts are not directly comparable. There has been much debate about the degree ofobjectivity in Bayesian estimation and we do not wish to dwell on the arguments here (see Bergerand Delampady, 1987; Eaton, 1987). While we follow Eaton’s (1987) argument that Bayesianestimation is essentially subjective, it is still useful to assess the relative performance of bothestimation methods since to do so aids our understanding of their forecasting performance underalternative VARs.

Data source and currenciesTo estimate our models, we obtained FX spot rates for seven major currencies against the UnitedStates (US) dollar. These are the: Swiss franc (CHF), Deutsche mark (DEM), French franc (FRF),British pound (GBP), Italian lira (ITL), Japanese yen (JPY) and Netherlands guilder (NLG). Therates are the daily bid-ask prices from the London FX market. They span the period 24 October1983 to 12 May 1997. The daily rates that are analysed comprise of observations for consecutiveweekdays in order to facilitate a proper application of our difference filter when testing for seasonalunit roots. Thus, London bank holidays and the observations surrounding those holidays wereexcluded. A separate analysis using all daily observations generated results that are similar to thosereported here. The weekly data sets, i.e. Monday to Monday, and so on, were generated from theoriginal data set but excluded pre- and post-bank holiday observations since they have been shownto behave in an unusual manner (see Joseph and Hewins, 1992). Finally, the monthly spot ratesare the observations for the last business day of each month. They span the period 31 October1983 to 30 May 1997. All the FX rates were obtained from Datastream. We perform the analysison the mid-point (in natural logarithms) of the bid and ask spot rates. Details of the number ofobservations in the estimation and forecast periods are shown in Appendix A.

EMPIRICAL RESULTS

Graphical plots and summary statisticsFigure 1 shows the plots for the level series over both the estimation and forecast periods. Theplots show that the series are non-stationarity. The trend in the series is not as pronounced overthe more recent period. If in fact the series are I(1) and are cointegrated, then the error-correction


458 N. L. Joseph

specification might well capture the DGP of the series. The plots for the differenced series appearstationary (see Appendix B). They suggest that heteroscedasticity and non-normality are likely tobe more severe for the daily data compared with weekly or monthly data. There seems to be noobvious sign of seasonal variation in the daily data but we choose to test for this formally.

The summary statistics for the differenced series are shown in Appendix C. The summary statis-tics are for the estimation period. The mean changes are negative and are approximately zero inmost cases. The magnitude of the mean changes does not appear to depend on the interval of thedata. The distributions of the daily and weekly rate changes exhibit strong positive peakedness, asmeasured by kurtosis. Kurtosis and skewness are more severe for both daily and weekly changescompared with monthly changes (see also Baillie and Bollerslev, 1989b). In particular, kurtosis iswithin the range of 12.161 for the ITL series, and �0.588 for the JPY series. The presence ofsignificant skewness and kurtosis suggests that the price changes are not normally distributed. Thiswas confirmed for all but five of the monthly series (p-value < 0.01), when a normality test inDoornik and Hendry (1994) was applied. Some of the non-normality may be due to the presenceof influential observations and outliers. We abstract for other sources of variability (e.g. structuralbreaks) but deal with the problem of outliers at a later stage.

Since the intervals at which the series were generated may affect forecasting performance, wetested whether the univariate series came from similar populations. Both the Kruskal–Wallis andthe van der Waerden statistics rejected the null hypothesis that the series came from populationswith identical distributions (p-value < 0.05; one-tailed test). This result is likely to have importantimplications for the forecasting performance of our models.

Daily seasonal unit roots and spectral estimatesTo formally test for seasonal unit roots in the univariate daily series, we estimated a version of thetest shown in Osborn (1990), thus

5Xt D ˛1S1t C ˛2S2,t C ˛3S3,t C ˛4S4,t C ˛5S5t C ˇ15Xt�1

C ˇ2Xt�5 Cm∑kD1

,k5Xt�k C �t �7�

where Sd,t is a zero/one dummy variable corresponding to day d. The hypothesis here is thatXt is I(1,1) and requires the coefficients of the ˛’s and ˇ’s to be (jointly) zero. The I(1,0)alternative hypothesis requires that ˇ1 D 0 and ˇ2 < 0, while the I(0,1) hypothesis requires thatˇ1 < 0 and ˇ2 D 0. The critical values shown in Osborn (1990) are for d D 1, 2, 3, 4, rather thand D 1, 2, . . . , 5, but they serve as a guide.

In brief, we found no evidence of seasonal unit roots in the daily series. Sarantis and Stewart(1993) were also unable to find seasonal unit roots in quarterly FX rates. To test whether the dailyseries contained both deterministic and stochastic seasonality, a version of Osborn’s (1990) secondtest was applied, thus

Xt D ˛0 C ˛1�S1,t � S5,t�C ˛2�S2,t � S5,t�C ˛3�S3,t � S5,t�

C ˛4�S4,t � S5,t�Cm∑kD1

,kXt�k C �t �8�

Here, k was initially set to 15 and insignificant coefficients were eliminated in order to obtain aparsimonious model. While we found very limited evidence of both deterministic and stochasticseasonality, the explanatory power of the model is very weak �R

2 � 0.013�.



(a) Levels of Monday exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

24/10/83 30/06/86 27/02/89 18/11/91 25/07/94 17/03/97

CHF DEM FRFGBP NLG

(b) Levels of Monday exchange rates

4.204.705.205.706.206.707.207.70

24/10/83 30/06/86 27/02/89 18/11/91 25/07/94 17/03/97

ITLJPY

(a) Levels of Tuesday exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

25/10/83 15/07/86 28/02/89 12/11/1991 26/07/94 1/4/1997

CHF DEM FRFGBP NLG

(b) Levels of Tuesday exchange rates

4.20

4.70

5.20

5.70

6.20

6.70

7.20

7.70

25/10/83 15/07/86 28/02/89 12/11/1991 26/07/94 1/4/1997

ITLJPY

(a) Levels of Wednesday exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

26/10/83 2/7/1986 22/02/89 16/10/91 8/6/1994 5/2/1997

CHF DEM FRFGBP NLG

(b) Levels of Wednesday exchange rates

4.204.70

5.205.70

6.206.70

7.207.70

26/10/83 2/7/1986 22/02/89 16/10/91 8/6/1994 5/2/1997

ITLJPY

(a) Levels of daily exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

24/10/83 27/02/87 20/07/90 22/11/93 14/04/97

LCHF LDEM LFRFLGBP LNLG

(b) Levels of daily exchange rates

4.204.70

5.205.70

6.206.70

7.207.70

24/10/83 27/02/87 20/07/90 22/11/93 14/04/97

LITLLJPY

(a) Levels of Thursday exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

27/10/83 17/10/85 29/10/87 26/10/89 24/10/91 4/11/1993 19/10/95

CHF DEM FRFGBP NLG

(a) Levels of Friday exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

28/10/83 6/12/1985 5/2/1988 30/03/90 15/05/92 15/07/94 13/09/96

CHF DEM FRFGBP NLG

(b) Levels of Thursday exchange rates

4.20

4.70

5.20

5.70

6.20

6.70

7.20

7.70

27/10/83 17/10/85 29/10/87 26/10/89 24/10/91 4/11/1993 19/10/95

ITLJPY

(b) Levels of Friday exchange rates

4.204.705.205.706.206.707.207.70

28/10/83 6/12/1985 5/2/1988 30/03/90 15/05/92 15/07/94 13/09/96

ITLJPY

(a) Levels of monthly exchange rates

0.00

0.50

1.00

1.50

2.00

2.50

31/10/83 30/04/86 31/10/88 30/04/91 29/10/93 30/04/96

CHF DEM FRFGBP NLG

(b) Levels of monthly exchange rates

4.204.705.205.706.206.707.207.70

31/10/83 30/04/86 31/10/88 30/04/91 29/10/93 30/04/96

ITLJPY

Figure 1. Plots of the levels of the foreign exchange rate series (in natural logarithms) over both the estimationand prediction periods. The daily and weekly series span the period 24 October 1983 to 12 May 1997 whilemonthly series span the period 31 October 1983 to 30 May 1997. The series in the left panel (from top tobottom) are: FRF, NLG, DEM and GBP. Similarly, for the right panel, the series are the ITL and FRF


460 N. L. Joseph

The standardized spectral density functions of the daily series were also examined. These areshown in Appendix D. The plots for the level series dominate the spectrum value at zero frequency(Panel A). This behaviour is generally consistent with a non-stationary process that has a unit root.The density functions of the first difference of the series appear stable (Panel B). At zero frequency,the spectra are just above the theoretical value of one. Most of the fluctuations in the fifth differencesof the series occur at zero frequency (Panel C). Thus there is hardly any evidence of seasonalpeaks. Also, the autocorrelation coefficients do not exhibit any seasonal effects.3 Furthermore, theautocorrelation coefficients of the residuals of VARL for the daily system did not provide anyevidence of seasonal variation (see Appendix E). In brief, the evidence for seasonal unit roots isvery weak although it may have been difficult to capture seasonal effects if the factors that causedthem are time-varying. Using the critical values in MacKinnon (1991), all of the series, i.e. daily,weekly and monthly, are stationary in first differences.

Testing for multivariate cointegrationSince the first difference of the univariate series appears stationary, we applied the Johansen FIMLtechnique (Johansen, 1988) to test for cointegration. The test was implemented by testing down froman initial lag structure of 20, 15 and 13 for daily, weekly and monthly series respectively, throughto one. The monthly series include pre- and post-bank holiday dummy variables (see footnote 2).We report the results for the lag structure that generated the largest Johansen test statistics providedthe associated residuals were white noise (p-value ½ 0.10, using the LM test). Figure 2 showsthe plots of the cointegrating vectors (CVs) for the daily system. The first three CVs and CV7appear fairly stationary, but CV4 and CV5 show means that vary with time, while CV6 is clearlynon-stationary. The features exhibited by CV4 to CV6 cast much doubt on the constancy of theparameter estimates of the system and therefore the reliability of the cointegration test. No seasonalpattern appears to be present in the CVs, except perhaps, for CV1.

The adjusted and unadjusted test statistics for the Johansen technique are show in Appendix F.The null hypothesis that r D 0 can be rejected for all systems. It appears that there are up to twoCVs for most of the daily and weekly series and, perhaps, one cointegrating vector for the monthlyseries. The trace statistic confirms that there is one CV for all the systems. Baillie and Bollerslev(1989a) also found one CV in a seven-variable system of monthly spot and forward FX rates.Our finding of cointegration implies that the stationary specification would be mis-specified forour data sets while the non-stationary specification would fail to explicitly account for the long-run information contained in the series. It follows that error-correction specification is the correctspecification for our data sets.

An interesting hypothesis to test is the degree of proportionality between certain pairs of cur-rencies. Such a test implies that the series xt and yt, for example, are cointegrated which in turnrequires their difference �xt � yt� to be stationary. The test was performed for the following pairsof currencies: (1) DEM and FRF; (2) DEM and GBP; (3) DEM and JPY; (4) FRF and GBP;(5) FRF and JPY; and (6) GBP and JPY. These combinations reflected our subjective assessmentof the importance of those currencies in international trade. Given the number of CVs identifiedin Appendix F, we imposed the restriction r D 1 on the parameters of the CVs. A LR test statisticrejected the (joint) null hypothesis that the parameters for the chosen pair of currencies belong to

3 Our experimentation with (only) pre- and post-bank holiday dummy variables in the univariate daily series provided veryweak evidence of deterministic and stochastic seasonality. Further, we could not apply a version of equations (7) and (8)to monthly systems because of the presence of multicollinearity.



Figure 2. Plots of the cointegrating vectors (CV) for daily foreign exchange rates using a lag structure of 11.The plots are for the estimation period

the space spanned by the CVs (p-value D 0.00). Using a procedure in Doornik and Hendry (1994),the null hypothesis was also rejected when a general restriction was imposed. Our failure to acceptthe null hypothesis has important implications for the exact causal link that might exist amongthe currencies and the particular linear combination of the level series that we use to generate theerror-correction terms. Of course, there is evidence of non-stationarity in plots of the CVs and wemay have been too restrictive in specifying the space spanned by the CVs. Finally, at r D 2, the(joint) null hypothesis on the coefficients of the dummy variables of the monthly series was rejected(p-value D 0.03). However, at r D 3, the null hypothesis could not be rejected (p-value D 0.29)


462 N. L. Joseph

and when tested as a single hypothesis, the null hypothesis was rejected for the coefficient of thepost-bank holiday variable (p-value D 0.01). Thus, at least one of the dummy variables appears tobe important in the monthly system. Since the LR test tends to have low power, our forecasts formonthly series contain both dummy variables. In brief, given the (adverse) behaviour of the CVsand the tendency for the LR statistic to reject the null too often, some caution is exercised in theinterpretation of our results.

Preliminary forecast results and forecast uncertaintyAn assessment of the degree of uncertainty that is associated with the forecasts under alternativespecifications and estimation methods is an important exercise in the evaluation of forecasts (seeBianchi et al., 1987). Here, we first evaluate the uncertainty in the forecasts of the alternative VARsfor each set of daily, weekly and monthly data; a discussion about forecasting accuracy will follow.

A comparison of the confidence bands for each specification and estimation method reveals threedistinct patterns that are similar for all intervals of the series:

(1) The confidence bands under the non-stationary specification display the usual monotonic in-crease with h. In most cases, realizations of the level series are above the forecasts. In contrast,the confidence bands for the stationary and error-correction specifications show almost noincrease in forecast uncertainty after (about) h D 5. Beyond this point, the confidence bandsconverge to a constant. The confidence bands of both specifications are more or less similar inmagnitude. They appear excessive given actual realizations of the series. Since the sets of seriesare cointegrated, the residuals of the non-stationary and stationary specifications are likely toexhibit autocorrelation thereby adversely affecting the estimation consistency of the confidencebands. The parameters of the error-correction specification are consistency estimated but thisspecification does not appear to generate superior forecast standard errors.

(2) Under the stationary and error-correction specifications, the forecasts converge to the mean ofthe actual series rather quickly. As expected, the forecasts under the non-stationary specificationdo not converge to the mean of the actual series; a non-stationary series does not have a constantmean. None of the forecasts appear to capture adequately realizations of the series.

(3) The patterns in (1) and (2) are very similar given the estimation method and the interval of theseries. Under the stationary and error-correction specifications, the Bayesian forecasts appearto converge to the mean of the actual series slightly more quickly than those of the OLSestimation method. Both estimation methods appear to contain a similar level of uncertainty inthe forecasts. Interestingly, the confidence bands of the OLS estimates exhibit more variabilityat very short horizons.

The patterns in the plots are illustrated in Figures 3(a) to 3(c). The plots are for the best forecastsfor each set of daily, weekly and monthly series for each specification and estimation method.4 The

4 The best out-of-sample forecasts were identified in terms of the lowest mean rank of the MAPEs and/or U-statisticsfor each set of daily, weekly and monthly series of all currencies and estimation methods. The MAPE for each forecasthorizon is:

1

N

h∑jD1

∣∣AtCjCh � FtCjCh∣∣

where At and Ft are respectively the actual and forecast values; j is the series being forecast for a given forecast horizon,i.e. for h D 1, 2, . . . 13 (48); and N is the number of forecast observations at each horizon. The MAPE is considered to bemore reliable than either the mean square error or root mean square error, and less sensitive to non-normality (see Meese



Figure 3(a). Plots of the actual data and the best daily forecasts with their š2 standard error bands underthe non-stationary (VARL and BVARL), stationary (VARD and BVARD) and error-correction (VECM andBECM) specifications. The 24 observations before the forecast period are also shown

patterns in the remaining plots are generally similar even though the forecasts shown by those plotsare not as good. Indeed, for the worse forecasts, the patterns in both the error bands and forecastsare maintained, but the forecasts occasionally lie outside the confidence bands. We now turn to adiscussion of our measures of forecasting accuracy.

Comparison of estimation methodsUnder the non-stationary specification, the superior performance of the Bayesian estimation methodis pronounced but there is not much to choose between the estimation methods for the other spec-ifications. Indeed, according to the Wilcoxon signed-rank W test, VARL and BVARL generatedU-statistics that are significantly different in 86% of the comparisons (p-value < 0.05; two-tailed

and Rogoff, 1983). The U-statistic is: RMSE (model)/RMSE (random walk) where the RMSE is given by

√√√√√ 1

N

h∑jD1

[AtCjCh � FtCjCh

]2

That is, the ratio of the h-step-ahead RMSE of each equation in the VAR to the corresponding h-step-ahead RMSE ofthe random walk forecast. Here, the one-step-ahead forecast is based on 48 observations (13 for monthly series) while thetwo-steps-ahead forecast is based on 47 observations (12, for monthly series), etc.


464 N. L. Joseph

Figure 3(b). Plots of the actual data and the best weekly forecasts with their š2 standard error bands underthe non-non-stationary (VARL and BVARL), stationary (VARD and BVARD) and error-correction (VECMand BECM) specifications. The last 24 observations of the estimation period are also shown

test). However, BVARL generated smaller U-statistics in 76% of those cases. While the U-statisticsfrom VARD and BVARD are significantly different in 39% of the cases (35% in mean absoluteprediction errors (MAPEs)), BVARD generated smaller U-statistics than VARD in 53% of thosecases (24% in MAPEs). Similarly, when the forecast measures under BECM and VECM are sig-nificantly different, BECM generated smaller U-statistics in 57% of the cases (29% in MAPEs).Furthermore, the performance of both estimation methods appears similar in terms of the best fore-casts. At relatively short horizons, the non-stationary specification generated U-statistics that are



Figure 3(c). Plots of the actual data and the best monthly forecasts with their š2 standard error bands underthe non-non-stationary (VARL and BVARL), stationary (VARD and BVARD) and error-correction (VECMand BECM) specifications. The last 12 observations of the estimation period are also shown

larger than those of the other specifications. Indeed, one-third of the average U-statistics (i.e. themean of U over all h’s) under BVARL is greater than one (p-value � 0.05; one-tailed test); forVARL, two-thirds of the average U-statistics are greater than one. Under the other specifications,the average U-statistic is typically less than one. Here, the finding that the U-statistics appear tooutperform the random walk is not impressive since the forecasts do not capture actual realizationsof the series well.

Forecasting accuracy of alternative specificationsAccording to Davies and Newbold (1980), mis-specified systems are likely to generate largersquare forecast errors. This prediction appears to hold for our worse forecasts. However, the bestforecasts under the non-stationary specification actually outperformed the best forecasts under boththe stationary and error-correction specifications, particularly at longer horizons (see Appendix G).The non-stationary specification clearly generated inferior forecasts at short horizons but the bestforecasts of the other specifications do not improve substantially over the h’s. The non-stationaryspecification generated less volatile forecast measures and the improvement in the forecasts issustained over time. It is noteworthy that the U-statistic and the associated MAPE at a given h do


466 N. L. Joseph

not always have the same ranking. Thus the reliability of our forecast measures across specificationsand data interval is in doubt. Partly to deal with this problem, we utilize both forecast measures(see Winkler, 1992) and the Wilcoxon signed-rank W test to assess forecasting performance. Sincethe standard random walk does not directly capture long-run effects, an evaluation based on theU-statistic should be interpreted with some caution.

Under the stationary specification, the JPY generated the worse forecasts particularly for theWednesday, Thursday and Friday series. However, at h D 9, VARL generated the lowest (ever)U-statistic of 0.1330 (associated MAPE is 0.0057) for the monthly GBP series although a high U-statistic of 1.2856 (associated MAPE D 0.0379), was also obtained at h D 2 for that series. Dailyseries consistently generated the lowest set of U-statistics. Here, VARL and BVARL respectivelygenerated low U-statistics of 0.2264 (associated MAPE D 0.0039) and 0.3276 (associated MAPE D0.0038) at h D 47.

Under the other specifications, the forecast measures at long horizons are dominated by extremevalues. For example, at h D 47, VECM generated its lowest U-statistic of 0.0756 (associatedMAPE D 0.0003) for the GBP Monday series, but at h D 48, VECM also generated its highestU-statistic of 5.7650 (associated MAPE D 0.0626) for the Wednesday ITL series. The forecastmeasures for BECM are also within that range. Similarly, VARD generated its lowest U-statistic of0.1222 (associated MAPE D 0.0005) at h D 48 for the Monday GBP series. Its highest U-statisticis 5.7345 (associated MAPE D 0.0493) at h D 48 for the Thursday ITL series. The tendency for thestationary and error-correction specifications to dominate the non-stationary specification at shorthorizons is consistent with Joutz et al.’s (1995) findings.

Table I shows the best forecast measures. In general, the three specifications have performedbadly. However, the best forecasts of the non-stationary specification dominated those of the otherspecifications, particularly at long horizons while there is hardly any difference between the bestforecasts of the other two specifications. This finding is more consistent with those of Clements andHendry (1995a) than with those of Engle and Yoo (1987). Thus aspects of Clements and Hendry’s(1995a) results appear to hold for large systems.

Market timing and alternative model specificationWe now focus on the extent to which the forecasts provide superior information for investmentdecisions. To do this, we applied Henriksson and Merton’s (1981) technique. Such a test providesa theoretical benchmark against which forecasting performance can be judged and avoids someof the problems of comparison that are inherent in standard accuracy measures. According to theHenriksson and Merton statistic, the predictive power of all the specifications is weak. Under thenon-stationary specification, the estimation methods successfully predicted FX rate increases inno more than 8% of the cases (p-value < 0.05; one-tailed test). The stationary specification faredslightly better (14% and 10% for VARD and BVARD, respectively). BECM exhibited predictiveability in 49% of the cases while the predictive ability of VECM is only 29%. Thus evidence on thepredictive ability of the VARs is not strong. Friday and monthly data accounted for more than half(57%) of the successful directional forecasts under VECM. Daily price increases were particularlydifficult to predict.

Some sub-period resultsTime series models tend to generate statistical results that are sample-specific, thereby resulting inpoor generalizations about forecasting performance (see also, Clements and Hendry, 1995b). Partlyto address this problem, the samples of the full estimation period were split into two sub-periods



Table I. Full period out-of-sample mean absolute prediction errors (MAPEs) and Theil U-statistics for OLSand Bayesian estimation methods

h 1 6 12 18 24 30 36 42 48 Mean

OLS daily forecast measuresNon-stationaryDEM MAPE 0.0043 0.0107 0.0114 0.0104 0.0092 0.0095 0.0073 0.0074 0.0107 0.0090

U 1.0108 0.9892 0.9489 0.8757 0.8206 0.6552 0.4407 0.4709 0.3947 0.7255ITL MAPE 0.0039 0.0081 0.0088 0.0088 0.0084 0.0087 0.0069 0.0073 0.0094 0.0079

U 1.0134 0.9797 0.9074 0.9753 1.0580 0.9181 0.4954 0.3299 0.2974 0.7891

StationaryITL MAPE 0.0039 0.0035 0.0033 0.0032 0.0033 0.0032 0.0026 0.0030 0.0010 0.0031

U 0.7174 0.6551 0.6696 0.6418 0.7759 0.8447 0.5126 0.5120 0.6413 0.5969

Error-correctionITL MAPE 0.0039 0.0035 0.0033 0.0032 0.0033 0.0032 0.0026 0.0030 0.0009 0.0031

U 0.7162 0.6534 0.6691 0.6412 0.7745 0.8409 0.5107 0.5085 0.6007 0.5949

OLS weekly forecast measuresNon-stationary TuesdayGBP MAPE 0.0090 0.0187 0.0246 0.0303 0.0414 0.0394 0.0339 0.0342 0.0539 0.0327

U 0.9662 0.9703 0.8340 0.8079 0.7720 0.6864 0.6223 0.5554 0.6802 0.7568Friday

GBP MAPE 0.0080 0.0182 0.0256 0.0296 0.0382 0.0339 0.0271 0.0277 0.0323 0.0279U 1.0109 0.9239 0.8228 0.7661 0.7420 0.6509 0.5602 0.4991 0.4565 0.7096

Stationary MondayGBP MAPE 0.0076 0.0073 0.0082 0.0091 0.0091 0.0090 0.0088 0.0071 0.0005 0.0080

U 0.6296 0.7437 0.7365 0.7008 0.7130 0.7804 0.6008 0.6987 0.1222 0.7041Friday

GBP MAPE 0.0079 0.0072 0.0081 0.0087 0.0095 0.0088 0.0078 0.0061 0.0015 0.0078U 0.6962 0.8378 0.7967 0.8003 0.8636 0.7837 0.7937 0.5561 0.4969 0.7618

Error-correction MondayGBP MAPE 0.0076 0.0074 0.0083 0.0092 0.0091 0.0089 0.0087 0.0070 0.0003 0.0080

U 0.6348 0.7517 0.7465 0.7089 0.7129 0.7761 0.6002 0.7036 0.0756 0.7052Tuesday

ITL MAPE 0.0073 0.0071 0.0073 0.0077 0.0087 0.0101 0.0104 0.0056 0.0025 0.0078U 0.6741 0.7563 0.7314 0.9024 0.8207 0.8408 0.8207 0.7852 0.7657 0.7897

OLS monthly forecast measuresh 1 3 4 6 7 9 10 12 13 Mean

Non-stationaryFRF MAPE 0.0165 0.0378 0.0440 0.0662 0.0726 0.0937 0.0931 0.0879 0.0757 0.0645

U 0.9146 0.9813 0.8660 0.8885 0.8683 0.8974 0.8312 0.7526 0.6828 0.8633StationaryGBP MAPE 0.0266 0.0216 0.0206 0.0240 0.0221 0.0196 0.0093 0.0084 0.0100 0.0178

U 0.8745 0.7581 0.7247 0.8396 0.8923 1.2511 0.4673 0.5256 0.4091 0.7241

Error-correctionGBP MAPE 0.0165 0.0378 0.0440 0.0662 0.0726 0.0937 0.0931 0.0879 0.0757 0.0645

U 0.9146 0.9813 0.8660 0.8885 0.8683 0.8974 0.8312 0.7526 0.6828 0.8633GBP MAPE 0.0265 0.0217 0.0220 0.0246 0.0223 0.0192 0.0095 0.0080 0.0104 0.0180

U 0.8776 0.7569 0.7330 0.8443 0.8891 1.2145 0.4841 0.5061 0.4249 0.7221

(continued overleaf )


468 N. L. Joseph

Table I. (Continued )

Steps 1 6 12 18 24 30 36 42 48 Mean

Bayesian daily forecast measuresNon-stationaryDEM MAPE 0.0043 0.0109 0.0115 0.0102 0.0090 0.0103 0.0089 0.0094 0.0133 0.0096

U 1.0029 1.0019 0.9490 0.8642 0.8121 0.7119 0.5225 0.5458 0.4890 0.7594ITL MAPE 0.0038 0.0078 0.0091 0.0085 0.0064 0.0073 0.0061 0.0080 0.0149 0.0078

U 1.0141 0.9939 0.9053 0.9300 0.8668 0.7550 0.4698 0.4323 0.4723 0.7582

StationaryITL MAPE 0.0038 0.0035 0.0033 0.0032 0.0033 0.0032 0.0026 0.0030 0.0010 0.0031

U 0.7046 0.6539 0.6679 0.6427 0.7761 0.8447 0.5126 0.5120 0.6414 0.5964

Error-correctionITL MAPE 0.0038 0.0035 0.0033 0.0032 0.0033 0.0032 0.0026 0.0029 0.0009 0.0031

U 0.7034 0.6525 0.6673 0.6420 0.7747 0.8410 0.5108 0.5085 0.6015 0.5944

Bayesian weekly forecast measuresNon-stationary TuesdayGBP MAPE 0.0093 0.0168 0.0239 0.0307 0.0421 0.0399 0.0325 0.0320 0.0507 0.0319

U 0.9983 0.9122 0.8274 0.8056 0.7687 0.6867 0.6005 0.5246 0.6405 0.7401Friday

GBP MAPE 0.0077 0.0175 0.0254 0.0316 0.0422 0.0386 0.0309 0.0321 0.0359 0.0307U 0.9983 0.9229 0.8444 0.8091 0.7966 0.7141 0.6267 0.5680 0.5070 0.7524

Stationary MondayGBP MAPE 0.0087 0.0076 0.0081 0.0091 0.0091 0.0090 0.0088 0.0071 0.0005 0.0081

U 0.7188 0.7693 0.7321 0.7011 0.7131 0.7807 0.6008 0.6989 0.1189 0.7087Friday

GBP MAPE 0.0086 0.0074 0.0081 0.0087 0.0095 0.0088 0.0078 0.0061 0.0015 0.0078U 0.7649 0.8475 0.7923 0.7991 0.8637 0.7841 0.7938 0.5565 0.4962 0.7636

Error-correction MondayGBP MAPE 0.0087 0.0077 0.0082 0.0092 0.0091 0.0089 0.0087 0.0070 0.0003 0.0081

U 0.7223 0.7795 0.7425 0.7097 0.7131 0.7762 0.6001 0.7035 0.0693 0.7100Tuesday

ITL MAPE 0.0074 0.0068 0.0072 0.0077 0.0087 0.0101 0.0104 0.0056 0.0026 0.0077U 0.7081 0.7465 0.7297 0.9014 0.8206 0.8412 0.8216 0.7842 0.7784 0.7900

Bayesian monthly forecast measuresh 1 3 4 6 7 9 10 12 13 Mean

Non-stationaryGBP MAPE 0.0186 0.0341 0.0326 0.0306 0.0326 0.0184 0.0227 0.0231 0.0321 0.0271

U 0.9772 0.9132 0.8410 0.6880 0.6768 0.3766 0.4182 0.3570 0.3802 0.6383StationaryGBP MAPE 0.0210 0.0174 0.0216 0.0228 0.0213 0.0204 0.0089 0.0083 0.0078 0.0166

U 0.7779 0.7037 0.7429 0.8210 0.8797 1.3028 0.4394 0.5187 0.3200 0.7044

Error-correctionCHF MAPE 0.0258 0.0305 0.0273 0.0289 0.0319 0.0309 0.0258 0.0273 0.0363 0.0288

U 0.7869 0.7213 0.6738 0.6292 0.7946 0.9594 1.9295 0.7396 0.4394 0.8167GBP MAPE 0.0214 0.0181 0.0226 0.0240 0.0221 0.0199 0.0093 0.0080 0.0087 0.0171

U 0.7793 0.7104 0.7545 0.8353 0.8773 1.2458 0.4789 0.4957 0.3540 0.7054

Out-of-sample forecast measures for best daily, weekly and monthly forecasts. Mean is the average of the MAPEs andTheil U-statistics over h D 1, 2, ..48 (13)-step ahead daily, weekly (monthly) forecasts. Measures for only certain h’s areshown.



of (approx.) equal lengths and the analyses were replicated for daily and weekly data. The last36 observations of each sub-period were reserved for the out-of-sample forecasts. The Johansentechnique revealed that the sets of series for both sub-periods are cointegrated. We used our earlierapproach to determine the lag structure and error-correction terms for the sub-period VARs. Thesame Bayesian priors were also used.

Our results indicated that the relative performance of the alternative specifications and estimationmethods are similar to those of the full period. Further, the Kruskal–Wallis statistic indicated that theforecast measures are not similar across all (three) periods (p-value < 0.05). All the specificationsand estimation methods performed better in the second sub-period (relative to the first and fullperiods) while the first sub-period generated the worse forecast measures. The forecast measures ofthe error-correction specification did not dominate in any period. This result is not surprising evenif the period up to 1987 was relatively turbulent. Related evidence suggests that a special case ofSETAR models may not have out-performed our linear models since the performance of SETARmodels depends on whether or not the forecasts are made over the peaks or troughs of the sampleperiod (see Tong and Noeanaddin, 1988).

Assessing the potential impacts of outliersWe now assess the impact of potential outliers on our forecasts. This focus is important sincesimulation results suggest that additive outliers can bias the OLS parameters of stationary first-orderautoregressive processes downwards (see Martin and Yohai, 1986). Other econometric applicationsindicate that additive outliers can cause the Johansen test to generate too many cointegrating vectors(see Franses and Haldrup, 1994), but since our primary concern is the potential impact on forecastingaccuracy, we focus on this aspect instead. We first assume that any outlier is of an additive typeand the time point of the disturbance is unknown. Our search for outliers focuses on the estimationperiod of the full sets of daily and weekly series. The identification and elimination of outliers aresimilar in spirit to the iterative methods found in empirical work (see e.g. Abraham and Chuang,1989).

To identify potential outliers we relied on the magnitude of the Studentized residuals which hasthe useful property of reflecting more precisely differences in the true variance of the error term(Belsley et al., 1980). The following procedure was used:

(1) Using the same lag structures that generated the forecasts for the full period, we computed theOLS estimates for VARL equation by equation, and the associated Studentized residuals. Thus,for t D 1, . . . n, we calculated A o�t� which represents the estimated Studentized residual ofthe outlier effect. So let 4t D MaxfjA o�t�jg.

(2) If 4t ½ C, the predetermined critical value of š3.0, an outlier was suspected. The suspiciousoutlier was removed by deleting the observations associated with 4t at the particular point t forthat equation. VARL was then re-estimated for the particular equation to determine whether ornot the desired effect was achieved. Otherwise, the observations (previously eliminated) werereplaced and another equation was estimated. The process was repeated equation by equation.When multiple outliers were detected, the observations associated with the largest 4t wereeliminated first. This consideration was party to reduce the estimation bias due to the influenceof adjacent outliers when applying sequential elimination (see Chen and Liu, 1993). Both theidentification and elimination of the outliers effect were terminated if 4t < C. The extent towhich 4t ½ C is illustrated in Appendix H for the daily series.

When steps (1) and (2) were completed, the VARs were used to generate the out-of-sample forecasts.The error-correction series were generated out of the level of the cleaned series in the usual manner.


470 N. L. Joseph

The same Bayesian priors of our earlier experiments were used and h was set to 1, 2, . . . 48, asbefore.

In general, the plots of the forecasts and their š2 standard error bands are similar to thoseof our earlier experiments. In most cases, the currencies that generated the best forecasts whenoutliers were included also generated that best forecast when outliers were eliminated. Our rankingof the alternative specifications and estimation methods is unaffected. Indeed, the best forecastsof the stationary and error-correction specifications are generally similar but the dominance of theBayesian estimation method over the OLS method is somewhat reduced. For example, the MAPEsand U statistics of both BECM and VECM are statistically different in 50% of the cases which isslightly lower than the percentage obtained when the data contained oultiers.

It is also useful to compare the gain/loss in forecasting accuracy when the series are cleaned ofoutliers and when outliers are retained. When outliers were eliminated, VARL generated the largestimprovement (over the untreated data) in the forecast measures (see Appendix I). Interestingly,the data without outliers generated poorer Bayesian forecast measures than the data with outliers.If outliers are in fact part of the DGP of the series, their elimination may have resulted in someloss of information. Indeed, outliers that are part of the DGP tend to be associated with volatileperiods (see Engle and Hakkio, 1995) and may well reflect agents’ behaviour during those periods.Perhaps, Sarno’s (1997) approach of retaining outliers by attributing weights to them rather thaneliminating them would have been more helpful here. Also, our treatment of multiple outliers mayhave introduced estimation bias, particularly to the detriment of the Bayesian forecasts. Finally,our results for market timing are generally unaffected by the elimination of outliers although thereis some variation in the performance across the specifications. For example, BECM was able tocorrectly predict price increases in 55% of the cases, while the predictive ability of VECM was25%. Predictive ability still does not appear to be strong.

CONCLUSIONS AND IMPLICATIONS

This study was primarily concerned with the forecasting accuracy of FX rates across differentdata intervals when alternative estimation methods and specifications are employed. We found veryweak evidence of seasonal unit roots in the daily data. Since the sub-samples appeared to havebeen drawn from different populations, we chose to generate separate forecasts for daily, weeklyand monthly observations. Not surprisingly, our out-of-sample forecasts depended on the intervalof the data but our main findings provided a consistent view of the extent to which forecastingaccuracy can be impacted by the estimation method and the specification of the model. While weacknowledge that our application of several tests on the same data sets is likely to increase ourchances of selecting a model that ”looks good”, our findings are broadly consistent thereby negatingany serious concern regarding our use of such tests.

Our key results are:

(1) The best forecasts under the non-stationary specification are superior, while the best forecastsunder the stationary and error-correction specifications are generally similar.

(2) The benefit of using Bayesian estimation (over OLS) is more pronounced for the non-stationaryspecification, particularly when the data contain outliers.

(3) None of the specifications captured actual realizations of the series well.



An important implication of those findings is that good model specification does not necessarilyimply superior forecasting performance. Of course, the use of a flat prior on the error-correctionterm may have restricted the performance of BECM. While there was variation in forecastingperformance across intervals and estimation methods, there are other aspects of our findings thatshould be emphasized:

(1) When daily and weekly data sets were cleaned of outliers, the superior performance of theBayesian estimation method was maintained, but there was some gain in the forecasting accu-racy of the OLS estimation method (over the Bayesian method). It is also worth noting thatthe inclusion of addition lagged terms in the OLS VAR can adversely affect its true parame-ter estimates. Thus, the variance of the OLS VAR will increase, thereby further affecting itsforecasting accuracy.

(2) The model specification and estimation methods generated U-statistics, which generally out-performed the random walk for both cleaned and untreated data sets. However, forecast uncer-tainty was still substantial and the forecasts lacked dynamism, thus suggesting that the randomwalk could not be relied upon to infer good forecasting performance. Our tests for markettiming further confirmed that the predictive ability of the models was weak.

Our finding that correctly specified models such as ECMs do not generate forecasts that uniformlydominate those of ‘non-causal models’ has important implications. The finding does not imply thatit is not useful to test for cointegration in forecast experiments since to do so would provideadditional information regarding the DGP as well as ensure estimation consistency. Besides, thefinding of a long-term relationship would allow meaningful interpretations from economic theoryas a steady state equilibrium condition would exist. Since the finding of cointegration only allowsone to infer steady state equilibrium until the system is disturbed, correctly specified systems maynot generate superior out-of-sample forecasts in the presence of (out-of-sample) structural change.This is because the ECM might not be able to anticipate future disequilibrium or structural breakswithin its own parameter estimation period. Perhaps, some form of intercept correction whichof itself implies model mis-specification, could have improved our forecasts. However, Clementsand Hendry (1999, p. 297) show that residual-based intercept corrections improved the short tomedium term forecasts of a VECM but the VAR in differences still dominated in terms of the tracemean-square forecast errors. Furthermore, since the parameters of the non-stationary and stationaryspecifications would have been estimated inconsistently, the forecasting performance of the bothspecifications should be viewed with caution.

The Bayesian estimation method can be useful particularly when forecasting VARs in levels.For the other specifications, the best forecasts under the Bayesian estimation method do not clearlydominate the best forecasts under the OLS estimation method. As we could not always rely onthe MAPEs or U-statistic to rank forecasting performance, an invariant model selection proceduresuch as GFESM would have been more useful. The use of an invariant model selection proceduretogether with adjustments for other known mis-specifications would have facilitated a more reliableinterpretation of our forecast results.

Finally, our finding of cointegration implies at least one causal link between a pair or combinationof currencies. Central banks often intervene in the FX market to bring about changes in monetarypolicy that can lead to co-ordinated attempts to maintain FX rates at some long-run target. Thusour use of seven major currencies increases the chance of finding cointegration. So, the resultthat the series are cointegration is not unexpected. However, we rejected the null hypothesis ofstationary when certain restrictions were imposed on the coefficients of the CVs. Indeed, the plots


472 N. L. Joseph

of the series for the error-correction terms did not always appear stationary. This feature wouldadversely affect the forecasting performance of the ECM and perhaps, cause it to behave more likethe stationary model. We did not analyse smaller systems of FX rates to assess the sensitivity of ourpreferred model to the size of the system. Future research should seek to incorporate those aspectsas well as employ models that seek to accommodate non-linearity within the specifications. Whilewe favour the performance of the non-stationary specification (and subject to the concerns raisedearlier), when compared on the basis of the best forecasts of all specifications, we do not suggestthat the forecasting performance of any specification cannot be improved (see e.g. Amisano andSerati, 1999).

APPENDIX A

Number of observations in estimation and forecast periods for thelevel specification

Interval Estimation period Forecast period T

Daily 3085 48 3133Monday 584 48 632Tuesday 581 48 629Wednesday 641 48 689Thursday 630 48 678Friday 584 48 632Monthly 151 13 164

The estimation periods are: (1) 24 October 1983 to 14 February 1997 fordaily series; (2) 24 October 1983 to April/May 1996 for weekly series; and(3) 31 October 1983 to 30 April 1996 for monthly series. T is the sum ofthe estimation and forecast periods. Daily and weekly data end on or near12 May 1997. Monthly data end on 30 May 1997. The frequencies are forthe non-stationary specification (ignoring lags).



APPENDIX B

Plots of the first difference of daily foreign exchange rate series over the estimation period (see Appendix A)


474 N. L. Joseph

APPENDIX C

Summary statistics for samples of full estimation period

Interval N Mean Variance Skewness Kurtosis

Swiss franc (CHF)Daily 3084 �0.000120 0.000066 �0.067 4.460a

Monday 583 �0.000906 0.000365 �0.086 0.576a

Tuesday 580 �0.000933 0.000315 0.002 0.746a

Wednesday 640 �0.000809 0.000305 �0.216b 1.067a

Thursday 629 �0.000835 0.000303 �0.272a 2.011a

Friday 583 �0.000934 0.000345 �0.105 0.938a

Monthly 150 �0.003634 0.001439 0.123 �0.024

Deutsche mark (DEM)Daily 3084 �0.000141 0.000055 �0.130a 4.090a

Monday 583 �0.000910 0.000306 0.032 0.323Tuesday 580 �0.000941 0.000269 0.049 0.916a

Wednesday 640 �0.000830 0.000266 �0.050 1.717a

Thursday 629 �0.000849 0.000257 �0.137 1.726a

Friday 583 �0.000926 0.000281 �0.037 1.105a

Monthly 150 �0.003607 0.001251 0.253 0.183

French franc (FRF)Daily 3084 �0.000109 0.000053 �0.146a 5.427a

Monday 583 �0.000736 0.000281 0.229b 1.112a

Tuesday 580 �0.000758 0.000244 0.049 0.932a

Wednesday 640 �0.000665 0.000253 �0.063 2.219a

Thursday 629 �0.000681 0.000245 �0.073 2.290a

Friday 583 �0.000750 0.000267 0.211b 2.565a

Monthly 150 �0.002916 0.001136 0.344 0.400

Great British pound (GBP)Daily 3084 �0.000026 0.000051 �0.169a 4.707a

Monday 583 �0.000019 0.000294 0.521a 3.090a

Tuesday 580 �0.000010 0.000279 0.849a 4.601a

Wednesday 640 �0.000015 0.000274 0.216b 3.129a

Thursday 629 �0.000020 0.000266 0.258a 4.848a

Friday 583 �0.000018 0.000258 0.195 3.488a

Monthly 150 �0.000046 0.001281 0.167 1.955a

Italian lira (ITL)Daily 3084 0.000013 0.000050 0.553a 8.158a

Monday 583 �0.000041 0.000285 0.556a 2.473a

Tuesday 580 �0.000010 0.000259 0.849a 4.601a

Wednesday 640 �0.000035 0.000248 1.277a 12.161a

Thursday 629 �0.000039 0.000237 0.987a 9.272a



APPENDIX C (Continued )

Interval N Mean Variance Skewness Kurtosis

Friday 583 �0.000037 0.000266 0.972a 8.027a

Monthly 150 �0.000190 0.001172 0.700a 1.786a

Japanese yen (JPY)Daily 3084 �0.000205 0.000044 �0.447a 5.275a

Monday 583 �0.001371 0.000268 �0.258b 0.832a

Tuesday 580 �0.001317 0.000231 �0.397a 1.721a

Wednesday 640 �0.001214b 0.000224 �0.313a 2.100a

Thursday 629 �0.001239b 0.000211 �0.714a 3.217a

Friday 583 �0.001356b 0.000237 �0.714a 3.617a

Monthly 150 �0.005358 0.001159 �0.156 �0.588

Netherlands guilder (NLG)Daily 3084 �0.000141 0.000053 �0.101b 4.233a

Monday 583 �0.000920 0.000300 0.097 0.465b

Tuesday 580 �0.000949 0.000260 0.066 0.930a

Wednesday 640 �0.000836 0.000262 0.003 1.533a

Thursday 629 �0.000855 0.000253 �0.105 1.866a

Friday 583 �0.000929 0.000273 �0.046 1.117a

Monthly 150 �0.003615 0.001235 0.231 0.190

a,b Statistical significance at the 1% and 5% levels, respectively. The estimation periods are shown inAppendix A.


476 N. L. Joseph

APPENDIX D

Standardized spectral density functions for consecutive weekdays of (daily) exchange rates in levels, first andfifth differences over the estimation period. The Bartlett window size is the usual 2 times the square root ofN. The plots are for four currencies



APPENDIX E

Autocorrelation coefficients of residuals from an OLS VAR (in levels) at 9 lags for dailyexchange rates

Lag CHF DEM FRF GBP ITL JPY NLG

1 �0.0004 �0.0015 �0.0009 �0.0003 �0.0012 0.0000 �0.00102 �0.0016 �0.0035 �0.0027 �0.0014 0.0017 �0.0005 �0.00233 �0.0038 �0.0026 �0.0018 0.0005 �0.0012 �0.0046 �0.00244 0.0027 0.0010 0.0027 0.0007 0.0008 0.0029 0.00235 �0.0019 �0.0020 0.0001 �0.0003 0.0035 �0.0041 �0.00096 0.0006 �0.0009 0.0016 0.0023 0.0051 �0.0019 0.00047 0.0021 0.0002 �0.0005 0.0021 0.0022 �0.0030 0.00028 �0.0047 �0.0043 �0.0047 �0.0032 �0.0004 �0.0027 �0.00299 �0.0244 �0.0089 �0.0199 0.0112 �0.0173 �0.0051 �0.0050

10 0.0050 �0.0162 �0.0242 �0.0113 �0.0062 0.0143 �0.013211 0.0056 0.0119 0.0212 0.0138 0.0122 0.0429 0.012212 0.0170 0.0099 0.0168 0.0128 0.0174 0.0086 0.017213 0.0337 0.0117 0.0195 0.0018 0.0490 0.0312 0.016514 0.0083 0.0050 �0.0063 0.0183 0.0410 �0.0079 0.004915 �0.0046 0.0038 0.0165 0.0101 0.0307 �0.0037 0.001016 0.0142 �0.0109 �0.0205 �0.0099 �0.0042 0.0044 �0.008717 0.0001 0.0122 0.0121 0.0250 �0.0181 �0.0154 0.006218 0.0012 �0.0174 �0.0221 �0.0285 �0.0174 0.0358 �0.014719 0.0126 0.0136 0.0062 0.0128 �0.0037 �0.0150 0.020520 0.0043 0.0143 0.0098 0.0133 0.0198 0.0067 0.008721 0.0098 �0.0037 0.0006 0.0066 �0.0120 �0.0084 �0.005526 �0.0260 �0.0353 �0.0250 �0.0189 0.0006 �0.0115 �0.0273

The autocorrelation coefficients are for the residuals of the VAR (in levels) for daily exchange rates. Theautocorrelation at lag one is for Friday of week three. The coefficients are not significant (p-value ½0.256).

APPENDIX F

Results of Johansen’s maximal eigenvalue (�max) and trace test statistics for cointegration of fullestimation period

Interval r D 0 r � 1 r � 2 r � 3 r � 4 r � 5 r � 6 r � 7 r � 8 Lag

DAILY

�max: (1) 62.88a 43.08b 14.88 10.84 5.754 3.619 1.619 11(2) 61.30a 42.00b 14.51 10.57 5.609 3.528 1.578

Trace: (1) 142.70a 79.79 36.71 21.83 10.99 5.237 1.619(2) 139.10a 77.79 35.79 21.28 10.72 5.106 1.578

(continued overleaf )


478 N. L. Joseph

APPENDIX F (Continued )

Interval r D 0 r � 1 r � 2 r � 3 r � 4 r � 5 r � 6 r � 7 r � 8 Lag

MONDAY

�max: (1) 63.17a 47.01a 17.34 11.53 5.597 2.761 2.079 4(2) 60.12a 44.74a 16.51 10.97 5.327 2.628 1.979

Trace: (1) 149.50a 86.32 39.31 21.97 10.440 4.840 2.079(2) 142.30a 82.15 37.41 20.91 9.934 4.604 1.979

TUESDAY

�max: (1) 72.27a 38.59 19.43 9.963 4.476 2.652 1.708 3(2) 69.65a 37.19 18.72 9.601 4.313 2.555 1.646

Trace: (1) 149.1a 76.82 38.23 18.800 8.836 4.360 1.708(2) 143.7a 74.02 36.84 18.120 8.515 4.201 1.646

WEDNESDAY�max: (1) 57.32a 43.58b 16.63 15.72 6.821 3.477 0.971 7

(2) 52.89a 40.22b 15.34 14.51 6.294 3.208 0.896Trace: (1) 144.50a 87.20 43.62 26.99 11.270 4.448 0.971

(2) 133.30a 80.46 40.25 24.90 10.400 4.105 0.896

THURSDAY�max: (1) 59.63a 47.73a 18.12 14.53 7.122 3.227 0.935 7

(2) 54.94a 43.98a 16.69 13.38 6.562 2.973 0.861Trace: (1) 151.30a 91.66 43.93 25.81 11.280 4.162 0.935

(2) 139.40a 84.45 40.47 23.78 10.400 3.834 0.861

FRIDAY

�max: (1) 62.36a 46.74a 17.48 13.57 6.834 2.891 1.562 9(2) 55.52a 41.62b 15.57 12.08 6.085 2.574 1.391

Trace: (1) 151.40a 89.07 42.33 25.85 11.29 4.453 1.562(2) 134.80a 79.31 37.70 22.13 10.05 3.965 1.391

MONTHLY

�max: (1) 85.21a 50.71 47.85b 30.01 24.78 17.76 9.923 3.960 3.197 5(2) 58.95 35.08 331.10 26.29 17.14 12.29 9.864 2.740 2.211

Trace: (1) 261.40a 196.20a 145.50a 97.63b 59.62 34.84 17.080 7.157 3.197(2) 194.70b 135.70 100.60 67.54 41.24 24.10 11.820 4.951 2.211

a,b Statistical significance at the 1% and 5% levels, respectively. r denotes the number of cointegrating vectors.The initial lag structure was 15 for weekly series and 20 for the daily series. (1) and (2) respectively denote theunadjusted (Johansen, 1988) and adjusted (Reimers, 1992) �max and trace statistics. The critical values wereobtained from Osterwald-Lenum (1992). The estimation periods are shown in Appendix A. The estimates formonthly data contain dummy variables for pre- and post-bank holiday observations.



APPENDIX G

Panel A: Plots of Thcil U-statistics for the best daily forecasts under the non-stationary ( NONSTAT), sta-tionary ( STAT) and error-correction ( ECM) specifications

Panel B: Plots of Thcil U-statistics for the best weekly forecasts under the non-stationary ( NONSTAT),stationary ( STAT) and error-correction ( ECM) specifications


480 N. L. Joseph

Panel C: Plots of Theil U-statistics for the best monthly forecasts under the non-stationary (NON-STAT),stationary ( STAT) and error-correction ( ECM) specifications

APPENDIX H

0

100

200

300

400

500

600

Freq

uenc

y

4.0 2.7 2.0 1.3 0.7 0.0 −0.7 −1.3 −2.0 −2.7 −4.0Standard deviation

ObservedExpected

DEM Studentized residuals

0

100

200

300

400

500

600

Freq

uenc

y


ObservedExpected

ITL Studentized residuals

0

100

200

300

400

500

600

700

Freq

uenc

y


ObservedExpected

GBP Studentized residuals

0

100

200

300

400

500

600

700

Freq

uenc

y


ObservedExpected

JPY Studentized residuals

Histograms showing OLS Studentized residuals for four daily series. The residuals are from a seven-variableOLS VAR (VARL) of daily rates, with a lag structure of 9



APPENDIX I

A comparison of the mean absolute prediction errors (MAPEs) and Theil U-statistics from forecastswith and without outliers

Panel A: OLS estimation method Non-stationary Stationary Error-correction

MAPE U-statistic MAPE U-statistic MAPE U-statistic

Cases where data without outliersgenerated larger forecastmeasures

10 12 15 17 10 10

Cases where data without outliersgenerated smaller forecastmeasures

27 24 10 13 5 8

Total number of cases where themeasures are significantlydifferent

37 36 25 30 15 18

Cases where sets of measures arenot statistically different

5 6 17 12 27 24

Total number of cases 42 42 42 42 42 42

Panel B: Bayesian estimation Non-stationary Stationary Error-correctionmethod

MAPE U-statistic MAPE U-statistic MAPE U-statistic

Cases where data without outliersgenerated larger forecastmeasures

31 30 27 26 22 19

Cases where data without outliersgenerated smaller forecastmeasures

10 9 13 11 8 17

Total number of cases where themeasures are significantlydifferent

41 39 40 37 30 36

Cases where sets of measures arenot statistically different

1 3 2 5 12 8

Total number of cases 42 42 42 42 42 42

A comparison of the frequency with which the data sets with and without outliers generated forecast measuresthat are statistically larger or smaller under each specification. The Wilcoxon signed-rank, W test was usedto compare the sets of measures. A result is statistically significant if the corresponding p-value is less than0.05 (two-tailed test). The comparisons are for daily and weekly data sets only.


482 N. L. Joseph

ACKNOWLEDGEMENTS

The author would like to thank two anonymous referees for their helpful comments and sugges-tions on this work. He is also grateful for the helpful comments which were received from theseminar participants at The Management School, University of Lancaster. The author takes fullresponsibility for any remaining errors.

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484 N. L. Joseph

Author’s biograpy :Dr. N. L. Joseph is a senior lecturer in accounting and finance at the Manchester School of Accounting andFinance, University of Manchester, United Kingdom. His main research interests focus on the forecastingfinancial time series, asset pricing, financial reporting and share price behaviour, and treasury management inmultinational companies. Dr Joseph is also an associate of the Chartered Institute of Management Accountants,United Kingdom.

Author’s address :N. L. Joseph, Manchester School of Accounting and Finance, University of Manchester, Manchester M139LP, UK.


model specification and forecasting foreign exchange rates with vector autoregressions

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