can cointegration-based forecasting outperform univariate models? an application to asian exchange...

Journal of ForecastingJ. Forecast. 21, 355–380 (2002)Published online 26 April 2002 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/for.824

Can Cointegration-based ForecastingOutperform Univariate Models? AnApplication to Asian Exchange Rates

MICHAEL McCRAE,1* YAN-XIA LIN,1 DANIEL PAVLIK2 ANDCHANDRA M. GULATI11 University of Wollongong, Australia2 Reserve Bank, Australia

ABSTRACTConventional wisdom holds that restrictions on low-frequency dynamicsamong cointegrated variables should provide more accurate short- tomedium-term forecasts than univariate techniques that contain no suchinformation; even though, on standard accuracy measures, the informationmay not improve long-term forecasting. But inconclusive empirical evidenceis complicated by confusion about an appropriate accuracy criterion and therole of integration and cointegration in forecasting accuracy. We evaluatethe short- and medium-term forecasting accuracy of univariate Box–Jenkinstype ARIMA techniques that imply only integration against multivariatecointegration models that contain both integration and cointegration fora system of five cointegrated Asian exchange rate time series. We use arolling-window technique to make multiple out of sample forecasts from oneto forty steps ahead. Relative forecasting accuracy for individual exchangerates appears to be sensitive to the behaviour of the exchange rate series andthe forecast horizon length. Over short horizons, ARIMA model forecastsare more accurate for series with moving-average terms of order >1. ECMsperform better over medium-term time horizons for series with no movingaverage terms. The results suggest a need to distinguish between ‘sequential’and ‘synchronous’ forecasting ability in such comparisons. Copyright 2002 John Wiley & Sons, Ltd.

KEY WORDS cointegration; forecasting; multivariate forecasting; exchangerate forecasting; ARIMA forecasting

INTRODUCTION

Univariate linear methods of financial time-series forecasting have a long history1 (e.g. Box andJenkins, 1970; Box and Tiao, 1976; Granger and Newbold, 1977; Engle and Yoo, 1987). But the

* Correspondence to: Michael McCrae, Department of Accounting and Finance, University of Wollongong, Wollongong,NSW, 2522, Australia. E-mail: [email protected] Three progressive stages in linear techniques are from univariate structures, such as the ARIMA models of Box–Jenkins,on to the multiple input, single output cases, such as ARMAX models, then through to VAR and VARMA models.

Copyright 2002 John Wiley & Sons, Ltd.

356 M. McCrae et al.

forecasting accuracy of univariate models in FER spot markets compared to Martingale forecastsremains a disputed issue (see Naidu, 1995; Tong, 1996 for useful reviews). In an early study,Meese and Rogoff (1983) compared various structural and time series exchange rate models. Theyfound that: (i) structural models performed poorly on the basis of their out-of-sample forecastingaccuracy, and (ii) random walk model performs no worse than the estimated autoregressive timeseries models based on out-of-sample forecasting.

Unfortunately, the differencing of non-stationary series required by univariate modelling resultsin a substantial loss of information about long-run trends. As an alternative, Baillie and Bollerslev(1989) applied cointegration to spot exchange rate time series for seven major currencies. Thiscreated interest in the statistical advantages of using cointegration based error correction models(ECM) over Box–Jenkins methods of forecasting. Cointegration avoids information loss due todifferencing while adding information on stationarity restrictions to the low-frequency dynamicscommonly present in systems of non-stationary time series2 (Bhar, 1996; see also Meese andRogoff, 1983; Chinn and Frankel, 1994; Diebold and Mariano, 1995; Holden and Thompson, 1996;Bhawnani and Kadiyala, 1997).

But empirical results on the forecasting dominance of cointegration remain inconclusive. Bail-lie and Bollershev (1994) found that cointegration-based exchange rate forecasts derived fromECM dominate univariate models. However, a re-examination of their data by Diebold et al.(1994) found that a Martingale model dominates ECM forecast performance, thus casting doubton the strength of the underlying cointegration relationship. In reply, Baillie and Bollerslev (1994)defended their interpretation, contending that the cointegrating relationship reflects fractional coin-tegration.

Christofferson and Diebold (1998) further limit the potential for forecasting improvement bydemonstrating that cointegration cannot improve the accuracy of very long-term forecasting forh-step-ahead forecasting where h ! 1. On standard mean squared error criteria, univariate fore-casts are just as accurate as cointegration-based forecasts.3 Contradictory empirical results arisethrough forecasting improvements due to integration rather than cointegration in time series sys-tems. But this qualification is limited to the situation of near-infinity forecasting. For more realisticshort- and medium-term forecast horizons, the restriction of long-term relationships to a stationaryprocess implied by cointegration should still provide extra forecasting information not available inunivariate time series models.

So far, cointegration studies of Asian spot foreign exchange markets tend to focus on marketefficiencies and the potential emergence of trade and economic ‘blocs’ (Rana, 1981; Lee, 1993;Glick and Hutchison, 1994; Aggarwal and Mougoue, 1998; Tse, 1997; Eichengreen and Bayoumi,1992; Janakiramanan, 1997). To our knowledge, there is no specific comparative analysis of theforecasting properties of ECM and Box–Jenkins type univariate models using actual exchange rateseries, and none in the Asian context. In particular, there is no previous comparison of ARIMAmodel forecasting with that of cointegration-based ECMs based on actual foreign exchange ratedata.

This study compares the forecasting accuracy for a system of exchange rates from a univariaterepresentation of integrated component series to that of a multivariate model that incorporates both

2 Short term and long term are, of course, relative terms which are context dependent; see Hakkio and Rush (1991).3 Christofferson and Diebold (1998) also emphasize that the evaluation of accuracy requires appropriate criteria to distinguishbetween forecasting improvements due to: (i) the reduction in deviations between actual and estimated forecasts, and (ii) thepreservation of cointegrating relationships within forecasts.

Copyright 2002 John Wiley & Sons, Ltd. J. Forecast. 21, 355–380 (2002)

Can Cointegration-based Forecasting Outperform Univariate Models? 357

integration and cointegration. Our objective is to compare the short- to medium-term forecastingaccuracy4 of a univariate ARIMA model with a cointegration-based ECM model for a vector offive Asian daily exchange rates time series in an emerging sphere of economic influence (Yenbloc). Univariate and ECM models are constructed from a time series vector of daily exchangesrate series for the Japanese Yen (JY), the Thai Baht (TB), the Singapore Dollar (SD), the MalaysianRinggit (MR) and the Philippine Peso (PP) over the period January 1985 to February 1997. Wethen use the models to provide h-step-ahead forecasts for each series over a one- to 40-day out-of-sample time horizon (h D 1, 2, . . . , T). We then use appropriate criteria to evaluate the forecastaccuracy improvement of ECMs for individual currencies and to comment on factors associatedwith forecasting dominance between models.5

Unlike Engle and Yoo’s (1987) comparison of UVAR with cointegrated forecasting, which usedan Engle–Granger cointegration approach, we estimate the cointegration vectors using the lessrestrictive Johansen (1991) technique which allows multiple cointegrating vectors to enter the ECM.We also use a moving-window forecasting technique to generate multiple samples for each h-step-ahead forecast and an improved metric for comparing forecast errors generated by the alternativemodels.

The next section reviews previous comparisons between the univariate and cointegration basedforecasting in spot exchange rate markets. The third section describes the univariate and ECMmodels under consideration. The fourth section describes the methodology used to compare themodels and data description. Discussion of results and implications for further study conclude theanalysis.

COINTEGRATION-BASED ECM FORECASTING

If the selected Asian exchange rates are related to one another and to the Japanese Yen alonga long-run equilibrium path, then cointegration-based ECM models have the potential to provideinformation not contained in univariate models. Univariate models only analyse short-run move-ments, whereas ECM models incorporate both short-term fluctuations and deviations of individualcurrencies from the cointegrated or long-run equilibrium path in the forecasting model.

Thus, in addition to information on short-run forecasts of individual currency exchange rate series,ECMs contain another dimension of forecasting—the long-run relationship between exchange rateseries. This relationship gives rise to three additional sources of information. First, the currentexchange rate for the forecast currency can be compared with the long-run equilibrium relationshipto establish whether it contains any over/undershooting of that equilibrium. The ECM will thendescribe the path of reversion to equilibrium in terms of time periods and coefficients of othercurrencies. Second, information contained in the time series of other cointegrated currencies nowenters into the reversion path to long-run equilibrium for the forecasted currency as well as itsown past history. Third, when given the forecasted level of a dominant currency, an ECM modelcan provide confidence interval information about the location of other (cointegrated) currency

4 Forecasting stability, robustness, plausibility and flexibility are other forecasting dimensions. We focus on accuracy as ameasure of forecasting efficiency.5 This study uses a relatively long data period in comparison with previous studies. Aggarwal and Mougoue (1998) alsouse a relatively long sample period from October 1983 to February 1992, but many other studies of cointegration betweenforeign exchange rates consider only relatively short time periods (e.g. Baillie and Bollerslev, 1989 March 1980 to January1985).



exchange rates at any point of time. We can also estimate the speed (in terms of time periods) ofcorrection in any over/undershooting of the long-run stationary equilibrium process. The amount ofinformation in these ‘relative position’ forecasts depends, of course, on the stability of the modelparameters over time, the size of the confidence intervals, and the accuracy of forecasts.

PREVIOUS COMPARISONS OF EXCHANGE RATE FORECASTING

As in many financial markets, the relative accuracy of multivariate, univariate and random modelforecasting in foreign exchange spot markets remains a disputed issue. Meese and Rogoff’s (1983)comparison of various structural and time series exchange rate models found that the structuralmodels perform poorly on the basis of their out-of-sample forecasting accuracy. Using a rootmean squared error (RMSE) criterion, they found that the random walk model performs no worsethan the estimated AR models in out-of-sample forecasting.6 Engle and Yoo (1987) compared unre-stricted UVAR model forecasts to forecasts from a restricted cointegrated system estimated using theEngle–Granger two-step (EG) error correction method (Engle and Granger, 1987). They found thatthe UVAR performed better for short horizon forecasts (up to five steps ahead) but the ECM modelperformed better for longer horizon forecasts (greater than five steps ahead) based on the meansquare error criterion (MSE). They concluded that imposing a long-run constraint (cointegration)allows increased long-term forecast accuracy.

Christofferson and Diebold (1998) identify two issues pertinent to conflicting empirical resultson forecasting accuracy of ECMs. They argue that improvements due to integration in a seriesare often attributed as improvements due to cointegration. Second, they show that for very longforecasting horizons, knowledge of cointegration is irrelevant to forecasting accuracy. Thus, Engleand Yoo (1987) mistakenly attribute the outcome of their Monte Carlo simulation to cointegra-tion because the UVAR imposed neither integration (differences) nor cointegration in the sys-tem while the EG two-step procedure imposed both. Christofferson and Diebold (1998) arguethat forecasting performance differences may be due to integration just as much as cointegra-tion. Their comparison indicates that forecasts from a VAR in differences (that is, integratedbut not cointegrated series) relative to cointegration forecasts performed equally well over longhorizons.

While Christofferson and Diebold (1998) demonstrate that knowledge of cointegration cannotimprove extremely long-term forecasting, this restriction is irrelevant to our study of the short- tomedium-term forecast horizons that dominate exchange rate forecasting. We concentrate on one- toten- day ahead forecasts with a maximum horizon of forty days. These horizons are well within thelimitations on cointegration usefulness identified by Christofferson and Diebold (1998), since theiranalysis only applies to extremely long h-step-ahead forecasts as h tends to infinity. Appendix Aderives a condition for determining whether or not cointegration can improve univariate forecastingaccuracy under Christofferson and Diebold’s (1998) restriction.

6 Bhar (1996) compares the performance of an ECM to a naive (no change) model based on out-of-sample forecasts ofbill futures trading on the Sydney Futures Exchange. He found modest forecast accuracy improvement relative to the naivemodel on the basis of the root mean squared error (RMSE) criterion.



THE MODELLING PROCESS

Development of ARIMA and cointegration models requires identification of the order of integrationand the presence/absence of unit roots for each series. Integration order may be defined as:

Definition 1 A series Xt with no deterministic component and which has a stationary, invertibleARMA representation after differencing d times, but which is not stationary after differencing�d � 1� times, is said to be integrated of order d, denoted Xt ¾ I�d�.

Unit roots are a closely related concept to integration. If a series Xt contains one unit root andits differenced series Xt D Xt � Xt�1 has a stationary ARMA representation, then Xt ¾ I�1�. Inorder to identify whether a series contains a unit root, the augmented Dickey-Fuller (ADF) test iscommonly used (Dickey and Fuller, 1981). These procedures are carried out before estimating themodels in this analysis.

Univariate Box–Jenkins type modelsSince 1970, Box and Jenkins progressively developed autoregressive-integrated-moving-average(ARIMA) models and a modelling approach based on diagnostic checking that emphasized amodel’s capacity to capture all serial correlation in the series (Box and Jenkins, 1970, 1994;Harvey, 1994). ARIMA models are widely used in forecasting economic and financial time seriessuch as exchange rates (eg. Meese and Rogoff 1983). Following Box et al. (1994), we brieflydescribe ARIMA models used in this analysis.

The mixed ARMA process Xt of order (p, q), commonly abbreviated as ARMA�p, q�, is describedby:

Xt D � C ��B�

��B�at �1�

where � is a constant, ��B� D �1 � �1B � Ð Ð Ð � �1Bq�, ��B� D �1 � �1B � Ð Ð Ð � �pBp�, at is a ran-dom shock with mean 0 and variance �2

a , and B is the back-shift operator.However, a non-stationary fXtg series integrated of order d must first be differenced d times

before a model can be fitted, denoted as ARIMA �p, d, q�. This is described by the followingmodel:

�1 � B�dXt D µ C ��B�

��B�at �2�

This model will be fitted to each of the five Asian exchange rate time series.

COINTEGRATION-BASED ECM MODELS

Johannson technique for identifying cointegrated series offers improved forecasting ability.7 Con-sider the case of two time series fXtg and fYtg, both of which are integrated of order d, or I�d�.Usually a combination of these two series will also be I�d�. However, a linear combination

Zt D Xt � ˛Yt �3�

7 This has significant implications for the study of efficient markets. Holden and Thompson (1996) state that “testing offorecasting accuracy also provides a test of market efficiency (p. 451)”.



may be integrated of order less than d, say Zt ¾ I�d � b�, b > 0 (Engle and Granger, 1987), where�1, �˛� is known as the cointegrating vector. Assume that the series fXtg and fYtg are both I�1�,so that d D b D 1. If Zt is I�0�, then the series Xt and Yt are cointegrated. The Engle and Granger(1987, p. 253) definition of cointegration is:

Definition 2 The components of the vector Xt are said to be cointegrated of order d, b, denotedXt ¾ CI�d, b�, if (i) all components of Xt are I�d�; and (ii) there exists a vector ˇ� 6D 0� so thatZt D ˇ0Xt ¾ I�d � b�, b > 0. The vector ˇ is called the cointegrating vector, where ˇ0 denotes thetranspose.

Johansen’s (1988) Maximum Likelihood Estimation (MLE) procedure to calculate cointegratingvectors among various times series is generally considered superior to the Engle–Granger two-stepprocedure since it avoids differencing, allows multiple cointegrating vectors in ECM representationsand is remarkably robust to non-normal data other deviations from classical assumptions (Aggarwaland Mougoue, 1998).

We outline the procedure for estimating the cointegrating vectors (for an expanded treatment,see Johansen, 1988; Doornik and Hendry, 1997). When the time series vector Xt 2 Rn are I�1�, theequilibrium-correction form is given by (see Engle and Granger, 1987; Johansen, 1988).

Xt Dm�1∑iD1

YiXt�i C P0Xt�m C vt �4�

By rearranging the coefficients,

Xt Dm�1∑iD1

υi C Xt�i C P0Xt�1 C vt �5�

where vt ¾ IN�0, n� and Xt�i D Xt�i � Xt�i�1�i D 1, 2, . . . , m�. From (5), when Xt is I�1�and Xt is I�0� then the system specification is balanced only if P0Xt�1 is I�0�. Obviously, P0

cannot be full rank since that would contradict the assumption that Xt was I�1�. Thus assumerank�P0� D p < n. Then there exists ˛ and ˇ (n ð p matrices) such that P0 D ˛ˇ0, and ˇ0Xt mustcomprise p cointegrating I�0� relations leading to the restricted I�0� representation

Xt Dm�1∑iD1

υiXt�i C ˛�ˇ0Xt�1� C vt �6�

The number of cointegrating vectors depends upon the rank of P0. The rank is usually estimatedusing Johansen’s (1988) MLE approach.

If the rank p satisfies the condition 0 < p < n, the reduced long-run matrix is given by O O 0, andO 0Xt�1 are the error-correction terms (following from Granger’s Representation Theorem).

However, equations (5) and (6) require an optimal lag choice to make the residuals vt ¾ IN�0, �.The process involves progressively increasing lag length until the residuals are uncorrelated. Thelaborious process may be eased by information criteria such as Akaike’s Information Criterion andSchwartz’s Information Criterion (Diebold et al., 1994).



ECMsCointegration implies a long-term ‘causal’ relationship in a system of series that can be representedby an ECM (Bhar, 1996). Following Granger’s (1981) suggestion of a relationship between cointe-gration and ECMs, Engle and Granger (1987) provided estimation procedures and tests to identifypotential stationary long-run equilibrium vectors. Subsequently, Johansen (1988, 1991) developeda maximum likelihood estimator approach that identified multiple cointegrating vectors within asystem.

The Granger Representation Theorem states that if variables are cointegrated then there is anECM representation of the form

Xt Dm�1∑iD1

υiXt�i C ˛�ˇ0Xt�1� C vt �7�

where vt ¾ IN�0, �. The advantage of using an ECM over, say, a standard vector autoregres-sion (VAR) is that it incorporates both long- and short-run dynamics. The short-run dynamics isrepresented by the Xt�i terms, which also arise in VARs, while the error correction term ˇ0Xt�1

represents the long-run dynamics, which do not arise in VARs (Hatanaka, 1996).Once a preferred method is chosen, the second stage would then be to make actual forecasts

using appropriate re-estimations.

FORECASTING ACCURACY METRICS

Forecast accuracy statistics for each currencyIn this analysis, out-of-sample forecast accuracy is measured by two statistics that are variants ofthose suggested by Meese and Rogoff (1983) and Bhawnani and Kadiyala (1997). These are themean absolute error (MAE) and the root mean square error (RMSE) between actual and forecast foreach h-step-ahead (h D 1, 2, . . . , 40) forecast over the entire forecast horizon. For each h, repeatedsamples are taken by rolling the forecast window forward one time period over the entire 40-dayforecasting horizon. Thus for 2-day-ahead forecasts we obtain 39 samples, while for 10-day-aheadforecasts we obtain 30 samples and so on. While the RMSE is the traditional accuracy measure,Meese and Rogoff (1983) suggest that the MAE is useful when the exchange rate distribution hasfat tails, even when the variance is finite. So we use both methods.

The error metrics are defined as:

MAEt0 �h� D 1

40 � h C 1

40�hC1∑iD1

jAt0Ci�h� � At0CiChj �8�

RMSEt0 �h� D[

1

40 � h C 1

40�hC1∑iD1

jAt0Ci�h� � At0CiChj2]1/2

�9�

where At�Ð� and At denote the forecast value and actual realisations (ex-post) respectively, and hdenotes the number of step-ahead forecasts in the prediction period.

Repeated samples of each h-step-ahead forecast come from rolling the forecast window forwardover the forecast horizon. This procedure in contrast to anchoring at t0 prevents calculation of anyaverage sample error since it allows only one forecast for each h-step-ahead error. The assumptions



made in this procedure are that: (i) the forecast period is similar to the model estimation period, and(ii) the data characteristics remains similar over the forecast period. Where either of these assump-tions changes then model re-estimation is required. However, re-estimation in this study woulddestroy the internal validity of forecast comparisons between models. Re-estimation is appropriateafter choice of the most accurate model.

As well as the absolute MAE we also give a relative or percentage error metric, defined as theratio of the MAEs for each method. This is given by:

MAEt0,ECM�h�

MAEt0,ARIMA�h�and

RMAEt0,ECM�h�

RMAEt0,ARIMA�h�

Accuracy metrics and implied loss functionsChristofferson and Diebold (1998) criticize the blanket use of traditional MSE as a measure offorecasting accuracy since it may not reflect the user’s intended loss function. A loss functionmay refer to either reduced deviation between actual and estimated forecasts (in the MSE sense)or to the strict preservation of the cointegration relationship when making forecasts, or to both.To measure accuracy in terms of these preservation effects, Christofferson and Diebold (1998)derive a second measure based on the MSE of the cointegrating combinations of the forecast errors˛0etCh. This criterion measures the total system error in the preservation of cointegration for eachforecast. They combine this criterion with the traditional trace MSE measure to form a total systemerror metric based on cointegration through triangulation called trace MSEtri. Under this compositemeasure, long-horizon cointegration-based forecasts dominate univariate forecasts since they areequally accurate in the MSE sense, while strictly preserving the cointegration relationship for eachforecast.

Relevance of trace MSEtri to our studyDespite these qualifications, there are several reasons why we retain the traditional MSE forecastaccuracy metrics rather than adopt Christofferson and Diebold’s (1998) alternative trace measures.The measurement of trace MSEtri requires re-estimation of our cointegrating vectors using Phillip’s(1991) triangulation approach. But lack of comparability between the re-estimated and the originalJohansen cointegrating vectors may invalidate accuracy comparison of the measure. Second, ouranalysis concerns the impact of cointegration on forecast errors of individual exchange rate timeseries, whereas Christofferson and Diebold’s (1998) three trace measures focus on total systemforecast error (see Appendix B). Trace MSEtri may indicate that cointegration reduces both typesof system-wide forecast errors for the five Asian exchange rates. But it tells us nothing about relativeaccuracy of ARIMA- and ECM-based forecasts for individual exchange rate series. Finally, ourprimary loss function relates to forecast accuracy in the MSE sense, rather than strict preservationof cointegration in forecasts. We focus on whether cointegration reduces the average deviationsbetween actual and ARIMA-forecasted values; not on whether cointegration-based forecasts do abetter job of preserving the cointegration relationship.

DESCRIPTION OF DATA, MODELS, AND ACCURACY CRITERIA

DataIn practice, exchange rates change almost continuously. As a proxy for these high-frequencychanges, this paper examines the log of daily exchange rates for five Asian currencies: the Japanese



Yen (JY), the Malaysian Ringgit (MR), the Philippines Peso (PP), the Thai Baht (TB) and theSingapore Dollar (SD) expressed against the United States (US) Dollar. The exchange rate seriesare denoted Xt1, Xt2, Xt3, Xt4, and Xt5 respectively.

The data period from 1 January 1985 to 28 February 1997 is considerably longer than manyother recent studies examining currency cointegration (e.g. Aggarwal and Mougoue, 1998; Baillieand Bollerslev, 1994; Diebold et al., 1994). The model is estimated on data from 1 January 1985to 19th February 1997. The last forty days (20 January 1997 to 28 February 1997) provide theout-of-sample forecast horizon.

The Dickey–Fuller test shows all five currencies integrated of order one and an eligible vectorfor cointegration analysis. Time series plots of the five exchange rates are in Appendix C confirmthat the data series required differencing once in order to calculate ARIMA models. The next stageis to estimate the univariate ARIMA models and ECMs for each of the five currencies.

ARIMA modelsOptimal ARIMA models of the form described in equation (2) are identified for the five exchangerate series (see Table I). Diagnostic checks on the residuals using the Ljung–Box (QŁ) statisticto test for model adequacy and autocorrelation indicated that the residuals in each model are notserially correlated (Doornik and Hendry, 1997). The optimal models for the five currencies are:

Japanese Yen: ARIMA�2, 1, 0�Malaysian Ringgit: ARIMA�4, 1, 0�Philippines Peso: ARIMA�1, 1, 2�Thai Baht: ARIMA�0, 1, 5�Singapore Dollar: ARIMA�0, 1, 1�

Cointegration-based ECM modelOur analysis identified two significant cointegrating vectors among the five exchange rate series(Table II).

The presence of cointegration implies a stable, stationary, long-run relationship between the fiveexchange rate series. Hence, information in one series should help in the forecasting of the otherseries (Holden and Thompson, 1996, p. 459). Table III presents the ECM used to provide forecasts.

The first vector is normalized on the Japanese Yen while the second is normalized on theMalaysian Ringgit. The optimal ECM contains two cointegrating vectors and five lagged differencesof the each of the exchange rates.

The next stage fits the ECM. Since all the series in equation (7) are I�0�, the coefficients can beestimated using the ordinary least squares (OLS) method. Again, it is imperative to ensure that theresiduals vt are uncorrelated. The fitted ECM involves two cointegration vectors ˇ1 and ˇ2. Fromthe cointegration model, the optimal ECM is of the form:

Xt D5∑

iD1

υiXt�i C ˛1ˇ01Xt�1 C ˛2ˇ0

2Xt�1 C vt �10�

where X0t D �Xt1, Xt2, Xt3, Xt4, 4t5�.

The optimal model contains two cointegrating vectors and five lagged difference terms, where ˇ1

and ˇ2 are the cointegrating vectors. Appendix D gives the coefficient estimates from the ordinaryleast squares method. This model is used to calculate out-of-sample forecasts for comparison withthe ARIMA forecasts under the two evaluation techniques used in this paper.



Table I. Arima model specifications for five exchange rate series

Japanese Yen: ARIMA�2, 1, 0� with QŁ�12� D 11.01, df D 10

O� D �0.0002469��B� D 1 � 0.034413B � 0.03004913B2

��B� D 1O�a D 0.00650672

Malaysian Ringgit: ARIMA�4, 1, 0� with QŁ�12� D 6.32, df D 8

O� D 0.00001211��B� D 1 C 0.3497B C 0.084365B2 C 0.056193B3 C 0.10502B4

��B� D 1O�a D 0.00419586

Philippines Peso: ARIMA�1, 1, 2� with QŁ�12� D 17.01, df D 9

O� D 0.00010372��B� D 1 � 0.61661B��B� D 1 � 0.9342B C 0.099342B2

O�a D 0.0100565

Thai Baht: ARIMA�0, 1, 5� with QŁ�12� D 14.52, df D 7

O� D �0.0000145��B� D 1��B� D 1 � 0.2528B � 0.19677B2 � 0.087658B3 � 0.16385B4 � 0.079657B5

O�a D 0.00667232

Singapore Dollar: ARIMA�0, 1, 1� with QŁ�12� D 19.39, df D 11

O� D �0.000142��B� D 1��B� D 1 � 0.39889BO�a D 0.00420345

Table II. Johansen–Juselius tests for cointegration between Japanese and ASEANcurrencies

Critical Values

Trace max

H0 : rank D p Trace max 90% 95% 90% 95%

p D 0 244.8ŁŁ 183.6ŁŁ 7.52 9.24 7.52 9.24p � 1 61.2ŁŁ 31.07Ł 17.85 19.96 13.75 15.67p � 2 30.13 19.77 32.00 34.91 19.77 22.00p � 3 10.36 6.575 49.65 53.12 25.56 28.14p � 4 3.79 3.79 71.86 76.07 31.66 34.40

ŁŁ and Ł indices significance at the 1% and 5% levels respectively. Critical values are obtained fromOsterwald-Lenum (1992).Eigenvalues are: (0.0570, 0.0099, 0.0063, 0.0021, 0.0012).



Table III. Analysis of cointegrating vectors

JY MR PP TB SD Constant

CV1 1.000 0.140 �0.202 �10.13 �0.627 28.77CV2 �0.108 1.000 �0.475 0.153 �0.381 0.789

The values in the table are the coefficients for each currency that indicate the stationary cointe-grating equation. Thus:CV1 D 28.77 C 1JY C 0.14MR � 0.202PP � 10.13TB � 0.627SDCV2 D 0.789 � 0.108JY C 1MR � 0.475PP C 0.153TB � 0.381SD

The Ljung-Box (QŁ-statistic) test and the Lagrange-multiplier test are used to test for residualserial autocorrelation (Doornik and Hendry, 1997). Both statistics are insignificant for the fiveequations considered.

Out-of-sample forecasts for each of the five exchange rate series are made from the ECMs asfollows:

OXt D5∑

iD1

Oυi OXt�i C O 1ˇ01

OXt�1 C O 2ˇ02

OXt�1 �11�

RESULTS

Exchange rate forecast accuracy for each currencyThe relative accuracy of ECM- and ARIMA-based forecasts is sensitive to length of forecast step.We discuss short-term horizons (up to five days) and medium-term horizons (six to 40 days).For short-term forecasts of less than five days ahead, the size and direction of relative forecastaccuracy for ECM- and ARIMA-based forecasts varies across currencies. Tables IV and V givethe absolute RMSE and MAE values associated with h-step-ahead forecasting for the ECM andARIMA models. They also show the ratio of errors between the ECM/ARIMA for both criteriafor each of the five currencies. The actual realizations and average forecast errors for each modelare given in Figures 1 to 5. For short-term horizons (up to 5 days), the RMSE (Table IV) showsARIMA based forecasts are more accurate for four of the five currencies (JY, MR, PP and TB);the exception being the SD. The results are similar on the MAE criteria (Table V), except that theshort-term accuracy of the MR also dominates ARIMA forecasts. However, on the RMSE criteria,the point of transition in relative accuracy from ARIMA- to ECM-based forecasts varies betweencurrencies For JY, MR and PP the transition points are 25 days, 35 days and 25 days, respectively.For the SD, ECM forecast errors are smaller than their respective ARIMA models at all horizons.The reverse situation applies to the TB where ARIMA-based forecasts are more accurate at allhorizons.

For medium-term forecast horizons from 6 to 40 days there is qualified support for the propositionthat ECMs dominate ARIMA forecasts except for the TB. In general, the ECM forecast errorsincrease at a slower rate than for the ARIMA models as the forecast step increases (Figures 1 to5). This is expected, since the ARIMA model incorporates only short-run components (representedby the differenced variables in the model) while the ECM also incorporates long-run adjustmentprocess, in the error correction terms (cointegrating vectors). Again, the long-run error behaviourfor the PP and the TB is interesting. For the Peso, as the forecast horizon lengthens, the ECM error



Table IV. Root mean square error statistics for ARIMA model and ECM-based h-step ahead daily forecastsfor five Asian currencies (20 January 1997 to 28 February 1997)

Days JY MR PP

ECM Arima Ratio ECM ARIMA Ratio ECM ARIMA Ratio(%) (%) (%)

1 0.00591 0.004752 1.24369 0.002509 0.001787 1.40403 0.002351 0.00129 1.822482 0.008795 0.0071 1.23873 0.002959 0.002361 1.25328 0.003629 0.001943 1.867733 0.01077 0.008955 1.20268 0.003812 0.00298 1.27919 0.004317 0.002481 1.740024 0.01268 0.0108 1.17407 0.004193 0.003073 1.36446 0.004845 0.002844 1.703595 0.01527 0.01374 1.11135 0.004349 0.003029 1.43579 0.005111 0.003114 1.6413

10 0.02758 0.02554 1.07987 0.004351 0.003458 1.25824 0.006264 0.0039 1.6061515 0.03948 0.03709 1.06444 0.003973 0.003353 1.18491 0.006826 0.00467 1.4616720 0.04886 0.04627 1.05598 0.004202 0.003465 1.2127 0.006693 0.005837 1.1466530 0.05714 0.06159 0.92775 0.004571 0.003883 1.17718 0.005239 0.01005 0.5212940 0.03497 0.04756 0.73528 0.01267 0.01657 0.76463 0.003214 0.08777 0.03662

Days TB SD

ECM Arima Ratio ECM ARIMA Ratio(%) (%)

1 0.002781 0.001121 2.48082 0.002059 0.002098 0.981412 0.004914 0.001821 2.69852 0.002615 0.003656 0.715263 0.00648 0.002225 2.91236 0.003134 0.005549 0.564794 0.007776 0.002549 3.05061 0.003561 0.007356 0.484095 0.008732 0.003087 2.82864 0.004064 0.009134 0.44493

10 0.01316 0.005471 2.40541 0.006969 0.0188 0.3706915 0.01642 0.007408 2.21652 0.009166 0.02794 0.3280620 0.01862 0.009388 1.98338 0.01104 0.03723 0.2965430 0.02056 0.01285 1.6 0.0152 0.05638 0.269640 0.01751 0.0109 1.60642 0.02027 0.07431 0.27278

Notes: The body of the table gives the root mean squared error between the out-of-sample forecast from the ECM andARIMA models respectively and the actual exchange rate. The value of the ratio of the two errors is also given. Errors andratios are calculated for each of the five currencies expressed against the US$. Repeated samples are taken over a 40-dayforecast period going from a one-day-ahead (forty samples) through to 40 days ahead (one sample). The error value is thesampling average for each h-day-ahead forecast.Ratios > 1 (<1) indicate that the ECM average forecast error is > �<� the ARIMA forecast error.

converges significantly while the ARIMA error increases exponentially (Figure 3). The ECM errorfor the TB is also greater than the ARIMA error in the very short run. But it does not show theusual behaviour of becoming less than ARIMA error in the medium term (Figure 4).

The results are also sensitive to the form of the univariate representation for a series. One reasonfor the relatively poor performance of the ECM for the Peso and Baht could be the absence ofmoving-average terms (Figures 3 and 4). The univariate models for both currencies contain moving-average terms of an order greater than one. The ECM excludes these terms since it is estimated ona vector autoregression VARMA systems are not used for cointegration due estimation complexityand associated software limitations.

For practical, short-term forecasting purposes up to five days, no one model dominates the other.However, the comparison of forecast accuracy is only one of many diagnostics that should beinvestigated when comparing various models (Diebold and Mariano, 1995). Further, the superiority



Table V. Mean absolute error statistics for ECM and ARIMA model-based h-step-ahead forecasts for fiveAsian Currencies (20 January 1997 to 28 February 1997)

Days JY MR PP

ECM Arima Ratio ECM Arima Ratio ECM Arima Ratio(%) (%) (%)

1 0.004768 0.004752 1.00337 0.001725 0.001787 0.9653 0.002165 0.00129 1.678292 0.007136 0.0071 1.00507 0.002262 0.002361 0.95807 0.003398 0.001943 1.748843 0.008916 0.008955 0.99564 0.00286 0.00298 0.95973 0.00404 0.002481 1.628384 0.01074 0.0108 0.99444 0.002984 0.003073 0.97104 0.004533 0.002844 1.593885 0.0136 0.01374 0.98981 0.002899 0.003029 0.95708 0.004789 0.003114 1.53789

10 0.02474 0.02554 0.96868 0.003177 0.003458 0.91874 0.00601 0.0039 1.5410315 0.03542 0.03709 0.95497 0.002965 0.003353 0.88428 0.006607 0.00467 1.4147820 0.04342 0.04627 0.93841 0.003242 0.003465 0.93564 0.006481 0.005837 1.1103330 0.0554 0.06159 0.8995 0.00316 0.003883 0.8138 0.005146 0.01005 0.5120440 0.03497 0.04756 0.73528 0.01267 0.01657 0.76463 0.003214 0.08777 0.03662

Days TB SD

ECM Arima Ratio ECM Arima Ratio(%) (%)

1 0.00249 0.001121 2.22123 0.001526 0.002098 0.727362 0.004532 0.001821 2.48874 0.001927 0.003656 0.527083 0.006043 0.002225 2.71596 0.002429 0.005549 0.437744 0.00732 0.002549 2.87171 0.002718 0.007356 0.369495 0.008315 0.003087 2.69355 0.003169 0.009134 0.34695

10 0.01278 0.005471 2.33595 0.006041 0.0188 0.3213315 0.01605 0.007408 2.16658 0.008198 0.02794 0.2934120 0.0183 0.009388 1.9493 0.0103 0.03723 0.2766630 0.02035 0.01285 1.58366 0.01507 0.05638 0.2672940 0.01751 0.0109 1.60642 0.02027 0.07431 0.27278

Notes: The body of the table gives the mean absolute (unsigned) error between the out-of-sample forecast from the ECMand ARIMA models respectively and the actual exchange rate. The value of the ratio of the two errors is also given. Errorsand ratios are calculated for each of the five currencies expressed against the US$. Repeated samples are taken over a40-days forecast period going from a one-day-ahead (forty Samples) through to 40 days ahead (one sample). The errorvalue is the sampling average for each h-day-ahead forecast.Ratios > 1 (<1) indicate that the ECM average forecast error is > �<� the ARIMA forecast error.

of one model over another does not necessarily imply that forecasts from the inferior model do notcontain any additional information.

Total system forecasting accuracyWhile our analysis compares the accuracy of short- and medium-term forecasts for daily exchangerates of each currency, we also emulated Christofferson and Diebold’s (1998) two measures oftotal system accuracy—trace MSE and trace MSE for cointegrating combinations of exchangerates (TMSECCE) (Tables VI and VII). The former criterion measures forecast accuracy for thetotal system in the traditional sense of error between forecasts and actual values. The latter refersto system accuracy in the sense of exactly preserving cointegrating relationships in forecasts.

On the first trace MSE accuracy criterion, the total system accuracy of ECM-based forecasts dom-inates ARIMA forecasts. Up to day 15, the trace MSEs of ECM-based h-step-ahead forecasts are64–70% of ARIMA-based forecasting errors. Progressive lengthening of the forecast horizons up



4.65

4.7

4.75

4.8

4.85

4.9

4.95

5

1 4 7 10 13 16 19 22 25 28 31 34 37 40

days ahead forecasts

actual

ecm

univar

mae

Figure 1. Japanese Yen: forecasts from 1 to 40 days ahead and ECM/ARIMA MAE bounds

0.875

0.885

0.895

0.905

0.915

0.925

0.935

1 4 7 10 13 16 19 22 25 28 31 34 37 40


ecm

univar

actual

mae

Figure 2. Malaysian Ringgit: forecasts from 1 to 40 days ahead and ECM/ARIMA MAE bounds

to 40 days brings substantial improvement in the relative accuracy of ECM forecasts. Furthermore,it appears that up to 10-day-ahead forecasts the ECM forecasts do not preserve the cointegrationrelationships within the system as well as the ARIMA forecasts; although this situation reversesdramatically for 11- to 40-day horizons. These system error results are not inconsistent with ourindividual currency results, although they do emphasize that the associated loss function is inap-propriate for our investigation into forecast accuracy for exchanges rates of individual currencies(Christofferson and Diebold, 1998). The apparent reversal of short-term results may reflect thesmall absolute size of the forecast errors and the relative distribution of MSEs across currencies.The dominance of ARIMA-over ECM-based forecast accuracy for the TB at all horizons is muchlarger than any of the reverse differences for other currencies.

In particular, the reversal of results between Tables IV and V and Tables VI and VII emphasizesthat total system forecast error gives little insight into the size or distribution of forecast errors



3.15

3.2

3.25

3.3

3.35

mae

univar

ecm

actual

1 4 7 10 13 16 19 22 25 28 31 34 37 40


Figure 3. Philippine Peso: forecasts from 1 to 40 days ahead and ECM/ARIMA MAE bounds

3.2

3.22

3.24

3.26

3.28

3.3

3.32

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39


mae

actual

ecm

arima

Figure 4. Thai Baht: forecasts from 1 to 40 days ahead and ECM/ARIMA MAE bounds

for individual currency exchange rates. The use of trace MSEs as substantive measures is notappropriate for our loss function. We are concerned only with accuracy in terms of minimizationof error differences between forecast and actual exchange rate values for individual currencies andnot with total system error or with strict preservation of cointegration in relationships.

DISCUSSION

Neither the ARIMA nor the ECM models perform particularly well in terms of forecasting accuracy.While most absolute forecast errors are small, there is insufficient information in either forecastingmodel to translate into financial significant information for profitable trading or lower hedging costs.The exception is the PP and long-run ECM forecasts.



univar

ecm

actual

1 4 7

0.5

0.45

0.4

0.35

0.3

0.25

0.210 13 16 19 22 25 28 31 34 37 40


mae

Figure 5. Singapore Dollar: forecasts from 1 to 40 days ahead and ECM/ARIMA MAE bounds

Table VI. Trace MSE of ECM and ARIMA model-basedh-step-ahead daily exchange rate forecasts for a system of fiveAsian Currencies for 1 to 40 days ahead (20 January 1997 to28 February 1997)

Days ECM ARIMA Ratio

1 0.0000587 0.0000879 0.6679526722 0.000130263 0.000199865 0.6517562653 0.000200973 0.000313966 0.6401101684 0.000274985 0.000427486 0.6432592395 0.000370973 0.000559717 0.662786126 0.000470561 0.000702855 0.6694996537 0.000597245 0.000873631 0.6836349518 0.000743093 0.001068608 0.6953836799 0.000896039 0.001280479 0.699768584

10 0.001040578 0.001487515 0.69954114815 0.001974681 0.002906498 0.67940221920 0.002918339 0.004622905 0.63127803225 0.003706528 0.006467167 0.57313014230 0.003967074 0.008133659 0.48773550535 0.003216564 0.009109748 0.35309037940 0.002111233 0.015880878 0.13294181

Notes: Columns two and three reflect Christofferson and Diebold’s(1998) Accuracy Measure 1: Trace MSE. They give the values for thetrace of the MSE for each h-step-ahead ECM and ARIMA forecastrespectively, where h D 1 to 40 days ahead. The term ‘trace’ indicatesthat each value in these columns is the sum of the diagonal elementsof the

∑h matrix (the forecasting error covariance matrix-h-step ahead

1 to 40 days) where K D I.



Table VII. Trace MSE for ECM and ARIMAh-step-ahead daily forecasts for cointegationcombinations of daily exchange rates for five Asiancurrencies (20 January 1997 to 28 February 1997)

Days ECM ARIMA Ratio

1 0.000734 0.000228 3.2193942 0.002256 0.000526 4.2857143 0.003851 0.001064 3.6193614 0.005450 0.001750 3.1142865 0.006756 0.002791 2.4206386 0.007927 0.004114 1.9268357 0.009280 0.005787 1.6035948 0.010770 0.007747 1.3902169 0.012270 0.010040 1.222112

10 0.013750 0.013130 1.0472215 0.019390 0.033560 0.57777120 0.022910 0.065740 0.34849425 0.025300 0.106800 0.23689130 0.026530 0.149400 0.17757735 0.022700 0.179200 0.12667440 0.025340 0.251000 0.100956

Notes: Columns two and three reflect Christofferson andDiebold’s (1998) Accuracy Measure 2: Trace MSE basedon cointegrating combinations of variables which is definedby Trace�K

∑h�, where K D ˛˛0; ˛0 is a r ð N matrix

given by cointegration vectors;∑

h is the forecasting errorcovariance matrix (h-step-ahead up to 40).The first column gives the trace MSE based on Cointe-grating Combinations of Variables for the relevant ECMforecasts.The second column gives the trace MSE based on CCVsfor the ARIMA forecasts.The third column is the ratio of the two trace MSEs.

There are two currencies where one model does predominate—ARIMA forecasts for the TBand ECM forecasts for SD. For the TB, error size is still too large in both the short and the longrun to give informationally significant forecasts. The only possible exception is the other case ofthe SD where the short-term ECM error up to 9 days is quite small and may provide significantimprovement over ARIMA forecasting.

For four of the five Asian exchange rates in this study, the presence of cointegration does notsignificantly improve the accuracy of ARIMA-based forecasts for short-term horizons, although itdoes improve forecasting accuracy from around the 30-day horizon onwards. Within this generalconclusion, some patterns are evident. The relative accuracy of ARIMA and ECM exchange rateforecasts for the five Asian currencies depend on two factors (i) the form of the ARIMA modelwhich fitted individual exchange rate series, and (ii) the length of forecast horizon. With the JY andMR, the ARIMA model is marginally best up to nine steps ahead, after which the ECM is better.The SD gave much less forecast error under the ECM while the TB is considerably better. Thus,ARIMA models perform relatively well for all short-term forecasts. For our data, the inclusion ofinformation about the restrictions which the stationarity of cointegrated vectors places on long-term



behaviour does not seem to improve short-term forecasting accuracy, with the notable exception ofthe SD.

However, for time series that can be fitted by an autoregression or ARIMA model containinglower-order moving-average terms, the ECM will also perform well for reasonably short periods.But the ECM seems to perform worse than the ARIMA models when the time series containedmoving-average terms of order greater than one, as in the case of the Philippines Peso and the ThaiBaht. In this circumstance, the ARIMA models for these two currencies exhibit smaller errors.

We also find that as the forecast horizon lengthens, the RMSE and MAE increases at a slower ratefor the ECM than ARIMA, except for the TB and SD. This slower increase reflects the long-termadjustment process included in ECMs but absent in ARIMA models. Thus, from a mathematicalviewpoint, it appears as though the ECM would outperform the ARIMA models in the longer termfor four out of the five currencies.

The advantages of ECMs in the long run would probably be negated by the more dynamicapproach of periodic re-estimation of the models for a moving ‘window’ of data. Re-estimationincreases current relevance through reducing the chance of mis-specification and updating coeffi-cients. Of course, it may also reduce test power and increase confidence bounds through reductionin sample points—a perennial conflict. There is much scope for further research in this area. Thecointegrating vectors could be examined for their degree of time invariance, that is, their stabilityover time. The practical implications of cointegrating relationships and their usefulness in predictingfuture values, of exchange rates also need investigation. As already mentioned, ECMs offer poten-tial in this regard since they provide three dimensions of forecasting information: (i) short-termpath prediction for individual series; (ii) relative prediction (confidence intervals) as to the likelyexchange rate for a sub-ordinate currency series at any particular future time given the rate of themajor currency; and (iii) the long-run ‘response path’ of a currency as it moves towards long-runequilibrium again when the major currency is given an initial ‘shock’.

Why don’t ECM forecasts always dominate univariate forecasts? After all, being able to specifya long-run stationary relationship among a set of exchange rates appears to have high forecastinginformation content.

There may be several classes of explanation. First, if exchange rates are substantially non-deterministic in the sense that the occurrence of future events that ‘shock’ exchange rates israndom, then ECMs cannot contain any extra information about future events and so do notimprove exchange rate forecasting ability at any time prior before events occur. However, afterthe random ‘shock’ has occurred, ECMs do contain information about the potential reaction of aspecific exchange rate as it readjusts towards a stationary equilibrium situation with other curren-cies. So in the strict forecasting sense of looking forward from a fixed point in time (t0) ECMsmay contain no extra forecasting information. But if forecasting is taken relative to the occurrenceof events, then ECMs can provide information for forecasting the reactions of exchange rates tomajor systemic ‘shocks’.

Second, the nature of the dynamics of readjustment to long-run equilibrium may contain littleextra predictive information for a particular exchange rate. Thus, the reversion process may beextremely long (over several time periods), so that reversion is swamped by other shocks andreadjustments, or the readjustment may occur through exogenous variables. For instance, althoughthere is good evidence of cointegration between price levels and exchange rates in PurchasingPower Parity (PPP) systems, the adjustment is often exogenous through price levels rather thanthrough exchange rates.



Third, the forecasting information potential in ECMs is mostly relevant to what we call ‘syn-chronous’ forecasting rather than sequential forecasting. That is, once the exchange rate levelof a dominant currency is known, ECMs may provide extremely strong information relative toforecasting the synchronous level of other, cointegrated exchange rates.

This sequential, synchronous forecasting dichotomy typifies exchange rate hedging situations,where the inherent lack of predictability and strong volatility of future spot rates contrasts with therelative stability of basis risk—the relationship between spot and future prices at future points intime. Even when cointegration-based ECM forecasts provide little extra information about futurespot risk, the models may still provide valuable information about forecasting issues connected with‘basis risk’. This occurs through their ability to identify stationary long-run equilibrium situationsand to provide information on the dynamics of the adjustment processes in a manner not previouslypossible under univariate models. We are currently investigating these issues.

APPENDIX A THE IRRELEVANCE OF CHRISTOFFERSON AND DIEBOLD’S (1998)RESTRICTION ON COINTEGRATION USEFULNESS FOR SHORT- AND MEDIUM-TERM

FORECAST HORIZONS

Christofferson and Diebold (1998) consider a N ð 1 vector process

�1 � B�Xt D � C C�B�εt

where � is a constant drift term, C�B� is an N ð N matrix lag operator polynomial of possibleinfinite order, and εt is a vector of i.i.d innovations.

Assume that the system is cointegrated and that the univariate representation for the Nth com-ponent of Xt is given by

�1 � B�xn,t D1∑

jD0

�n,jun,t�j

where �n,0 D 1 and un,t is white noise.Let OetCh and QetCh be the h-step-ahead forecast error for system and univariate forecasts respec-

tively. Christofferson and Diebold (1998) show that

QetCh D OetCh C ��t C h� Ct∑

iD1

tCh�1∑jD0

Cjεi � ��h C Xt C ��B�ut��

Therefore

Var �QetCh� D Var �OetCh� C Var

t∑

iD1

tCh�1∑jD0

Cjεi � ��h C Xt C ��B�ut�

� Cov

OetCh,

t∑iD1

tCh�1∑jD0

Cjεi � ��h C Xt C ��B�ut�



Christofferson and Diebold (1998) focus on long-horizon forecasting where h ! 1. Now, when his large enough such that:

t∑iD1

tCh�1∑jD0

Cjεi ³ C�1�t∑

iD1

εi

then,

QetCh ³ OetCh � CŁ�B�εt C ��B�ut

where C�B� D C�1� C �B � 1�CŁ�B�. By noting that εt’s are serially uncorrelated that the ut’sdepend only on current and past εt’s, we have that:

Var �QetCh� D Var e�OetCh� C O�1�

From the above equation, Christofferson and Diebold (1998) concluded that in terms of traceMSE criteria, univariate forecasts were as accurate as system forecasts over long-forecast hori-zons. Since the conclusion holds only as h goes to infinity, the restriction is likely to have littlepractical implication since such long-horizon forecasting is rare in practice. However, for short- tomedium-term forecasting, Christofferson and Diebold (1998) do provide an analytical condition fordetermining the maximum forecast horizon over which cointegration information is useful and therate of decay of that usefulness. The ability of cointegration to improve forecasts will depend onwhether or not the follow expression is positive:

�Cov�OetCh,t∑

iD1

tCh�1∑jD0

Cjεi � ��h C Xt C ��B�ut�� > 0

APPENDIX B HOW CHRISTOFFERSEN AND DIEBOLD’S ALTERNATIVE MEASURESAPPLY TO SYSTEM-WIDE ERRORS

Three accuracy measures are discussed by Christoffersen and Diebold: (i) trace MSE, (ii) traceMSE for cointegrating combinations of variables, and (iii) MSEtri. In general, these measurementsare defined by

E�e0tChKetCh�

where K is a matrix related to different measurements.

The three measures are defined in terms of the above representation as follows:(i) Trace MSE is defined when is K an identity matrix.

(ii) Trace MSE for cointegrating combinations of variables applies where the matrix K is definedby the system’s cointegrating vectors matrix ˛ (i.e. K D ˛˛0).

(iii) When the system of interest is a cointegrated system with r linear independent cointegratingvectors and there exists a triangular representation for the system cointegrating vector (ther ð N cointegrating vectors matrix ˛0 has form ˛0 D �I � 0�), then, the trace MSE from thetriangular presentation is given by K with the following form

K D(

Ir 0� 1 � B

)(Ir �00 1 � B

)



The above definitions imply that the measurements are defined for whole system and the lastmeasurement is conditionally held.

APPENDIX C TIME SERIES PLOTS OF DAILY EXCHANGE RATES FOR FIVE ASIANCURRENCIES (US$) (3 OCTOBER 1983 TO 28 FEBRUARY 1997)

Log Japanese Yen

4.2

3.373.353.333.313.293.273.253.233.213.193.173.15

3.353.4

3.33.253.2

3.153.1

3.052.952.9

2.852.8

2.752.7

0.90.850.8

0.750.7

0.650.6

0.550.5

0.450.4

0.350.3

0.250.2

1/1/

8523

/07/

8511

/2/8

62/

9/86

24/0

3/87

13/1

0/87

3/5/

8822

/11/

8813

/06/

892/

1/90

24/0

7/90

12/2

/91

3/9/

9124

/03/

9213

/10/

924/

5/93

23/1

1/93

14/0

6/94

3/1/

9525

/07/

9513

/02/

963/

9/96

1/1/

851/

8/85

3/3/

861/

10/8

61/

5/87

1/12

/87

30/0

6/88

30/0

1/89

30/0

8/89

30/0

3/90

30/1

0/90

30/0

5/91

30/1

2/91

29/0

7/92

26/0

2/93

28/0

9/93

28/0

4/94

28/1

1/94

28/0

6/95

26/0

1/96

27/0

8/96

1/1/

8529

/07/

8521

/02/

8618

/09/

8615

/04/

8710

/11/

876/

6/88

30/1

2/88

27/0

7/89

21/0

2/90

18/0

9/90

15/0

4/91

8/11

/91

4/6/

9230

/12/

9227

/07/

9321

/02/

9416

/09/

9413

/04/

958/

11/9

54/

6/96

30/1

2/96

1/1/

856/

8/85

11/3

/86

14/1

0/86

19/0

5/87

22/1

2/87

26/0

7/88

28/0

2/89

3/10

/89

8/5/

9011

/12/

9016

/07/

9118

/02/

9222

/09/

9227

/04/

9330

/11/

935/

7/94

7/2/

9512

/9/9

516

/04/

9619

/11/

96

1/1/

856/

8/85

11/3

/86

14/1

0/86

19/0

5/87

22/1

2/87

24/0

7/88

28/0

2/89

3/10

/89

8/5/

9011

/12/

9016

/07/

9118

/02/

9222

/09/

9227

/04/

9330

/11/

935/

7/94

7/2/

9512

/9/9

516

/04/

9619

/11/

96

4.4

4.6

4.8

5

5.2

5.4

5.6

Log Value Malaysian Ringitt

0.850.870.890.910.930.950.970.991.011.031.05

Log Value Philipine Peso Log Value Thai Baht

Log Value Singapore Dollar

Notes: The time period illustrated is 3 October 1983 to 28 February 1997. The time period used to estimate the models in the analysis is 1 January 1985 to 31 December 1996. This is a total of 3131 observations. The remaining 40 days represent the out-of-sample forecast period.

.



APPENDIX D ECM COEFFICIENT ESTIMATES FOR SYSTEM OF FIVE ASIANEXCHANGE RATES (5-LAG)

URF Equation 1 for dJYVariable Coefficient Std.Error t-value t-prob

dJY 1 0.025243 0.020087 1.257 0.2090dJY 2 0.026719 0.020228 1.321 0.1866dJY 3 �0.0054133 0.020264 �0.267 0.7894dJY 4 0.0033984 0.020176 0.168 0.8663dJY 5 �0.0040644 0.019856 �0.205 0.8378dMR 1 0.037952 0.039134 0.970 0.3322dMR 2 0.12032 0.039943 3.012 0.0026dMR 3 �0.0075931 0.039948 �0.190 0.8493dMR 4 �0.0016780 0.039917 �0.042 0.9665dMR 5 �0.021102 0.038936 �0.542 0.5879dPP 1 �0.010510 0.013050 �0.805 0.4207dPP 2 �0.016075 0.013612 �1.181 0.2377dPP 3 0.0016537 0.013733 0.120 0.9042dPP 4 �0.0095006 0.013574 �0.700 0.4840dPP 5 �0.0065518 0.013011 �0.504 0.6146dTB 1 0.0065769 0.022368 0.294 0.7688dTB 2 0.017699 0.021946 0.806 0.4200dTB 3 0.020389 0.021846 0.933 0.3507dTB 4 �0.0090019 0.021179 �0.425 0.6708dTB 5 �0.013671 0.020388 �0.671 0.5026dSD 1 0.021400 0.040265 0.531 0.5951dSD 2 �0.065355 0.042155 �1.550 0.1212dSD 3 0.016957 0.042366 0.400 0.6890dSD 4 0.0061698 0.042323 0.146 0.8841dSD 5 0.042795 0.040486 1.057 0.2906CIa 1 0.0018198 0.0013648 1.333 0.1825CIb 1 0.0075300 0.0032187 2.339 0.0194

nsigma D 0.00650861 RSS D 0.1312375672

URF Equation 2 for dMRVariable Coefficient Std.Error t-value t-prob

dJY 1 0.064105 0.012670 5.060 0.0000dJY 2 0.033430 0.012759 2.620 0.0088dJY 3 �0.016406 0.012782 �1.284 0.1994dJY 4 �0.010384 0.012726 �0.816 0.4146dJY 5 �0.0034665 0.012525 �0.277 0.7820dMR 1 �0.23218 0.024684 �9.406 0.0000dMR 2 �0.0010215 0.025195 �0.041 0.9677dMR 3 �0.064720 0.025198 �2.569 0.0103dMR 4 �0.074240 0.025178 �2.949 0.0032dMR 5 �0.021323 0.024560 �0.868 0.3853dPP 1 �0.034247 0.0082316 �4.160 0.0000dPP 2 �0.024034 0.0085857 �2.799 0.0052dPP 3 0.0019234 0.0086621 0.222 0.8243



(Continued )

URF Equation 2 for dMRVariable Coefficient Std.Error t-value t-prob

dPP 4 �0.014986 0.0085620 �1.750 0.0802dPP 5 6.0841e–005 0.0082066 0.007 0.9941dTB 1 0.028211 0.014109 2.000 0.0456dTB 2 0.018621 0.013843 1.345 0.1787dTB 3 0.0087908 0.013780 0.638 0.5235dTB 4 0.017209 0.013359 1.288 0.1978dTB 5 0.015580 0.012860 1.212 0.2258dSD 1 �0.21244 0.025398 �8.364 0.0000dSD 2 �0.17088 0.026590 �6.426 0.0000dSD 3 �0.019107 0.026723 �0.715 0.4747dSD 4 �0.037724 0.026696 �1.413 0.1577dSD 5 0.0060027 0.025537 0.235 0.8142CIa 1 0.0028630 0.00086088 3.326 0.0009CIb 1 �0.0059539 0.0020302 �2.933 0.0034

nsigma D 0.0041054 RSS D 0.05221460125

URF Equation 3 for dPPVariable Coefficient Std.Error t-value t-prob

dJY 1 �0.087037 0.030589 �2.845 0.0045dJY 2 0.011135 0.030805 0.361 0.7178dJY 3 �0.00021707 0.030859 �0.007 0.9944dJY 4 �0.011894 0.030726 �0.387 0.6987dJY 5 �0.027125 0.030238 �0.897 0.3698dMR 1 0.011751 0.059596 0.197 0.8437dMR 2 0.15239 0.060828 2.505 0.0123dMR 3 �0.032995 0.060835 �0.542 0.5876dMR 4 �0.029627 0.060789 �0.487 0.6260dMR 5 �0.077239 0.059295 �1.303 0.1928dPP 1 �0.30717 0.019874 �15.456 0.0000dPP 2 �0.18438 0.020729 �8.895 0.0000dPP 3 �0.093986 0.020913 �4.494 0.0000dPP 4 �0.083366 0.020671 �4.033 0.0001dPP 5 �0.035057 0.019813 �1.769 0.0769dTB 1 0.083558 0.034063 2.453 0.0142dTB 2 0.0031588 0.033422 0.095 0.9247dTB 3 �0.058070 0.033268 �1.746 0.0810dTB 4 �0.047392 0.032252 �1.469 0.1418dTB 5 �0.086501 0.031049 �2.786 0.0054dSD 1 �0.039674 0.061319 �0.647 0.5177dSD 2 �0.17353 0.064197 �2.703 0.0069dSD 3 0.0078659 0.064518 0.122 0.9030dSD 4 �0.027596 0.064452 �0.428 0.6686dSD 5 0.062042 0.061655 1.006 0.3144CIa 1 0.013853 0.0020784 6.665 0.0000CIb 1 0.0080530 0.0049016 1.643 0.1005

nsigma D 0.00991176 RSS D 0.3043564861

(continued overleaf )



(Continued )

URF Equation 4 for dTBVariable Coefficient Std.Error t-value t-prob

dJY 1 0.030005 0.020341 1.475 0.1403dJY 2 0.068322 0.020485 3.335 0.0009dJY 3 �0.011583 0.020521 �0.564 0.5725dJY 4 0.0066734 0.020432 0.327 0.7440dJY 5 0.014735 0.020108 0.733 0.4637dMR 1 0.11210 0.039630 2.829 0.0047dMR 2 0.15338 0.040449 3.792 0.0002dMR 3 �0.015391 0.040454 �0.380 0.7036dMR 4 �0.080156 0.040423 �1.983 0.0475dMR 5 0.026073 0.039430 0.661 0.5085dPP 1 0.0067674 0.013216 0.512 0.6086dPP 2 �0.0069122 0.013784 �0.501 0.6161dPP 3 �0.022727 0.013907 �1.634 0.1023dPP 4 0.020791 0.013746 1.512 0.1305dPP 5 �0.0040566 0.013175 �0.308 0.7582dTB 1 �0.12156 0.022651 �5.367 0.0000dTB 2 �0.11671 0.022225 �5.252 0.0000dTB 3 �0.023390 0.022123 �1.057 0.2905dTB 4 �0.12564 0.021447 �5.858 0.0000dTB 5 �0.071621 0.020647 �3.469 0.0005dSD 1 �0.025200 0.040776 �0.618 0.5366dSD 2 �0.072862 0.042689 �1.707 0.0880dSD 3 0.050153 0.042903 1.169 0.2425dSD 4 0.036027 0.042859 0.841 0.4006dSD 5 0.0013550 0.040999 0.033 0.9736CIa 1 0.016731 0.0013821 12.105 0.0000CIb 1 �0.00082172 0.0032595 �0.252 0.8010

nsigma D 0.00659109 RSS D 0.1345846741

URF Equation 5 for dSDVariable Coefficient Std.Error t-value t-prob

dJY 1 0.064207 0.012803 5.015 0.0000dJY 2 0.047808 0.012893 3.708 0.0002dJY 3 0.0074682 0.012916 0.578 0.5632dJY 4 0.0012195 0.012860 0.095 0.9245dJY 5 �0.0016763 0.012656 �0.132 0.8946dMR 1 �0.068811 0.024943 �2.759 0.0058dMR 2 0.024009 0.025459 0.943 0.3457dMR 3 0.0058229 0.025462 0.229 0.8191dMR 4 �0.063479 0.025442 �2.495 0.0126dMR 5 �0.044913 0.024817 �1.810 0.0704dPP 1 �0.029103 0.0083180 �3.499 0.0005dPP 2 �0.023865 0.0086758 �2.751 0.0060dPP 3 0.0011431 0.0087530 0.131 0.8961dPP 4 �0.018040 0.0086518 �2.085 0.0371dPP 5 �0.010212 0.0082927 �1.231 0.2183dTB 1 0.041732 0.014257 2.927 0.0034dTB 2 0.024453 0.013988 1.748 0.0805



(Continued )

URF Equation 5 for dSDVariable Coefficient Std.Error t-value t-prob

dTB 3 0.021603 0.013924 1.551 0.1209dTB 4 0.018699 0.013499 1.385 0.1661dTB 5 0.022573 0.012995 1.737 0.0825dSD 1 �0.39181 0.025664 �15.267 0.0000dSD 2 �0.21217 0.026869 �7.897 0.0000dSD 3 �0.12102 0.027003 �4.482 0.0000dSD 4 �0.045414 0.026976 �1.684 0.0924dSD 5 0.029491 0.025805 1.143 0.2532CIa 1 0.0032864 0.00086991 3.778 0.0002CIb 1 �0.0052714 0.0020515 �2.569 0.0102

nsigma D 0.00414846 RSS D 0.05331581374

REFERENCES

Aggarwal R, Mougoue M. 1998. Common stochastic trends among Asian currencies: cointegration betweenJapan, ASEANS, and the Asian tigers. Review of Quantitative Finance and Accounting 10: 193–206.

Baillie RT, Bollerslev T. 1989. Common stochastic trends in a system of exchange rates. Journal of Finance44: 167–181.

Baillie RT, Bollerslev T. 1994. Cointegration, fractional cointegration, and exchange rate dynamics. Journalof Finance 49: 737–745.

Bhar R. 1996. Cointegration in interest rate futures trading on the Sydney future exchange. Applied FinancialEconomics 6: 251–257.

Bhawnani V, Kadiyala KR. 1997. Forecasting foreign exchange Rates in developing economies. AppliedEconomics 29: 51–62.

Box GEP, Jenkins GM. 1970. Time Series Analysis, Forecasting and Control. Holden Day: San Francisco,CA.

Box GEP, Tiao GC. 1976. Comparison of forecast and actuality. Applied Statistics 25: 195–200.Chinn M, Frankel J. 1994. Patterns in exchange rate forecasts for twenty-five currencies. Journal of Money,

Credit, and Banking 26: 759–767.Christofferson PF, Diebold FX. 1998. Cointegration and long-horizon forecasting. Journal of Business and

Economic Statistics 16: 450–458.Dickey DA, Fuller WA. 1981. Likelihood ratio statistics for autoregressive time series with a unit root.

Econometrica 60: 1057–1072.Diebold FX, Mariano RS. 1995. Comparing predictive accuracy. Journal of Business and Economic Statistics

13: 253–263.Diebold FX, Gardeazabal J, Yilmaz K. 1994. On cointegration and exchange rate dynamics. Journal of Finance

49: 727–735.Doornik JA, Hendry DF. 1997. Modelling Dynamic Systems Using PcFml 9.0 for Windows. International

Thomson Press: London.Eichengreen B, Bayoumi T. 1992. Shocking aspects of European Monetary Unification. NBER Working Paper

Number 3949, Cambridge, MA.Engle FE, Yoo BS. 1987. Forecasting and testing in cointegrated systems. Journal of Econometrics 35:

143–159.Engle RF, Granger CWJ. 1987. Co-Integration and error correction: representation, estimation and testing.

Econometrica 58: 251–276.Glick R, Hutchinson M. 1994. Exchange Rate Policy and Interdependence: Perspectives from the Pacific Rim.

New York, Cambridge University Press: New York.Granger CWJ. 1981. Some properties of time series data and their use in econometric model specification.

Journal of Econometrics 16: 121–130.



Granger CWJ, Newbold P. 1977. Forecasting Economic Time Series. Academic Press: New York.Hakkio C, Rush M. 1991. Cointegration: how short is the long run? Journal of International Money and

Finance 10: 571–81.Harvey A. 1994. Time Series (Vol. 1). Elgar Publishing: Aldershot.Hatanaka M. 1996. Time-series-based Econometrics: Unit roots and cointegrations. Oxford University Press:

Oxford.Holden K, Thompson JL. 1996. Forecasting interest rates from financial futures markets. Applied Financial

Economics 6: 449–461.Janakiramanan S. 1997. An empirical examination of linkages between Pacific-Basin stock markets. Working

paper, Dept. of Accounting and Finance, University of Melbourne.Johansen S. 1988. Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control

12: 231–154.Johansen S. 1991. Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive

models. Econometrica 59: 1551–1581.Lee J. 1993. News and volatility of the foreign exchange market. In Advances in Pacific Basin Financial

Markets, Bos T, Fetherston T (eds). JAI Press: Greenwich, CT.Meese RA, Rogoff K. 1983. Empirical exchange rate models of the seventies. Journal of International

Economics 14: 3–24.Naidu P. 1995. Modern Spectrum Analysis of Time Series. CRC Press: London.Osterwald-Lenum M. 1992. A note with quantiles of the asymptotic distribution of the maximum likelihood

cointegration rank test statistics. Oxford Bulletin of Economics and Statistics 54: 461–471.Phillips PCB. 1991. Optimal inference in cointegrated systems. Econometrica 59: 283–306.Rana P. 1981. Exchange rate risk under generalised floating exchange rates: Eight Asian currencies. Journal

of International Economics 11: 459–66.Tong H. 1996. Non-linear Time Series: A Dynamical System Approach. Clarendon Press: Oxford.Tse Y. 1997. The cointegration of Asian currencies revisited. Japan and the World Economy 9: 109–114.

Authors’ biographies:Michael McCrae (PhD, Australian National University, 1986) is Professor of Finance, Department of Account-ing and Finance at Wollongong University. A Fellow of the Australian Banking and Finance Institute, hepublishes in the area of econometric analysis of international markets and statistical methods in investmentperformance measurement.

Chandra Gulati (PhD, Carnegie-Mellon) is senior lecturer in the School of Mathematics and Applied Statis-tics, Wollongong University. His research interests are in the area of time series and decision theory.

Yan-Xia Lin (PhD, Australian National University) is senior lecturer in the School of Mathematics andApplied Statistics, Wollongong University. Her research and publication interests include stochastic processanalysis, inference and application, time series estimation methods and cointegration analysis.

Daniel Pavlik is senior research officer with the Reserve Bank of Australia.

Authors’ addresses:Michael McCrae, Department of Accounting and Finance, University of Wollongong, Wollongong, NSW,2522, Australia.

Chandra Gulati and Yan-Xia Lin, School of Mathematics and Applied Statistics, University of Wollongong,Wollongong, NSW, 2522, Australia.

Daniel Pavlik, Senior Analyst (Portfolio), International Division, Reserve Bank of Australia.


can cointegration-based forecasting outperform univariate models? an application to asian exchange...

Documents