Cointegration, error-correction models, and forecasting using realigned foreign exchange rates

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<ul><li><p>Journal of Forecasting, Vol. 14,499-522 (1 995) </p><p>Cointegration, Error-correction Models, and Forecasting Using Realigned Foreign Exchange Rates </p><p>NATHAN LAEL JOSEPH Manchester University </p><p>ABSTRACT </p><p>This study employs error-correction models (ECMs) to forecast foreign exchange (FX) rates where the data-sampling procedures are consistent with the rules governing the settlement (delivery) of FX contracts in the FX market. The procedure involves matching (aligning) the forward rate to the actual realized (future) spot rate at the settlement (delivery) date. This approach facilitates the generation of five different sets of subsamples of FX rate series for each currency. For comparative purposes, non-aligned month-end rates are also examined. The results indicate that the moments of the realized forecast errors for the same currency are not similar. Further, the ECMs derived are unstable, and their forecasting performance vary. The forecasting performance of the ECMs appear to be affected by the choice of the interval in which the sets of subsamples are observed. These results are attributed to the observed seasonal variation in FX rates. </p><p>KEY WORDS unbiasedness hypothesis; unit roots; cointegration; error- correction models; forecasting </p><p>INTRODUCTION </p><p>There is a large body of research on the informational efficiency of the foreign exchange (FX) market. Most empirical work has focused on the unbiased forward rate hypothesis (UFRH), also often called the unbiusedness hypothesis, and has either provided support for it or provided mixed results (see Hansen and Hodrick, 1980; Levich; 1979; Longworth; 1981). Where the hypothesis has been rejected, the authors generally conclude that the forward rate differs from the realized spot rate by a time-varying risk premium. More recent empirical studies (see Hakkio and Rush, 1989; Baillie and Bollerslev, 1989a; Barnhart and Szakmary, 1991; Leachman and El Shazly, 1992) have employed tests for a unit root initially to determine the order of integration, and then cointegration tests to examine the informational efficiency of the FX market. These studies indicate that the logarithms of FX rates contain a unit root (see also Meese and Singleton, 1982), and where considered, the realized spot and forward rates as well as the current spot and forward rates for the same currency are cointegrated (see Barnhart and Szakmary, 1991) with a cointegrating vector of one. It should be noted that cointegration is a CCC 0277-6693195lO60499-24 Received May 1994 0 1995 by John Wiley &amp; Sons, Ltd. Revised June 1995 </p></li><li><p>500 Journal of Forecasting Vol. 14, Iss. No. 6 </p><p>necessary but not sufficient condition for accepting the unbiasedness hypothesis. However, using error-correction equations, Hakkio and Rush (1989) were further able to reject the joint hypothesis of no risk premium and the efficient use of information by market participants. Recently, Copeland (1993) employed realigned data to examine the forward market efficiency for day of the week and month of the year observations. Copeland (1993, p. 80) asserts that . . . the basic market efficiency hypothesis meets with very differing degrees of acceptability, depending on ... the currency and interval at which the FX rates are observed. Similar results have also been briefly reported for realigned day of the week forward and realized spot rates (see Joseph and Hewins, 1992, p. 70). Tests for the cointegration of FX rates across currencies have also provided mixed results (see, for example, MacDonald and Taylor, 1989; Alexander and Johnson, 1992) presumably due to differences in methodologies and/or other factors. In this context, the test is based on the notion that prices from two efficient markets for different assets cannot be cointegrated (see Granger, 1986). </p><p>The empirical evidence that the realized spot and forward rates as well as the current spot and forward rates for the same currency are cointegrated implies that error-correction models (ECMs) can be employed to model the dynamic structure of FX rates. The connection between ECMs and cointegration was first suggested by Granger (1981). If linear combinations of 1(1) variables happen to be I(0) the variables are said to be cointegrated. If the variables are cointegrated, there exists an error-correction representation and implies that movement away from equilibrium in one period may be (proportionately) corrected in the next (see Engle and Granger, 1987). </p><p>In this paper we employ ECMs to model the dynamic relationships among the current spot, realized spot and forward rates of five currencies where the current spot and forward rates-at the start of the forward contracts-are observed on specific days of the week, and the forward rates are matched to their respective maturity dates in the next month. This approach is intended to eliminate measurement errors and capture seasonal variation in the data. A more detailed justification is provided in the next section. ECMs are also employed to assess the forecasting accuracy of the sets of subsamples. For comparative purposes, non-aligned month-end rates are also examined. Engle and Yo0 (1987) have shown that when the ECM is correctly specified it provides improved long-term forecasting performance over (correctly specified) unrestricted vector autoregression (VAR). Further, Granger (1986, p. 226) asserts that ECMs should produce better short-run forecasts and will certainly produce long-run forecasts that hold together in economically meaningful ways. </p><p>This study contributes to current empirical work in the following ways. We demonstrate that the moments of the realigned samples for a given currency are dissimilar. This is consistent with the empirical results on seasonal variation in FX returns. We also compare our results with non-aligned month-end data and further indicate that significant autoconelation appears persistent at certain intervals, although not systematically. An interesting result which follows from our approach is that the error-correction term in the final ECMs varies across currencies and sets of subsamples. The out-of-sample forecasting accuracy of the ECMs also appears to be affected by the choice of the interval in which the FX rates are observed. </p><p>The remaining sections of this paper are as follows. The next section provides a justification for the methodology adopted in this study. The data-sampling procedure is also described. The discussion emphasizes the need to apply data-sampling procedures which are consistent with the settlement (delivery) of FX contracts and the need to account for seasonal variation in the data. </p><p>This author became aware of Copelands study after an earlier draft of this paper was submitted for refereeing. The author would like to thank an anonymous referee of this journal for also bringing the study to his attention. </p></li><li><p>Nathan Lael Joseph Cointegration , Error-correction models 50 1 </p><p>The third section provides a brief theoretical discussion of the ECM we employ to forecast the FX rates. The results on unit roots and cointegration are briefly described in the fourth section and the fifth section presents the results on the out-of-sample forecasting performance of the ECMs. A summary of the results and their implications is provided in the final section. </p><p>REALIGNMENT, SEASONAL VARIATION, AND THE DATA SET </p><p>Many empirical studies which use FX data fail to employ data-sampling procedures which are consistent with the rules governing the delivery of FX contracts (see, for example, Longworth, 1981; Baillie and Bollerslev, 1989a). A few exceptions are the empirical studies of Levine (1989), and Copeland (1991, 1993). Where the rules are observed, such that the forward rate is matched (realigned) with the appropriate realized spot rate, researchers for the most part tend to concentrate on month-end rates, though sometimes also weekly rates, and ignore possible seasonal effects. For example, although Bekaert and Hodrick (1993) found that the point estimates of realigned (correctly sampled) Friday data were more negative compared to the non-aligned data, they ignored the possible effects of seasonal variation across other days of the week. However, Copeland (1991) employed realigned daily forward and spot rates but did not find significant day-of-the-week effects in the data. </p><p>The relative importance of realignment can be seen as follows. Assume that today, a one- month forward contract for the purchase (sale) of the United States dollar (British pound) is executed in London. To determine the correct rate in the future the one-month forward rate is predicting, todays spot value date (which is the next two business days in London and New York) is first found and then extended to the same date of the next month, if a business day in both trading centres. If such a date is not a business day, the next available date is chosen without moving into another new month. Otherwise one moves back to the last eligible value date of the specific month. The relevant spot value date would be two days earlier (Riehl and Rodriguez, 1977). If the spot value date is the last business day of the current month, the eligible forward value date is the last business day of the next month. These rules imply that although FX contracts involving the United States dollar and the British pound can be executed in London and/or New York when either one of these financial centres is open, the spot and forward contracts can only be settled on similar business days when both centres are open. The use of non-aligned (mismatched) data is therefore likely to introduce measurement errors into the analysis (Cornell, 1989) although the impact may not be significant (see Bekaert and Hodrick, 1993). Further, unless the forward rate is matched with the appropriate realized spot rates, the true return on the forward contract is not being measured. </p><p>Seasonal variation may also have a significant impact on the empirical results. Ignoring seasonal variation may therefore severely distort the dynamic structure of the time series models (see Osborn et al., 1988) for several reasons. First, the empirical studies of McFarland et al., (1982) and Joseph and Hewins (1992) indicate that FX rates exhibit strong seasonal variation in daily and monthly returns. Seasonal variation in this context is defined as the systematic inter- day or inter-month movement in prices (returns). Further, the empirical distribution of FX returns appears to vary at certain intervals. Indeed, several studies have indicated that FX rate changes may be described by the non-normal stable paretian distribution, the Student t or the mixture of normals distributions (see for example, So, 1987; McFarland et al., 1982; Boothe and Glassman, 1987) depending on the interval the price changes are observed. These findings suggest that the results of tests which employ FX rates may be unreliable where the density functions of the series under consideration are dissimilar and/or when the data exhibit strong </p></li><li><p>502 Journal of Forecasting Vol. 14, Iss. No. 6 </p><p>seasonal variation. These factors therefore have important implications not only in terms of their likely impacts on forecast results and tests of market efficiency but also for the choice of the methodological approach to modelling the FX series. </p><p>Secondly, realignment is intended to replicate the eligible value dates as practised in the FX markets and to facilitate the generation of several different sets of subsamples for each currency. Further, recent analyses of FX rates have employed a variety of unit root and cointegration tests whose power vary across empirical circumstances. The econometric specification, lag length, sample size, period of study (see Barnhart and Szakmary, 1991; Sephton and Larsen, 1991; Harris, 1992), and the choice of the interval in which the time series are observed, may affect empirical results. Time-dependent heteroscedasticity (see Baillie and Bollerslev, 1989b) may also have an impact. Thus the availability of several sets of subsamples for each currency facilitates extensive and consistent testing, and circumvents some of the ad hoc approaches of earlier studies. </p><p>Finally, some empirical studies have shown that the FX daily bid-ask spread exhibits seasonal variation (Joseph and Hewins, 1992). Indeed, Allen (1977) indicates that a risk-averse trader who intends to reverse a position would widen the spread, that is, lower buying price and raise the selling price (or the reverse) in order to maximize profit. The bid-ask spread may also be affected by the level of liquidity in the market. Thus the variability of the spread may result in measurement errors and may therefore affect the empirical results. </p><p>Taking the above factors into account, the data were constructed from daily bid and ask spot and one-month forward rates for the Austrian schilling, the Deutsche mark, the French franc, the Canadian dollar and the United States dollar. All currencies were quoted against one British pound. The observations were obtained from DATASTREAM on-line database and span the period from January 1976 to June 1993. To comply with the rules governing the delivery of FX contracts in the market, the eligible value dates were identified and realigned by referring to back issues of the Europa Year Book. The holiday conventions and a sample of official holidays were further validated by contacting the respective countries embassies and various commercial banks in London. Thus, five sets of subsamples for each currency were generated, where the current spot and forward rates-at the start of each FX forward contract-were consistently observed on a specific day of the week for each sample set. The forward rates were then matched to the realized spot rates at the maturity date of the FX contracts. That is, the realized spot rate within each subsample set corresponds to the maturity date of the forward contract executed one month earlier. The observed realized spot rates were often on different days relative to the forward rate. Thus, although we refer to a given set of subsamples by a specific day of the week, only the current spot and forward rates-at the start of each forward contract-were consistently observed on the specific day of the week over the entire period of the study. </p><p>The observations within each set of subsamples were non-overlapping. Indeed, for each subsample set of a given day of the week there are at least two weeks between the realized spot rate (at maturity of the forward contract) and the forward rate and current spot rate record of the next new contract. Based on observations of the last working day of the month, a further set of (non-aligned) subsamples of month-end series was generated for each currency. All computations are performed on the natural logarithm of the FX rates. </p><p>The sampling procedure employed included observations on pre- and/or post-bank holidays of the United Kingdom (UK) and the corresponding foreig...</p></li></ul>