Cointegration, error-correction models, and forecasting using realigned foreign exchange rates
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Journal of Forecasting, Vol. 14,499-522 (1 995)
Cointegration, Error-correction Models, and Forecasting Using Realigned Foreign Exchange Rates
NATHAN LAEL JOSEPH Manchester University
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.
KEY WORDS unbiasedness hypothesis; unit roots; cointegration; error- correction models; forecasting
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 & Sons, Ltd. Revised June 1995
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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).
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).
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.
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.
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.
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.
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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.
REALIGNMENT, SEASONAL VARIATION, AND THE DATA SET
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.
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.
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
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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.
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.
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.
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.
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.
The sampling procedure employed included observations on pre- and/or post-bank holidays of the United Kingdom (UK) and the corresponding foreign country, but excluded all trading holiday observations. Trading holiday observations were excluded because the FX contracts cannot be settled unless both trading centre are opened on the same day. To account for the possible effects of pre- and/or post-bank holidays of the UK and the corresponding foreign country, two dummy variables were created and were added to the right-hand side of equation
Nathan Lael Joseph Cointegration, Error-correction models 503
(2) for each subsample set. One dummy variable for pre- and/or post-bank holiday observations was included for each country. Where holidays were shared between the UK and the corresponding foreign country, the pre- and/or post-bank holiday observations were attributed (perhaps erroneously) to the UK. The dummy variables were not consistently significantly different from zero for each subsample set. Thus a further dummy variable was used to separately account for the combined effect of pre- and/or post-bank holiday observations for both countries. This final dummy variable was significantly different from zero in only five cases and is therefore incorporated into subsequent analyses at those intervals only.
UNIT ROOTS, COINTEGRATION AND ALTERNATIVE SPECIFICATIONS
Tests of market efficiency often focus on the relationship between the realized (future) spot and forward rates of the same currency. This approach is consistent with the empirical studies of Hakkio and Rush (1989) and Baillie and Bollerslev (1989a) among others. The efficiency of the FX market in conjunction with rational expectations implies that economic agents expectations of the determinants of future FX values are fully reflected in the forward rate. Investors therefore cannot earn unusual profits by exploiting available information. If the forward rate at t is an unbiased predictor of the realized spot rate at t + 1, then the cointegration of the forward and realized spot rates is implied (see Hakkio and Rush, 1989) although this is not a sufficient condition for unbiasedness. Assuming no risk premium, the unbiasedness of the forward rate can be represented by
Fi = Er(RSi+,) (1)
where F , is the forward rate and E, (RS ,+ , ) is the rational expectation of the realized spot rate at t + 1, based on the information set at t . If both the realized spot and forward rates are both integrated of first order, that is, I ( l ) , then the two series are cointegrated if a linear of both series is I(0). Where the series have unit roots, cointegration is consistent with unbiasedness (see Dwyer and Wallace, 1992) assuming risk neutrality.
Standard tests of unbiasedness involve regressing the realized spot rate, RS, + on the forward rate F , in the form
In the absence of a risk premium, equation (2) is used to test the joint null hypothesis that a = 0 and /? = 1 where E , is white noise. However, assuming that RS, + , and F , are I ( 1) series the error E , is not a stationary white noise process unless the two variables are cointegrated.. . (Maddala, 1992, p. 600). Therefore, if RS,,, and F , are not cointegrated, in that the error p r = F , - R S , + , is also I ( l ) , then the long-term relationship between F , and RS,,, is spurious and therefore non- existent. Thus, a further test of unbiasedness is to test for the stationarity of the restricted residual ,LA,. If the residual is stationary, then RS,., and F , are cointegrated of order one which is unique when it exists.
In an attempt to avoid the non-stationary problem of equation (2) researchers have regressed the rate of change of the realized spot rate RS,., and current spot rate S, on the forward premium/discount ( F , - S,). This is often referred to as the percentage change specification and is represented by
RS,,, = a + BF, + E ,
RS,., - S t = a + P ( F , - S , ) + ~ , (3)
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This specification also requires that a = 0 and /3 = 1 in the absence of a risk premium. However, this formulation is only valid provided ( F , - S,) is stationary. Thus, while (RS,,, - S,) is stationary if it is a white noise process, ( F , - S,) is not necessarily white noise, in which case one would be regressing variables of different orders of integration.
If RS,,, and F , are cointegrated then they can be written in an error-correction form
R S , , ] - S , = a + / 3 ( F r - , - S , ) + i l ( F , - F , _ , ) + & , (4)
where the right-hand side variables are the decomposed form of F, - S, from equation ( 3 ) . That is, F , - S, = ( F , - ] - S, ) + ( F , - F , - ] ) . This is the ECM which will be used for forecasting. The usual interpretation of an ECM is that the change in RS,, , is due to the immediate short-run effect from the change in F, plus last periods error, F,- , - S,, which is a measure of error or deviation from equilibrium.
TESTING FOR UNIT ROOTS AND COINTEGRATION
Some preliminary results As a preliminary test of possible variation in the forecasting performance of the forward rate at specified intervals, we test whether the realized forecast error, p, is zero and free of significant autocorrelation, where p , = F , - RS,,, is implied from equation (1). The results also have important implications for the unbiasedness hypothesis. If the moments of the errors are similar across sets of subsamples, then the forecasting accuracy of F , would be similar such that the choice of the interval at which the sets of subsamples are observed would be of no economic consequence.
Table I presents summary statistics for the realized forecast errors p, for three currencies. All statistical results not presented are available from the author. The results are based on the mid- point of the realized spot and forward rates. These results are preliminary and therefore include pre- and post-bank holiday observations which appear to have a significant effect on the results of some subsample sets. As the FX rates are expressed in the natural logarithms, the statistical measures are unit free and comparable across currencies and sets of subsamples. The table indicates that the mean forecast errors and their variances appear to vary across sets of subsamples. Thus Wednesdays mean forecast errors tend to be positive while those of Thursday tend to be negative. The null hypothesis that the means forecast errors are zero is accepted in all cases. The Ljung-Box statistic indicated (persistent) significant autocorrelation for the Wednesday, Friday, and month-end intervals of the Austrian schilling, North American currencies and the French franc respectively. This result questions the extent to which the rationality of the forecasts can be maintained at those intervals.
A further examination of Table I, indicates that European currencies exhibit significant skewness (non-symmetry) and kurtosis, confirming the non-normality of the forecast errors. The Studentized Range (SR) test, that is, the range of the series divided by the standard deviation, further rejected normality. Skewness was less significant among the North American currencies although tests of normality remained significant (to some extent). These results therefore suggest that the level series may themselves be non-normal and that the parameters of our estimates may not be constant. To minimize the impact of non-normality on the results we employ the mean absolute forecast error which is more robust to fat-tailed distributions and outliers (Meese and Rogoff, 1983). The results on the right-hand side of Table I indicate that the
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Table I. Summary statistics of subsamples' for one month realized forward rate forecast errors'
Currency N Mean Variance Skewness Kurtosis SR Abs. mean
Austrian schilling Monday 129
Deutsche inark Monday 130
United States dollar Monday 130
Wednesday 13 1
0.0458 (0.2135) 0.0343
(0.2575) 0.003 1
(0.2250) -0.0002 (0.2234)
-0.2059 (0.2209) 0.1110
(0.2272) -0.0636 (0.2504)
1.8479" (0.1375) 2.0735"
(0.1724) 1.9 100"
2.0739" (0.1702) 1.9067"
(0.1455) 1 .8907a
(0.1562) 2.068 1 "
(0.1532) 2.0935" (0.1351)
3.0775" (0.2164) 2.7748"
'For day-of-the-week results only the current spot and forward rates at the start of the forward contract are systematically observed on the specific day. The realized spot rate for that interval corresponds to the maturity of the forward contract which was executed one month earlier. Month-end results are based on observations of the last business day of each month. 2The realized forecast error is defined as p , =F, -RS,_, where F, and RS,,, are, respectively, the natural logarithm of the forward and realized spot rate for the subsample set of the particular interval. The realized forecast errors are not adjusted for effects of pre- and/or post bank holidays of the UK and the corresponding foreign country. "-'The appropriate test statistic is significant at a 1%, 5% or 10% level, respectively. Standard errors are in parentheses ( ). The means, variances, Abs. (absolute) mean and standard errors are multiplied by 100; SR is the Studentized Range test, that is, the range of the series divided by the standard deviation.
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mean absolute forecast errors are significantly different from zero at a 1% level in every case. Tests of normality, skewness, and kurtosis remained highly significant.
To test whether the realized forecast errors are similar across sets of subsamples, the non- parametric Kruskal-Wallis test is employed. Only the significant results are shown in Table 11. The null hypothesis that the samples came from the same population or from identical populations with the same median was rejected for 45% of the pair-wise comparisons; 76% of these are at a I % level. Of the total number of cases where the null is rejected, 44% involved Monday subsample sets while a further 41% involved Thursday subsample sets. The parameters of month-end subsample of the Deutsche mark are not significantly different from those of other intervals. The overall results suggest that the choice of the interval at which the sets of subsamples are observed may affect the forecasting performance of ECMs as well as empirical tests of unbiasedness since the moments of the realized forecast errors are dissimilar.
Tests of unit roots and structural change Empirical tests of the stationarity of FX rates must precede cointegration tests in order to determine the order of integration of the individual series. We employ the Augmented
Table II. Kruskal-Wallis one-way analysis of variance by ranks for pair-wise comparisons of the sets of subsamples' realized forecast errors2
Currency MON BY TUE BY WED BY THU BY FRI BY MTH BY
Austrian Flu TUE MON THU THU schilling 26.299" 28.55 1" 39.319" 24.699" 27.949" Deutsche inark TUE WED WED MON
22.205" 25.87Sa 62.804" 37.860" THU THU
64.8 10" 33.846" French franc THU MTH TUE TUE MON
22.566" 22.250b 16.17 1' 37.260" 26.724" FFU
28.574" Canadian TUE WED MON TUE WED dollar 21.306b 33.087" 26.453" 22.779' 26.427"
FRI THU MTH THU 17.614' 85.771" 17.563' 27.549"
United States MTH MON MON MON MON dollar 19.850' 41.430" 35.717" 22.829b 30. 177a
WED FRI 37.516" 27.764"
' As defined in Table I and the text. *As defined in Table I and the text. a-E are as defined in Table I. MON to FRI are for the subsample sets of day-of-the-week intervals. MTH is for the subsample set of the month-end interval. The statistics indicated are the observed values of the Kruskal-Wallis test. Only the significant values are shown. The Kruskal-Wallis test is well-approximated by the x 2 distribution with k - 1 degrees of freedom.
Nathan Lael Joseph Cointegration, Error-correction models 507
Dickey-Fuller (ADF) test for unit root, since it may be possible to capture autocorrelation in the error term of the model which may affect the distribution of the test statistics and therefore invalidate the tests. To perform unit root tests on the variable x f , we employ the model
A x f = a. + a , T + a 2 x , - + , 6 ;Axf - , + Ef i = 1
where Ax, = x, - x f - , is the first-order time difference for the series and m (the lag order) is large enough to ensure that the residual E , is white noise. Our choice of m is based on the modified Lagrange multiplier (LM) statistic since Kiviets (1986, p. 257) simulation results indicate that the modified LM statistic (F-statistic) test is comparatively more relatively invariant to sample size, order of serial correlation, true coefficient values, and redundant regressors in models involving lagged (dependent) variables. The ADF test procedure as well as the suitability of the test for various data generating processes (see DeJong et af., 1992; Schwert, 1989; Said and Dickey, 1984) are well known. To account for the combined impact of pre- and/ or post-bank holiday observations of the UK and the corresponding foreign country, a dummy variable is included in equation (5) and subsequent analyses, where earlier results indicated that the dummy variable was significant.
To test for the presence of a unit root, the regression for equation (5) inclusive of the pre- and/or post-bank holiday dummy variable (where appropriate), was first run on a small number of subsamples of bid and offer realized spot and forward rates. The appropriate critical values are from Fuller (1976) and MacKinnon(l991). The results are generally similar to those based on the mid-point of the current spot, realized spot, and forward rates. All subsequent results are therefore based on the mid-point of the FX rates. The results of tests for unit roots were similar to those of Meese and Singleton (1982) and Baillie and Bollerslev (1989a) among others, in that the null hypothesis of a unit root could not be rejected for any subsample. To test for the effects of structural changes (breaks) on the tests for unit roots, dummy variables were used to span three important periods when the value of the pound relative to other currencies appeared to have undergone dramatic shifts. For European currencies, the first dummy variable D, spans the period June 1979 to October 1982 which coincides with the early years of the Conservative government administration. For the North American currencies, D, spans the period October 1978 to October 1982. The periods spanned by the other two dummy variables were similar for all currencies. Thus, D, coincides with the period of the coal miners strike, April 1984 to June 1985, while D, spans the period of the sterling crisis, September 1992 to June 1993. The tests for unit roots were again performed but with the three dummy (0,l) variables included for structural changes for the periods indicated. That is, for the period of suspected structural change the dummy variable was 1 but 0 otherwise. The inclusion of the three dummy variables for structural changes substantially increased the test statistics. Based on the critical values in MacKinnon(l991), the null hypothesis of a unit root for ?, in the level series is rejected at less than a 10% level for only 28% of the subsamples. This involved most subsamples except those of Tuesday and Friday, and all currencies except the French franc and Canadian dollar. The test statistic for ?,, was also significant in 14% of the cases, primarily for the United States dollar. These latter results suggest that the segmented trend model may be a feasible alternative to the difference stationary model for these subsamples (but see Nelson and Plosser, 1982). The results based on both D, and D, were not appreciably different from those of the three-segmented model. Rappoport and Reichlin (1989) indicate that the critical values for the two-segment trend model do not appear to depend substantially on the point of change in regime.
ard rates o