Download - Cointegration, error-correction models, and forecasting using realigned foreign exchange rates
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
ABSTRACT
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
INTRODUCTION
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 Copeland’s 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.
Nathan Lael Joseph Cointegration , Error-correction models 50 1
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, today’s 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
502 Journal of Forecasting Vol. 14, Iss. No. 6
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
(2)
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)
504 Journal of Forecasting Vol. 14,Iss . No. 6
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 period’s 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 Wednesday’s 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
Nathan Lael Joseph Cointegration, Error-correction models 505
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
Tuesday 130
Wednesday 132
Thursday 129
Friday 128
Month-end 209
Deutsche inark Monday 130
Tuesday 131
Wednesday 132
Thursday 130
Friday 129
Month-end 209
United States dollar Monday 130
Tuesday 130
Wednesday 13 1
Thursday 129
Friday 128
Month-end 209
0.0458 (0.2135) 0.0343
(0.2575) 0.003 1
(0.25 12)
(0.2525) 0.1097
(0.2250) -0.0002 (0.2234)
-0.2013
-0.0438 (0.2496)
-0.2059 (0.2209) 0.1110
(0.2272) -0.0636 (0.2504)
-0.0264 (0.2327)
-0.0932 (0.1982)
-0.4426
-0.1673 (0.3446)
(0.3126) 0.2681
(0.3014)
(0.3334)
(0.3016)
(0.2538)
-0.1229
-0.0617
-0.1007
0.0588
0.0862
0.0833
0.0823
0.0648
0.1044
0.0810
0.0639
0.068 1
0.0815
0.0698
0.0821
0.1544
0.1270
0.1190
0.1434
0.1164
0.1346
0.751"
0.686"
0.25 1
0.520"
0.229
-0.040
0.858"
0.034
0.628"
0.565"
0.395'
0.380b
-0.201
0.008
0.335
0.395'
0.250
0.072
1.067
2.582"
2.498"
1.322"
0.538
3.162"
2.625"
0.724'
2.303a
1.491"
0.91Sb
1.515"
0.619
0.586
0.706
3.38ga
0.135
0.854"
5.290
7.046a
7.339"
5.960'
5.452
7.788"
6.734"
5.345
7.309"
6.151
5.815
6.498b
6.024'
6.242b
6.193b
8.238"
5.259
7.264"
1.8479" (0.1375) 2.0735"
(0.1817) 2.0311"
(0.1778) 2.0964"
(0.1724) 1.9 100"
(0.1482) 2.2349"
(0.1610)
2.0739" (0.1702) 1.9067"
(0.1455) 1 .8907a
(0.1562) 2.068 1 "
(0.1720) 1.9820"
(0.1532) 2.0935" (0.1351)
3.0775" (0.2164) 2.7748"
(0.1955) 2.7575"
(0.1813) 2.8142"
(0.2223) 2.7022"
(0.1830) 2.8328"
(0.1608)
'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.
506 Journal of Forecasting Vol. 14, Iss. No. 6
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"
MTH 20.794b
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
m
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 Kiviet’s (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.
Tabl
e 11
1. Jo
hans
en's
max
imal
eig
enva
lue a
nd tr
ace t
est s
tatis
tics f
or co
inte
grat
ion o
f cu
rren
t and
real
ized
spot
and
forw
ard rates o
f ea
ch su
bsam
ple s
et'
Pane
l A
Pane
l B
r=O
r<
l rs
2
rs3
V
AR
r=
O
rs
l rs
2
rc3
rs
4
rs5
r
~6
VA
R
Aus
tria
n sc
hilli
rig
Mon
day
lMax
65
.081
b Tr
ace
78.2
79b
Tues
day
* lM
ax
20.5
44
Trac
e 47
.622
W
edne
sday
lM
ax
45.7
89b
Trac
e 61
.763
b Th
ursd
ay
lMax
76
.666
b Tr
ace
86.0
55
Frid
ay
lMax
63
.201
Tr
ace
79.5
78b
Mon
th-E
nd
lMax
23
.107
b Tr
ace
24.6
39b
Deu
tsch
e m
ark
Mon
day
lMax
68
.049
b Tr
ace
80.3
Ub
Tues
day
AMax
30
.107
b Tr
ace
39.4
92
Wed
nesd
ay
lMax
64
.259
b Tr
ace
73.3
42
12.0
33
13.1
98
19.0
21'
27.0
78'
15.7
65
15.9
74b
8.70
1 9.
389
15.7
75b
16.3
77b
1.53
3 1.
533
11.6
44
12.2
95
9.16
1 9.
385
9.07
7 9.
083
1.16
5 1.
165
7.99
6 0.
612
8.05
7 0.
612
0.20
9 0.
209
0.68
8 0.
688
0.60
3 0.
603
0.65
1 0.
65 1
0.22
4 0.
224
0.00
6 0.
006
Aus
tria
n sc
hilli
ng
Mon
day
AMax
66
.298
b 19
.720
14
.119
7.
693
3.15
5 0.
062
Trac
e 11
1.04
7b
44.7
49
25.0
29
10.9
10
3.21
7 0.
062
Tues
day
AMax
43
.748
' 36
.463
25
.354
15
.275
8.
238
2.55
0 O.
OO0
Trac
e 13
1.62
3b
87.8
75
51.4
12
26.0
58
10.7
82
2.55
0 O.
OO0
Wed
nesd
ay
Ma
x
49.4
20b
30.0
26
12.2
29
7.15
7 3.
647
0.54
6 Tr
ace
103.
026b
53
.606
23
.580
11
.351
4.
194
0.546
Thur
sday
AM
ax
Trac
e Fr
iday
M
ax
Tr
ace
Mon
th-E
nd
80.1
27b
17.4
74
9.81
9 6.
148
2.98
4 0.
224
16.7
76b
36.6
50
19.1
76
9.35
7 3.
209
0.22
4
72.7
56b
23.1
13
11.9
60
5.83
3 2.
867
0.06
9 16
.598
b 43
.841
20
.728
8.
768
2.93
5 0.
069
AMax
30
.539
13
.387
6.
942
2.97
6 0.
529
Trac
e 54
.374
23
.835
10
.448
3.
505
0.52
9
Deu
tsch
e mar
k M
onda
y lM
ax
85.0
83b
21.2
60
13.8
12
7.35
7 3.
194
0.09
5 Tr
ace
130.
800b
45.
717
24.4
57
10.6
45
3.28
8 0.
095
Tues
day
AMax
32
.276
22
.675
17
.511
13
.212
2.
905
0.65
8 Tr
ace
89.2
36
56.9
60
34.2
85
16.7
74
3.56
2 0.
658
Wed
nesd
ay
lMax
68
.706
b 18
.264
14
.941
7.
007
3.11
3 0.
174
Trac
e 11
2.20
6b
43.5
00
25.2
36
10.2
95
3.28
7 0.
174
Thu
rsda
y lM
ax
62.4
16’
8.61
0 0.
352
Tra
ce
71.3
79‘
8.96
3 0.
352
Frid
ay
lMax
70
.562
’ 7.
586
0.45
2 T
race
78
.60O
b 8.
038
0.45
2 M
onth
-End
AM
ax
9.41
4 0.
228
Tra
ce
9.64
3 0.
228
Uni
ted S
tate
s dol
lar
Mon
day
lMax
T
race
Tue
sday
lM
ax
Tra
ce
Wed
nesd
ay
LMax
T
race
Thu
rsda
y LMax
Tra
ce
Frid
ay ’
LMax
T
race
103.
889b
14
.077
b 3.
586‘
12
1.55
2’
17.6
63’
3.58
6‘
90.9
29b
11.8
35
4.43
6’
107.
199’
16
.270
4.
436’
95.1
94
14.1
83
3.78
5 11
3.16
2’
17.9
68’
3.78
5’
78.7
11’
12.6
72‘
3.20
1‘
94.5
84b
15.8
73’
3.20
1‘
71.9
11b
37.3
96’
12.4
51‘
4.25
8’
126.
016b
54.
105’
16
.709
b 4.
258b
Mon
th-E
nd2
AMax
69
.334
17
.571
3.
500
Trac
e 51
.763
14
.071
3.
500
Thu
rsda
y lM
ax
60.6
50b
Tra
ce
103.
878b
Fr
iday
L
Max
35
.404
T
race
84
.295
M
on th
-End
A
Max
18
.567
Tr
ace
44.1
53
Uni
ted S
tate
s dol
lar
Mon
day
lMax
11
3.62
7b
Tra
ce
169.
338b
Tue
sday
A
Max
94
.357
T
race
14
9.59
7 ’ W
edne
sday
A
Max
94
.833
T
race
14
9.48
5 T
hurs
day
AM
ax
79.8
23 ’
Tra
ce
132.
988b
Frid
ay
lMax
90
.997
’ T
race
18
6.07
5 ’ M
onth
-End
2 lM
ax
60.1
21
Trac
e 11
1.53
5
19.8
97
43.2
28
25.6
90
48.8
91
14.6
61
25.5
86
23.7
59
55.7
1 1
24.6
16
55.2
40
21.7
69
54.6
52
23.0
46
53.1
65
41.2
01’
95.0
78’
21.5
73
51.4
14
12.9
75
23.3
31
12.6
78
23.2
0 1
7.45
4 10
.925
20.6
14
31.9
52
18.7
33
30.6
25
20.8
06
32.8
83
19.4
30
30.1
19
23.0
61
53.8
76
18.6
37
29.8
41
7.02
4 10
.356
8.01
6 10
.523
3.16
2 3.
472
7.71
4 11
.338
7.67
2 11
.892
8.74
5 12
.077
7.00
1 10
.689
18.8
21
30.8
16
8.08
6 11
.204
3.18
6 0.
146
3.33
3 0.
146
2.50
7 0.
000
2.50
7 0.
000
0.31
0 3
0.31
0
3.51
4 0.
110
3.62
4 0.
110
4.01
0 0.
210
4.22
0 0.
210
3.33
0 0.
003
3.33
3 0.
003
3.68
8 O.
OO0
3.68
8 0.
000
7.30
8 4.
626
0.06
1 11
.995
4.
687
0.06
1
2.99
1 0.
127
3.11
8 0.
127
~~
~~
___.
6’
The
test
sta
tistic
s in
Pane
ls A a
nd B
are
, res
pect
ivel
y, fo
r the
tim
e se
ries
with
out a
nd w
ith th
e in
clus
ion
of d
umm
y va
riabl
es fo
r stru
ctur
al c
hang
es (
brea
ks).
2s ’ As
def
ined
in T
able
I an
d th
e te
xt.
21nc
lude
s a d
umm
y va
riabl
e fo
r th
e ef
fect
s of
pr
e- a
nd/o
r po
st-b
ank
holid
ays
of
the
UK
and
the
cor
resp
ondi
ng f
orei
gn c
ount
ry (
com
bine
d). T
radi
ng h
olid
ay
obse
rvat
ions
are
not
inc
lude
d in
the
anal
ysis
sin
ce in
pra
ctic
e, th
e tra
ding
cen
tres
of t
he a
ppro
pria
te c
ount
ries
need
to
be o
pene
d fo
r del
iver
y of
fo
reig
n cu
rren
cies
to
r de
note
s th
e nu
mbe
r of
co
inte
grat
ing
vect
ors.
The
LMax
sta
tistic
tes
ts t
he n
ull
hypo
thes
is o
f ex
actly
r c
oint
egra
ting
vect
ors
agai
nst
the
alte
rnat
ive
of
r + 1
coin
tegr
atin
g ve
ctor
s, w
hile
the
Tra
ce s
tatis
tic te
sts
the
null
hypo
thes
is o
f ex
actly
r c
oint
egra
ting
vect
ors.
VAR
is th
e la
g or
der o
f th
e ch
osen
VAR
.
8- ta
ke p
lace
. c 8
In
5 10 Journal of Forecasting Vol. 14, Iss. No. 6
Tests of cointegration and structural change Since the current spot, realized spot and forward rates for the most part are I(1), we next test whether they are cointegrated. We first employ the Engle and Granger (1987) two-step procedure as a preliminary test for cointegration of the realized spot and forward rates. This procedure involves regressing RS,,,, the realized spot rate on F,, the forward rate (and the reverse) with a constant term (equation (2)), and then testing the residuals for unit roots. Again, the pre- and/or post-bank holiday dummy variable is included where appropriate. Cointegration requires that the pseudo f-statistic be negative and greater than the critical values in MacKinnon( 199 1). We also employ the multivariate cointegration procedure of Johansen (1988) which is now well known (see also Johansen and Juselius, 1990). This procedure is superior to that of Engle and Granger (1987) particularly when there are more than two I( 1) variables under consideration. Further, the Johansen (1988) procedure provides a test statistic which has an exact limiting distribution and facilitates identification of possible variation in the number of cointegrating vectors among the sets of subsamples.
For the Johansen test, the number of cointegrating vectors is determined sequentially. The tests employed are: (i) the AMax statistic which tests the null hypothesis of exactly r cointegrating vectors against the alternative of r + 1 cointegrating vectors; and (ii) the Trace statistic which tests the null hypothesis that there are, at most r cointegrating vectors. The relevant critical values are found in Osterwald-Lenum (1992). To implement the Johansen procedure the lag length of the VAR was initially set to 2 and the residuals of the VAR were tested for whiteness. The order of the VAR was increased by one (to a maximum of eight) until the null that the residuals were white noise was not rejected at a 10% level. Six sets of subsamples exhibited significant residual autocorrelation for the chosen VAR at higher lags. In those cases, the VAR chosen was the one with less severe residual autocorrelation.
Based on the residuals of the realized spot and forward rates, the Augmented Dickey-Fuller statistic (ADF) at four lags indicated that cointegration could not be accepted for four sets of subsamples. The values of the test statistic varied by subsample set. As the results may be sensitive to structural instability, the dummy variables (for structural changes) were also included in the basic model and the regressions were rerun. The inclusion of the dummy variables caused the test statistics to be more negative, thus accepting cointegration. Although both sets of cointegrating regressions indicated that the realized spot and forward rates were cointegrated, the residuals from the regressions exhibited significant residual autocorrelation at several intervals, although not systematically. Significant autocorrelation may be evidence of irrationality in the forward market at certain intervals or an ARMA process in the risk premium (see Copeland, 1991). As a further test of cointegration, we assess the forecast error. If RS,., and F , are both I(1), in that they have unit roots, and the difference F , - RS,,, which represents the forecast error, p , is I(O), then we may conclude that RS,,, and F, are cointegrated (see Liu and Maddala, 1992). Otherwise the forward and realized rates would drift apart over time if the error was I(1). Further, in the absence of a risk premium, the forward rate should be equal to the realized spot rate. Thus p, should have a mean of zero and should be uncorrelated with all of the variables in the information set I,. The results indicated that the test statistics for this restricted cointegration test were of a similar magnitude to the Engle-Granger two-step approach, thus accepting cointegration. The realized forecast errors are not significantly different from zero (although not in absolute terms (see Table I)), but the errors are significantly autocorrelated in some cases. The overall results therefore provided further support for the cointegration of the realized spot and forward rates.
The results of the cointegration test using Johansen’s multivariate procedure are presented in Table 111. Only the results for three currencies are shown. The Johansen procedure is performed
Nathan Lael Joseph Cointegration, Error-correction models 5 1 1
on the current spot, realized spot, and forward rates with and without dummy variables for structural changes. The dummy variable for pre- and/or post-bank holiday observations was also included in the model where appropriate. The results in panel A indicate that the hypothesis that there are no cointegrating vectors could be rejected for a minority of the cases. The null hypothesis that there are up to three cointegrating vectors for the North American currencies and up to two for the European currencies in the three variable-system cannot be rejected. The larger number of cointegration vectors for the North American currencies suggests a more ‘stable’ system. However, there is generally strong variation in the number of cointegrating vectors among the sets of subsamples. The finding that there is at least one cointegrating vector for all currencies implies that the variables are tied together in the long run. However, the inclusion of dummy variables for structural changes (panel B) reduces the number of cointegrating vectors to one in most cases.
We also tested for pair-wise cointegration between (1) the realized spot and the forward rates and (2) the current spot and the forward rates. The number of cointegrating vectors linking the realized spot and forward rates exhibited similar variation by interval and currency. These results are consistent with those of Copeland (1993) who also employed the Johansen procedure. The current spot and forward rates were generally cointegrated but the residuals from the VAR were always significantly different from zero for all selected lag structures.
ERROR-CORRECTION MODELS AND FORECASTING PERFORMANCE
Error-correction model and alternative specification Before considering the correctly specified ECM we examine the stationarity of the remaining variables in the ECM as well as the forward premium/discount for the five currencies. We do not test for structural shifts. Ideally, a correctly specified model should involve variables of the same order of integration, although Pagan and Wickens (1989) indicate the conditions for variation on this theme.
An assessment of the stationarity of the above variables is useful, since the results will have implications for the ECM we employ and well as the percentage change specification of equation (3). As the Dickey-Fuller test is based on the assumption of at most one unit root in the process, this test may not be theoretically justified if more than one unit root is present. The unit root test we employ in this section is therefore due to Dickey and Pantula (1987). This test amounts to sequential testing for the null of three unit roots, then two unit roots, and finally a single unit root, based on the coefficient of the most recently added variable. The test is performed via ordinary least squares (OLS) regressions in the form:
DP3: A3yr = a, + BlA2yt - 1 + P ~ A Y ~ - I + B 3 Y t - l + ~ 2 , (8) where A is the difference operator and DP,, DP2 and DP, are the sequential OLS testing procedures for the null of three, two, and one unit root respectively. Again the pre- and/or post- bank holiday dummy variable is included where necessary. The appropriate critical values are from Fuller (1 976) and MacKinnon (1 991). The results from the three regressions indicated that the error correction term (F,-l - S,) and the difference between the realized spot and current spot rates, (RS,,, - S,) for all the currencies are I(O), implying that these variables meet the necessary conditions for stationarity. That is, all the variables in the ECM we employ are of the
5 12 Journal of Forecasting Vol. 14, Iss. No. 6
same order of integration and we could therefore expect to achieve a stationary I(0) error term. Except for the Deutsche mark, all the forward premium/discount ( F , - S,) series were also I(0). The finding that ( F , - S,) is I(1) for the Deutsche mark implies that the percentage change specification of equation (3) may be inappropriate for this currency. Overall, the results suggest that the ECM appears to be correctly specified since all the variables it employs are of the same order of integration.
Final error-correction model Based on equation (4), the initial ECM for each subsample set was estimated by OLS regression inclusive of the first difference of the current spot rate, 12 lagged differences of the current spot and forward rates, and the (appropriate) dummy variable for pre- and/or post-bank holiday observations. The final (reduced) models were derived by eliminating insignificant coefficients as long as this reduced the standard error of the regression (SER) and minimized the Akaike Information Criterion (AIC). We also relied on the Encompassing test (see Mizon and Richard, 1986) and J-test in selecting the final ECM. The SERs of the final models were always smaller than those of the original (full) models. However, some of the coefficients and their standard errors were abnormally large. Suspiciously large residuals were identified using the leverage measures of each regression (see Cook and Weisberg, 1982). To assess the impact of the residuals, the test statistic shown in Snedecor and Cochran (1989, pp. 351-353) was employed but the test results were not found to be significant. Tests for autocorrelation in the regression residuals uniformly indicated that the errors were not significantly correlated. Further, the ‘modified Lagrange multiplier (LM) statistic’ indicated that the null of a constant residual variance against the autoregressive conditional heteroscedasticity (ARCH) alternative could not be rejected. We feel that the combination of diagnostic results suggests that the final ECMs indicate good performance although the Jarque-Bera (J-B) test generally rejected the null hypothesis that the regression residuals are normal.
Table IV presents the results of the final ECM for each subsample set. An interesting feature of the results is the variability of the lag length. This feature appears to be a function of both the subsample set and the currency and indicates that disequilibrium ‘shocks’ are elimin- ated from the system at different rates. Most of the coefficients of the final ECMs are significant and very large especially for the North American currencies. The error- correction term (F , - , - s,), where present, is always negative and is generally larger for the Friday subsample sets. The negative error-correction term indicates that the imme- diate period’s current spot rate exceeds last period’s forward rate by an ‘error’ which is partially corrected in the future. A further important feature of the results is the statistical insignificance or non-existence of error-correction terms particularly for Monday (and Thursday to some extent) ECMs of all European currencies. This feature is also exhibited for Wednesday ECM of the Canadian dollar and for the Thursday ECM of the United States dollar. This result indicates severe problems in adjusting to equilibrium at these intervals and has implications for the extent to which the series are tied together in the long run, even if earlier results suggested that the variables were co-integrated. Further, the constant term generally appears unimportant in the final ECMs for both the French franc and the Canadian dollar. The inclusion of significant constant terms for the other sets of subsamples is inappropriate when all the observations lie close to the steady-state growth path (Harvey, 1981).
For comparative purposes, the final ECMs for month-end subsample sets are also shown in Table IV. Except for the Austrian schilling, all the ECMs for the month-end subsample sets include significant error-correction terms. The explanatory power of the day-of-the-week ECMs is generally greater than those of the month-end. It therefore appears that the explanatory power
Tabl
e IV
. T
he fi
nal e
rror
-cor
rect
ion
mod
el (
ECM
) of
the
diff
eren
t set
s o
f su
bsam
ples
'
Coe
ffic
ient
s of
ECM
Ausir
iari
schi
llirig
M
onda
y a
-0.0
05
(0.0
02)
Tues
day2
AF,
-ll
1.79
1 (I
,179
)
0.16
0h
(0.0
78)
AFt
-1"
2.98
9h
(1.4
89)
AS,
-"
3.06
7h
(1.4
15)
AFt
-I"
AS,
., A
S,.,
AS(
-7
-0.0
99
2.59
2h
-2.1
72'
(0.0
63)
(1.2
42)
(1.2
32)
-2.2
26h
-0.1
37'
0.09
0 (0
.990
) (0
.075
) (0
.081
)
AS!
., A
S,-2
M
-5
AF
t-1
1
AF
1-1
2
ASr
-5
-0.1
21
2.24
1 2.
289
(0.0
86)
(1.4
42)
(1.5
76)
ASu-9
A
st-1
0 -0
.086
-
1.62
2 (0
.070
) (1
.414
)
AS,
- 1
.273
' (0
.725
) U
t-
11
A
S, A
S,_,
(0.0
71)
(0.8
07)
(0.9
88)
0.14
0h
-1.2
93
-1.7
60'
Coe
ffic
ient
s of
ECM
AS,
-"
-0.0
83
(0.0
63)
ASI
-8
-0.1
33'
(0.0
78)
AS
,-I"
-2.9
27h
(1.4
96)
ASt-Y
-0
.064
(0
.061
)
AS,
-,,
-0.1
64h
(0.0
76)
-2.3
00
( 1.4
55)
AS,
-I2
ASt
-11
- 1.
896
(1.1
76)
HO
L
(0.0
06)
-0.0
19a
K2
0.04
06
F,-
I - S
I
-0.0
61
(0.0
61)
F,-
i - S,
-0.1
61h
(0.0
7 5 )
F,-
I - s,
-0.1
77h
(0.0
78)
F,-
1-
s, -0
.108
(0
.072
)
AF,
.4
-2.5
9Ih
(1.2
44)
AF,
- I -2
.1 8
6h
(0.9
98)
AF,
- 2
(0.0
79)
AF#
-"
-2.9
67h
(I .3
97)
AF,
- 1
-0.1
09'
(0.0
66)
U,
-4
-0.0
91
(0.0
69)
-0.1
48'
AF,
-7
2.15
0'
( 1.2
40)
AF,
-4
-0.1
69h
(0.0
78)
AF
t-5
-2
.184
( 1
S69)
1.56
7 (1
.396
)
AF,
- 1"
K2
0.14
40
K2
0.07
08
Wed
nesd
ay
a -0
.004
(0
.003
)
Thur
sday
K2
0.
0416
R2
0.04
73
Frid
ay
n -0
.010
" (0.003)
-0.0
07h
(0.0
03)
Mon
th-e
nd
a
F,-
l -
s, -
1.29
7' A
Ft-
4
-0.0
87
AFf-1
1 -0
.120
(0
.718
)
F,-
l - s,
-1
.219
(0
.800
)
(0.0
66)
AF,
- 0
1.84
4'
(0.9
94)
(0.0
66)
AF!
- 7
-0.0
95
(0.0
70)
K2
0.02
83
Deu
tsche
rna
rk
Mon
day
n -O
.Wh
(0
.003
) Tu
esda
y
K2
0.10
23
Ft-I - S
I
-0.0
78
(0.0
69)
Fl-I - S
I
-0.2
00"
(0.0
67)
AF
I -
2 -0
.132
' (0
.069
)
AF
I-I
2.96
3'
(I .6
58)
4.26
1 ( 1
.998
)
AF,
-2
-4.7
07h
(2.4
17)
AF
I-3
AF,
- 3
-3.0
20"
(1.1
85)
AF,
- ' 3.
18Sh
(1
.626
)
AF
l-4
-5.7
41"
(2.0
94)
AF,
- H
-0.1
09
(0.0
70)
AF,
-4
-5.0
90"
( 1.6
26)
AF
t-1
0
5.01
7"
( 1.9
62)
4.64
3' (2
.428
)
AS,
-2
AF
t-1
2
- 1.8
96'
(1.1
49)
AF
t-5
3.
66gh
(1
.679
)
AS,
-2
-0.0
88
(0.0
70)
ASI
- 10
(0.0
66)
-0.0
89
AS,
-' 2.
979"
(1
.192
)
Ut
-1
0
4.59
8"
( 1.6
25)
AF,
- ' -4
.312
h (2
.005
)
ASt
-11
-0.1
07'
(0.0
64)
AL
ll
-0
.201
" (0
.068
)
AS,
-, -3
.002
' ( 1
.672
)
AS,
-4
5.72
9"
(2.1
06)
AS,
-l2
1.88
6 (1
.158
)
ASt
-3
- 3.2
59
( 1.6
35)
AS,
4 0.
151'
(0
.078
)
K2
0.19
71
AS,
-4
5.05
5"
(1.6
40)
ASI
-I"
-4.9
89"
( 1.9
74)
ASt
-5
AL
I"
AS,
-,,
-3.5
51h
-4.5
31"
( 1.6
93)
(I .6
42)
-0.1
52h
(0.0
73)
K2
0.14
29
Fl-I -
s, -0
.086
W
edne
sday
a -0.0
06"
(0.0
02)
Thur
sday
a -0.0
1 3"
(0
.005
)
(0.0
68)
Fl-I -
SI
-2.1
66h
(1.0
84)
R2
0.08
86
AS,
-2.1
25
(1.0
96)
2
(con
tinue
d)
w
Tab
le I
V.
(Con
tinue
d)
ul P
c.
Frid
ay
a F,
-l -s
, -0
.019
" -3
.314
" (0
.005
) (1
.112
)
AF,.l
AF,.5
AF
!. 6
4.76
3"
2.84
7' 5.
496"
(1
.739
) (1
.587
) (1
.671
)
AFJ-
7 AF
r-1,
) AF
t-ii
4.07
0"
2.55
4' 4.
143"
(1
.509
) (1
.515
) (1
.495
)
AS,
-3
.306
" (1
.123
)
&6 5.38
7"
( 1.6
63)
AS,.l
AS,-'
AS
z.4
M-
5
-4.6
30"
-0.1
36'
-0.155h
-2.7
65'
(1.7
26)
(0.0
75)
(0.0
72)
(1.5
69)
ASt-7
AS
t-iu
ASt-
ii AS
,-,*
-4
.188
" -2
.448
-4
.084
a -0
.146
h (1
.505
) (1
.504
) (1
.491
) (0
.075
)
8'
1 0.
1422
Mon
th-e
nd
a F,
- I - s,
(0
.004
) (0
.747
) -0
.01
1"
-2.0
32"
AF
, AF
,-4
AF
, -7
-1.9
63"
-0.1
19'
-0.0
91
(0.7
57)
(0.0
66)
(0.0
67)
AF,-"
AS
,+,
AS,
_,
(1.5
97)
(0.0
67)
(1.6
11)
2.75
1' 0.
115'
-2
.817
'
Coe
ffic
ient
s of
ECM
ii'
0.06
39
Fren
ch fr
anc
Mon
day
F,- I -
s, -0
.844
(0
.529
) Tu
esda
y
AF
, -0
.866
(0
.544
)
F,
-2.1
36h
(0.9
04)
AF
, - 3
2.12
0"
(0.8
37)
AF
, -4
-3.2
95"
(1.0
77)
AF,
-1.1
49'
(0.6
74)
AFj-7
2.
361"
(0
.81 1
)
AFt-5
-
1.36
2' (0
.780
)
AF
, ~ 5
(0.8
82)
- 1.
639'
AF
,-,
-1.7
87'
(1.0
86)
AFr-5
-
1.56
8 (0
.984
)
AF
, - w
1.70
4h
(0.7
88)
2.33
9"
(0.9
09)
AF,-
6
(0.9
30)
AFr-
9 -
1.97
4' (1
.041
)
ASr-
4 -0
.059
(0
.068
)
0.13
3' (0
.071
)
AS,
- 1.
423
AFr-6
AFt-1
0 -0
.187
" (0
.067
)
AS,-2
-0
.077
(0
.068
)
AF
, - 7
1.11
0 (0
.892
)
AFt.1
1 AS
t-1
0.11
4'
1.52
2h
(0.0
66)
(0.7
70)
AS,-5
1.
407'
(0
.798
) A
S,-,
A
S,_,
(0.8
34)
(0.8
78)
-2.1
64>
1.73
5h
ASt-7
0.
10s'
(0
.059
)
AS,-"
-2
.306
a (0
.793
)
AS,
-,,
ii' K' 0.09
54
-1.7
10h
-1.6
56h
0.11
51
(0.7
82)
(0.7
63)
Wed
nesd
ay
F,. I
- s,
-0
.128
' (0
.071
)
AS,-,
-1
.004
(0
.898
)
ASt-5
1.
875'
(1
.080
)
8' 0.12
36
AS,-,
1.
506
(0.9
29)
Thur
sday
Ft
.1-
s, 1.
073
AFt-1
" AS
, -1
.529
1.
203'
-0.0
98
ASf-
9 As
t-10
A
s,-1
2 ii'
1.85
5'
1.73
3' -0
.022
0.
1230
(1
.041
) (0
.985
) (0
.072
)
RZ
0.08
07
(0.6
68)
F,.
I - s
, -1
.251
'
(0.9
81)
(0.6
71)
(0.0
74)
AS,
_,
HO
L
(0.9
89)
(0.0
05)
1.66
4 -0
.114
h
(1.0
58)
Frid
ay2
(0.6
62)
Mon
th-e
nd
F,-I
- s,
-0.1
59h
(0.0
67)
AF,-4
1.
229'
R2
0.09
5 1
AFt-5
-0
.133
' A
S, -9
0.11
4' (0
.070
) (0
.660
) (0
.070
) (0
.659
) (0
.651
) (0
.659
)
Coe
ffic
ient
s of
ECM
C
anad
ian
dolla
r M
onda
y n
F,., - S,
Tues
day
F,-I
- s,
-
3.24
5 (1
.302
)
-0.0
04
-3.7
29h
(0.0
04)
(1.5
21)
8'
0.05
64
AF
, AF
,-7
AF
, - n
-3.7
43h
-0.1
18
-2.2
74
(1.5
41)
(0.0
74)
(2.0
03)
AF
, A
F,-
4 AF
,.,
(2.0
54)
(0.0
64)
(0.0
66)
-8.0
29"
-0.0
68
-0.2
17"
AS,
-,
2.17
4 (2
.020
)
AS,
4.87
2h
(2.0
60)
4
Q
R2
0.16
58
n
Tab
le I
V.
(Con
tinu
ed)
2
Wed
nesd
ay
Thua
day
n -0.0
05
(0.0
04)
Frid
ay
Mon
th-e
nd'
Uni
ted S
tate
s dol
lar
Mon
day
a -0.0
06
(0.0
04)
-0.0
06
(0.0
04)
-0.0
1 I a
(0.0
04)
-0.0
05
(0.0
04)
-0.0
06'
(0.0
04)
Tues
day
a
Wed
nesd
ay
a
Thur
sday
a
Frid
ay'
a
Mon
th-e
nd'
F,-
,- s,
(1
303)
F,.
, - s,
-3
.570
h (1
.780
)
-2.1
25
F,-I - s.
-4
.264
, (1
.393
)
Ft-I
- s,
-3.5
96"
(0.9
85)
Ft-
I - s,
-3.7
56"
( 1.3
44)
Fl-I - $1
(1.2
05)
F,-I - S,
( 1.2
14)
Ft-1
- s,
-2.6
86h
-2.8
24h
- 1.
939
(1.2
25)
Ft-
I - s,
-3.7
05"
(1.1
56)
F,-I
- s,
- 1.
543h
(0
.758
)
AF,
(1.3
19)
AF,
-3
.574
h (1
.802
)
AF
I
-2.1
62"
-4.2
63"
(1.4
11)
AF,
( 1.0
02)
-3.4
31"
AF,
-3
.715
" (1
.367
)
AF,
( 1.2
21)
AF,
-2.8
26h
( 1.2
30)
MI
-5.0
04"
(2.0
23)
AF,
-)
2.81
5 ( 1
.845
)
AF,
-4
.276
h (2
.092
)
-2.6
46h
AFC.1
-4
.123
' (2
.209
)
AF,
- I
(2.6
10)
AF,-3
3.
608'
(2
.201
)
AFt
-5
2.63
5 (1
.897
)
-4.7
88'
AF,-,
-0
.145
(0
.074
)
AF1-
2 -3
.555
' (1
.883
)
AFr-6
3.
285'
(1.9
67)
AFt
-1
-0.1
20'
(0.0
73)
-5.3
97"
( 1.9
13)
AFt
-4
M1-
4
-2.9
1 I
(1.9
39)
AFt
-4
(2.1
06)
AF,
-3
(0.0
76)
4.16
5h
(2.1
08)
AF,
-7
-5.5
73"
(1.9
05)
-5.4
61"
-0.1
42'
AF,-6
AFt
-8
-3.2
62'
(1.9
89)
3.33
2' ( 1
.883
)
0.10
7 (0
.074
)
MI
-,
AFt
-7
AFt
- 3
-0.1
13
(0.0
72)
AF1-
6 4.
096h
(1
.872
)
AF
,4
-0.1
42h
(0.0
69)
AS
,-,
(2.2
1 I)
AF
,-,
-6.9
14"
(2.4
80)
4.10
4'
AF,
-*
@,-
,o
-0.1
94"
(0.0
68)
3.52
1'
(1.9
23)
AS,-,
5.
442"
(2
.105
)
AF,-f
i 3.
364
(2.5
29)
ASt-1
0.
145h
(0
.069
)
AS,-,
-2.7
66
( 1.9
08)
AFt-1
2 0.
072
(0.0
75)
-3.5
95
(2.2
07)
ASt-7
5.
594"
(1
.916
)
AS, -
3
Coe
ffici
ents
of E
CM
AS1-
4 0.
074
(0.0
74)
MI
-,
2.13
2 ( 1
.895
)
AF
t-11
-3
.273
' ( 1
.903
)
AF,-'
4 0.
034
(0.0
71)
3.17
4'
( 1.8
50)
AF1-7
AF,
- 7
(1 37
2)
-5.5
58"
AS,-"
3.
184
(1.9
95)
AFI
-8
ASI-2
-0
.243
" 3.
500'
(0
.W)
(1.8
83)
AFt-1
2 AS,-,
(1.8
81)
(1.9
70)
3.08
3'
-3.2
77'
AS,
3.
179
(2.0
64)
AFr-8
AS
1 -0
.227
" -3
.613
" (0
.066
) (1
.171
)
0.17
4h
2.83
0 (0
.072
) (2
.082
)
AFt-1
, AS
,
AS,.,
ASt-6
(2.1
14)
ASt.9
-0
.100
(0
.067
)
4.66
1 '
(2.6
1 1)
-4.1
15h
AS1 -
3
ASt-8
-0
.258
" (0
.072
)
-3.2
80'
(1.8
84)
AS,-)
AS, -
2
-2.7
63
(1.8
36)
-0.0
76
(0.0
69)
AS,-,
AS,-h
(2.4
76)
(2.5
31)
6.98
4*
-3.1
95
AS,-,
" H
OL
(1.9
38)
(0.0
05)
-3.3
29'
-0.0
14"
AS14
-2
.061
(1
.898
)
AS
I-I1
3.38
4' (1
.909
)
AS,-4
5.
331"
( 1
.904
)
2.97
0 (1
.941
)
AS,-4
ASt-1
2 -2
.987
(1
.889
)
3
8'
0.09
03 :
L.
K' B
0.16
30
$' R' 0.13
73
K' 0.17
09
RZ
0.08
33
K2
0.12
72
8
AS,_
, H
OL
R2 r,
-3.9
11
-3.2
40"
-0.0
177h
0.
2553
3 AS
,-7
HO
L 82
5
(1.8
74)
(0.0
05)
2
(1.8
67)
(1 34
3)
(0.0
08)
5.64
6"
-0.1
40"
0.10
93
3 ii
rg
I As
def
ined
in T
able
I an
d th
e te
xt.
*HO
L is
the
coef
ficie
nt o
f th
e du
mm
y va
riab
le fo
r pre
- and
/or
post-
bank
hol
iday
obs
erva
tions
. Thi
s is d
efin
ed in
Tab
le II
I and
the
text
. '-
'AS
def
ined
in T
able
I.
The
coef
ficie
nts i
n th
e ta
ble
are
for
the
ECM
of
equa
tion
(4)
incl
usiv
e of
the
fir
st d
iffer
ence
of
the
curr
ent s
pot r
ate
and
12 la
gged
diff
eren
ces o
f cu
rren
t spo
t and
for
war
d
The s
tand
ard
erro
rs a
re in
par
enth
eses
( ).
z-
2 ra
tes,
for e
ach
subs
ampl
e of
a sp
ecifi
c in
terv
al, w
here
A is
the d
iffer
ence
oper
ator
. u
5 16 Journal of Forecasting Vol. 14, Iss. No. 6
of the models vary and different lags are required for adjustment to long-run equilibrium. These results further suggest that choice of interval at which the sets of subsamples are observed appears important for tests of equilibrium models.
To further test the parameters of the basic ECM of equation (4), a multivariate analysis of variance was performed conditioned on the intervals of the series. To reduce the impact of non- normality, the test was performed on the absolute deviation from the mean of each variable in the basic ECM. In all cases the usual F-statistic rejected the null hypothesis that the parameters of the sets of subsamples are similar across intervals. However, the null hypothesis that the slopes of the regressions are similar across intervals was only rejected for the Canadian dollar and the Austrian schilling.
Error-correction models and parameter stability The notion of parameter stability is important both for the forecasting performance of the final ECMs as well as the adequacy of the estimated coefficients over the period. To test the alternative hypothesis that the parameters of the final ECM follow (smooth) linear trends (see Farley and Hinich, 1970; Farley et al., 1975), we run OLS regressions in the form
RS,, , - S , = OL+ /?(F,- , - S,)+ A(F,- F,-I) + a‘T+ P’T(F,_l - S, ) + A ’ T ( F , - F , - , ) + E , (9)
The exact form of the linear trend model depends on the variables in each ECM in Table IV and involves testing whether a’ , B’ and A ’ are significantly different from zero, where T is the linear time trend. A significant result implies that the relevant parameter is not stable or constant over the sample period.
The results indicated that the parameter of the ECMs were not stable for 30% of the intervals and involved all currencies except the United States dollar. Both a’ and p’ were significantly different from zero at less than a 10% level for four sets of subsamples. Overall, the significant coefficients were generally among the Tuesday and month-end subsamples. Bleaney (1990) has shown that the linear trend test is more powerful than the midpoint Chow test against smooth trends in the parameters.
To further validate the time varying properties of the coefficients, the multi-period Chow test was also applied to the final ECM. The multi-period Chow test is more powerful relative to the cumulative sum (CUSUM) and CUSUM of squares (CUSUMSQ) tests (Ashley, 1984). The month-end subsample sets were therefore subdivided into eight (approximately) equal subgroups. All the other sets of subsamples were subdivided into four subgroups. This generated a reasonable number of observations within each subsample set for empirical analysis. The results in Table V indicate that the null hypothesis that the coefficients are invariant can be rejected for most sets of subsamples. Overall, the results suggest that the constant coefficient hypothesis cannot be accepted for the currencies.
Forecasting performance and structural change In order to assess the out-of-sample forecasting adequacy and performance of the final ECMs at various intervals, the last 24 observations of the sets of subsamples are used for post-sample forecasting. For the European currencies, the Predictive Failure test which tests the adequacy of the predictions was significant at only two intervals. This test was significant for the North American currencies in most cases. The two-period Chow test which tests for structural change was only significant among the North American currencies. Thus the two-period Chow test rejected the null hypothesis for 50% of the North American subsample sets. The error
Nathan Lael Joseph Cointegration, Error-correction models 5 17
Table V. Multi-period Chow test for structural stability in final error-correction model (ECM) of the sets of subsamples'
Ill 111
ECM model: RS,+ , - S,= a0 +Po(F,- I - S,) +Ao(F, - F , - 1 ) + AjAFf- j + d;As,- , + E , .-4@ 1 i = 1 i = o
Currency Monday Tuesday Wednesday Thursday Friday Month-end
Austrian schilling x2(33)
Deutsche inark x2(27)
162.784"
52.585"
245.587"
18.332
United Slates dollar ~ ~ ( 2 1 ) 29.128
French franc ~ ~ ( 3 6 )
Canadian dollar x 2 ( W
X2(33)* 91 ,674"
61.780a
25.29~5~
40.630"
64.502"
x2(36)
x 2 ( W
~ ~ ( 1 8 )
~ ~ ( 3 0 )
X2(3O)
X2(30)
X2(30)
X2W)
X2(33)
77,919"
65.794"
61.689"
33.072b
104.877 "
X 2 W
x2(21)
x2(39)
X2(30)
X2(21)
30.860'
21.992
124.459"
91.591 "
42. 139a
X2(W
XV4)
X 2 (2
X2(27)
X2(39)2
57.718a
7404.2 13"
44.907 a
159.528a
149.358"
X2(48)
X*(48)
x2(54)
X 2 ( W 2
X2(72)*
1 14.67fia
134.891"
126.5OOa
313.982"
235.529'
' As defined in Table I and the text. 'As defined in Table III and the text. "'As in Table I. ,yz test is adjusted for heteroscedasticity. The numbers in parentheses are the degrees of freedom. i, ... 4 (8) indicates that the multi-period Chow test was estimated with 4 (approx. equal) sub-periods for day of the week subsample sets and 8 (approx. equal) sub-periods for the month-end subsample sets.
autocorrelations were only significant for Monday and Friday forecast errors of the United States dollar, the month-end forecast errors of the Austrian schilling, and the Monday and Thursday forecast errors of the Canadian dollar. The error autocorrelations for the Tuesday and month-end forecast errors of the Deutsche mark were also significant. Summary statistics for the mean prediction error (MPE), mean absolute prediction error (MAPE), mean square prediction error (MSPE), and root mean square prediction error (RMSPE) were also examined. All four measures could be taken to imply a quadratic loss (cost) function which would lead to the selection of the subsample set with the minimum value. The MPEs were generally negative except for those of the Deutsche mark and the Austrian schilling (to some extent). The MPEs for the Deutsche mark were significant different from zero in two cases while those for the United States dollar were significantly different from zero at most intervals. The preponderance of negative MPEs suggests a consistent bias across sets of subsamples. Skewness (generally negative) and kurtosis tended to be significantly different from zero, except for Monday and Friday forecast errors (to some extent). The more robust MAPE tended to be larger for the Canadian and United States dollars and was significantly different from zero at a 1 % level for all currencies. Skewness and kurtosis remained significant. This may therefore cast some doubts on the validity of the test of forecasting adequacy.
To further assess these results, the non-parametric Wilcoxon signed-ranks test for two related samples was applied to the forecast errors. The null hypothesis is that there is no difference in the direction and relative magnitude of the forecast errors for the pair-wise forecast errors of different intervals. The null hypothesis that the absolute prediction errors are similar was rejected for five pair-wise subsamples. Taking the sign of the forecast errors into account, the
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null hypothesis was also rejected for seven pair-wise subsamples. In those instances where the null hypothesis was rejected this was primarily for the Deutsche mark forecast errors. Except for the Deutsche mark, the overall non-parametric Friedman test indicated that the null hypothesis of no difference in the forecast errors of the sets of subsamples of each currency could be accepted.
To test the equality of the expected square forecast errors, we employ a more robust procedure (see Granger and Newbold, 1986). This involves the Spearman rank-order correlation test on pairs of random variables s, and d, which are respectively the sum and difference of two forecast errors. That is, s, = e , , , + er,2 and d, = e, , , - e,,? where e, , , and e,,? are pairs of bivariate forecast errors. The null hypothesis is that there is no association between the ranks of expected square forecast errors, so that no subsample set used for forecasting is consistently better or worse than another. The results indicated that the null hypothesis was rejected in 48% of the cases. This was primarily for the Monday, Friday, and month-end square forecast errors relative to those of other intervals.
To assess the impact of structural changes on the out-of-sample forecasting performance of the ECMs, the first two dummy variables (for structural changes) were included in the final ECMs. The MPEs obtained also appeared to vary across subsamples and were of a similar magnitude relative to the MPEs without the dummy variables. Overall, these results suggest that the forecasting performance may be affected by the interval at which the sets of subsamples are observed.
As a final test of the forecasting performance of the final ECM at various intervals, we employ the non-parametric test (based on conditional probabilities) of Henriksson and Merton (1981). The test focuses on the direction rather than the magnitude of FX rate changes and therefore, facilitates the assessment of differing forecasting ability based on FX rate increases (upward directional changes) instead of decreases. The null hypothesis of no forecasting ability
Table VI. Directional forecasting performance of final error-correction model (ECM) for the last 24 observations’ based on conditional probabilities
.x=.y* \ * I \
Austrian French Canadian United Stares sdiilling Deutsche mark franc dollar dollar
Subsample [ 1)
Monday 0.0013 Tuesday 0.0170’ Wednesday 0.0853 Thursday 0.0382 Friday 0.0098 Month-end 0.0075
(2)
0.0013 0.01702 0.0853 0.0382 0.0098 0.0075
(1) (2) (1) (2) (1) (2) (1)
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0092 0.0137 1.0000 1.0000 0.5102 0.5102 0.0297 0.5667 0.1228 1.0000 1.OOOO 0.0137 0.0272 0.9634 0.0382 0.0382 1.0000 1.0000 0.0092 0.0092 0.0137 0.0046 0.0013 1.0OOO2 0.0853* 0.0046 0.0191 0.00942 1.0000 1.0000 0.0686 0.0686 0.00702 0.0075’ 0.1228’
(2)
1 .0000 0.0294 0.9634 0.0460 0.0046’ 0.1228’
’ As defined in Table I and the text.
The values given are the probabilities of successful directional forecasting. For example, the month-end series of the Austrian schilling indicates a significance value of 0.0075 which implies a confidence level of 0.9925. (1) The forecasts generated from the final ECM did not include the first two dummy variables for structural breaks. (2) The forecasts generated from the final ECM included the first IWO dummy variables for structural breaks.
As defined in Table I11 and the text.
Nathan Lael Joseph Cointegration, Error-correction models 5 19
is rejected if n , 5 x ( a ) where x*( a ) is the solution to: *
2 (:I) ( n:x) / ( z ) = 1 - confidence level (a ) y'=x
where
N , =the number of observations where the FX rates actually rose N 2 =the number of observations where the FX rates did not actually rise N = N , + N2 = the total number of observations n, = the number of successful predictions given that the FX rates rose n2 = the number of unsuccessful predictions given that the FX rates did not rise n = n, + n2 = total number of times FX rate rises were forecasted.
The results in Table VI indicate good out-of-sample forecasting performance for most currencies. For example, for the month-end subsample set of the Austrian schilling, the significance value of 0.0075 implies a confidence level of 0.9925. The results indicate that, except for the French franc, it is not too difficult to predict FX rate increases for the series under consideration, although the directional changes for Mondays were generally unpredictable. The absence of significant error-correction terms in the final ECMs does not appear to significantly affect the ability to forecast directional changes. These forecast results were not substantially different from those based on in-sample forecast errors.
SUMMARY AND CONCLUSIONS
This study has employed ECMs to forecast FX rates where the data-sampling procedures were consistent with the rules governing the delivery of FX contracts. For comparative purposes, the non-aligned month-end rates were also examined. The preliminary results indicated that the moments of the realized forecast error were dissimilar. Also, the number of cointegrating vectors appeared to vary by interval. This latter result would have strong implications for the forecasting performance of the ECMs as well as the extent to which the series within the sets of subsamples are tied together in the long run. Seasonal variation in the data appeared to vary by currency and subsample set. Empirical work involved with FX returns has generally reported strong systematic variation across currencies, and has partly attributed such variation to institutional arrangements in the FX and stock markets. Given the results on the seasonality of FX returns, one would expect that the realigned sets of subsamples would exhibit strong seasonal variation across currencies, even if the realized spot rates within each subsample set were not consistently observed on the same day as the forward and current spot rates. The variation in our results across currencies may be due (in part) to variation in the length of the maturity of FX contracts, as a result of differences in the number of trading holidays for each country. However, an important feature of our results is the statistical insignificance or non- existence of the error-correction term in the final ECMs for certain subsample sets. This has important implications for the extent to which disequilibrium shocks are eliminated from the system. Although the final ECMs exhibited strong structural breaks, it was relatively easy to accurately forecast the direction of FX rate changes even if the forecasting accuracy varied by subsample set. Overall, it appears therefore that the results of empirical work which involves FX data can be affected by sampling procedures adopted. Future work should attempt to adopt sampling procedure which are consistent with the nature of the FX data.
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Author’s biography: Dr Nathan Lael Joseph is a lecturer in accounting and finance at the Department of Accounting and Finance, University of Manchester, Manchester, UK. His research interests include international financial markets and the behaviour of multinational corporations.
Author’s address: Dr Nathan Lael Joseph, Department of Accounting and Finance, Manchester University, Manchester M13 9PL, UK.
