# Is the forward rate for the Greek drachma unbiased? A VECM analysis with both overlapping and non-overlapping data.

AbstractThis paper uses cointegration techniques to test the hypothesis that the forward rate for the Greek drachma-US dollar exchange rate is an unbiased predictor of the future spot rate. The relationship between cointegration and error correction models is exploited, and a full Vector Error Correction Model (VECM) is estimated with a priori assumptions. It is demonstrated that the unbiasedness hypothesis imposes restrictions not only on the elements of the cointegrating vector but also on the coefficients of the error correction equations, and both sets of restrictions are explicitly tested. These tests are conducted for the Greek drachma after the abolition of Greek capital controls and before entry to the ERM. Both overlapping and non-overlapping observations are used, and the results from these different data sets are compared. Some possible explanations are provided for the empirical failure of the forward rate unbiasedness hypothesis for Greece.

Key words : Forward rate unbiasedness, Foreign exchange rates, Cointegration, Greek drachma, Overlapping and non overlapping data

JEL Classification : F31, C4

Introduction

The question of forward rate unbiasedness touches on a number of important issues in financial markets--forecasting spot exchange rates, market efficiency, the expected cost of hedging and the presence of a risk premium in the forward market. If the forward exchange rate is an unbiased predictor of the future spot rate, exchange risk will have no effect on the expected profits of a hedged firm. However, if there is a bias, hedging alters the expected profits of a firm, and may involve the payment of a risk premium. It also means that the forward rate cannot be used directly as a forecast of the future spot rate, and may mean that the market is inefficient.

The forward rate unbiasedness hypothesis states that the foreign exchange forward rate is an unbiased predictor of the future spot rate. Thus, the forward rate at time t for delivery at time t+1 differs from the spot rate realised at time t+1 only by a random error :

[S.sub.t+1] = [f.sub.t] + [u.sub.t+1] (1)

where all the variables are in natural logarithms, and [f.sub.t] is the forward rate at time t for delivery at time [t.sub.t+1], [s.sub.t+1] is the spot rate at time t+1 and [u.sub.t+1] is a zero-mean forecast error which is serially uncorrelated and orthogonal to the information set [I.sub.t] available to investors at time t.

The unbiasedness hypothesis in (1) is a joint hypothesis based on two assumptions. First, we assume that market participants at time t set the forward rate for time t+1 equal to the expected future spot rate at time t+1:

E([s.sub.t+1]) = [f.sub.t] (2)

where E([s.sub.t+1]) is the market's expectation of the spot rate at time t+1, conditional on the information set [I.sub.t]. As long as market participants are risk-neutral and there are no transactions costs, competition will ensure that (2) holds continuously. Second, we assume that expectations are formed rationally and thus:

[s.sub.t+1] = E([s.sub.t+1]) + [u.sub.t+1] (3)

i.e., the spot rate at time t+1 differs from its expected value only by a zero-mean forecast error [u.sub.t+1], which is serially uncorrelated and orthogonal to the information set, I, available to investors at time t. Combining the risk neutrality and rational expectations hypotheses we obtain the forward rate unbiasedness condition (FRUC), as expressed in (1)

Early Tests of the FRUC

Researchers have been conducting empirical tests of the FRUC for more than twenty years. Early studies estimated simple regressions of the form :

[s.sub.t+1] = [alpha] + [beta][f.sub.t] + [u.sub.t+1] (4)

by OLS and then tested the joint hypothesis that [alpha] equals zero, [beta] equals unity, and the forecast error ut+1 is serially uncorrelated, e.g., Frenkel (1) Frankel (2), Longworth (3), MacDonald (4) and Murfin and Ormerod (5). In the 1980s, researchers noted that exchange rates are non-stationary variables, and most restated the FRUC by subtracting [S.sub.t] from both sides of (1) to induce stationarity:

[s.sub.t+1] - [s.sub.t] = [f.sub.t] - [s.sub.t] + [u.sub.t+1] (5)

Equation (5) states that the forward premium is an unbiased predictor of the future change in the spot rate and may be examined by running a regression of the form

[s.sub.t+1] - [s.sub.t] = [alpha] + [beta]([f.sub.t] - [s.sub.t]) + [u.sub.t+1] (6)

and testing the joint hypothesis at [alpha] equals zero, [beta] equals unity, and the forecast error [u.sub.t+1] is serially uncorrelated. Unbiasedness tests based on (6) have been conducted by a large number of researchers for a variety of currencies and time periods, e.g., Agmon and Amihud, (6) Bilson, (7) Boothe and Longworth, (8) Cumby and Obstfeld (9) and Fama. (10) The vast majority of these studies not only rejected the joint hypothesis that [alpha] = 0 and [beta]=1, but also found that the slope coefficient [beta] was negative for nearly all currencies and time periods studied.

Finally, in the late 1980s the concept of cointegration fundamentally changed the way we treat non-stationary variables. Widely used econometric techniques, such as first-differencing both the regress and the regressors to induce stationarity are not valid if these variables are cointegrated. Therefore, the existence of a cointegrating relationship between spot and forward exchange rates casts doubt on the validity of the 'traditional' tests described above (a). When researchers tested for cointegration between spot and forward rates they found [s.sub.t+1] and [f.sub.t] to be cointegrated for almost every currency and time period studied, e.g., Baillie and Bollerslew, (14) Hakkio & Rush and Copeland. (15) Hence, new tests taking into account the cointegration relationship between spot and forward rates were needed.

Cointegration-Based Tests of the FRUC

Cointegration is a property of non-stationary variables. Two non-stationary variables (e.g. [s.sub.t+1] and [f.sub.t]) are said to be cointegrated if they are integrated of the same order (e.g., first-order integrated, I(1)) (b) and there exists a linear combination of [s.sub.t+1] and [f.sub.t] that is stationary. This unique linear combination is known as the 'cointegrating equation' or the 'cointegrating vector', and can be thought of as a long-run equilibrium relationship that forces spot and forward rates to move together. Eagle and Grange (16) showed that, if [s.sub.t+1] and [f.sub.t] are cointegrated and their cointegrating equation is [s.sub.t+1] [alpha] + [beta] [f.sub.t], then they have an error correlation representation of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7b)

The above specification is known as a Vector Error Correction Model (VECM). It resembles a Vector Autoregression (VAR) model in first differences with an error correction term added to the right hand side of each equation. This term respresents the long-run dynamics of the model. Coefficients [gamma].sub.s] and [[gamma].sub.t] denote the fraction of the deviation from long-run equilibrium observed at time t-1 which is 'corrected' at time t through changes in the spot and forward rates respectively. The lagged difference terms represent the short-run dynamics of the model, and their coefficients, [[kappa].sub.i], [[lambda].sub.i], [[mu].sub.i], [[upsilon].sub.i], i =1 standard interpretation of the coefficients of a VAR model.

Finally, the error terms [[epsilon].sub.t+1] and [[eta].sub.t+1] are, by definition, serially uncorrelated. (c)

Two Testable Conditions

Given that both the spot and forward exchange rates are non-stationary variables, a randomly chosen linear combination of these variables, e.g., their difference or realised forecast error, [u.sub.t+1] = [s.sub.t+1] - [f.sub.t], will probably also be non-stationary. Being non-stationary, [s.sub.t+1]-[f.sub.t] will have a non-constant mean and an infinite variance, so that [s.sub.t+1] and [f.sub.t] may drift infinitely far apart, which is inconsistent with the FRUC.

However, if [s.sub.t+1] and [f.sub.t] are cointegrated, then there will exist a linear combination of [s.sub.t+1] and [f.sub.t] that is stationary; and the two exchange rates will generally move together. Therefore, cointegration between [s.sub.t+1] and [f.sub.t] is a necessary condition for forward rate unbiasedness (Condition 1). However, the sole existence of a cointegrating vector between [s.sub.t+1] and [f.sub.t] is not sufficient for the FRUC to apply. In fact, the elements [alpha] and [beta] of this cointegrating vector must equal zero and unity respectively, so that the forecast error [s.sub.t+1] is stationary (Condition 2).

Conditions 1 and 2 have been tested by a number of researchers in the 1990s. For instance, Moore (13) used monthly data on the DM-$ and [pounds sterling]-$ exchange rates and found [s.sub.t+1] and [f.sub.t] to be cointegrated for both currencies (Condition 1). However, his dam produced a clear rejection of the hypothesis [alpha] = 0 and [beta] = 1 (Condition 2), again for both currencies. Luintel and Paudyal (17) found that [s.sub.t+1] and [f.sub.t] were cointegrated for all five currencies in their study of C$, FFr, DM, [yen], $, all against [pounds sterling] (Condition 1). However, they rejected Condition 2 for all except the C$.

Orthogonality Condition

Imposing Conditions 1 and 2 on (7) and subtracting (7b) from (7a) we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

where [u.sub.t+1] = [[epsilon].sub.t+1] - [[eta].sub.t+1], the FRUC requires that the forecast error [s.sub.t+1] - [f.sub.t] be non autocorrelated and orthogonal to any variable whose value is known at time t (the 'orthogonality' condition). Given (8), this implies:

[[gamma].sub.s] - [[gamma].sub.f] + 1 = 0 and [[kappa].sub.i]] - [[mu].sub.i] = 0 and [[lambda].sub.t] - [[upsilon].sub.i] = 0, i=1, ..., [[rho].sub.1] (Condition 3).

Previously, the most common way to test the orthogonality condition has been to regress [s.sub.t+1] - [f.sub.t] on a set of variables whose value is known at time t, usually past forecast errors, and test whether their coefficients are jointly equal to zero. Researchers have used either OLS (Geweke and Feige, (18) Frankel, and Delcoure, et. al., (19)), or GMM (Hansen and Hodrick, (20) Cumby & Obstfeld and Hsieh, (21)). Another group of researchers have tested the orthogonality condition using a VAR model, e.g., Hakkio, (22) Baillie, et al. (23) and Levy and Nobay. (24) The results obtained by the majority of these papers were unfavourable to the FRUC.

The main advantage of the test we propose (Condition 3) over the ones mentioned above is that it orginates directly from the bivariate error correction model, thus fully exploiting the cointegration property of [s.sub.t+1] and [f.sub.t]. There have also been a couple of attempts to develop a test of the orthogonality condition directly from an ECM: Hakkio and Rush, using monthly data on the DM-$ and [pounds sterling]-$ exchange rates, estimated the following single-equation ECM:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and conducted--among other tests--test of the hypothesis:

H0: [[gamma].sub.s] = -1 and [[gamma].sub.i] = [[delta].sub.i] = 0, i = 1, ..., [rho] - 1 (10)

However, as noted by Johansen, a single-equation model like (9) is valid when [[gamma].sub.f] = 0, i.e., when the forward rate is weakly exogenous and displacement from long-run equilibrium is corrected only through changes in the spot rate. (d) Moore, although estimating the complete bivariate model (7), proposed the same test--hypothesis (10). (e) It is easy to see that Condition 3 reduces to hypothesis (10), if we assume forward rates are exogenous. However, recent evidence (f) suggests that it is the spot rather than the forward rate that is exogenous. The latter finding casts doubt on the validity of models based on the assumption of exogenous forward rates, e.g. single equation ECMs--Hakkio and Rush, Barnhart and Szakmary (25)--and the 'traditional' tests described above. To sum up, Condition 3 is superior to earlier tests since it exploits the cointegration property of [s.sub.t+1] and [f.sub.t], without making any a priori assumptions regarding the exogeneity of either the spot or the forward rate.

Tests of the FRUC on Data from the Greek Foreign Exchange Market

Until the early 1990s, when liberalisation began, the Greek foreign exchange market was characterised by strict capital controls. This paper is the first to use data from this recent period of a more llberalised regime (1993-1998), and four samples of forward and spot data on the Greek drachma (GRU)-US$ exchange rate are used. (g) Sample A contains 112 weekly observations of the spot and one-week forward GRD-US$ exchange rates for the period January 24th 1996 to March 13th 1998. (h) Samples B and C each contain 1,125 daily observations for the spot and forward GRD-US$ exchange rate for the period November 22nd 1993 to March 13th 1998. The length of the forward contracts is one month for sample B and three months for sample C. Finally, sample D uses the same forward contracts and covers the same period as sample B, but contains only one observation per month. Therefore, observations in samples B and C are overlapping, while those in A and D are non-overlapping.

The data for sample A were supplied by Commercial Bank of Greece S.A. and come from Reuters. The data for samples B, C and D were collected from the Financial Times and their source is also Reuters. We matched up the current forward rate, ft, with the appropriate future spot rate, st+1, using the procedure described in Bekaert and Hodrick. (27) The use of both overlapping and non-overlapping data may look unfamiliar. Indeed, overlapping data in standard OLS regressions induces serial correlation, leading to spurious rejection of the null hypothesis. However, the use of overlapping data with a VAR or VECM model presents no difficulties because these models include sufficient lagged terms to remove all serial correlation. As a consequence, the researcher estimating a VECM is free to choose whether to use overlapping or non-overlapping data. Nevertheless, most previous cointegration-based papers have used non-overlapping data (which greatly reduces the sample size), (i) a few have used overlapping observations, (j) and none has used both. We included samples with overlapping data in our tests for comparative purposes, in order to investigate the effects of using such data, relative to using non-overlapping data.

Research Findings

Tests for Stationarity

Before conducting cointegration tests, we must verify that [s.sub.t+1] and [f.sub.t] are integrated of the same order. The Phillipe-Perron statistic tests whether a variable is non-stationary, as against the alternative hypothesis that it is stationary; and the results of this test are reported in Table 1. The null hypothesis that the levels of the spot and forward rates are non-stationary is not rejected for any sample. However, this hypothesis is easily rejected for the first differences of the spot and forward rates; hence, the first differences of both rates are stationary. Taken together, the above results imply that both the spot and forward rates in all the samples are first-order integrated, or I(1).

Determining the VAR Order

To determine the appropriate VAR order for each sample, VAR models of up to the fourth order were estimated. For each model the Akaike Information Criterion and the Schwarz Bayesian Criterion were calculated. The selection procedure involves choosing the VAR order ([rho]) that produces the highest value for these criteria. This choice is then tested against alternative lag lengths with a likelihood ratio (LR) test.

To test the null hypothesis that the order of VAR is [rho] against the alternative that it is [rho] +1, the relevant LR-statistic is given by:

LR([rho], [rho] + 1 ) = 2 [LL([rho] + 1) - LL([rho])]

where LL([rho]) refers to the maximised value of the log-likelihood function for the [[rho].sub.th]-order VAR@. Using the above procedure we determined that the appropriate VAR order is 1 for samples A and D (the non-overlapping samples), and 2 for samples B and C (the overlapping samples).

Tests for Cointegration Between [s.sub.t=1] and [f.sub.t] : Condition 1

The existence of a cointegrating vector between [s.sub.t=1] and [f.sub.t] was investigated within the framework developed by Johansen. (28) In general, the number of cointegrating vectors, r, between two variables may range from zero to two. If there are zero cointegrating vectors, then [s.sub.t+1] and [f.sub.t] are both I(1) and not cointegrated with each other. If there is one cointegrated vector, then [s.sub.t+1] and [f.sub.t] are both I(1) and not cointegrated with each other. Finally, the existence of two cointegrating vectors implies that [s.sub.t+1] and [f.sub.t] are both I(0), i.e., stationary. Table 2A reports the LR-statistics for the hypothesis that r = 0 against the alternative that r [greater than or equal to] 1, and for the hypothesis that r [less than or equal to] l against the alternative that r = 2. The results of the LR-tests are qualitatively the same for all samples. At the 5 per cent significance level, the null hypothesis that r = 0 is rejected in favour of the alternative that r [greater than or equal to] 1, while the hypothesis that r [less than or equal to] 1 is not rejected in favour of r = 2. This result indicates the existence of exactly one cointegrating vector between [s.sub.t+1] and [f.sub.t]; therefore all samples are consistent with Condition 1 of the FRUC.

Because cointegration tests may be sensitive to the specification of the model regarding the intercept term, [alpha], we repeated the above tests using an unrestricted intercept, i.e., placing [alpha] outside the cointegrating vector. Results of these tests are reported in Table 2B and are not qualitatively different from those reported in Table 2A. We also tested the null hypothesis that the intercept is restricted, against the alternative that it is unrestricted. The relevant LR-statistic ([LR.sub.5]) is the difference between the maximal eigenvalue statistic obtained with a restricted intercept ([LR.sub.1] statistic) and that obtained with an unrestricted intercept. ([LR.sub.3] statistic). [LR.sub.5] is not significant for any sample at the 5 per cent level, which indicates that the restricted intercept model (7) is not rejected.

Tests of Restrictions on the Elements of the Cointegrating Vector : Condition 2

The second cointegration-based test of the FRUC involves a test of the hypothesis that the coefficients [alpha] and [beta] of the cointegrating vector equal zero and unity respectively. Although this hypothesis cannot be tested using a standard F-test because [s.sub.t+1] and [f.sub.t] are non-stationary, a procedure developed by Johansen and Juselius (29) provides a valid test. Table 3 reports the estimated values of [alpha] and [beta] and their standard errors. Coefficient [beta] is very close to its theoretical value of unity in all the samples. The LR-statistic for the joint hypothesis that [alpha] = 0 and [beta] = 1 is given in the last row. This joint hypothesis cannot be rejected for the overlapping data samples (B and C), but is easily rejected for the two samples with non-overlapping observations (A and D). Thus, only the samples with overlapping observations satisfy Condition 2, and so Condition 3 is only tested on samples B and C.

Error Correction Equations and Diagnostic Tests

Table 4 reports the estimates of Equations (7a) and (7b). Since [rho] = 1 for samples A and D, no lagged terms are included in the error correction equations for these samples, while the equations for samples B and C include first order lags because [rho] = 2 for these samples. Normality of the residuals is tested first by means of the Jarque-Bera statistic. The null hypothesis that the residuals are normally distributed is rejected at the one per cent significance level for samples B and C and the forward equation for sample A. Rejection of normality implies that we must exercise care when interpreting the results of the various tests of coefficient restrictions. However, in the subsequent tests, whenever there is a rejection of the null hypothesis, this rejection is at the one per cent significance level.

Ramsey's 'RESET' statistic tests the functional form of the model by adding the square of the fitted values to the right-hand side of the regression. The null hypothesis is not rejected for any sample, which indicates no problems with the functional form of the model. Finally, the LM-statistic for heteroscedasticity rejects the null hypothesis of homoscedasticity for samples B and C (the overlapping samples). In consequences, White's (30) heteroscedasticity-consistent covariance matrix was used to calculate the standard errors for these samples.

The LM-statistics for up to third order serial correlation are reported in Table 4B. This statistic was calculated for orders of serial correlation up to 12 and serial correlation was found only for the forward Equation (7b) of sample D. This latter finding, which is not caused by the use of overlapping data, is of course inconsistent with the FRUC, but--since the FRUC has already failed for this particular sample--the general conclusions do not change.

Exogeneity

Given that spot and forward rates are cointegrated, one of three alternatives must apply. First, both spot and forward rates adjust to long-run disequilibrium situations ([[gamma].sub.s] [not equal to] 0, [[gamma].sub.s] [not equal to] 0). Second, spot rates weakly exogenous and all adjustment is via changes in the forward rate ([[gamma].sub.s]=0, [[gamma].sub.f] [not equal to] 0). Third, forward rates are weakly exogenous and all adjustment is via changes in the spot rate ([[gamma].sub.s] [not equal to] 0, [[gamma].sub.f] [not equal to] =0). The Wald tests reported in the first two rows of Table 4C indicate that the hypothesis [[gamma].sub.s]=0 cannot be rejected for any sample at the 5 per cent significance level, while the hypothesis [[gamma].sub.f] = 0 is easily rejected at the one per cent level for all samples. Thus, although many earlier tests of the FRUC have been based on the assumption that the forward rate is exogenous, it is the spot rate which is exogenous.

Tests of Restrictions on the Coefficients of the Error Correction Equation :Condition 3

Since only samples B and C satisfy Condition 2, Condition 3 may only be tested on these samples. Wald tests of the hypothesis [H.sub.0]: [[gamma].sub.s] - [[gamma].sub.f+1] = 0, reported in the third row of Table 4C, indicate rejection of the hypothesis at the one per cent level for both samples. Finally, the joint hypothesis [H.sub.0]: [[gamma].sub.s] - [[gamma].sub.f+1] = 0 & [[kappa].sub.1] - [[mu].sub.1] = 0 & [[lambda].sub.1] - [[upsilon].sub.1] = 0 is also rejected for both samples at the one per cent level, as the statistics reported in the last row of the table indicate. Thus, neither sample satisfies Condition 3.

The above tests for the GRD-US$ show that the forward rate is cointegrated with the spot rate, i.e., there is a long-run relationship between the two exchange rates that tends to keep them close together (Condition 1). This is consistent with the FRUC for all four samples. The second finding is that the hypothesis that the coefficients [alpha] and [beta] of the cointegrating vector equal zero and unity respectively (Condition 2) is easily rejected when non-overlapping data is used, but cannot be rejected when overlapping data is used. This supports the FRUC for overlapping data, but not for non-overlapping data. Third, the orthogonality condition (Condition 3)--which is tested only for overlapping data--is easily rejected. This latter finding leads to the rejection of the FRUC for overlapping data as well. Therefore, none of the samples support the FRUC.

Possible Explanations for the Empirical Findings

Three possible explanations for the bias in the Greek forward market will be offered. First, if the rational expectations hypothesis holds continuously (i.e., Equation 3), the rejection of the FRUC could be due to the presence of a risk premium. The forward rate bias ([s.sub.t+1] - [f.sub.t]) has generally been negative for the period covered by the samples. Assuming rational expectations, this implies that market participants set the forward rate above their expectation of the future spot rate, i.e., they required a positive risk premium in order to purchase drachmas forward. Although this explanation may seem logical, earlier attempts to relate the failure of the FRUC to the presence of a risk premium in exchange rates have generally failed. (k)

Second, investors may set the forward rate equal to the expected future spot rate, but their expectations are biased. A bias in expectations may be caused by the 'learning hypothesis'. (33) If investors are in the process of learning about a new exchange rate regime or monetary policy, they may make systematic forecast errors. In the sample period, the drachma operated under a liberalised regime that differed from the previous regime of capital controls.

Third, the 'peso' hypothesis (34) states that, although expectations are formed rationally, forecast errors may appear to have a systematic pattern if market participants anticipate an 'event' (e.g., a large shift in monetary policy), which does not occur in-sample. From the early 1990s and until ERM entry on March 16th, 1998 Greek monetary policy was strict, and the exchange rate policy announced by the Greek authorities may not have been viewed as fully credible. (l) Each year the Greek monetary authorities adopted a target devaluation rate for the drachma that was well below the expected inflation rate differential between Greece and its main trading partners, and participants in the Greek foreign exchange market may rationally have expected the authorities to devalue the drachma at any time. While rational, their behaviour appears irrational because the long anticipated devaluation occurs out-of-sample.

Conclusions

This paper is the first to test the forward rate unbiasedness hypothesis for the Greek drachma after the abolition of capital controls, and this was done using a full VECM analysis, including a new test. The data comprised weekly and monthly forward rates for the period November 1993 to March 1998, with both overlapping and non-overlapping observations. Three conditions, which must jointly hold for the forward rate to be an unbiased predictor of the future spot rate, were tested. The forward rate was cointegrated with the future spot rate for all four samples (Condition 1). However, the samples with non-overlapping data rejected the restrictions imposed on the elements of the cointegrating vector (Condition 2), while the samples with overlapping data rejected the restrictions imposed on the coefficients of the error correction equations (Condition 3). Hence, no sample satisfies all three conditions, and forward rate unbiasedness is rejected for the Greek drachma-US$ exchange rate. Another important finding is that spot rates are exogenous, which supports the findings of Moore and Norrbin and Reffett and contradicts the implicit assumption made in many earlier papers that forward rates are exogenous.

Three possible explanations for the bias in the forward rate were considered: a risk premium, the learning hypothesis and the peso problem; of which only the second questions market rationality. Finally, this paper has investigated how the use of overlapping and non-overlapping data affects the results. While the FRUC is rejected for both types of data, the detailed results differ, suggesting that in other situations the analysis could be affected by the type of data used.

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(26.) Norrbin, S. C. and Reffett, K. L., Exogeneity and Forward Rate Unbiasedness, Journal of International Money and Finance 05: 1996)

(27.) Bekaert, G. and Hodrick, R. J., On Biases in the Measurement of Foreign Exchange Risk Premiums, Journal of International Money and Finance (12: 1993)

(28) (i) Johansen, S., Statistical Analysis of Cointegrating Vectors, Journal of Economic Dynamics Control (12: 1988) (ii) Johansen, S., Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica (59:1991)

(28) (iii) Johansen, S., Co-Integration in Partial Systems and the Efficiency Single Equation Analysis, Journal of Econometrics (52: 1992) (29) (i) Johansen, S. and Juselius, K., Maximum Likelihood Estimation and Inference on Cointegration with Applications to the Demand for Money, Oxford Bulletin of Economics and Statistics (52: 1990)

(ii) MacKinnon, J., Critical Values for Cointegration Tests, in R.F. Engle and C. Granger (eds.), Long-Run Economic Relationships, (Oxford : 1991) (30.) White, H., A Heteroscedasticity-Consistent Covariance Estimator and A Direct Test for Heteroscedasticity, Econometrica (48: 1980)

(31.) Domowitz, I. and Hakkio, C. S., Conditional Variance and the Risk Premium in the Foreign Exchange Market, Journal of International Economics (19: 1995)

(32.) Christodoulakis, N. M. and Kalyvitis, S. C., Efficiency Testing Revisited : A Foreign Exchange Market with Bayesian Learning, Journal of International Money and Finance (16: 1997)

(33.) Lewis, K., Changing Beliefs and Systematic Rational Forecast Errors with Evidence from Foreign Exchange, American Economic Review (79:1989)

(34.) Krasker, W. S., The Peso Problera in Testing the Efficiency of Forward Exchange Markets, Journal of Monetary Economics (6:1980).

The authors wish to thank Jan Podivinsky of University of Southampton for his very helpful advice in performing some of the joint hypothesis tests.

The authors own full responsibility for the contents of the paper.

(a) See Hakkio and Rush (11), Backus, et al. (12) and Moore (13).

(b) A non-stationary variable is said to be I(1) if its first difference is stationary.

(c) An appropriate VAR order (p) must be chosen so that serial correlation is eliminated.

(d) The forward rate is weakly exogenous when [y.sub.t] = 0 and strongly exogenous when there are no lagged terms in the model and [[gamma].sub.f] = 0.

(e) Moore assumed [s.sub.i] - [alpha] - [beta][f.sub.t] (instead of [s.sub.t+1] - [alpha] - [beta][f.sub.1]) is the cointegrating vector. However, it is the future spot rate that should be cointegrated with the current forward rate.

(f) e.g., Norrbin and Reffett (24), and the present paper.

(g) Christodoulakis and Kalyvitis tested the FRUC for the GRD-US$ rate on older data and rejected the unbiasedness hypothesis. Their tests were based on Equation (6), and so assumed that spot and forward rates are not cointegrated and that forward rates are exogenous.

(h) On Monday March 16th 1998, the Greek Drachma entered the Exchange Rate Mechanism (ERM) of the European Monetary System and was devalued by 14 per cent. Including the post-ERM period in the data set would lead to highly non-normal errors and unstable estimates, and so the data period was truncated at this point.

(i) e.g., Hakkio and Rush, Moore, Barnhart and Szakmary, Norrbin and Reffett.

(j) e.g., Luintel and Paudyal.

(k) See e.g. Frankel and Domowitz and Hakkio (31).

(l) See Christodoulakis and Kalyvitis (32) for a relevant discussion.

NIKOLAOS ZACHARATOS, ESQ. and PROFESSOR CHARLES SUTCLIFFE Accounting and Finance Division School of Management University of Southampton Southampton, U.K.

TABLE 1 PHILLIPS-PERRON(PP) TESTS FOR STATIONARITY Sample A Sample B PP-statistic on [s.sub.t] -0.373 -0.049 PP-statistic on [DELTA][s.sub.t] -11.241 * -35.921 * PP-statistic on [f.sub.t] -0.364 -0.129 PP-statistic on [DELTA][f.sub.t] -10.963 * -36.729 * 5% critical value ** -2.887 -2.864 1% critical value ** -3.490 -3.439 Sample C Sample D PP-statistic on [s.sub.t] -0.362 -0.141 PP-statistic on [DELTA][s.sub.t] -35.466 * -0.750 * PP-statistic on [f.sub.t] -0.115 -0.137 PP-statistic on [DELTA][f.sub.t] -36.976 * -6.630 * 5% critical value ** -2.864 -2.920 1% critical value ** -3.439 -3.565 * Denotes rejection of the null hypothesis that the variable is non-stationary at the 1 % level. ** The critical values shown are those computed by MacKinnon. (21) The PP-statistic tests the null hypothesis that a variable is non-stationary, as against the alternative that the variable is stationary. For all samples, both [s.sub.t] and [f.sub.t] are found to be non-stationary, while [DELTA][s.sub.t] and [DELTA][f.sub.t] are stationary; thus both [s.sub.t] and [f.sub.t] are first-order integrated, or 1(I). TABLE 2 JOHANSEN COINTEGRATION TESTS: CONDITION 1 A. RESTRICTED INTERCEPT Null Alternative Statistic Hypothesis Hypothesis [LR.sub.1] r = 0 r [greater than or equal to] 1 [LR.sub.2] r [less than or equal r = 2 to] 1 5 % critical Statistic Sampe A Sample B Sample C Sample D value [LR.sub.1] 294.500 * 52.685 * 15.998 * 178.907 * 15.870 [LR.sub.2] 2.857 1.515 0.978 1.078 9.160 * Denotes rejection of the null hypothesis at the 5% level. The cointegration tests exhibited in Table 2A are based on a system with a restricted intercept (Model 7). The LR-statistics are based on the maximal eigenvalue of the stochastic matrix. Results indicate the existence of exactly one cointegrating vector (r = 1) B. UNRESTRICTED INTERCEPT [LR.sub.3] r = 0 r [greater than or equal to] 1 [LR.sub.4] r [less than r = 2 or equal to] 1 [LR.sub.5]~[chi square] (1) constant is constant is (= [LR.sub.1]-[LR.sub.3]) restricted unrestricted [LR.sub.3] 292.114 * 52.012 * 15.495 * [LR.sub.4] 0.001 0.058 0.011 [LR.sub.5]~[chi square] (1) (= [LR.sub.1]-[LR.sub.3]) 2.386 0.673 0.503 [LR.sub.3] 177.813 * 14.880 [LR.sub.4] 0.009 8.070 [LR.sub.5]~[chi square] (1) (= [LR.sub.1]-[LR.sub.3]) 1.094 3.841 * Denotes rejection of the null hypothesis at the 5% level. The cointegratinn tests exhibited in Table 2B are based on a system with an unrestricted intercept. The LR-statistics are based on the maximal eigenvalue of the stochastic matrix. Results again indicate the existence of exactly one cointegrating vector (r = 1). The [LR.sub.5] statistic tests the null hypothesis that the intercept is restricted, against the alternative that it is unrestricted. LRs is distributed [chi square] with (n-r) degrees of freedom, where in denotes the number of variables under consideration (i.e., m = 2), and r is the number of cointegrating vectors (i.e., r = 1). Since none of the LRs statistics is significant, Model (7) and not the unrestricted Model is the approach representation. TABLE 3 JOHANSEN-JUSELIUS TESTS OF RECTRICTIONS ON THE ELEMENTS OF THE COINTEGRATING VECTOR: CONDITION 2 Sample A Sample B Sample C Sample D Intercept [alpha] * 0.0021 -0.0643 -0.2395 -0.0422 (0.0199) (0.2443) (0.9308) (0.0425) Coefficient [beta] * 0.9992 1.0104 1.0397 1.0063 (0.0035) (0.0442) (0.1684) (0.0077) [H.sub.0]: [alpha] = 0 45.032a 5.183 3.194 81.506 ** & [beta]-1 *** (0.000 (0.075) (0.202) (0.000) * Estimated values of a and [beta]. Standard errors appear in the parentheses ** Denotes rejection of the null hypothesis that [alpha] = 0 and [beta] = 1 at the I% level. *** An LR-test of the joint hypothesis [alpha] = 0 and [beta] = 1. This statistic was suggested by Johansen and Juselius (29) and is distributed [chi square] with 2 degrees of freedom. Marginal significance levels appear in the parentheses. The null hypothesis is not rejected for samples B and C (overlapping data), while it is easily rejected at the 1% level for samples A and D (non-overlapping data) @ Under the null hypothesis, LR (p, p+1) is distributed asymptotically as a [chi square] variable with lit' degrees of freedom, where lit is the number of variables under consideration, i.e., m = 2 in our case. TABLE 4A ESTIMATES OF THE ERROR CORRECTION EQUATIONS (7A-7B) Spot Equation (7A) Sample Sample Sample Sample A B C D Adjustment -0.015 -0.004 0.0001 0.074 Coefficients (0.092) (0.008) (0.005 (0.138) ([[gamma].sub.s], [[gamma].sub.f]) * Coefficients on -0.078 -0.082 [][s.sub.t] NA (0.041) (0.042) NA ([[kappa].sub.1], [[mu].sub.1]) * Coefficients on -0.025 0.026 [][f.sub.t-1] NA (0.031) (0.029) NA ([[lambda].sub.1], [[upsilon.sub.1]) * Log-likelihood 353.9 4,164.2 3,996.8 123.2 Jarque-Bern 2.515 329.7 ** 331.3 ** 1.534 Normality test *** (0.284) (0.000) (0.000) (0.464 RESET functional 0.833 0.833 2.157 0.174 form test * (4) (0.361) (0.361) (0.142) (0.676) Heteroscedasticity 0.167 24.680 ** 34.613 ** 0.286 LM-test @ (0.683) (0.000) (0.000) (0.593) Forward Equation (7B) Sample Sample Sample Sample A B C D Adjustment 0.952 0.053 0.017 0.970 Coefficients (0.024) (0.007) (0.004) (0.023) ([[gamma].sub.s], [[gamma].sub.f]) * Coefficients on -0.080 -0.067 [][s.sub.t] NA (0.033) (0.035) NA ([[kappa].sub.1], [[mu].sub.1]) * Coefficients on -0.076 -0.116 [][f.sub.t-1] NA (0.039) (0.040) NA ([[lambda].sub.1], [[upsilon.sub.1]) * Log-likelihood 498.4 4,177.0 3,928.4 211.5 Jarque-Bern 46.1 ** 294.9 ** 1,188.2 ** 4.433 Normality test *** (0.000) (0.000) (0.000) (0.109) RESET functional 0.947 1.115 2.007 0.121 form test * (4) (0.330) (0.291) (0.157) (0.727) Heteroscedasticity 0.990 14.708 ** 4.447 * (3) 0.882 LM-test (@) (0.320) (0.000) (0.035) (0.348) TABLE 4B LM-TESTS FOR SERIAL CORELATION @@ Spot Equation (7a) Order of Serial Sample Sample Sample Sample Correlation (n) A B C D 1 1.105 0.238 0.163 1.371 (0.293) (0.625) (0.686) (0.242) 2 2.929 0.606 1.038 1.467 (0.231) (0.739) (0.595) (0.480) 3 4.884 1.984 2.568 1.643 (0.180) (0.576) (0.463) (0.650) Forward Equation (7b) Order of Serial Sample Sample Sample Sample Correlation (n) A B C D 1 0.046 0.904 0.615 23.945 ** (0.830) (0.342) (0.433) (0.000) 2 0.215 0.932 0.754 24.026 ** (0.898) (0.627) (0.686) (0.000) 3 0.549 4.059 1.353 25.228 ** (0.908) (0.255) (0.716) (0.000) TABLE 4C HYPOTHESIS TESTS Sample A Sample B [H.sub.0]: [[gamma].sub.s] = 0 (@2) 0.027 0.291 (0.869) (0.589) [H.sub.0]: [[gamma].sub.f] = 0 (@2) 1,475.8 ** 47.147 ** (0.000) (0.000) [H.sub.0]: [[gamma].sub.s]-[[gamma].sub.f] N/A 8,173.8 ** + 1 = 0 (@3) (0.000) [H.sub.0]: [[gamma].sub.s]-[[gamma].sub.f] N/A 8,540.5 ** + 1 = 0 & [[kappa].sub.1]-[[mu].sub.1] = (0.000) 0 & [[lambda].sub.1]-[[upsilon].sub.1] = 0 (@4) Sample C Sample D [H.sub.0]: [[gamma].sub.s] = 0 (@2) 0.0001 0.291 (0.995) (0.589) [H.sub.0]: [[gamma].sub.f] = 0 (@2) 17.612 ** 1,690.8 ** (0.000) (0-000). [H.sub.0]: [[gamma].sub.s]-[[gamma].sub.f] 28,572.7 ** N/A + 1 = 0 (@3) (0.000) [H.sub.0]: [[gamma].sub.s]-[[gamma].sub.f] 28,951.9 ** N/A + 1 = 0 & [[kappa].sub.1]-[[mu].sub.1] = (0.000) 0 & [[lambda].sub.1]-[[upsilon].sub.1] = 0 (@4) * Estimates of the relevant VECM coefficients. Standard errors appear in the parentheses There are no estimates of [[kappa].sub.1], [[lambda].sub.1], [[mu].sub.1], and [[upsilon].sub.1] for samples A and D because these samples are first-order VARs. ** Denotes rejection of the null hypothesis at the 1% level. *** A test of the hypothesis that the errors are normally distributed. Marginal significance levels appear in the parenthesis. Normality is rejected for samples B and C and for the forward equation (7b) of sample A. **** Denotes rejection of the null hypothesis at the 5% level. ***** Ramsey's functional form test. The null hypothesis is not rejected for any sample, which indicates no problems with the functional form of the model. (@) An LM-test for heteroscedasticity. The null hypothesis of homoscedasticity is rejected for samples B and C. Therefore, White's heteroscedasticity-consistent covariance matrix was used to compute the standard errors and Wald statistics for these samples. (@@) Lagrange Multiplier (LM)-tests for serial correlation, distributed [chi square] with n degrees of freedom, where n is the order of serial correlation. Marginal significance levels appear in the parentheses. (@2) LR-tests of statistical exogeneity, distributed [chi square] with one (I) degree of freedom. The hypothesis [[gamma].sub.s] = 0 is not rejected for any sample, while the hypothesis [[gamma].sub.f] = 0 is rejected for all samples at the I% level. This indicates that spot rather than forward rates are exogenous. (@3) An LR-test--distributed [chi square] with one (1) degree of freedom--of the hypothesis [[gamma].sub.s]-[[gamma].sub.f] + 1 = 0. The null hypothesis is easily rejected at the 1 % level. (@4) LR-test--distributed [chi square] with three (3) degrees of freedom--of the hypothesis HO: [[gamma].sub.s-[[gamma].sub.f] + 1 = 0 & [[kappa].sub.1]--[[mu].sub.1] = 0 & [[lambda].sub.1]-[[upsilon].sub.1] = 0 (Condition 3). The null hypothesis is easily rejected at the 1% level.

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Title Annotation: | vector error correction model |
---|---|

Author: | Zacharatos, Nikolaos; Sutcliffe, Charles |

Publication: | Journal of Financial Management & Analysis |

Geographic Code: | 4EUGR |

Date: | Jan 1, 2012 |

Words: | 7653 |

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