Do currency futures prices follow random walks?

Journal of Empirical Finance 4 Ž1997. 1–15

Do currency futures prices follow random walks? Ming-Shiun Pan

a,)

, Kam C. Chan b, Robert C.W. Fok

c

a

Department of Finance, Management Science, and Information Systems, Shippensburg UniÕersity, Shippensburg, PA 17257, USA b Department of Accounting, Finance, and Information Systems, UniÕersity of Wisconsin-Parkside, Kenosha, WI 53141, USA c Department of Finance, National Chung Cheng UniÕersity, Chia-Yi 621, Taiwan Accepted 15 July 1996

Abstract This paper examines the random walk process for four currency futures prices for the period 1977–1987 by using the variance ratio test. The random walk hypothesis is tested through asymptotic standardized statistics as well as by computing the significance level based on the bootstrap method. Both long time-series prices and individual contract prices for four currency futures, the British pound, the German mark, the Japanese yen, and the Swiss franc are analyzed. The results provide little evidence against the random walk null hypothesis, though non-randomness is documented in the Japanese yen. Additionally, the currency futures markets apparently become more efficient as markets mature over time. JEL classification: G13 Keywords: Random walk process; Currency futures

1. Introduction The issue of whether currency futures prices follow a random walk process has received considerable attention. A random walk process for currency futures has several important implications on testing the financial models of currency futures. First, the studies on the cross-hedging performance of currency futures and )

Corresponding author.

0927-5398r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 5 3 9 8 Ž 9 6 . 0 0 0 1 0 - 2

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futures-forward parity require a stationary price process Že.g. Denis, 1976; Cornell and Reinganum, 1981; Park and Chen, 1985.. The literature assumes taking a log first-difference of futures prices will yield a stationary currency futures series. If the currency futures prices do not follow a random walk, the findings in these studies are questionable. Second, an efficient foreign currency futures market suggests that the current price reflects all available information; i.e., the currency futures price changes are independently distributed Žand maybe identically distributed as well., and there is no risk premium in currency futures prices. Samuelson Ž1965. argues that futures prices in an efficient market follow a martingale process which implies that the futures prices are unbiased predictors of future spot rates. McCurdy and Morgan Ž1987. test the martingale hypothesis for currency futures by using a regression model: Ž f tq1 y f t . s a q b Ž f t y f ty1 . q e tq1; where f t is the log futures price at time t and e tq1 is the error term with mean zero and is not autocorrelated. Under the null hypothesis of a martingale, the coefficients a and b should be zero. 1 However, a result of coefficients of a and b are significantly different from zero does not necessarily imply inefficient currency futures markets, but may merely indicate the existence of time-varying risk premium since Samuelson’s argument is built on the hypothesis of risk neutrality. Thus, one way of testing the martingale hypothesis is examining the significance of the first order serial correlation of changes in currency futures prices. In addition, the random walk process of currency futures prices implies that prices have a unit root, which plays a crucial role in testing the unbiased futures rate hypothesis Že.g. Barnhart and Szakmary, 1991.. Usually, the literature tests the unbiased hypothesis with a regression model: Ž st y sty1 . s a q bŽ f t y f tq1 . q e t , where st is the log spot rate at time t. The regression model implicitly assumes that prices have a unit root and stationarity will be obtained by taking the log first-difference. Despite the importance of the random walk hypothesis in the study of currency futures, the empirical findings are mixed. Naidu and Shin Ž1979. document that weekly currency futures prices display some serial correlation in small lags, though a concrete conclusion cannot be made due to the lack of a joint test of randomness. By using autocorrelation, spectral, and non-parametric rank tests on contract by contract daily data, Cavanaugh Ž1987. contends that most currency futures prices follow random walks. Glassman Ž1987. also reports similar results, though her results indicate that the random walk process tends to be rejected for contracts before 1976. Applying Dickey–Fuller unit root tests, Doukas and Rahman Ž1987. find a unit root in currency futures prices, indicating that currency futures prices follow

1

Note that when a s 0 and b sy1, i.e., f tq 1 s f ty1 q e tq1 for t g 2,4, . . . 4, the process is also a martingale. We thank Franz C. Palm, the editor, for pointing this out.

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random walks. 2 Using a generalized method of moments regression analysis, Hodrick and Srivastava Ž1987. reject the hypothesis of zero risk premium and hence, all currency futures prices do not follow a random walk. McCurdy and Morgan Ž1987. also document similar results of rejecting the martingale hypothesis with daily data. More recently, using a methodology based on the arc-cosine property of Brownian motion, Barrett and Rodriguez Ž1990. claim that most of the futures contracts follow random walks. Although the empirical results of previous studies are inconclusive, it bears emphasizing that the results of most of these studies are not robust to the specification of the currency futures prices distribution. More specifically, the conclusion of the hypothesis of a random walk drawn by using the autocorrelation test, the Dickey–Fuller unit root test, or the test based on arc-cosine property of Brownian motion is justified only if the underlying variables Že.g. currency futures price changes. are normally distributed. However, it is well documented that the distribution of futures price changes is not normally distributed. Instead, the distribution exhibits excess kurtosis relative to a normal distribution. 3 Moreover, it is well known that non-normally distributed variables can contain statistical dependence even though the autocorrelations are insignificant. Hence, the results of supporting a random walk process based on the assumption of normality could be spurious. Although non-parametric tests, such as the runs test, do not require the assumption of normality, they have extremely low statistical power against complicated movements in futures prices Žsee Leuthold, 1972.. Therefore, a further examination of the random walk hypothesis on currency futures is warranted. The purpose of this study is to reexamine whether currency futures prices follow random walks. The initial impetus for a renewal of interest in the empirical study on currency futures prices develops from the recent innovation in time series techniques which overcomes the flaws of techniques used in previous studies. In particular, this study improves previous research in at least three aspects. First, the variance ratio test of random walk, which is due to Cochrane Ž1988. and Lo and MacKinlay Ž1988., is employed in this study. This test allows heteroscedasticity in the data, and, more importantly, does not require the assumption of normality. 4 Thus, the test is robust relative to the tests used in the literature. In addition, under

2 Chan et al. Ž1992. also document a unit root in currency futures prices. They apply the unit root test proposed by Phillips Ž1987. and Perron Ž1988.. The Phillips–Perron unit root test permits heteroscedasticity in the autoregression error terms, while the Dickey–Fuller test assumes homoscedasticity. 3 There are mounting studies showing that the distribution of futures price changes is not normally distributed Že.g. Cornew et al. Ž1984., So Ž1987. and Hall et al. Ž1989., among others.. 4 In fact, the specifications for the error terms in Lo and MacKinlay’s variance ratio test are general enough so that the distribution of the error terms can be a variety of forms Že.g. mixture of normals and the Autoregressive Conditional Heteroscedasticity ŽARCH. model..

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the null of a heteroscedastic random walk process, Lo and MacKinlay Ž1989. show that the variance ratio test is more powerful than the Box–Pierce Q test and the Dickey–Fuller unit root test against several alternative hypotheses. 5 Therefore, the results provided in this study will be more reliable. Second, we utilize the bootstrap method, which was developed by Efron Ž1979., to compute the significance level for the variance ratio statistic, along with the test statistic derived by Lo and MacKinlay Ž1988, 1989.. The bootstrap technique is desirable because it allows for the possible serial dependence in the data. Furthermore, the bootstrap method allows us to calculate a joint significance level for the variance ratio test. Third, both contract by contract and long time-series data are employed in the study. The study of both data sets can provide us with the information regarding the possible gradual efficiency of currency futures markets over time. The remainder of the paper is organized as follows. Section 2 explains the variance ratio test and the bootstrap method. Section 3 describes the data and presents the empirical results. A brief conclusion is contained in Section 4.

2. Methodology 2.1. Variance ratio test The variance ratio test exploits the fact that the variance of returns should be proportional to the length of the sample interval, if the logarithms of asset prices follow a random walk. 6 For instance, the variance of annual returns should be twelve times as large as the variance of monthly returns if monthly prices follow a random walk. The variance ratio test is used in this study to test the hypothesis that the increments in currency futures prices are uncorrelated. The variance ratio at lag q, VRŽ q ., is defined as: VR Ž q . s

Var Ž D q X t . Var Ž D X t .

=

1 q

Ž 1.

where VarŽP. is the unbiased estimation of the variance of ŽP., X t is the natural 5

It is noteworthy that the Phillips–Perron unit root test permits heteroscedasticity and strong-mixing form of dependence in the autoregression error terms. As Lo and MacKinlay Ž1989. point out, the variance ratio differs from the unit root test in several aspects. First, the null hypothesis of the variance ratio test differs from that of the unit root test in the context of Beveridge and Nelson Ž1981., implying that the unit root test cannot detect some departures from a random walk. Second, the variance ratio is asymptotically normally distributed and independent of any nuisance parameters, while the distribution of the unit root t statistics depends on the nuisance parameters. Since the theme of the study is on the uncorrelatedness of currency futures price changes, the variance ratio test is preferred. 6 Since there is no initial investment associated with trading futures, in this study returns actually mean price changes.

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logarithm of currency futures price at time t, and D q is the qth difference operator. Under the random walk null hypothesis, VarŽ D q X t . is equal to qs 2 . Thus, the variance ratio between D q X t and D X t , VRŽ q ., reduces to one. That is, if VRŽ q . is equal to one across q, then currency futures prices follow a random walk. The unbiased estimators of VarŽ D X t . and VarŽ D q X t . in Eq. Ž1. and two alternative test statistics, Z1Ž q . and Z2 Ž q ., have been derived by Lo and MacKinlay Ž1988, 1989. to test the significance of the null hypothesis of a random walk. According to Lo and MacKinlay, Z1 Ž q . s Ž T y 1 .

1r2

Z2 Ž q . s Ž T y 1 .

1r2

VR Ž q . y 1 2 Ž 2 q y 1 . Ž q y 1 . r3qy1r2 , VR Ž q . y 1 V Ž q .

y1r2

Ž 2a . Ž 2b .

where T is the number of observations of X t and V Ž q . is the consistent estimate of the variance of VRŽ q .. 7 The first test statistic, Z1Ž q ., assumes an independent and identically distributed normal error term. The second test statistic, Z2 Ž q ., allows the error term to having a general heteroscedastic behavior and relaxes the assumption of normality. Both test statistics are shown to be asymptotically standard normally distributed. 2.2. Significance leÕel from bootstrap method Although Lo and MacKinlay show that the variance ratio statistics are asymptotically standard normally distributed, the sample variance ratios computed from a finite sample might not be distributed as a standard normal due to the possibility of dependence in the sample data. In a study on mean-reversion in stock prices, Kim et al. Ž1991. demonstrate that the distribution of the sample variance ratios is not a standard normal. They attribute this to the dependence in the sample data. In this study, we apply the bootstrap technique to provide the p-value of the VRŽ q . statistics in comparison with the p-value from the Z-statistics. The bootstrap resampling scheme provides a truly independent specification of the error term. In addition, the bootstrap method makes no assumption about the distribution of the error terms. Thus, if the data exhibits some degree of dependence, the results from the bootstrap method would be more reliable than those from the Z-statistics. Consider a sample X s Ž X 1 , X 2 , . . . , X T . from an unknown probability distribution F. The bootstrap method estimates the sampling distribution of a statistic Y Ž X, F ., on the basis of the observed data X. It proceeds as if the sample is the population for the purpose of estimating the sampling distribution of the test statistic, Y Ž X, F .. For example, in this study we have 2,771 daily returns on the 7 See Lo and MacKinlay Ž1988, p. 50. for a complete formula of V Ž q .. Note that there is a mistake in the expression of V Ž q . in Lo and MacKinlay Ž1988., and a correction has been made in The Review of Financial Studies ŽNo. 1, 1990 issue..

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British pound currency futures, a random sample of 2,771 is drawn Žwith replacement. from the 2,771 observations for each replication. This is done to remove possible dependence in the data. Then VRŽ q . is computed for each replication. This procedure of shuffling and computation of VRŽ q . is repeated N times to form the bootstrapped distribution of VRŽ q .. Finally, the p-value of the sample VRŽ q . is determined from the frequency table of the bootstrap distribution. The results reported in the next section are obtained from bootstrapping the data 1,000 times. 8

3. Data and empirical results 3.1. Data Four currency futures, the British pound ŽBP., the German mark ŽGM., the Japanese yen ŽJY., and the Swiss franc ŽSF., are the subjects of the analysis. These four currency futures together count for about ninety percent of currency futures trading. The daily settlement currency futures prices for the period from January 2, 1977 to December 31, 1987 are taken from the International Monetary Market ŽIMM. of the Chicago Mercantile Exchange. The IMM futures contracts are traded for delivery on the third Wednesday of March, June, September, and December. In order to have a long time series and also to minimize differences in maturity, we use the daily settlement prices in December, January, and February for each March contract, and prices in March, April, and May for each June contract, and so on. By constructing data in this way, all price data within the delivery month were excluded to avoid the possibility of noise created by options of redemption during the delivery month. Following the common practice, the return Žor price change. is computed as the difference in logarithms of daily settlement prices. Sample statistics for the four currency futures returns series are reported in Table 1. The Kolomogorov–Smirnov D-statistics are significant at the 5% level for all four currency futures, suggesting that currency futures returns are not normally distributed. Furthermore, the kurtosis in all cases is statistically significantly greater than 3 Žas required by the normal distribution., which indicates that the empirical distributions of the currency futures returns have fat tails and sharp peaks at the center. Table 1 also provides the sample autocorrelations of the four currency futures returns at different lags. Most of the autocorrelations are not significantly different from zero, which usually leads to the conclusion of independence and, hence, a random walk. However, this conclusion based on the 8

As indicated by Efron Ž1987., a replication of 1,000 times is sufficient to provide an accurate result.

M.-S. Pan et al.r Journal of Empirical Finance 4 (1997) 1–15 Table 1 Sample statistics of daily currency futures return series, 1977–1987

Mean Žin %. t Žmeans 0. Standard deviation Skewness Kurtosis D-statistic Observations ŽT y1. Autocorrelation: Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 8 Lag 10 Lag 15 Lag 20 QŽ12. QŽ24. Adjusted QŽ12. Adjusted QŽ24. Squared returns: QŽ12. QŽ24.

7

a

British pound

German mark

Japanese yen

Swiss franc

0.0043 0.3124 0.0073 0.1362 8.4889 0.0704 2771

0.0147 1.0981 0.0071 0.3295 5.4021 ) ) 0.0650 ) ) 2775

0.0323 2.3312 0.0073 0.5092 6.7589 0.0660 2766

0.0236 1.4886 0.0084 0.2894 4.7477 0.0509 2774

y0.013 0.004 y0.001 y0.024 y0.021 0.028 0.020 y0.019 0.041 0.029 11.09 26.11 7.51 17.27

y0.001 0.040q 0.038 y0.036 0.003 0.025 0.042q 0.000 0.024 0.022 19.53 33.29 13.40 24.28

y0.020 y0.014 0.043q 0.010 0.019 0.022 0.024 0.042q y0.009 0.032 22.28 ) ) 33.03 17.57 26.86

0.010 0.043q 0.019 y0.022 y0.009 0.008 0.008 0.000 0.034 0.019 10.78 25.51 8.42 20.16

333.36 ) ) 514.01 ) )

330.13 442.67

))

149.91 ) ) 183.74 ) )

215.83 324.35

)) ))

))

))

)) ))

)) ))

)) ))

a

Ži. t-statistic is for testing mean equal to zero. Žii. The critical value of the Kolomogorov–Smirnov D-statistic for normality is 0.023 for the 5% significance level. Žiii. Kurtosis is coefficient of excess kurtosis. Živ. QŽ q . and adjusted QŽ q . are the Ljung–Box Q-statistic and the heteroscedasticity-adjusted Q-statistic developed by Diebold Ž1986., respectively, for q order serial correlation, which distribute as a chi-square variate with q degrees of freedom. Žv. ) ) indicates statistically significant at the 5% level. Žvi. q indicates coefficient is two times larger than its computed standard error, which is approximated as Ž1rŽT y1..1r 2 .

autocorrelation test should be interpreted with caution since the assumption of normality is not valid. We also compute the Ljung–Box Q-statistic for the raw and squared returns for checking heteroscedasticity. As can be seen in the table, the Ljung–Box Q-statistics for the squared returns are substantially larger than those for the raw returns, indicating the existence of heteroscedasticity. Given the existence of heteroscedasticity, we also compute the heteroscedasticity-adjusted Q-statistics as described in Diebold Ž1986. for a further check on serial correlation. The results from the heteroscedasticity-adjusted Q test suggest insignificant serial correlation. Nevertheless, non-normality, heteroscedasticity, and serial dependence in the data indicate the importance of using the Z2-statistic and the bootstrap methods.

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3.2. Results of the long time-series data The sample variance ratios, VRŽ q .’s, and the corresponding standardized Z-statistics for the four currency futures are given in the second, third, and fourth columns of Table 2. For the British pound, the variance ratio estimates are less than one and decrease as q increases. For the other three currency futures ŽGM,

Table 2 Sample variance ratios, test statistics, and the bootstrap distribution of variance ratios for daily currency futures daily return series, 1977–1987 a q

VRŽ q . Z1Ž q .

Panel A: BP 2 0.987 4 0.985 8 0.958 10 0.965 15 0.942 20 0.939

Z2 Ž q .

Mean Median S.D.

p-value a

1%

5%

95%

99%

1.000 0.999 1.001 0.998 0.992 0.995

0.019 0.037 0.058 0.066 0.083 0.097

0.954 0.924 0.870 0.855 0.815 0.785

0.970 0.944 0.908 0.895 0.866 0.842

1.033 1.064 1.100 1.111 1.140 1.164

1.048 1.091 1.138 1.159 1.191 1.237

0.236 0.350 0.239 0.291 0.244 0.275

Panel B: GM 2 0.999 y0.074 y0.057 0.999 1.000 4 1.058 1.639 1.307 0.998 0.997 8 1.085 1.514 1.226 0.998 0.997 10 1.116 1.805 1.468 0.998 0.997 15 1.156 1.932 1.589 0.998 0.999 20 1.168 1.781 1.479 0.997 0.997

0.019 0.036 0.056 0.064 0.080 0.094

0.954 0.913 0.875 0.862 0.833 0.799

0.967 0.939 0.905 0.893 0.867 0.846

1.030 1.059 1.094 1.108 1.134 1.155

1.044 1.079 1.132 1.153 1.189 1.226

0.481 0.051 0.064 0.033 0.028 0.033

Panel C: JY 2 0.979 y1.099 y0.762 1.000 1.000 4 0.977 y0.641 y0.478 0.999 0.998 8 1.032 0.565 0.449 1.000 0.998 10 1.070 1.087 0.882 1.000 0.997 15 1.174 ) 2.154 1.810 0.999 0.994 20 1.235 ) ) 2.483 2.135 0.997 0.996

0.020 0.037 0.058 0.066 0.082 0.096

0.951 0.911 0.863 0.845 0.818 0.793

0.968 0.939 0.909 0.897 0.872 0.845

1.031 1.059 1.097 1.109 1.136 1.160

1.046 1.090 1.140 1.160 1.199 1.239

0.126 0.254 0.276 0.147 0.019 0.013

Panel D: SF 2 1.010 4 1.070 8 1.085 10 1.101 15 1.135 20 1.151

0.019 0.035 0.056 0.063 0.079 0.092

0.955 0.920 0.876 0.856 0.816 0.797

0.968 0.942 0.906 0.893 0.871 0.849

1.031 1.060 1.089 1.101 1.131 1.152

1.041 1.084 1.130 1.148 1.192 1.222

0.294 0.027 0.057 0.050 0.043 0.051

a

y0.690 y0.419 y0.753 y0.552 y0.725 y0.644

0.542 1.965 1.515 1.579 1.674 1.602

y0.394 y0.264 y0.524 y0.395 y0.541 y0.493

0.438 1.627 1.295 1.360 1.456 1.403

1.001 1.001 1.000 1.000 0.998 0.998

Fractiles q

1.000 1.000 0.999 0.998 0.998 0.996

1.000 0.999 0.998 0.999 0.996 0.994

Ži. Variance ratios, VRŽ q .’s, are calculated according to Eq. Ž1.. Žii. ) Ž ) ) . indicates that the variance ratio is statistically different from one at the 10% Ž5%. level based on Z2 Ž q .. Žiii. q reports lower x% of VRŽ q . from the bootstrap distribution. Živ. a reports the probability that the variance ratio from bootstrap distribution is less Žlarger. than the sample variance ratio if the sample value is less Žlarger. than the median of bootstrap distribution.

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JY, and SF., the variance ratios generally start from a value close to one and then increase as q increases. The pattern of the variance ratio estimates indicates that the British pound futures returns are negatively serially correlated, and that the German mark, the Japanese yen, and the Swiss franc futures returns are positively serially correlated. Furthermore, all of the absolute values of Z2 , except Z2 Ž20. of the Japanese yen, are less than 1.96, which suggests that the random walk hypothesis should not be rejected at the 5% significance level Žtwo-tailed test.. For the Japanese yen, there is evidence to reject the random walk hypothesis for higher values of q Že.g., q s 20.. 9 It is important to note that while weaker results Ži.e. the rejection of the random walk. are obtained based on the Z1-statistics, the non-normality and the heteroscedasticity found in these currency futures returns recommend the use of the Z2-statistic. All four currency futures returns exhibit some serial correlation, this implies that statistical inference based on the Z2-statistic could be biased. To address this possibility we provide the p-value of the VRŽ q . estimated by the bootstrap method. The mean, median, standard deviation, and lower fractiles of the distribution of VRŽ q ., and the estimated p-value are reported in Table 2. The p-value reports the probability that the variance ratio from the bootstrap distribution is less Žlarger. than the sample variance ratio if the sample value is less Žlarger. than the median of the bootstrap distribution. 10 Using these p-values, we reject the null hypothesis that the currency futures prices of the German mark, the Japanese yen, and the Swiss franc follow random walks at the 5% significance level. However, there is no evidence of rejecting the random walk null hypothesis for the British pound. There is an important caveat to the above findings, however. The significance of the results based on the Z2-statistic or the bootstrap p-value at any given lag might be overstated. This is because the individual Z2-statistic and p-value fail to include joint tests over multiple lags Žsee Richardson and Stock Ž1989., Kim et al. Ž1991., and Richardson Ž1993. for discussions.. Accordingly, we calculate the probability of finding at least one individually significant VR among the six lags under bootstrapping, using the procedure recommended by Kim et al. Ž1991.. For instance, the lowest p-value Ži.e. 0.236. for the British pound futures returns is at lag 2, and the experiment shows that the probability of having at least one significant VR at that level or better is 0.436. The probabilities of getting at least

9

It is noteworthy that the use of Z test for the higher values of q might lead to low power since higher-order autocorrelations are more imprecisely estimated. Accordingly, the significance of rejecting the random walk hypothesis should be adjusted downward. We thank an anonymous referee who suggested this interpretation. 10 The estimated p-value can be interpreted as the significance level for rejecting the random walk null hypothesis. Since testing the random walk hypothesis is a two-tailed test, the estimation of the p-value depends on whether the sample variance ratio lies in the left- or right-hand tail of the bootstrapped distribution.

10

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one significant VR at the lowest p-value are 0.083 for the German mark, 0.036 for the Japanese yen, and 0.073 for the Swiss franc. 11 Therefore, the random walk hypothesis is rejected at the 5% significance level only for the Japanese yen futures. 3.3. Results of contract by contract data This section presents the results from examining contract by contract data. Testing the contract by contract data permits us to investigate whether dependence changes over time. We focus on the September currency futures contracts from 1977 to 1987. We use the daily settlement prices from February through August, with each contract containing about 145 daily prices. Data before February and during September are excluded because of thin trading before February and possible noise within the delivery month. The ranges of the variance ratio estimates and of the associated Z2-statistics for the British pound, the German mark, the Japanese yen, and the Swiss franc are summarized in Table 3. 12 In summary, the random walk process of currency futures returns cannot be rejected at the 5% significance level for most of the contracts. However, we find that the Z2-statistic is significant at the 5% level for the following contracts: 1. German mark: 1979 at q s 2; 1981 at q s 2. 2. Swiss franc: 1978 at q s 2, 4, 8, 10, and 15; 1979 at q s 2. Thus, of the 66 sample variance ratios for each currency futures, there are only eight variance ratios that are significant at the 5% level. In other words, only 3% Ž8r264. of the collective variance ratios are significant, providing little evidence for rejection of the random walk hypothesis. Finally, it is interesting to note that most of the significant Z2-statistics cluster in the early contracts. This suggests that prices of recent contracts tend to follow random walks. This phenomenon could be due to the more effective processing of information as the markets became more mature in recent years. Significance tests based on the bootstrap method are also conducted on selected contracts. Two contracts for each currency futures are selected – the one that shows a relatively larger Z2-value Že.g. BP 1977 contract. and the one that shows a relatively smaller Z2-value Že.g. BP 1985 contract..

11

We are grateful to Myung Jig Kim for guiding us in calculating this ‘pseudo-’joint significance level. 12 The detailed results of the variance ratio estimates and the Z2-statistics are available upon request. We also compute the homoscedasticity-robust Z1 -statistics. The Z1 -statistic results are qualitatively the same as those of the Z2 -statistic. The similar results between Z1 - and Z2 -statistics could be due to the insignificance of non-normality and heteroscedasticity in contract by contract data.

0.793–1.089 0.705–1.200 0.641–1.400 0.607–1.456 Ž0.064–2.087. Ž0.273–1.828. Ž0.023–1.433. Ž0.020–1.381. Number of significant variance ratios at the 5% Ž10%. levels 2 Ž4. 0.850–1.058 0.768–1.113 0.618–1.427 0.604–1.450 Ž0.065–1.316. Ž0.057–1.283. Ž0.035–1.406. Ž0.108–1.296. Number of significant variance ratios at the 5% Ž10%. levels 0 Ž1. 0.738–1.333 0.759–1.595 0.576–1.998 0.539–2.118 Ž0.108–3.370. Ž0.017–3.524. Ž0.208–3.759. Ž0.068–3.714. Number of significant variance ratios at the 5% Ž10%. levels6 Ž2.

GM

JY

SF

0.482–2.023 Ž0.009–2.739.

0.528–1.458 Ž0.007–1.206.

0.504–1.777 Ž0.171–1.754.

0.447–1.365 Ž0.022–1.165.

15

0.358–1.825 Ž0.106–1.914.

0.515–1.720 Ž0.218–1.668.

0.427–1.879 Ž0.319–1.709.

0.445–1.742 Ž0.016–1.898.

20

The ranges of variance ratio estimates are given in main rows, with the ranges of the absolute values of heteroscedasticity-robust Z2 Ž q . statistics given in parentheses.

a

0.871–1.122 0.653–1.127 0.516–1.146 0.487–1.365 Ž0.033–1.568. Ž0.093–1.167. Ž0.087–1.260. Ž0.023–1.322. Number of significant variance ratios at the 5% Ž10%. levels 0 Ž1.

BP

10

Ranges of VRŽ q . and heteroscedasticity-robust test statistics 2 4 8

Currency futures

Table 3 Summary of sample variance ratios and heteroscedasticity-robust test statistics for the September futures contracts Ž1977–1987. a

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Table 4 Sample variance ratios and the bootstrap distribution of variance ratios for selected currency futures contracts q

VRŽ q .

Z2 Ž q .

Mean

Median

S.D.

Fractiles q

p-value a

1%

5%

95%

99%

Panel A: BP 1977 2 0.878 y0.598 4 0.653 y0.993 8 0.516 y0.934 10 0.487 y0.905 15 0.447 y0.962 20 0.445 y0.808

0.995 0.997 0.995 0.995 0.992 0.984

0.998 0.993 0.975 0.969 0.936 0.915

0.080 0.155 0.252 0.290 0.372 0.441

0.778 0.652 0.475 0.443 0.349 0.292

0.859 0.741 0.611 0.575 0.478 0.402

1.119 1.245 1.437 1.498 1.635 1.766

1.184 1.375 1.650 1.752 2.059 2.330

0.074 0.011 0.016 0.019 0.041 0.072

Panel B: BP 1985 2 0.969 y0.330 4 1.060 0.337 8 1.024 0.087 10 1.062 0.197 15 1.030 0.076 20 0.993 y0.016

1.000 0.998 0.999 0.999 0.998 0.995

0.997 0.991 0.976 0.964 0.962 0.935

0.084 0.154 0.240 0.276 0.353 0.421

0.821 0.677 0.538 0.487 0.394 0.330

0.862 0.763 0.635 0.585 0.501 0.424

1.146 1.267 1.421 1.467 1.623 1.753

1.205 1.409 1.634 1.780 2.023 2.246

0.371 0.321 0.427 0.367 0.411 0.443

Panel C: GM 1981 2 0.829 y2.023 4 0.705 y1.828 8 0.641 y1.433 10 0.607 y1.381 15 0.504 y1.402 20 0.427 y1.408

0.999 1.001 0.994 0.988 0.980 0.976

1.001 0.993 0.980 0.961 0.925 0.894

0.081 0.151 0.239 0.276 0.353 0.425

0.801 0.670 0.508 0.472 0.400 0.324

0.865 0.750 0.639 0.586 0.496 0.429

1.132 1.263 1.409 1.494 1.623 1.775

1.187 1.376 1.631 1.718 1.969 2.213

0.020 0.019 0.051 0.069 0.057 0.048

Panel D: GM 1982 2 1.022 0.241 4 1.084 0.496 8 0.994 y0.023 10 1.055 0.180 15 1.090 0.243 20 1.137 0.319

0.997 0.996 0.997 0.998 0.997 0.997

0.996 0.990 0.976 0.967 0.934 0.907

0.084 0.162 0.261 0.298 0.377 0.449

0.791 0.672 0.558 0.499 0.397 0.319

0.858 0.747 0.634 0.607 0.511 0.443

1.137 1.278 1.481 1.537 1.680 1.826

1.187 1.408 1.729 1.865 2.138 2.456

0.367 0.277 0.458 0.379 0.343 0.318

Panel E: JY 1982 2 1.009 0.102 4 1.037 0.236 8 1.035 0.144 10 1.079 0.281 15 1.161 0.469 20 1.166 0.418

1.005 1.006 1.004 1.002 1.003 1.002

1.008 0.997 0.981 0.972 0.939 0.922

0.081 0.154 0.240 0.277 0.360 0.432

0.807 0.686 0.547 0.497 0.398 0.345

0.873 0.771 0.646 0.600 0.516 0.452

1.137 1.275 1.433 1.516 1.671 1.803

1.193 1.392 1.628 1.757 2.054 2.335

0.493 0.397 0.409 0.360 0.295 0.292

Panel F: JY 1985 2 0.850 y1.178 4 0.789 y0.973 8 0.618 y1.253 10 0.604 y1.172

1.000 1.002 1.001 1.001

0.999 0.996 0.978 0.977

0.085 0.158 0.255 0.291

0.804 0.684 0.543 0.506

0.863 0.762 0.627 0.584

1.146 1.272 1.458 1.549

1.196 1.395 1.668 1.732

0.036 0.077 0.040 0.067

M.-S. Pan et al.r Journal of Empirical Finance 4 (1997) 1–15

13

Table 4 Žcontinued. q

VRŽ q .

Z2 Ž q .

Mean

Median

Fractiles q

p-value a

1%

5%

95%

99%

0.368 0.433

0.389 0.329

0.500 0.443

1.674 1.775

2.088 2.332

0.071 0.123

0.997 0.986 0.979 0.978 0.945 0.913

0.086 0.156 0.246 0.284 0.371 0.444

0.803 0.664 0.545 0.485 0.382 0.282

0.856 0.750 0.642 0.592 0.495 0.421

1.136 1.259 1.418 1.484 1.661 1.802

1.193 1.363 1.654 1.831 2.218 2.488

0.000 0.000 0.001 0.001 0.016 0.044

0.997 0.987 0.970 0.962 0.925 0.908

0.083 0.153 0.246 0.280 0.354 0.417

0.797 0.655 0.511 0.450 0.350 0.304

0.870 0.756 0.632 0.586 0.485 0.412

1.135 1.251 1.435 1.486 1.649 1.792

1.202 1.426 1.654 1.761 1.994 2.223

0.305 0.424 0.443 0.494 0.471 0.460

Panel F: JY 1985 15 0.528 y1.173 20 0.536 y1.024

1.000 0.996

0.948 0.914

Panel G: SF 1978 2 1.333 3.730 4 1.595 3.524 8 1.998 3.759 10 2.118 3.714 15 2.023 2.739 20 1.825 1.914

0.998 0.995 0.996 0.997 0.999 0.999

Panel H: SF 1985 2 0.956 y0.522 4 1.011 0.068 8 0.935 y0.257 10 0.966 y0.117 15 0.948 y0.142 20 0.954 y0.106

0.998 0.993 0.985 0.984 0.977 0.972

a

S.D.

See notes to Table 2.

The results from the bootstrap method are given in Table 4, with a similar reporting format as in Table 2. Using the bootstrap distribution leads to the rejection of randomness for the four contracts which have larger Z2-values: BP 1977, GM 1981, JY 1985, and SF 1978. For the other four contracts, which have smaller Z2-values, the smallest p-value is 0.28 and the others are in the range of 0.30 to 0.45. Thus, the hypothesis of randomness could not be rejected for these four contracts. We also calculate the probability of getting at least one individually significant variance ratio among the six lags under bootstrapping for these eight contracts. The probabilities are 0.027 ŽBP 1977., 0.739 ŽBP 1985., 0.055 ŽGM 1981., 0.503 ŽGM 1982., 0.511 ŽJY 1982., 0.107 ŽJY 1985., 0.000 ŽSF 1978., and 0.837 ŽSF 1985.. Again, the results from the collective test confirm that little evidence of departure from the randomness can be found in the currency futures prices.

4. Conclusion This paper examines the random walk hypothesis for four currency futures prices for the period 1977 to 1987 by the variance ratio test. In addition to the asymptotic standardized test statistic, we also apply the bootstrap method to construct the individual and joint significance levels of the test statistic.

14

M.-S. Pan et al.r Journal of Empirical Finance 4 (1997) 1–15

Both the heteroscedasticity-robust Z2-statistic and the p-value from the bootstrap distribution yield similar results. Little evidence of non-randomness is found in the currency futures returns, except for the Japanese yen futures. The joint significance levels of variance rations also confirm this finding. The results of examining contract by contract data are very similar to those for the entire time series. Of the 264 sample variance ratios, only 3 percent of them are significant at the 5 percent level, a result that could be explained by chance. In summary, the evidence provided in this study is robust with respect to the methodology used and the time horizon of the data. Certainly, the few cases of a departure from a random walk do not necessarily imply that currency futures markets are not efficient. These departures could be an indication of that currency futures price containing a time-varying risk premium. Finally, the finding of a random walk in currency futures prices implies that the futures price is not an unbiased estimator of the future spot rate and supports the notion of taking the log first-difference to achieve stationarity in the examination of various hypothesis in currency futures studies.

Acknowledgements We thank an associate editor, two anonymous referees, the editor, Franz C. Palm, and John Bishop for helpful comments and suggestions on the paper. We are responsible for any remaining errors.

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