Econometric fragility of market anomalies: evidence from weekday effect in currency markets.
This study examines the sensitivity of day-of-the-week effect, also known as weekday effect, to the specification of error distribution assumption in both mean and conditional variances of several foreign currencies daily returns. There is overwhelming evidence that weekday returns vary with the day of the week across various types of assets and markets. Numerous explanations have been put forward to justify the existence of this anomaly. This includes the release of adverse information over the weekend, thin trading, settlement procedures, inventory control costs and measurement errors. However, none of the suggested explanations can consistently and fully explain the empirical results. Moreover, as noted by Hansen, Lunde and Nason (2005), these theoretical explanations have only been developed after the empirical "discovery" of the anomalies. We argue that one reason for this failure is likely due to the methods used, which fail to account for the stylized facts of financial time series.
In fact, several empirical studies show that financial time series have fatter tails than the normal distribution and exhibit volatility clustering. For instance, Fama (1965) reports departure from normality in stock returns. Akgiray and Booth (1988), among others, find similar results in foreign exchange price changes. A number of studies, including Andersen and Bollerslev (1998), show that information clustering induces volatility clustering in foreign exchange markets. Virtually all previous studies ignore these stylized facts and use standard methods, such as ANOVA, F tests or t tests, to investigate the day-of-the-week effect. This approach raises serious doubt on the reliability of their results and questions the existence of such an effect, given that one of the fundamental assumptions of such tests is normality.
Furthermore, there is an increasing number of studies questioning the economic significance, the persistence and even the very existence of the calendar effects. For instance, Yamori and Kurihara (2004) examine the daily returns of 29 foreign exchange rates and report that evidence of day-of-the-week effect in the 1980s disappears for almost all currencies in the 1990s. Yamori and Kurihara's results are in line with earlier findings of Corhay et al. (1995), who show that evidence of higher Wednesday returns in five currencies disappear by mid-1981. They argue that their results are likely to be due to a change in the settlement procedure which took effect on October 1, 1981. More relevant to our paper is Hsieh (1988), who cautions that evidence of the day-of-the-week effect documented in spot foreign exchange rate distributions may be illusory if it does not properly account for the non-normality and volatility clustering. (1) Hsieh hints at potential fragile inferences in the day-of-the-week effect, but he does not explicitly deal with extending the neighborhood of assumptions as we do in this paper. In fact, we employ a GARCH (p,q) type model under normal distribution as well as two error distributions known to better depict financial time series: student-t and generalized error distribution (GED). (2)
We hypothesize that evidence of day-of-the-week effect in foreign currency rates is not robust to error distributional assumptions in both returns and conditional variance. Given the stylized facts of financial time series, examining the weekday effect in mean and variance under the assumption that returns are normally distributed and/or homoskedastic (as postulated in most prior studies) may not be appropriate. Violations of normality, such as those arising from excess kurtosis in the population, introduce biases in traditional hypothesis testing. In fact, traditional tests require homoscedasticity, which does not hold for financial time series.
This paper complements and extends an earlier study by Baker, Rahman and Saadi (2008) showing that weekday effect in one stock market index (i.e., S&P/TSX Canadian index) is sensitive to the specification of error distribution. The present study is different in several ways. First, it examines foreign currency markets rather than a stock market index. The foreign exchange market is as important as the stock market since it plays a major role in portfolio investment as well as in a country's financial stability and economic growth. Foreign exchange market (forex) provides participants with the possibility to engage in currency conversion, currency hedging, currency arbitrage and also currency speculation. Forex is considered the largest financial market in the world. It is also highly liquid. In fact, according to recent BIS Triennial Survey (April 2013), the average daily trading volume in global foreign exchange markets is US$5.3 trillion per day, compared to US$4.0 trillion in April 2010 and US$3.3 trillion in April 2007. Second, this study analyzes several financial time series rather than one so that the results are more robust. Third, this study directly tests the prediction of Hsieh (1988) about foreign currency time series. Fourth, it employs more advanced and appropriate econometric techniques to limit the possibility of spurious results. For instance, unlike Baker, Rahman and Saadi (2008), we test for the presence of nonlinear dependencies using the powerful BDS test to determine whether the GARCH error terms are independent and identically distributed (i.i.d.). (3) This assumption is of crucial importance for an appropriate examination of the day-of-the-week effect. A rejection of i.i.d. assumption indicates the existence of a hidden unexplained structure in the residual terms. If it is not accounted for, it may spuriously increase the statistical significance of dummy variables in the model specification, thus leading to an erroneous conclusion of evidence of a weekday effect. Fifth, we account for shifts in monetary regimes; the presence of structural breaks that could arise due to, for instance, changes in settlement procedures; as well as other major structural transformations that affect the liquidity and operational efficiency of the foreign exchange market (e.g., Barker 2007). Accounting for non-linear dynamic and structural breaks is crucial to properly examine the sensitivity of weekday effect to the specification of error distribution. In fact, Corhay et al. (1995) show that evidence of weekday effect for six currencies disappears after October 1, 1981, which corresponds to the date on which a change in the settlement procedures took effect.
Weekday Effect in Foreign Exchange Market: A Brief Review of the Literature
Though the literature on weekday effect is dominated by studies on stock market returns, there are a number of papers that have extended the analysis to foreign exchange markets. (4) The first paper to document weekday effect in foreign exchange currencies is McFarland et al. (1982). They examine seven major currencies (i.e., the British pound, the German mark, the Japanese yen, the Swiss franc, the Australian dollar, the Spanish peseta and the Swedish krona) and find that, in both the spot and forward market, average returns for Monday and Wednesday are higher than for Thursday and Friday. Hsieh (1988) examines the statistical property of daily rates of change of five foreign currencies and find evidence of significant difference in mean and variance across days of the week. Tang (1997) investigates the interaction between diversification and day-of-the-week effect on exchange risks in six foreign currencies. Evidence by Flannary and Protopapadakis (1988), Gay and Kim (1987) and Gesser and Poncet (1997) indicate that the return distribution of futures foreign exchange markets also varies by day of the week. Using both parametric and nonparametric tests, Thatcher and Blenman (1998) report evidence of weekday effect in the sterling/dollar spot and forward market. Breuer (1999), however, finds statistically insignificant evidence in favour of weekday effect in forward foreign exchange markets. Corhay et al. (1995) report evidence of day-of-the-week effect over the period 1973 through 1992 for the British pound, Canadian dollar, Deutsch mark, French franc, Japanese yen and Swiss franc but then demonstrate that their results are driven by pre-October 1981 period. Aydogan and Booth (2003) document weekday effect in the daily depreciation of Turkish lira against US dollar, a finding that was later confirmed by Berument et al. (2007) over the period 1986-1994. Yu, Chiou and Wagner (2008) employ panel probability distributions techniques and uncover day-of-the-week effect in the yen spot market over the period 1994-2003. More recently, Popovic and Durovic (2014) show that there is evidence of weekday as well as intraday anomalies in euro/dollar market using ANOVA and t-test. Bessembinder (1994) motivates day-of-the-week effects in the foreign exchange market through inventory-carrying costs. Similar to Glassman (1987), Bessembinder (1994) proposes that bid-ask spreads in the spot and forward market are higher on Fridays and prior to holidays. Relatedly, Bossaerts and Hillion (1991) suggest a role for weekend effects since central banks are more likely to intervene over the weekend. However, as indicated earlier by Corhay et al. (1995), the suggested explanations are not unanimous and cannot fully explain the documented weekday effect.
Data and Some Preliminary Statistical Tests
Our data consists of daily closing spot exchange rates of five currencies expressed as per US dollar obtained from the New York Foreign Exchange market through DataStream. These currencies are: the Canadian dollar, British pound, Australian dollar, Indian rupee and Brazilian real. Our sample period is from January 2, 1976 through December 31, 2007. Our main sample ends in 2007 to avoid our results being affected by the recent financial turmoil. But as part of the robustness check analysis, we extend the sample to December 31, 2012. For brevity, we present a detailed discussion only for Canadian dollar spot exchange rate (CAD/USD). We report the main results for the remaining foreign currencies as a check of robustness. By doing so we are able to see whether our results are driven by the choice of the foreign currency.
The foreign exchange market closes at 5pm EST on Friday to reopen at 5pm EST on Sunday with no possible trade during that time. Yet, unlike securities, currencies continue to be traded 24 hours a day around the world during weekdays owing partly to electronic communication networks and different time zones. That is, if a currency trader in New York decides to trade currency at 4am, she/he will not be able to do so through foreign exchange dealers located in New York, but she/he can do so through foreign exchange dealers located, for instance, in London or Paris. Even though it is often considered a 24-hour market, a daily closing price is frequently reported in financial media and literature. Depending on the currency, a closing price refers to the closing price of the most important market in one of the three main regions: North America, Asia and Europe. As for the US dollar, the main market is New York (in Asia it is Tokyo, and in Europe it is London). Thus, in our paper, a closing price for the US dollar refers to the closing price reported at 5pm EST Friday at the New York Foreign Exchange market.
We compute the daily rate of change as the natural logarithmic first difference of the daily price of the exchange rate: [f.sub.t]=100(1n[S.sub.t] - 1n[S.sub.t-1]), where [S.sub.t] denotes the daily spot exchange rate. Table 1 provides descriptive statistics of CAD/USD daily changes for each day of the week and for the entire period of study. The average daily changes are positive with a relatively high kurtosis, indicating that the series is non-symmetric with higher peaks and fatter tails than the normal distribution. A closer look at each day's statistics shows that all daily summary statistics are likely to be shaped by those of Monday. In comparison with the remaining weekdays, Monday has the highest return, variance, kurtosis, skewness and range. We observe the lowest return on Thursday and the lowest variance on Friday. The high variance on Monday can be explained by greater volatility on the day following the exchange's weekend. The kurtosis and skewness being significantly different than three and zero, respectively, suggest that we reject the null hypothesis of the series being normally distributed. This assertion is confirmed by a Jarque-Bera (JB) test statistic of 3,787.
We use some common tests to further validate the initial observations drawn from the statistics in Table 1. First, we test for the constancy of variance. We do so prior to comparing the mean across different days, because the choice of the test of comparison of mean depends on whether the variance is homogeneous across different days. If the variance is constant across the days, then we use a test such as the Bonferroni correction. Otherwise, we use a test such as Games-Howell, which does not assume equal variances across different days.
To test for homogeneity of variance we use the Brown-Forsythe test. We choose the Brown-Forsythe test because of its robustness to departure from the normality, an assumption that is strongly rejected in our data. In doing so we follow Conover et al. (1981), who list and compare 60 methods for testing the homogeneity of variance assumption, and find that the Brown-Forsythe test outperforms all the procedures that are robust to departure from normality. Let [R.sub.ij] be the ith observation in the jth group, let [M.sub.j] be the sample median for the jth group and let
[Z.sub.ij] = [absolute value of ([R.sub.ij] - [M.sub.j])] (1)
The Brown-Forsythe test consists of performing an F test on the Zij's. A significant F test indicates that the variances are not equal. The Brown-Forsythe statistic equals 3.59 with a p-value of 0.006, indicating rejection of the null hypothesis at the 1% level of significance that the variances are the same across different days of the week. Accordingly, we use the Games-Howell test to compare the mean of each day to the remaining days. The results, reported in Table 2, reject the null hypothesis that the mean is constant over the week. Monday's mean change is significantly different from Tuesday's and Thursday's mean changes at the 5% and 1% levels, respectively.
Testing for Day-of-the-Week Effect in Mean Under Different Error Distributional Regimes
Although Tables 1 and 2 suggest that there is a day-of-the-week effect in the mean for the CAD/USD exchange rate, this result could be spurious. The test for constancy of the mean across time is best examined in a regression context and accounts for linear dependence (autocorrelation), volatility clustering (heteroscedasticity) and departure from normality. Similar to Hsieh (1988), to deal with linear dependence, we include k lag values of the daily rates of change to the equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
where [r.sub.t] is the daily rate of change at time t; a is a constant term; and [D.sub.Mt], [D.sub.Tt], [D.sub.Wt], and [D.sub.Ht], are dummy variables for Monday, Tuesday, Wednesday and Thursday, respectively. The choice of the lag length (k) is based on the lowest Akaike Information Criterion (AIC). To test for remaining non-captured linear dependencies, we use the Ljung-Box test to the residuals from the above model.
To deal with volatility clustering, we allow the variances of errors to be time dependent. Although some previous studies on calendar anomalies have considered volatility clustering in financial time series, they have not considered the strong departure from normality. Here, we examine whether evidence of the day-of-the-week effect is robust to modeling volatility clustering and to the choice of alternative distributions to the normal, both of which better portray financial series. A variety of volatility models have been proposed. Particular classes of models that demonstrate great flexibility in capturing multiplicative dependence in a series are the ARCH models. Assuming that the rate of changes expressed in Equation (2), conditional on information set up to time t-1, [[epsilon].sub.t] is an i.i.d. random variable with mean zero and variance oh we can express a GARCH (p,q) model as follows:
[[sigma].sup.2.sub.t] = [eta] + [lambda](L)[[omega].sup.2.sub.t] + [??](L)[[sigma].sup.2.sub.t-1] (3)
where L is the lag operator, [lambda](L)=[p.summation over (i=1)][[lambda].sub.i][L.sup.i] and [??](L)=[q.summation over j=1] [[??].sub.j][L.sup.i] with constraints:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Based on BIC and AIC criteria, GARCH (1,1) outperforms all our attempts (available on request) to model the data series with other ARCH-type models, such as PGARCH, GARCH-M and FIEGARCH, and also bilinearity models. This result is consistent with the results of Hansen and Lunde (2005), who compare 330 ARCH-type models in terms of their ability to describe the conditional variance. Hansen and Lunde find that a GARCH (1,1) outperforms all tested models, particularly when modeling exchange rates.
The second column in Table 3 contains the results of GARCH (1,1) specification under normal error distribution. It shows that the rate of change on Monday is not statistically significant; instead Tuesday's, Thursday's and the constant term are. Monday and Thursday rates of change are the highest and the lowest, respectively, among the weekday changes. In addition, ARCH (X) and GARCH (0) coefficients are significant at the 1% level with a sum (0.0946 + 0.9016) close to unity, which indicates that shocks to the conditional variance are persistent over future horizons. The sum of [[lambda].sub.1] and 0, is also an estimation of the rate at which the response function decays on a daily basis. Since the rates are quite high, the response functions to shocks are likely to die slowly. Moreover, although GARCH (1,1) captures all the volatility clustering (LM statistic of 15.76), the JB test suggests that the normality assumption is rejected at the 1% level. The qq-plot illustrated in Figure I corroborates the results of the JB test: the tails of the residuals of GARCH model are indeed fatter than the normal distribution. This finding suggests using a distribution with a fatter tail to fit our data.
To examine whether evidence of the day-of-the-week effect is sensitive to the error distributional assumption and to capture the fat tail in our series, we allow for two types of error distribution known to better represent financial time series: the student-t and the generalized error distribution (GED) proposed by Nelson (1991). (5) The results are reported in Table 3. Although under normal distribution GARCH estimations show that Monday has the highest rate of change and Thursday has the lowest, under fatter tails distributions the results tend to change. When we use student-t and GED distributions, Friday seems to have the highest rate of change and Thursday the lowest. The instability of the results seems to be limited to the day on which the highest rate of change occurs. Further, the statistical significance of the constant term is lower when we consider fatter tail distributions. Linder the normal distribution, the constant term is significant at the 1% level, but it is such at 5% level under student's t and GED.
To examine whether each GARCH model succeeds in capturing all the nonlinear structure in the data, we use BDS to test their standardized residuals. A rejection of the i.i.d. hypothesis will imply that the conditional heteroskedasticity is not responsible for all the nonlinearity in series and that there is some other hidden structure in the data. The BDS statistics, which we report in Table 4, fail to reject the null hypothesis that the standardized residuals are i.i.d. random variables. Thus, each model captures all the nonlinearity in the data used and conveys the conditional heteroskedasticity is the cause of the nonlinearity structure. This result shows that the nonlinear dependence caused by volatility clustering is the reason for the difference in the results of standard tests (reported in Section 3) and those of GARCH models. Hence, the failure to model volatility clustering, to consider fat tail distribution and to test for i.i.d. assumption may cause spurious evidence of day-of-the-week effect.
The results of the diagnostic tests show that both GARCH models are not misspecified. The LB statistics up to lag 100 cannot reject the null hypothesis of no autocorrelation. The LM tests are also insignificant, indicating that the GARCH models (i.e., under student-t and GED distributions) are successful in modeling the conditional volatility. The JB test for normality rejects the null hypothesis that the standardized residuals are normally distributed. This result corroborates results from qq-lot shown in Figure I. However, we note that for both models the sum of the parameters estimated by the variance equation is close to one. A sum of [[lambda].sub.i] and [[THETA].sub.i.] near one indicates a covariance stationary model with a high degree of persistence and long memory in the conditional variance. We also note that both AIC and BIC model-selection criteria favor GARCH with student-t error distribution, followed by GARCH-GED and then GARCH-normal models. This shows the superiority of models that account for fat tails in financial time series over those that assume normality.
Testing for Day-of-the-Week Effect in Variance Under Different Error Distributional Regime
To examine whether volatility changes across the days of the week, we introduce week-day dummies into the mean and conditional variance equation of the GARCH (1,1) model:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [[sigma].sup.2.sub.t] is the conditional variance; q is a constant term; and [D.sub.Mt], [D.sub.Tt], [D.sub.Wt] and [D.sub.Ht] are the dummy variables for Monday, Tuesday, Wednesday and Thursday, respectively. Equations (2) and (4) are estimated jointly. If all the coefficients for dummy variables in Equation (4) are not significantly different from zero, then we reject the day-of-the-week effect assumption in volatility. Again, we consider the normal distribution but also the student-t and GED to capture the fat tails observed earlier in the data.
Hence, in this sub-section, we examine whether the day-of-the-week effect in conditional volatility, if any, is sensitive to the particular specification of the underlying distribution. Table 5 displays the results of introducing dummy variables into the mean and conditional variance equations under normal, student-t and GED. The results for the mean equation are similar to those in Section 4, except that when we introduce the dummy variables into the conditional variance equation, the intercept in GARCH-normal and GARCH-GED becomes significant at only the 5% level. Accordingly, we focus on the variance equation where several interesting observations emerge.
First, although the GARCH (1,1) normal model suggests that there is a very strong evidence of day-of-the-week effect in conditional variance, GARCH (1,1)-student-t and GARCH (1,1)-GED show weak, if any, such evidence. Precisely under normal distribution, the Monday, Tuesday and Thursday dummy variables are statistically different from zero. However, under student-t distribution, there is hardly any evidence of day-of-the-week effect in conditional variance. We report similar result when we consider GED. We note that, again, both model selection criteria (AIC and BIC) rank the GARCH-student-t as the best model to fit the data.
In other words, when we consider error distribution that better fits the fat tails displayed by the data used, indications of day-of-the-week in variance tend to disappear. Thus, evidence of day-of-the-week effect in variance could be spurious, probably arising from the existence of fat tails in the data, and cast doubt on the relevant findings of previous studies. We note that our findings for GARCH normal are similar to those of Hsieh (1988), who applies an AR (k)-GARCH (1,1) normal model t for the CAD/ USD daily rate of change series for the period January 1974 to December 1983, and finds that the Tuesday and Thursday dummy variables are significant. However, only Monday and Thursday dummy variables are significant in the conditional variance equation.
Second, when we examine the mean and conditional variance equation parameters in Table 5, it is evident that the nature of relation between returns and volatility depends on the error distributional assumption. In the GARCH normal model, the mean equation estimation shows that the highest change takes place on Monday and the lowest occurs on Thursday, but the conditional variance equation demonstrates that the volatility peak is reached on Thursday and that the all-week low volatility occurs on Tuesday. This finding suggests a negative relation between returns and volatility in the series. If we consider the GARCH-student-t model, we see that the highest return occurs on Friday and lowest on Thursday, but the highest conditional variance occurs on Monday and lowest conditional variance on Tuesday. There is a similar pattern in the GARCH-GED model. Thus, there is no evidence of a relation between returns and conditional volatility. This result implies that studies investigating the link between asset price and conditional volatility should take into account the error distributional assumption.
Table 5 also presents the diagnostics tests used to check how well the models fit the data. As in Table 3, evidence of non-normality is confirmed by the JB test at the 1% level for all the GARCH models. The LB and LM test statistics show that the AR(k)-GARCH(p,q) models are successful in capturing the linear dependencies and volatility clustering in data. The sum of the variance parameters for each model, [lambda] + [[THETA].sub.i], is close to one, suggesting a very high magnitude of persistence and implying that the conditional variance is nearly integrated. The BDS statistics (available on request) on the standardized residuals from each GARCH process fail to reject the null hypothesis that the standardized residuals are i.i.d. random variables at the 5% and 1% degrees of significance. Thus all forms of dependency in the series are captured, and the dummy variable coefficients are not contaminated by any hidden structure.
Robustness Check and the Choice of the Error Distribution
ACCOUNTING FOR OUTLIERS, MONETARY REGIMES SHIFTS, PRESENCE OF STRUCTURAL BREAKS AND RECENT FINANCIAL CRISIS (6)
Galai et al. (2008) show that evidence of day-of-the-week effect in stock returns is sensitive to the presence of outliers. Because the problems associated with outliers are common in virtually any financial time series, we examine whether our conclusions are due to potential outliers in our foreign currency data. To do so, we repeat the analysis in Sections 4 and 5, but now we exclude the returns with higher absolute values, thus reducing the sample by 0.5, 1, 2 and 3%. Our results are similar to those obtained from the full sample. To further reduce potential outlier problems, we also exclude ten daily observations from the sample before and after the October 19, 1987 and October 27, 1997 stock market crashes, as well as the September 11, 2001 event. Again, our results are robust to outliers. Consequently, we conclude that our findings are not driven by outliers.
Many studies show that evidence day-of-the-week effect has gradually disappeared. To see whether our results hold over time, we divide our sample into k equal subsamples and repeat our analysis for each period. There is no clear-cut rule on choosing the value of k. Therefore, we perform our analysis for several values of k, but to not adversely affect the power of our tests we set the shortest subsample to four years (i.e., k = 8).
We also re-examine our hypothesis for subsamples that are based on monetary regime shifts. To do so, we partition the full sample of CAD/ USD exchange rate into subperiods based on U.S. monetary policy regime shifts. The rationale underlying this choice is the evidence that monetary regime shifts may have a significant impact on the conditional volatility of several financial time series. For example, Lastrapes (1989) notes that monetary policy regimes significantly affect the mean and variance of nominal exchange rates. Choi and Kim (1991) find that the foreign exchange risk premium depends on changes in the monetary regime. In the same vein, Hsieh (1991) finds that changes in operating procedures of the U.S. Federal Reserve Board (FRB) can shift the volatility of financial markets. In identifying the monetary regimes over the period 1976-2007, we use U.S. data as our proxy for shifts in Canadian monetary policy. Several authors have pointed out the high correlation between Canadian and U.S. interest rates (e.g., Mittoo 1992 and Yamada 2002). For instance, Yamada (2002) reports a one-to-one long-run linkage between the U.S. and Canadian real interest rates.
Lastrapes (1989) identifies three non-overlapping monetary regimes over the period 1976-1986, each characterized by different operating procedures (e.g., January 7, 1976 to October 3, 1979; October 10, 1979 to October 1, 1982 and October 6, 1982 to November 19, 1986). (7) Apart from these three time periods, there is evidence that Canadian monetary policy also experienced significant shifts in the post-1986 period. Shambora et al. (2006) find that January 1991 and January 1999 represents structural shifts in monetary policy. For example, in January 1991, the Bank of Canada adopted a policy of inflation targeting. It shifted away from this policy in January 1999 to a less restrictive monetary policy in order to boost corporate earnings growth. Consequently, we partition the full sample period into six subperiods representing the three U.S. monetary regime shifts ending in 1986 and the two shifts in Canadian monetary policy in the period January 1991 and January 1999.
We note that monetary regime shifts that we consider above may not coincide with structural breaks in our exchange-rate series that could arise due to, for instance, changes in settlement procedures. To see whether our findings are robust to the presence of structural breaks, we employ unit root test. If our data exhibits structural break(s), then we partition our sample based on the structural break date(s) and repeat our analysis for each subsample. We first use a widely used unit root test developed by Zivot and Andrews (1992). (8) But one major limitation with the Zivot-Andrews test is that only one-time structural change is assumed to exist. However, Kim et al. (2000) demonstrate that Zivot-Andrews approach may provide flawed results when a true second structural break is not accounted for. Imposing a number of structural changes without prior knowledge of their number may bias the results of the test and the estimation of the dates of the structural breaks. To deal with this issue we also use the Bai and Perron (1998) procedure, which accounts for multiple structural breaks.
The results for repartitioning the sample based on equal time periods, monetary regime shifts, structural breaks and extended our sample to December 31, 2012 (available upon request) are qualitatively similar to our results based on the full sample: evidence of day-the-week effect is sensitive to the particular specification of the underlying distributions.
Although we have shown that evidence of day-of-the-week effect in logarithmic changes in spot CAD/USD rate is sensitive to the particular specification of the underlying distributions, these results could be limited only to the Canadian exchange rate. Thus, they cannot be generalized to other foreign currencies. We address this issue by examining the weekday anomaly in the mean and the conditional volatility for other currencies.
Table 6 reports the results of introducing dummy variables into the mean and conditional variance equations under normal, student-t and GED for the Australian dollar, British pound, Indian rupee and Brazilian real. Our findings for all these currencies suggest that evidence of day-of-the-week effect in returns varies according to the assumption made on the series distribution, thus calling into question the seasonal pattern described in the extant literature. For instance, if we consider the Australian dollar (Panel A), we see that while there is no evidence of weekday effect under normal and generalized error distributions, the Wednesday coefficient is significant at 1% level when we consider student-t distribution. The results from the conditional volatility are also sensitive to the error distribution assumption. In fact, for the Australian dollar, while the Monday and Tuesday coefficients are statistically significant under normal and generalized error distributions, they are not when we assume a student-t distribution. The evidence for the British pound is more striking with respect to conditional volatility. Under normal distribution, all the coefficients are significant at the 1% level, but under the student-t distribution none of the coefficients is significant at any level. We observe the same pattern for the Brazilian real.
The results of our diagnostic tests show that each of the GARCH model is not misspecified. The LB statistics up to lag 100 could not reject the null hypothesis of no autocorrelation. The LM tests are also significant, indicating that three GARCH processes are successful at modeling the conditional volatility. The JB test for normality rejects the null hypothesis that the standardized residuals are normally distributed. However, we note that for all the GARCH models the sum of the parameters estimated by the variance equation is close to one. A sum of [[lambda].sub.1] and [[THETA].sub.1] near one indicates a covariance stationary model with a high degree of persistence and long memory in the conditional variance.
Finally, based on the model-selection criteria (e.g., AIC and BIC), all our results show that GARCH (1,1) with student-t error distribution outperforms the two other distributions. Therefore, we recommend the use the student-t distribution when investigating the day-of-the-week effect in mean and conditional volatility.
We provide a comprehensive analysis of the-day-of-the-week effect in returns and volatility for several foreign currencies. Our objective is to scrutinize the overwhelming support for what is considered calendar regularity (i.e., anomaly), since many studies stand on assumptions that are strongly rejected in financial time series. Consistent with our central hypothesis, we find that evidence of day-of-the-week effect in returns and conditional variance are not robust to heteroskedasticity or to our selected error distributional assumptions with weaker support under fat-tail distributions. Most strikingly, the day-of-the-week effect in variance disappears completely for some currencies when we control for autocorrelation, heteroskedasticity and non-normality. In addition, our results recommend choosing student-t distribution when investigating day-of-the-week effect. Our findings hold over time as well as when we take into account outliers, shifts in monetary regime and the presence of structural breaks. They also demonstrate the apparent fragility of previous empirical studies on the day-of-the-week effect and shed light on the importance of the choice of error distribution in model adequacy tests.
Because financial time series are known to have fatter tails than normal distributions and they exhibit volatility clustering, our findings cast doubt on the various theoretical explanations put forward in the extant literature to explain the day-of-the-week effect. In addition, the findings in this paper suggest a new avenue of research to identify the real cause of this market anomaly. As noted by Hansen, Lunde and Nason (2005), theoretical explanations have been suggested only subsequent to the empirical "discovery" of the anomalies. Because the previously reported evidence is likely to change under distributions that better fit financial time series, future researchers should direct their attention to carefully examining the calendar effects under fatter tail distributions and providing new economic explanations for them. Another promising line of research is to study how the profound structural changes that Forex witnessed over the last two decades (see Barker 2007) influence the dynamic of return anomalies in currency markets.
Akgiray, V., and G.G. Booth. "Distribution Properties of Latin American Black Market Exchange Rate." Journal of International Money and Finance 7.1(1988): 37-48.
Andersen, T.G., and T. Bollerslev. "Deutsche Mark-Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies." Journal of Finance 53.1(1998): 219-265.
Andolfatto, D., and P. Gomme. "Monetary Policy Regimes and Beliefs." International Economic Review 44.1(2003): 1-30.
Aydogan, K., and G. Booth. "Calendar Anomalies in the Turkish Foreign Exchange Markets." Applied Financial Economics 13.5(2003): 353-360.
Bai, J., and P. Perron. "Estimating and Testing Linear Models with Multiple Structural Changes." Econometrica 66.1(1998): 47-79.
Baker, H.K., A. Rahman, and S. Saadi. "The Day-of-the-Week Effect and Conditional Volatility: Sensitivity of Error Distributional Assumptions." Review of Financial Economics 17.4(2008): 280-95.
Barker, W. "The Global Foreign Exchange Market: Growth and Transformation." Bank of Canada Review Autumn(2007): 3-12.
Barnett, W.A., A.R. Gallant, M.J. Hinish, J. Jungeilges, D. Kaplan, and M.J. Jensen. "Robustness of Nonlinearity and Chaos Tests to Measurement Error, Inference Method, and Sample Size." Journal of Economic Behaviour and Organization 27.2(1995): 301-320.
Berument, H., M.N. Coskun, and A. Cahin. "Day of the Week Effect on Foreign Exchange Market Volatility: Evidence from Turkey." Research in International Business and Finance 21.1(2007): 87-97.
Bessembinder, H. "Bid-ask Spreads in the Interbank Foreign Exchange Markets." Journal of Financial Economics 35.3(1994): 317-348.
Bossaerts, R, and P. Hillion. "Market Microstructure Effects of Government Intervention in the Foreign Exchange Markets." Review of Financial Studies 4.3(1991): 513-541.
Brenner, R.J., R.H. Harjes, and K.F. Kroner. "Another Look at Models of the Short-term Interest Rates." Journal of Financial and Quantitative Analysis 31.1(1996): 85-107.
Breuer, J. "Day-of-Week Effects in Tests of Forward Foreign Exchange Rate Unbiasedness." International Journal of Finance and Economics 4.3(1999): 193-204.
Brock, W.A., W. Dechert, and J. Scheinkman. "A Test for Independence Based on the Correlation Dimension." Working Paper, University of Wisconsin at Madison, 1987.
Brock, W.A., W. Dechert, J. Scheinkman, and B. LeBaron. "A Test for Independence Based on the Correlation Dimension." Econometric Reviews 15.3(1996): 197-235.
Brock, W.A., D.A. Hsieh, and B. LeBaron. Nonlinear Dynamics, Chaos and Instability: Statistical Theory and Economic Evidence. Cambridge: MIT Press, 1991.
Choi, S., and B.J.C. Kim. "Monetary Policy Regime Changes and the Risk Premium in the Foreign Exchange Markets: A GARCH Application." Economics Letters 37.4(1991): 447-452.
Conover, W.J., M.E. Johnson, and M.M. Johnson. "A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data." Technometrics 23.4(1981): 351-361.
Corhay, A., A.M. Fatemi, and A. Tourani-Rad. "On the Presence of a Day-of-the-Week Effect in the Foreign Exchange Market." Managerial Finance 21.8(1995): 32-43.
Cross, F. "The Behavior of Stock Prices on Fridays and Mondays." Financial Analysts Journal 29.6(1973): 67-69.
Duffy, J., and J. Engle-Warnick. "Multiple Regimes in US Monetary Policy? A Nonparametric Approach." Journal of Money Credit and Banking 38.5(2006): 1363-1378.
Fama, E. "The Behavior of Stock Market Prices." Journal of Business 38.1(1965): 34-105.
Flannary, M. J. and A. A. Protopapadakis. "From T-bills to Common Stocks: Investigating the Generality of Intra-week Return Seasonality." Journal of Finance 43.2(1988): 431-449.
Galai, D., H. Kedar-Levy, and B.Z. Schreiber. "Seasonality in Outliers of Daily Stock Returns: A Tail that Wags the Dog?" International Review of Financial Analysis 17.5(2008): 784-792.
Gay, G., and T. Kim. "An Investigation into Seasonality in the Futures Market." Journal of Futures Markets 7.2(1987): 169-181.
Gesser, V., and P. Poncet. "Volatility Patterns: Theory and Some Evidence from the Dollar-mark Option Market." Journal of Derivatives 5.2(1997): 46-61.
Glassman, D. "Exchange Rate Risk and Transactions Costs: Evidence from Bid-Ask Spreads." Journal of International Money and Finance 6.4(1987): 479-490.
Grassberger, P, and I. Procaccia. "Measuring the Strangeness of Strange Attractors." Physica 9.1-2(1983): 189-208.
Hansen, PR., and A. Lunde. "A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH(1,1)?" Journal of Applied Econometrics 20.7(2005): 873-889.
Hansen, P.R., A. Lunde, and J.M. Nason. "Testing the Significance of Calendar Effects." Working Paper, Federal Reserve Bank of Atlanta, 2005
Hsieh, D.A. "The Statistical Properties of Daily Foreign Exchange Rates: 1974-1983." Journal of International Economics 24.1-2(1988): 129-145.
Hsieh, D.A. "Testing for Nonlinearity Dependence in Daily Foreign Exchange Rate." Journal of Business 62.3(1989): 339-368.
Hsieh, D.A. "Chaos and Nonlinear Dynamics: Application to Financial Markets." Journal of Finance 46.5(1991): 1837-1877.
Kim, T.H., S. Leybourne, and P. Newbold. "Spurious Rejections by Perron Tests in the Presence of a Break." Oxford Bulletin of Economics and Statistics 62.3(2000): 433-44.
Lastrapes, W.D. "Exchange Rate Volatility and U.S. Monetary Policy: An ARCH Application." Journal of Money, Credit and Banking 21.1(1989): 66-77.
McFarland, J.W., R.R. Pettit, and S.K. Sung. "The Distribution of Foreign Exchange Price Changes: Trading Day Effects and Risk Measurement." Journal of Finance 37.3(1982): 693-715.
Mittoo, U.R. "Additional Evidence on Integration in the Canadian Stock Market." Journal of Finance 47.5(1992): 2035-2054.
Nelson, D.B. "Conditional Heteroskedasticy in Asset Returns: A New Approach." Econometrica 59.2(1991): 347-370.
Perron, P. "The Great Crash, the Oil Price Shock, and the Unit Root Piypothesis." Econometrica 57.6(1989): 1361-1401.
Philips, C.B., and P. Perron. "Testing for Unit Roots in Time Series Regression." Biometrika 75.2(1988): 335-346.
Philpot, J., and C.A. Peterson. "A Brief History and Recent Developments in Day-of-the-Week Effect Literature." Managerial Finance 37.9(2011): 808-816.
Popovic, S., and A. Durovic. "Intraweek and Intraday Trade Anomalies: Evidence from FOREX Market." Applied Economics 46.32(2014): 3968-3979.
Said, S.E., and D. Dickey. "Testing for Unit Roots in Autoregressive Moving-average Models with Unknown Order." Biometrika 71.3(1984): 599-607.
Scheinkman, J.A., and B. LeBaron. "Nonlinear Dynamics and Stock Returns." Journal of Business 62.3(1989): 311-337.
Shambora, W.E., K. Choi, and C. Jung. "Credibility and Effectiveness in Canadian Inflation Targeting." Working Paper, Ohio University, 2006
Stiglitz, J.E. "The Current Economic Crisis and Lessons for Economic Theory." Eastern Economic Journal 35.3(2009): 281-296.
Tang, G.Y.N. "Impact of the Day-of-the-Week Effect on Diversification and Exchange Rate Risks." International Business Review 6.1(1997): 35-51.
Thatcher, J.S., and L.P. Blenman. "Synthetic Trades and Calendar Day Patterns: The Case of the Dollar/Sterling Markets." The Financial Review 36.2(1998): 177-200.
Yamada, H. "On the Linkage of Real Interest Rates between the U.S. and Canada: Some Additional Empirical Evidence." Journal of International Financial Markets, Institutions and Money 12.3(2002): 279-289.
Yamori, N., and Y. Kurihara. "The Day-of-the-Week Effect in Foreign Exchange Markets: Multi-currency Evidence." Research in International Business and Finance 18.1(2004): 51-57.
Yu, H.C., I. Chiou, and J.J. Wagner. "Does the Weekday Effect of the Yen/ Dollar Spot Rates Exist in Tokyo, London, and New York? An Analysis of Panel Probability Distribution." Applied Economics 40.20(2008): 353-375.
Zivot, E., and D.W.K. Andrews. "Further Evidence on the Great Crash, the Oil-price Shock, and the Unit-root Hypothesis." Journal of Business and Economic Statistics 10.3(1992): 251-270.
To investigate the existence of nonlinear dependency, we test whether the residuals in Equation (2) are i.i.d. To do so, we use a powerful test (BDS), originally proposed by Brock et al. (1987) and designed by Brock et al. (1996). The BDS test is a nonparametric test with the null hypothesis that the series in question is i.i.d. against an unspecified alternative. The test is based on the concept of a correlation integral, a measure of spatial correlation in n-dimensional space originally developed by Grassberger and Procacccia (1983).
Thus, we consider a vector of m histories of the CAD/USD rate of change,
[r.sup.m.sub.t] = ([r.sub.t], [r.sub.1+1], ..., [r.sub.t+m-i]) (a)
the correlation integral measures the number of m vectors within a distance of [epsilon] of one another. We define the correlation integral as:
[C.sub.m]([epsilon], T) = 2/[T.sub.m]([T.sub.m] - 1) [summation over (t<s)] [I.sub.[epsilon]] ([r.sup.m.sub.t], [r.sup.m.sub.s] (b)
where the parameter m is the embedding dimension; T is the sample size; [T.sub.m] = T - m +1 is the maximum number of overlapping vectors that we can form with a sample of size T; and [I.sub.[epsilon]] is an indicator function that is equal to one if [parallel][r.sup.m.sub.t], - [r.sup.m.sub.s][parallel] < [epsilon], and zero otherwise. A pair of vectors [r.sup.m.sub.t] and [r.sup.m.sub.s] is said to be [epsilon] apart if the maximum-norm [parallel]. [parallel] is greater or equal to [epsilon]. Under the null hypothesis of independently and identically distributed random variables, [C.sub.m]([epsilon]) = [C.sub.1][([epsilon]).sup.m]. Using this relation, we define the BDS test statistic as:
BDS(m, [epsilon]) = [C.sub.m]([epsilon], T)-[[[C.sub.1]([epsilon])].sub.m]/[[sigma].sub.m]([epsilon], T)/[square root of T] (c)
where [[sigma].sub.m]([epsilon], T)/[square root of]T is the standard deviation of the difference between the two correlation measures [C.sub.m]([epsilon],T) and [[[C.sub.1]([epsilon])].sup.m]. For large samples, the BDS statistic has a standard normal limiting distribution under the null of i.i.d. If asset price changes are not identically and independent random variables, then [C.sub.m]([epsilon]) > [C.sub.1][([epsilon]).sup.m].
We note that the BDS test statistic is sensitive to the choice of the embedding dimension m and the bound e. As mentioned by Scheinkman and LeBaron (1989), if we attribute a value that is too small for [epsilon], then the null hypothesis of a random i.i.d. process will be accepted too often regardless of whether it is true or false. As well, it is not safe to choose too large a value for [epsilon]. To deal with this problem, Brock et al. (1991) suggest that for a large sample size (T > 500), [epsilon] should equal 0.5, 1, 1.5 and 2 times standard deviations of the data. For the choice of the relevant embedding dimension m, Hsieh (1989) suggests considering a range of values from two to 10 for this parameter. Following Barnett et al. (1995), we implement the BDS test for the range of m-values from two to an upper bound of eight. In general, a rejection of the null hypothesis is consistent with some type of dependence in the returns that could result from a linear stochastic process, non-stationarity, nonlinear stochastic process or non-linear deterministic system.
(1) The existence of calendar effects is also questioned by studies examining equity returns. Hansen, Lunde and Nason (2005), for instance, examine several calendar effects using a test statistic that limits data-mining bias and find that evidence of calendar effects in equity returns has diminished over time except in small-cap stock indexes.
(2) The student-t and GED are heavy-tailed distributions with positive excess kurtosis relative to a normal distribution, which has excess kurtosis of 0. Excess kurtosis is 6/ (df - 4), where df = degrees of freedom, for the student-t distribution. For the GED, the value of the shape parameter determines the thickness of the tail. When this parameter has a value less than two, the distribution is thick-tailed. When the value is two, the result is a normal distribution. The extant literature guides the appropriate choice of these distributions as error terms. For example, Brenner et al. (1996) show that a student-t distribution better characterizes short-term interest rates. Nelson (1991) popularized GARCH models with GED errors.
(3) See Appendix for detailed description of BDS test.
(4) The earliest evidence of weekday effect in stock returns was reported by Fama (1965) and Cross (1973). They find that both lowest mean return and highest variance occur on Monday, offering a poor risk-returns relationship compared to those of the other days of the week. Since these influential papers, several empirical studies emerged supporting the day-of-the-week effect in stock returns in the U.S. but also in other developed as well as developing markets. See Philpot and Peterson (2011) for a brief review of studies on the weekend effect published prior to 2003 along with some of the recent work on the subject.
(5) Besides the student-t distribution and the generalized error distribution (GED), we also use a third heavy-tailed distribution named the double exponential distribution (DED). The results, available upon request, corroborate our main hypothesis that evidence of weekday effect is sensitive to choice of the underlying distribution specification.
(6) We do not report these results; however, they are available from the authors upon request.
(7) Andolfatto and Gomme (2003) find evidence that October 1979 represented a significant shift in monetary policy in Canada. Duffy and Engle-Warnick (2006) confirm this finding.
(8) Until recently, most research dealing with structural breaks and random walk hypothesis uses the classics unit root tests of Said and Dickey (1984) (known as Augmented Dickey Fuller, ADF). However, Perron (1989) demonstrates that ADF is subject to misspecification bias and size distortion when the series involved undergo structural shifts that lead to spurious acceptance of the unit root hypothesis. The Philips-Perron test (PP) (1988) overcomes this limitation by allowing for a one-time structural break. But the PP test has also been criticized because it assumes that the breakpoints are exogenously determined. In fact, Zivot and Andrews (1992) demonstrate that endogenously determining the time of structural breaks may reverse the results of the unit root hypothesis previously suggested by earlier conventional tests, such as the ADF and PP tests. Consequently, many papers use the procedure advocated by Zivot and Andrews (1992), which allows an endogenously-estimated structural change.
TABLE 1 Summary Statistics of Log CND/USD Daily Price Changes Note: We compute the daily return as the natural logarithmic first difference of the daily price of CND/USD exchange rate: [r.sub.1] = 100 x 1n([S.sub.t] / [S.sub.t-1]), where [S.sub.t] and [S.sub.t-1] are the exchange rates at date t and t-1, respectively. The sample period is January 2, 1976 to December 31, 2007. Jarque-Bera test statistic is 3786.77, which rejects the null hypothesis of the series being normally distributed. Std. Error Std. Weekday N Mean Median of Mean Deviation Monday 1467 0.0287 0.0084 0.0000 0.3103 Tuesday 1599 -0.0062 0.0076 0.0000 0.2930 Wednesday 1602 0.0044 0.0077 0.0000 0.2978 Thursday 1601 -0.0098 0.0076 -0.0062 0.2952 Friday 1568 0.0034 0.0076 0.0000 0.2912 All Days 7837 0.0037 0.0035 0.0000 0.2976 Std. Error Std. Error Weekday Kurtosis of Kurtosis Skewness of Skewness Monday 4.1449 0.1326 0.4264 0.0664 Tuesday 3.1940 0.1266 -0.0201 0.0633 Wednesday 3.7628 0.1265 -0.3386 0.0633 Thursday 3.5592 0.1265 0.3117 0.0633 Friday 2.8060 0.1279 -0.0744 0.0640 All Days 3.5274 0.0573 0.0701 0.0287 TABLE 2 Testing for Day of the Week Effect in the Mean of CND/USD Daily Price Changes: Games & Howell's Test Note: We compute the daily return as the natural logarithmic first difference of the daily price of CND/USD exchange rate: [r.sub.t] = 100 x 1n([S.sub.t]/[S.sub.t-1]), where St and SM are the exchange rates at date t and t-1, respectively. The sample period is January 2, 1976 to December 31, 2007. 95% Mean Confidence (I) (J) Difference Std. P- Interval Upper Weekday Weekday (I-J) Error value Lower Bound Bound Monday Tuesday 0.0349 0.0113 0.0178 0.0040 0.0658 Wednesday 0.0243 0.0114 0.2061 -0.0068 0.0555 Thursday 0.0385 0.0114 0.0063 0.0075 0.0695 Friday 0.0253 0.0113 0.1682 -0.0056 0.0563 Tuesday Monday -0.0349 0.0113 0.0178 -0.0658 -0.0040 Wednesday -0.0106 0.0108 0.8656 -0.0401 0.0189 Thursday 0.0036 0.0108 0.9972 -0.0257 0.0330 Friday -0.0096 0.0107 0.9008 -0.0389 0.0198 Wednesday Monday -0.0243 0.0114 0.2061 -0.0555 0.0068 Tuesday 0.0106 0.0108 0.8656 -0.0189 0.0401 Thursday 0.0142 0.0108 0.6853 -0.0154 0.0438 Friday 0.0010 0.0108 1.0000 -0.0286 0.0306 Thursday Monday -0.0385 0.0114 0.0063 -0.0695 -0.0075 Tuesday -0.0036 0.0108 0.9972 -0.0330 0.0257 Wednesday -0.0142 0.0108 0.6853 -0.0438 0.0154 Friday -0.0132 0.0108 0.7373 -0.0426 0.0162 Friday Monday -0.0253 0.0113 0.1682 -0.0563 0.0056 Tuesday 0.0096 0.0107 0.9008 -0.0198 0.0389 Wednesday -0.0010 0.0108 1.0000 -0.0306 0.0286 Thursday 0.0132 0.0108 0.7373 -0.0162 0.0426 TABLE 3 Introducing Dummy Variables into the Mean Equation Only Note: AR(5) - GARCH(1,1): [r.sub.t] = [alpha] + [B.sub.M] [D.sub.Mt] + [B.sub.T] [D.sub.Tt] + [B.sub.W] [D.sub.Wt] + [B.sub.H] [D.sub.Ht] + [5.summation over (i=1)] [[alpha].sub.i] [r.sub.t-i] + [[epsilon].sub.t]; [[sigma].sup.2.sub.t] = [eta] + [lambda](L)[[omega].sup.2.sub.t] + [theta](L) [[sigma].sup.2.sub.t-1] where [r.sub.t] is the daily return at time t; a is a constant term; and [D.sub.Mt], [D.sub.Tt], [D.sub.Wt], [D.sub.Ht] are the dummy variables for Monday, Tuesday, Wednesday and Thursday, respectively. [lambda] and [theta] are the ARCH and GARCH parameters, respectively. Garch (1,1) with Normal Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0141 0.0059 0.0089 Monday 0.0011 0.0082 0.4484 Tuesday -0.0187 0.0084 0.0127 Wednesday -0.0053 0.0081 0.2567 Thursday -0.0320 0.0082 0.0000 [eta] 0.0009 0.0001 0.0000 [lambda] 0.0946 0.0043 0.0000 [theta] 0.9016 0.0037 0.0000 Jarque-Bera 2051 0.0000 Ljung-Box (20) 9.86 0.9706 Ljung-Box (30) 12.83 0.9974 Ljung-Box (40) 19.86 0.9968 Ljung-Box (50) 24.12 0.9993 Ljung-Box (100) 75.29 0.9691 LM Test 15.76 0.2026 AIC 1386 BIC 1703 Garch (1,1) with Student-t Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0112 0.0055 0.0211 Monday -0.0037 0.0077 0.3172 Tuesday -0.0125 0.0078 0.0547 Wednesday -0.0048 0.0077 0.2671 Thursday -0.0285 0.0077 0.0001 [eta] 0.0006 0.0001 0.0000 [lambda] 0.0927 0.0072 0.0000 [theta] 0.9088 0.0062 0.0000 Jarque-Bera 2392 0.0000 Ljung-Box (20) 19.30 0.5025 Ljung-Box (30) 20.96 0.8892 Ljung-Box (40) 27.58 0.9318 Ljung-Box (50) 31.78 0.9792 Ljung-Box (100) 82.21 0.9021 LM Test 16.38 0.1744 AIC 918 BIC 1242 Garch (1,1) with GED Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0104 0.0053 0.0242 Monday -0.0018 0.0075 0.4053 Tuesday -0.0108 0.0075 0.0743 Wednesday -0.0031 0.0073 0.3353 Thursday -0.0266 0.0074 0.0002 [eta] 0.0007 0.0001 0.0000 [lambda] 0.0890 0.0069 0.0000 [theta] 0.9082 0.0062 0.0000 Jarque-Bera 2288 0.0000 Ljung-Box (20) 24.30 0.2295 Ljung-Box (30) 27.48 0.5980 Ljung-Box (40) 34.48 0.7165 Ljung-Box (50) 38.54 0.8809 Ljung-Box (100) 88.82 0.7807 LM Test 16.43 0.1725 AIC 928 BIC 1252 TABLE 4 BDS Statistics for the Standardized Residuals of fhe GARCH (1,1) Models Note: m is embedding dimension, and e is the bound. * Significant at the 5% level. ** Significant at the 1% level. The critical values for BDS test are 1.96 for 5% and 2.58 for 1%. Panel A: Student-t Error Distribution [epsilon]/ [epsilon]/ m [sigma] [sigma] 1 0.5 0.52 1 0.49 2 0.5 1.07 1 0.89 3 0.5 1.38 1 1.24 4 0.5 1.59 1 1.40 5 0.5 1.77 1 1.58 6 0.5 1.77 1 1.70 7 0.5 1.63 1 1.97 8 0.5 1.88 1 1.86 [epsilon]/ [epsilon]/ m [sigma] [sigma] 1 1.5 0.56 2 0.11 2 1.5 0.72 2 0.20 3 1.5 0.93 2 0.24 4 1.5 0.96 2 0.37 5 1.5 1.03 2 0.44 6 1.5 1.03 2 0.56 7 1.5 1.11 2 0.55 8 1.5 1.06 2 0.62 Panel B: GED Error Distribution [epsilon]/ [epsilon]/ m [sigma] [sigma] 1 0.5 0.56 1 0.54 2 0.5 1.16 1 0.98 3 0.5 1.51 1 1.36 4 0.5 1.73 1 1.52 5 0.5 1.93 1 1.72 6 0.5 1.94 1 1.85 7 0.5 1.78 1 2.15 8 0.5 1.84 1 1.83 [epsilon]/ [epsilon]/ m [sigma] [sigma] 1 1.5 0.61 2 0.12 2 1.5 0.78 2 0.22 3 1.5 1.01 2 0.26 4 1.5 1.05 2 0.40 5 1.5 1.13 2 0.49 6 1.5 1.12 2 0.61 7 1.5 1.21 2 0.60 8 1.5 1.27 2 0.74 TABLE 5 Introducing Dummy Variables into the Mean Equation as Well as Conditional Variance Equation Note: AR(5)- GARCH(1,1): [r.sub.t] = [alpha] + [B.sub.M][D.sub.Mt] + [B.sub.T][D.sub.Tt] + [B.sub.W][D.sub.Wt] + [B.sub.H][D.sub.Ht] + [5.summation over (i=t)][[alpha].sub.t][r.sub.t-1] + [[epsilon].sub.t]; [[sigma].sup.2.sub.t] = [eta] + [lambda](L)[[omega].sup.2.sub.1] + [theta](L)[[sigma].sup.2.sub.t-1] + [B.sub.M][D.sub.Mt] + [B.sub.T][D.sub.Mt] + [B.sub.T][D.sub.Tt] + [B.sub.W][D.sub.Wt] + [B.sub.H][D.sub.Ht] where [r.sub.t] is the daily return at time t; [alpha] is a constant term; and [D.sub.Mt], [D.sub.Ft], [D.sub.Wt], [D.sub.Ht] are the dummy variables for Monday, Tuesday, Wednesday and Thursday, respectively. [lambda] and [theta] are the ARCH and GARCH parameters, respectively. Garch (1,1) with Normal Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0122 0.0060 0.0209 Monday 0.0030 0.0086 0.3644 Tuesday -0.0169 0.0083 0.0208 Wednesday -0.0036 0.0081 0.3280 Thursday -0.0303 0.0084 0.0001 [eta] -0.0001 0.0013 0.4799 [lambda] 0.0887 0.0041 0.0000 [theta] 0.9076 0.0036 0.0000 Monday 0.0034 0.0020 0.0466 Tuesday -0.0029 0.0020 0.0754 Wednesday 0.0000 0.0020 0.4985 Thursday 0.0041 0.0022 0.0302 Jarque-Bera 2055 0.0000 Ljung-Box (20) 9.34 0.9786 Ljung-Box (30) 12.14 0.9984 Ljung-Box (40) 18.84 0.9982 Ljung-Box (50) 23.04 0.9996 Ljung-Box (100) 74.38 0.9742 LM Test 16.68 0.1619 AIC 1385 BIC 1729 Garch (1,1) with Student-t Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0108 0.0055 0.0242 Monday -0.0024 0.0080 0.3794 Tuesday -0.0114 0.0077 0.0680 Wednesday -0.0042 0.0076 0.2908 Thursday -0.0279 0.0077 0.0001 [eta] 0.0001 0.0021 0.4880 [lambda] 0.0910 0.0072 0.0000 [theta] 0.9101 0.0062 0.0000 Monday 0.0041 0.0032 0.0993 Tuesday -0.0039 0.0033 0.1229 Wednesday 0.0011 0.0031 0.3569 Thursday 0.0016 0.0039 0.3388 Jarque-Bera 2442 0.0000 Ljung-Box (20) 18.93 0.5263 Ljung-Box (30) 20.53 0.9022 Ljung-Box (40) 27.32 0.9366 Ljung-Box (50) 31.43 0.9815 Ljung-Box (100) 81.93 0.9059 LM Test 16.38 0.1743 AIC 920 BIC 1272 Garch (1,1) with GED Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0098 0.0052 0.0308 Monday -0.0013 0.0076 0.4349 Tuesday -0.0094 0.0072 0.0976 Wednesday -0.0024 0.0072 0.3687 Thursday -0.0260 0.0073 0.0002 [eta] 0.0000 0.0021 0.4954 [lambda] 0.0899 0.0068 0.0000 [theta] 0.9080 0.0061 0.0000 Monday 0.0042 0.0032 0.0929 Tuesday -0.0043 0.0033 0.0922 Wednesday 0.0008 0.0030 0.3969 Thursday 0.0031 0.0036 0.1987 Jarque-Bera 2392 0.0000 Ljung-Box (20) 24.26 0.2311 Ljung-Box (30) 27.62 0.5908 Ljung-Box (40) 35.15 0.6880 Ljung-Box (50) 39.14 0.8662 Ljung-Box (100) 90.38 0.7440 LM Test 16.66 0.1627 AIC 930 BIC 1281 TABLE 6 Introducing Dummy Variables into the Mean Equation and Conditional Variance Equation: Others Currencies Note: AR(3)- GARCH(1,1): [r.sub.t] = [alpha] + [B.sub.M][D.sub.Mt] + [B.sub.t][D.sub.Tt] + [B.sub.W][D.sub.Wt] + [B.sub.H][D.sub.Ht] + [3.summation over (i=1)] [[alpha].sub.t][r.sub.t-1] + [[epsilon].sub.t]; [[sigma].sup.2.sub.t] = [eta] + [lambda](L)[[omega].sup.2.sub.t] + [theta](L)[[sigma].sup.2.sub.t-1] + [B.sub.M][D.sub.Mt] + [B.sub.T][D.sub.Tt] + [B.sub.W][D.sub.Wt] + [B.sub.H][D.sub.Ht] where [r.sub.t] is the daily return at time t; a is a constant term; and [D.sub.Mt], [D.sub.Ft], [D.sub.Wt], [D.sub.Ht] are the dummy variables for Monday, Tuesday, Wednesday and Thursday, respectively. [lambda] and [theta] are the ARCH and GARCH parameters, respectively. Panel A: Australian Dollar/US Dollar Garch (1,1) with Garch (1,1) with Normal Error Student-t Error Distribution Distribution Std. Std. Coefficients Value Error P-value Value Error P-value [alpha] 0.0353 0.0149 0.0183 -0.0028 0.0055 0.6137 Monday -0.0243 0.0195 0.2129 0.0022 0.0078 0.7818 Tuesday -0.0163 0.0212 0.4434 0.0053 0.0079 0.5046 Wednesday 0.0007 0.0229 0.9763 0.0191 0.0067 0.0046 Thursday -0.0136 0.0192 0.4788 0.0102 0.0070 0.1429 [eta] 0.0956 0.0018 0.0000 0.0034 0.0006 0.0000 [lambda] 0.1362 0.0050 0.0000 0.2848 0.0181 0.0000 [theta] 0.7921 0.0063 0.0000 0.8176 0.0059 0.0000 Monday -0.1077 0.0019 0.0000 -0.0015 0.0013 0.2293 Tuesday -0.0172 0.0025 0.0000 -0.0002 0.0013 0.8679 Wednesday -0.0271 0.0022 0.0000 -0.0076 0.0012 0.0000 Thursday -0.1682 0.0004 0.0000 -0.0037 0.0004 0.0000 Jarque-Bera 8018.1 0.0000 8343,3 0.0000 Ljung-Box (20) 60.52 0.4354 26.75 0.1422 Ljung-Box (30) 68.95 03223 34.61 0.2568 Ljung-Box (40) 94.48 0.5675 61.61 0.0956 Ljung-Box (50) 97.83 0.2123 65.53 0.0692 Ljung-Box (100) 150.20 0.9879 116.85 0.1195 LM Test 0.1922 1.0000 0.0604 1.0000 AIC 15631 7734 BIC 15752 7861 Garch (1,1) with GED Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0003 0.0059 0.9569 Monday -0.0004 0.0076 0.9631 Tuesday -0.0003 0.0086 0.9701 Wednesday 0.0062 0.0084 0.4603 Thursday 0.0027 0.0083 0.7462 [eta] 0.0020 0.0002 0.0000 [lambda] 0.1415 0.0030 0.0000 [theta] 0.8415 0.0059 0.0000 Monday -0.0179 0.0001 0.0000 Tuesday 0.0150 0.0010 0.0000 Wednesday 0.0012 0.0002 0.0000 Thursday 0.0080 0.0005 0.0000 Jarque-Bera 8326.3 0.0000 Ljung-Box (20) 22.05 0.3374 Ljung-Box (30) 29.75 0.4780 Ljung-Box (40) 56.90 0.1403 Ljung-Box (50) 60.92 0.1384 Ljung-Box (100) 112.02 0.1936 LM Test 0.0319 1.0000 AIC 9353 BIC 9480 Panel B: British Pound/US Dollar Garch (1,1) with Garch (1,1) with Normal Error Student-t Error Distribution Distribution Std. Std. Coefficients Value Error P-value Value Error P-value [alpha] -0.0293 0.0130 0.0239 -0.0021 0.0036 0.5528 Monday 0.0429 0.0180 0.0172 0.0092 0.0054 0.0908 Tuesday 0.0217 0.0169 0.2010 -0.0023 0.0054 0.6738 Wednesday 0.0692 0.0162 0.0000 0.0318 0.0049 0.0000 Thursday 0.0141 0.0161 0.3820 -0.0151 0.0047 0.0014 [eta] 0.0631 0.0008 0.0000 0.0000 0.0003 0.8497 [lambda] 0.0987 0.0043 0.0000 0.1904 0.0127 0.0000 [theta] 0.8844 0.0042 0.0000 0.8664 0.0055 0.0000 Monday -0.0448 0.0034 0.0000 0.0004 0.0004 0.3610 Tuesday -0.0997 0.0041 0.0000 -0.0002 0.0004 0.5738 Wednesday -0.0771 0.0039 0.0000 -0.0003 0.0003 0.2914 Thursday -0.0542 0.0028 0.0000 0.0000 0.0003 0.8884 Jarque-Bera 5976 0.0000 1825 0.0000 Ljung-Box (20) 16.98 0.6538 38.40 0.1179 Ljung-Box (30) 35.15 0.2372 56.48 0.0924 Ljung-Box (40) 44.32 0.2943 64.56 0.2382 Ljung-Box (50) 61.26 0.1321 80.85 0.4537 Ljung-Box (100) 123.79 0.0535 143.0 0.0031 LM Test 6.03 0.9143 0.17 1.0000 AIC 13461 11701 BIC 13616 11864 Garch (1,1) with GED Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0004 0.0064 0.9563 Monday 0.0085 0.0087 0.3281 Tuesday -0.0078 0.0068 0.2571 Wednesday 0.0312 0.0082 0.0002 Thursday -0.0156 0.0070 0.0252 [eta] 0.0143 0.0002 0.0000 [lambda] 0.1389 0.0073 0.0000 [theta] 0.8848 0.0046 0.0000 Monday -0.0130 0.0003 0.0000 Tuesday -0.0269 0.0005 0.0000 Wednesday -0.0105 0.0006 0.0000 Thursday -0.0177 0.0004 0.0000 Jarque-Bera 14742 0.0000 Ljung-Box (20) 47.59 0.3225 Ljung-Box (30) 65.94 0.4452 Ljung-Box (40) 74.09 0.6788 Ljung-Box (50) 90.38 0.6654 Ljung-Box (100) 152.19 0.1226 LM Test 4.49 0.9727 AIC 11891 BIC 12054 Panel C; Indian Rupee/US Dollar Garch (1,1) with Garch (1,1) with Normal Error Student-t Error Distribution Distribution Std. Std. Coefficients Value Error P-value Value Error P-value [alpha] -0.0730 0.0451 0.0541 -0.0011 0.0020 0.5733 Monday 0.0941 0.0019 0.0202 0.0031 0.0027 0.2599 Tuesday 0.0601 0.0539 0.1865 0.0077 0.0028 0.0050 Wednesday 0.2194 0.0631 0.0002 0.0006 0.0025 0.8142 Thursday 0.0063 0.0732 0.5532 -0.0018 0.0023 0.4246 [eta] 0.0092 0.0018 0.0000 0.0006 0.0001 0.0000 [lambda] 0.0994 0.0096 0.0000 0.4175 0.0539 0.0000 [theta] 0.9078 0.0079 0.0000 0.8007 0.0084 0.0000 Monday -0.0448 0.0034 0.0000 -0.0004 0.0002 0.0537 Tuesday -0.0997 0.0041 0.0000 0.0001 0.0001 0.6443 Wednesday -0.0771 0.0039 0.0000 -0.0013 0.0002 0.0000 Thursday -0.0542 0.0028 0.0000 -0.0008 0.0001 0.0000 Jarque-Bera 2405 0.0000 3022 0.0000 Ljung-Box (20) 31.62 0.0876 1945.21 0.0000 Ljung-Box (30) 36.94 0.1790 2178.45 0.0000 Ljung-Box (40) 45.41 0.2569 2252.34 0.0000 Ljung-Box (50) 49.32 0.5005 2352.04 0.0000 Ljung-Box (100) 100.58 0.4648 2747.03 0.0000 LM Test 16.63 0.1641 0.18 1.0000 AIC -6432 -5120 BIC -6219 -4965 Garch (1,1) with GED Error Distribution Std. Coefficients Value Error P-value [alpha] -0.0001 0.0009 0.9481 Monday 0.0002 0.0013 0.9032 Tuesday 0.0035 0.0017 0.0367 Wednesday 0.0000 0.0011 0.9694 Thursday 0.0000 0.0011 0.9741 [eta] 0.0015 0.0001 0.0000 [lambda] 0.2617 0.0209 0.0000 [theta] 0.8127 0.0100 0.0000 Monday -0.0014 0.0003 0.0000 Tuesday -0.0010 0.0004 0.0137 Wednesday -0.0019 0.0003 0.0000 Thursday -0.0019 0.0002 0.0000 Jarque-Bera 8336 0.0000 Ljung-Box (20) 1792.84 0.0000 Ljung-Box (30) 2017.08 0.0000 Ljung-Box (40) 2086.22 0.0000 Ljung-Box (50) 2174.69 0.0000 Ljung-Box (100) 2535.32 0.0000 LM Test 2.11 0.9992 AIC -5279 BIC -5125 Panel D: Brazilian Real/US Dollar Garch (1,1) with Garch (1,1) with Normal Error Student-t Error Distribution Distribution Std. Std. Coefficients Value Error P-value Value Error P-value [alpha] 0.0071 0.0328 0.8277 0.0263 0.0064 0.0000 Monday -0.0060 0.0514 0.9068 -0.0132 0.0086 0.1229 Tuesday 0.0120 0.0524 0.8197 -0.0006 0.0085 0.9407 Wednesday 0.0621 0.0596 0.2974 0.0033 0.0081 0.6869 Thursday 0.0228 0.0481 0.6352 -0.0134 0.0082 0.1014 [eta] -0.1023 0.0035 0.0000 0.0014 0.0011 0.1908 [lambda] 0.1149 0.0087 0.0000 0.2664 0.0295 0.0000 [theta] 0.8401 0.0106 0.0000 0.8154 0.0130 0.0000 Monday 0.1886 0.0250 0.0000 -0.0016 0.0019 0.4038 Tuesday 0.1462 0.0270 0.0000 -0.0002 0.0018 0.8898 Wednesday 0.2904 0.0184 0.0000 -0.0018 0.0017 0.3135 Thursday 0.0880 0.0142 0.0000 -0.0023 0.0017 0.1722 Jarque-Bera 5708 0.0000 1756 0.0000 Ljung-Box (20) 22.44 0.3166 15.04 0.0000 Ljung-Box (30) 31.70 0.3814 16.90 0.0998 Ljung-Box (40) 58.10 0.0319 19..22 0.1123 Ljung-Box (50) 65.05 0.0747 20.25 0.3421 Ljung-Box (100) 128.42 0.0292 27.23 0.2334 LM Test 29.07 0.0038 0.01 1.0000 AIC 5229 2989 BIC 5453 3218 Garch (1,1) with GED Error Distribution Std. Coefficients Value Error P-value [alpha] 0.0046 0.0129 0.7177 Monday -0.0033 0.0187 0.8590 Tuesday 0.0344 0.0175 0.0500 Wednesday 0.0186 0.0208 0.3701 Thursday 0.0144 0.0215 0.5039 [eta] -0.0419 0.0001 0.0000 [lambda] 0.1258 0.0161 0.0000 [theta] 0.8773 0.0133 0.0000 Monday 0.0498 0.0094 0.0000 Tuesday 0.0408 0.0090 0.0000 Wednesday 0.0903 0.0105 0.0000 Thursday 0.0661 0.0117 0.0000 Jarque-Bera 2532 0.0000 Ljung-Box (20) 36.65 0.0129 Ljung-Box (30) 49.11 0.0153 Ljung-Box (40) 75.93 0.0005 Ljung-Box (50) 83.64 0.0020 Ljung-Box (100) 148.63 0.0012 LM Test 1.50 0.9999 AIC 3692 BIC 3922
|Printer friendly Cite/link Email Feedback|
|Author:||Chkir, Imed; Chourou, Lamia; Rahman, Abdul; Saadi, Samir|
|Publication:||Quarterly Journal of Finance and Accounting|
|Date:||Dec 22, 2014|
|Previous Article:||Takeovers and the size effect.|
|Next Article:||The determinants of corporate debt mix.|