Robert A. Connolly - An Examination of the Robustness of...

An Examination of the Robustness of the Weekend EffectAuthor(s): Robert A. ConnollySource: The Journal of Financial and Quantitative Analysis, Vol. 24, No. 2 (Jun., 1989), pp.133-169Published by: Cambridge University Press on behalf of the University of WashingtonSchool of Business AdministrationStable URL: http://www.jstor.org/stable/2330769Accessed: 07-08-2017 16:10 UTC

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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 24, NO. 2, JUNE 1989

An Examination of the Robustness of the Weekend Effect

Robert A. Connolly*

Abstract

This paper analyzes the robustness of the day-of-the-week (DOW) and weekend effects to alternative estimation and testing procedures. The results show that sample size can distort the interpretation of classical test statistics unless the significance level is adjusted down? ward. Specification tests reveal widespread departures from OLS assumptions. Hypothe? sis tests results are reported using robust econometric methods and a GARCH model. The strength of the DOW and weekend effect evidence appears to depend on the estimation and testing method. Both effects seem to have disappeared by 1975.

I. Introduction

Since French (1980) and Gibbons and Hess (1981) first documented unusual stock returns over weekends, there have been a number of additional studies of

the weekend effect.1 Researchers have uniformly reported evidence of anomal? ous returns between the market close on Fridays and the market close on Mon? days. There have been some disagreements in the literature about the size, tim? ing, or stability of the weekend effect, however. Efforts to explain the weekend effect as a consequence of settlement practices have met with only mixed suc? cess.2 Consequently, researchers regard the pattern of negative Monday stock returns as an anomaly.

Much of the empirical work on the weekend effect rests on a foundation of simple econometric models with strong statistical assumptions. These founda- tions are rarely, if ever, evaluated systematically. The importance of the stock

* Graduate School of Management, University of California, Irvine, Irvine, CA 92717'. Receipt ofthe American Assoeiation of Individual Investor's Award at the 1988 Western Finance Assoeiation

Meetings is gratefully acknowledged. The author would like to thank Tim Bollerslev, John Clarke, JFQA Managing Editor Peter Frost, Chuan-Yang Hwang, Roger Koenker, Edward Leamer, Kwang Lee, Henry McMillan, Laura Starks, and participants in the Econometrics Workshop at the UCLA Economies Department for their assistance with this paper. The author also is grateful to Stuart Allen who helped find a programming error. Remaining errors are his own.

1 Among the many studies are Lakonishok and Levi (1982), Keim and Stambaugh (1984), Jaffe and Westerfield (1985), Rogalski (1984), Mclnish and Wood (1985), Harris (1986), and Smirlock and Starks (1986).

2 Flannery and Protopapadakis (1988) investigate intra-week return seasonality using stock re? turn and Treasury debt returns. They find the pattern of negative Monday returns is related to the maturity of the asset and are skeptical that an explanation can be found by examining institutional aspects ofthe various asset markets.

133


134 Journal of Financial and Quantitative Analysis

return anomalies issue for finance research certainly justifies a healthy suspicion of any untested assumption.

In this paper, we examine the sensitivity of inferences about day-of-the- week (DOW) and weekend effects to an extended analysis of the statistical foun- dations ofthe existing research. The analysis concentrates on three issues. First, we consider the problem of interpreting classical test statistics with very large samples. Standard F-statistics, used in virtually every study ofthe weekend ef? fect, will distort sample evidence if the significance level is not adjusted as the sample size increases. Bayesians are very familiar with this problem and refer to it as the Lindley Paradox.3 This problem is potentially quite severe in empirical finance work, given the enormous quantities of data available.

Second, we consider the impact on inferences of relaxing the Gaussian error distribution assumption. Error terms from regressions involving stock returns are almost certainly not normally distributed.4 The problem created by fat-tailed dis? tributions, for example, is that test statistics based on nonrobust standard error estimates cannot be interpreted in the usual way. What is not known is whether the problem is severe enough to reverse inferences. We demonstrate several ways to construct robust hypothesis tests and apply them to the DOW and week? end effect problem.

Third, there is mounting evidence that stock returns have time-varying vola? tility. Studies ofthe DOW and weekend effect that standardize returns use a vo? latility estimate for each day of the week, if at all. What is not known is whether inferences are sensitive to alternative heteroskedasticity corrections. As a first step in answering this question, this paper tests for DOW and weekend effects using a volatility model due to Bollerslev (1986) and Engle (1982).

The analysis shows that the maintained statistical assumptions are rejected by the data. It also shows that inferences depend on these maintained assump? tions. Treating each of the three problems separately reveals the evidence for DOW and weekend effect ranges from very strong in a few instances, to weak in most cases, and to nonexistent for recent time periods. The findings reported here suggest that claims of a financial market anomaly should be treated skeptically until the data are subjected to extensive sensitivity analysis. The procedures ap? plied in this paper are a useful starting point for this sort of analysis.

Section II of the paper applies the tests used in earlier papers and reports coefficient estimates and hypothesis test results. We show ways in which infer? ences about the existence of DOW and weekend effects depend on how the sig? nificance level is adjusted with the sample size, and also show the sensitivity of results to alternative sample periods.

Section III documents the extent of nonnormality and the seriousness of the related outlier problem using box and whisker plots. Corroborating evidence from several error distribution specification tests is presented, and tests for serial correlation and heteroskedasticity also are reported. Results from this battery of

3 See Leamer ((1978), Ch. 4) for a discussion of this problem. Shanken (1987) explores the implications of the Lindley Paradox for multivariate tests of portfolio efficiency. Connolly (1988a) reports an extensive Bayesian analysis of the weekend effect and shows the Lindley Paradox may explain most of the anomalous Monday return behavior.

4 See Mandelbrot (1963), Fama (1963), (1965), and Blattberg and Gonedes (1974) for evidence onthis issue.


Connolly 135

tests suggest the casual attitude of some studies toward these problems may be inappropriate.5

The failure of the normality, autocorrelation, and heteroskedasticity as? sumptions prompts a search for an alternative econometric approach to finding consistent estimates of coefficient values and particularly their standard errors.6 In Section IV, several methods for constructing hypothesis tests that are robust to failure of the normality assumption and the presence of outliers are considered. These distribution-free econometric methods are based on L-estimators and re?

lated to the regression quantiles (RQ) approach of Bassett and Koenker (1978), (1982). This section documents the impact of robust hypothesis testing methods on inferences about DOW and weekend effects.

Section V implements another robust estimation approach that relies on M- estimators. Because the weighting scheme used to reduce the impact of outliers is arbitrary, some analysis of the sensitivity of results to alternative weighting schemes is desirable. Using a generalized least-squares framework, Leamer (1984) has developed a technique that shows how coefficient estimates are af? fected by re weighting particularly influential observations. This work is espe? cially interesting because the M-estimator of Huber (1981) is most easily imple? mented in a weighted least-squares framework. Since there are arbitrary elements to the weighting scheme used to form this estimator, Leamer's results are an especially attractive way of assessing the impact of different weighting schemes on coefficient estimates.

A recent innovation in modeling time series, the Generalized Autoregres? sive Conditional Heteroskedasticity (GARCH) model (Bollerslev (1986)) is in? troduced in Section VI. This model is particularly useful because it can incorpo? rate autocorrelated returns, time-varying return volatility, and the fat-tailed error distribution parsimoniously. Statistical tests reported in this section leave little doubt about the importance of GARCH effects in empirical weekend effect mod? els. The results from this estimation method differ in some ways from those based on other robust methods. A final section summarizes the paper.

II. Interpreting Evidence on DOW and Weekend Effects A. Basic Results

The work reported here uses daily return data drawn from the CRSP Daily Tape for the S&P 500 index (S&P), the equal weighted CRSP index (EW), and the value weighted CRSP index (VW). Daily returns are multiplied by 100, and the sample period runs from the first trading day in 1963 through the last trading day in 1983.

5 Some researchers have been aware of the problems created by heteroskedasticity and auto? correlation and have incorporated adjustments into their statistical work (see Gibbons and Hess and Flannery and Protopapadakis for examples). There is little evidence on whether these corrections are entirely appropriate.

6 Failure of the standard normality, serial correlation, and heteroskedasticity assumptions cre? ates no asymptotic difficulties with Gaussian estimators of the coefficients. The real problems arise because standard error estimates are not consistent. Thus, standard hypothesis testing procedures are suspect.



The model used to evaluate the DOW hypothesis is

(1) Rt = <*0 + aiM, + a2Tt + a3TH, + a4F, + Et '

where Rt is the market return index, Mt, TU,, TH,, F, are dummy variables iden? tifying Monday, Tuesday, Thursday, and Friday observations, respectively, and et is an error term assumed to be normally distributed with zero mean and finite variance. This corresponds to the trading time model in which returns are gener? ated only when there is active trading.

In Table 1, estimates of this model for the entire period and seven three-year subsamples are reported. For the entire sample period, the results for each return measure confirm what other researchers have found. For the entire sample pe? riod, the F-statistic rejects the null of no DOW effects. The same result emerges for the first four subperiods, too. In the 1975-1977 and 1981-1983 subperiods, however, there is no evidence of DOW effects except for the CRSP value weighted return series.

The average Monday return is significantly negative while the returns for other days are generally positive.7 For the first four subperiods, the same pattern emerges and the size of the negative Monday return grows. Just as for the DOW tests, the results change for the 1975-1977 subperiod where Monday's estimated return is positive. The pattern of negative Monday returns reappears for the 1978-1980 and 1981-1983 subperiods, but the estimate is significantly different from zero only for the VW return.

These results suggest that there may have been DOW effects until the mid- 1970s but the effect then seems to disappear. At a minimum, the effect appears to be unstable in view of the changes in the return differential estimates across the subperiods.

The following model is useful for focusing on the weekend effect question

(2) R, = P0+P,M, + e,.

This model has been used in earlier studies because hypothesis tests generally found that return differentials for Tuesday, Thursday, and Friday were zero. Table 2 reports estimates of (2). The pattern of results is similar to that found in Table 1. Monday returns appear to be negative for the entire sample and the first four subsamples. Monday returns become indistinguishable from zero in the 1975-1977 sample, are negative in the 1978-1980 sample, but are only margi? nally significant and negative in the 1981-1983 sample.

B. Sample Size and the Lindley Paradox

Interpretation ofthe F-statistics in Tables 1 and 2 is conditioned on selection of a significance level. As Lehman ((1959), p. 61) points out, The choice of a level of significance a will usually be somewhat arbitrary since in most situations there is no precise limit to the probability of an error of the first kind than can be tolerated. It has become customary to choose for a one of a number of

7 Point estimates are significantly negative for Tuesday returns during the 1969-1971 period and for Thursday returns during the 1978-1980 period.


Connolly 137

TABLE 1

Estimates of Model 1 for 1963-1983

Rt = ao + aiMf+a2Tf+a3THf+a4Ff+ef

Note Absolute value of f-statistics is reported in parentheses below the coefficient estimates. Under F- Test Value are the F-statistics for the hypothesis a1 = a2 = a3 = a4 = 0 The p-value for the F- statistic is in the next column Under P0/Pi are the prior odds at which the F-statistic leaves us indifferent to the choice between the null and alternative hypotheses



Note Absolute value of f-statistics is reported in parentheses below the coefficient estimates Under F- Test Value are the F-statistics for the hypothesis a1 = 0. The p-value for the F-statistic is in the next column. Under P0/P1 are the prior odds at which the F-statistic leaves us indifferent to the choice between the null and alternative hypotheses.


Connolly 139

standard values such as 0.005, 0.01, or 0.05. There is some convenience in such standardization since it permits a reduction in certain tables needed for carrying out various tests. Otherwise, there appears to be no particular reason for selecting these values.

In practice, statisticians and econometricians who work with large data sets know that F-tests have a tendency to reject the null hypothesis too often unless the significance level is adjusted downward.8

The usual F-statistic is given by

where the 1 and 0 subscripts indicate conditions under the null and alternative hypotheses, k{ is the number of parameters estimated under the null or alternative hypothesis, and T is the sample size. With the kt fixed, any difference in the R 2s can be made "significant" if the sample size can be increased enough, holding the significance level fixed. For example, \fR$ = 0.06, Rx2 = 0.05, kQ = 5, kx = 1, and T = 250, then the F-statistic is 0.65. If T = 1000, the F-statistic is 2.65. Thep-values of the two test statistics are very different. In equity market research, the sample size challenge has been successfully met with the CRSP daily return tapes (with approximately 6000 observations, on average, per stock) and by some researchers using intraday data. Consequently, it is important to recognize the impact of sample size on hypothesis testing procedures.

In classical statistical terms, the justification for reducing the significance level is that some of the increased sample size should be used to reduce type II errors (Brown and Klein (1984)). As the quote from Lehman indicates, there are no guidelines for doing this, however.

Bayesians have understood this problem at least since Lindley (1957). Lind? ley 's analysis ofthe problems of testing an hypothesis with varying sample sizes but a fixed significance level is fairly straightforward. His work shows that, using conventional sampling theory methods, a statistician can reject the null hypothe? sis even if the posterior odds favor the null when the sample size is increased but the significance level is fixed. In the example, the posterior probability ofthe null hypothesis relative to the alternative is large in both cases. The /?-value of the calculated test statistic can be a misleading indication of the probability associ? ated with a particular hypothesis.9

Bayesian analysis is directed at the posterior odds of the two hypotheses. Using the same F-values as are conventionally calculated and reported in most studies, posterior odds can be computed easily. More specifically, assuming diffuse priors for the regression coefficients and variance and even prior odds (for the moment), the posterior odds may be written

(4) p(h^)/p(hx) = 7<*o"*0/2(ESSo/ESS1)(7'/2), 8 Wallace (1972) makes this observation about the usual F-test in his work on weaker MSE test

criterion.

9 This point is often made in Bayesian analyses of hypothesis testing issues. Shanken (1987) and Connolly (1988a) contain further analysis of these issues and extensive references.



where P(H0) gives the posterior probability of the null hypothesis, P(HX) gives the posterior probability of the alternative hypothesis, ESS0 and ESSj are the unexplained sums of squares under the null and alternative hypotheses, and k0, kx, and T are as defined earlier. Klein and Brown show that (4) is simply a trans? formation of the usual F-statistic so the attractive properties of standard software and inference procedures are retained by this approach.10

Leamer (1978) extended Lindley's analysis and provided an easy way to construct hypothesis tests that are not subject to the sample size distortion prob? lem. He shows that an F-statistic implies posterior odds less than one (favors the alternative hypothesis) if

(5) F> [{T-kx)/p\.[T"IT-\], where p = k0 ? kl, the number of restrictions being tested.11 Sample size-adjusted critical F- and f-values for use in conventional hypothesis tests can be calculated using

(6) F* = [(T-kx)/p\[T'"T-\-\ for the critical F-values and

(7) f* = (T-kf-5(TVT-i)

for critical r-values. If a calculated test statistic exceeds the appropriate critical value from (6) or (7), the sample evidence is said to favor the alternative hypoth? esis. If the prior odds are instead, say, 3 to 1, the right-hand side of (6) or (7) should be multiplied by 3, since for the posterior to favor the alternative should take at least three times the F-statistic (sample evidence) as in the case of even prior odds.

There are at least two advantages of the Bayesian approach. First, the, hy? pothesis testing procedures automatically adjust for sample size so inferences are not distorted. Second, no special software is required to implement this ap? proach.12

In the case of the DOW and weekend effects, posterior odds results and inferences based on the conventional F-test are inconsistent in many cases. The

10 Connolly (1988a) contains an extensive posterior odds analysis of the weekend effect and includes a lengthy discussion of the various approaches to calculating posterior odds and the relation? ships between posterior odds and classical test statistics.

11 The form of (5) is not due to special assumptions. Zellner (1984) reports a small sample approximation to posterior odds that yields a similar result on the connection between F-statistics and posterior odds. In personal communication, he said that including the higher-order expansion terms will produce an odds expression identical to Leamer's. The prior distributions (weighting functions) in the Leamer and Zellner analyses are not identical, however.

12 If the researcher has prior opinions about the hypotheses under study, these opinions can be incorporated directly into the analysis using the posterior odds approach.


Connolly 141

right-hand columns of Tables 1 and 2 show this divergence. F-statistics for the following hypotheses are reported:

Model 1: Rf = aQ + a1M/ + a2T? + a3TH? + a4F? + e, ,

0 ,

For Model 1, the F-statistics for these tests, along with their associatedp-values, are reported in Table 1. The classical test rejects the null in all but four cases. Using the sample size-adjusted F-values, the same tests would only reject the null in 4 of 32 cases. The column labeled P0IPX gives the prior odds at which the F-test statistic would exactly equal the sample size-adjusted critical F-value. From a Bayesian perspective, only a small increase in prior odds is required to conclude that the posterior probabilities associated with H0 and Hx are the same.

The right-hand side of Table 2 contains the same results for the Monday- only Model (2). At conventional significance levels, the null is rejected in 28 out of 32 cases. Using sample size-adjusted critical F-values, this rejection rate only falls to 26 out of 32 cases. The sample size adjustment makes a much smaller difference for this test.13

In summary, using sample size-adjusted critical F-values, researchers would probably not have concluded that stock returns behaved anomalously us? ing Model 1. The same cannot be said unambiguously of the evidence from Mo? del 2. Here, a researcher must have prior odds of better than two to one to con? clude the evidence favors the null hypothesis of no difference in Monday returns.

III. Testing Assumptions on the Error Term

The results discussed in Section II depend on a number of statistical as? sumptions regarding the error terms in (1) and (2). This section examines these assumptions more closely through a set of specification tests.

Normality is important in sampling theory-based inference procedures be? cause the properties of test statistics depend on normality. Failure of the nor? mality assumption can lead to misleading inferences. This is because the calcu? lated standard error based on the normality assumption is unreliable by comparison with an estimate that accounts for, say, a leptokurtic distribution. Fat-tailed error distributions have been noted in financial market econometric re?

search many times. Early studies basically concluded there seemed to be no im? pact on inferences.14

13 Connolly (1988a) argues that efficient market theory implies prior odds in favor of the null for this particular case. If the prior odds are, say, five to one, the rejection rate fails to three out of 32 cases.

14 See the references in footnote 4.



There are at least two approaches to assessing whether the normality as? sumption is reasonable. Box plots are a graphic data analysis device that can be very useful in identifying long-tailed distributions due, for example, to outli? ers.15 Figure 1 portrays standard box plots for normally distributed random series with mean zero and variances of one, two, and three. The distance from the

lower to upper extreme indicates the range of the data. The interquartile range is from one edge of the box to the other edge. The line inside the box indicates the median value. When the median is exactly in the center ofthe box, the mean and median are the same. The box plot indicates a symmetric distribution if the whiskers (the lines outside the box) have approximately the same length and the median approximates the mean closely. As the box width narrows, the distribu? tion is growing more peaked; as box width increases, the distribution flattens.

Note: The top figure has a mean of zero and a variance of one. The middle figure has zero mean and a variance of two. The bottom figure has zero mean and variance of three.

FIGURE 1

Sample Box Plots for Normally Distributed Random Variable

Figures 2-4 are box plots of stock return data for each of the three-year samples for each stock return measure. The relatively long whiskers in every box

15 Tukey (1977) reports pioneering work in this area. Other useful references are McGill, Tu- key, and Larsen (1978) and Velleman and Hoaglin (1981).


Connolly 143

plot for all samples and all return measures testify to the existence of outliers. These box plots are also consistent with the long-tailed distribution commonly ascribed to stock returns. The relatively narrow width of boxes implies the distri? butions are leptokuptic relative to the normal distribution. Thus, graphical evi? dence suggests the return data are nonnormally distributed. This makes it un? likely the error distribution will be normal.

The other approach to evaluating nonnormality is more rigorous. There are a number of formal statistical tests for normality of residuals. They are all com- plicated, however, by the need to specify the error distribution under the alterna? tive hypothesis. Since there are no completely general alternative distributions, econometric theorists have specified families of distributions for the alternative hypothesis. In Bera and Jarque (1982), the alternative hypothesis relies on the Pearson family. By contrast, the Poirier, Tello, and Zin (1986) test specifies the exponential family for the alternative hypothesis. Finally, Maddala ((1977), p. 46) describes a general chi-square goodness-of-fit test. Test statistics from these procedures all have chi-square distributions.

Table 3 reports results for these tests when applied to residuals from esti? mating Model (1). The Jarque-Bera and Poirier-Tello-Zin tests reject the null hypothesis in virtually every case regardless of sample period or return mea? sure.16 The data reject the null hypothesis in most of the goodness-of-fit tests as well. Thus, there is substantial evidence of nonnormality in the estimated residu? als for Model (1). The same results hold for Model (2) but are not reported to save space.

The next four columns contain the skewness and kurtosis estimates for the

empirical error distributions along with the respective standard errors. If the er? rors are normally distributed, skewness should equal zero and kurtosis should be three. Inspection of the appropriate columns shows that 95 confidence intervals for skewness and kurtosis estimates often exclude these values.1?

White's (1980) heteroskedasticity test was used to check the constant vari? ance assumption and results are reported in the column marked White. This is a chi-square test. The test indicates heteroskedasticity in regressions over the entire sample period and weaker evidence of misspecification for the 1966-1968 sam? ple period. Section VI reports strong evidence of time-conditional heteroskedas? ticity using a test from Engle (1982). Of course, heteroskedasticity invalidates standard hypothesis test procedures because they are based on constant error variance.

Finally, autocorrelation diagnostic tests were computed. The two rightmost columns of Table 3 contain Durbin-Watson statistics and Box-Pierce g-statistics for each regression. The calculated Box-Pierce statistics have a x2 distribution. These diagnostic tests indicate there is an untreated autocorrelation problem, particularly with the VW returns. This compromises inferences in a sampling-theory

16 Rather than using critical values for a more standard 0.05- or 0.01-level test, a smaller signifi? cance level is selected to offset the impact of the very large sample sizes. Critical values are from Lindley and Scott (1984).

17 The box plots portray the distributional characteristics of the raw return data. The skewness and kurtosis estimates in Table 3 are for the realized errors. In general, the distributional characteris? tics of the two samples need not be the same.



framework, of course, because of the downward bias in the reported standard errors. Bias in this direction would favor the anomalies hypothesis.

In summary, this section of the paper has shown that the regressions under? lying the sampling theory-based tests for weekend effects are flawed. There is considerable evidence that the error distribution assumptions are violated, thereby casting doubt on all inferences. The next three sections implement sev? eral hypotheses, testing which are robust to departures from normality, serial correlation, and heteroskedasticity.

Note Under x2 are results for a x2 goodness-of-fit test with 53 degrees of freedom. Under J-B and PTZ are results for the Jarque-Bera and Poirier-Tello-Zin normality tests. They have 2 and 1 degree(s) of freedom, respectively. Under Skew and Kurt are the estimated skewness and kurtosis of the empirical error distribution. Results for White's heteroskedasticity test are reported under White Under DW are the Durbin-Watson statistics for each regression and Box-Pierce O-statistics are reported under O-Stat. Critical values for each test statistic are listed at the bottom of the table.

IV. Distribution-Free Estimation: L-Estimators A. Introduction

The normality test results reported in Section III are not surprising. For at least 25 years, researchers have regarded data on speculative prices and rates of return as nonnormally distributed. Researchers in empirical finance have tended to downolav the imnortance of these statistical results.18 In the case of svmme-

18 Virtually no empirical studies in finance over the past ten years have used distribution-free linear model techniques. The most notable exception to this is Cornell and Deitrich (1978). They evaluated the relative forecasting performance of least-absolute deviations (LAD) and OLS estima? tors of symmetric risk (stock market beta).


Connolly 145

-3-11 3 5 -4-2 0 2 4 6 -4-2 0 2 4 6

FIGURE 2

Box Plots for CRSP Equal-Weight Market Return, 1963-1983

-3-2-10 1 2 3

Sample Period-

1963-1965

-4-2 0 2 4 -10-7-4-1 2 5

-3-11 3 5

1975-1977

,,,,!,.Minnl,,,,!,,,,! 10-7-4-1 8 5 8 4 -a 0 8 4

1981-1983 1978-1980

FIGURE 3

Box Plots for CRSP Value-Weight Market Return, 1963-1983

tric, fat-tailed distributions, the common claim is that a t-score is so large that using a correctly calculated standard error would not change inference.

However, t- and F-tests can be very misleading. More specifically, in the case of fat-tailed distributions, the outliers are often very influential in determin-



Sample Period:

1963-1965

-3-2-10 1 2 3 M.j.ml,,,,!,,,,!,,

-3-11 3 5 7 -4 -2 0 2 4 6

...I.M.i, -4-2 0 2 4

1981-1983

FIGURE 4

Box Plots for S&P 500 Market Return, 1963-1983

ing results.19 Because the outliers may change from sample to sample, it can be difficult to get reliable estimates of coefficients and their standard errors. Thus, reliable inferences are potentially rare.

Distribution-free methods are one approach to the outlier, fat-tailed distribu? tion problem. The impact and usefulness of distribution-free methods in empiri? cal finance research has not been established. Evaluation ofthe DOW and week?

end effect evidence from a distribution-free estimation perspective is appropriate and provides an important test case.

B. Theory of /.-Estimators

There are three major classes of distribution-free estimators for linear mod? els, M-, L-, and /^-estimators. Each estimator class constructs parameter esti? mates in a different way.20 This section describes some of the fundamental re? sults for L-estimators based on the regression quantiles research of Koenker and Bassett (1978). The discussion emphasizes several particularly useful special cases.

In the .simple location problem, the Oth quantile is given by the solution to the following linear program

(8) min 2>l*-*| + X o-?>|?,-*| t\yt s* b t\yt < b

19 Belsley, Kuh, and Welsch (1981) analyze the impact of outliers on estimates using several influence measures.

20 See Joiner and Hall (1983) for a discussion of the similarities of these estimators. Koenker (1982) is an excellent survey.


Connolly 147

(ignoring nonuniqueness problems for the moment). When 6 = 0.5, (8) gives the special case of median estimation. If we suppose e, = yt?xp has a distribu? tion function F, the 6th regression quantile is the solution to the following linear program

(9) min

t\yt s* xtb t\yt < xtb

where again, if 0 = 0.5 (9) yields median regression (also called least absolute deviation (LAD) regression) results. Solving (9) with 0 < 0 < 1 involves fitting m hyperplanes with a set of estimates for each hyperplane. Koenker and Bassett (1978) show that, while the intercept estimate varies by quantile,21 the slope esti? mates are alternative estimates of the same vector, p. Koenker and Bassett dis? cuss the conditions for unique estimates of the sequence of coefficient vector estimates, p*(6). Defining 8 = [p*^), p*(62),..., p*(6J] and 8 = [p*(01), P*(62),..., p*(6J], Koenker and Bassett show that

(10) (T{h - h)^N(09Cl?Q~') , where the typical element of fl is given by

di) ??- [min(^j)-^j\/f(Qf(Q. Here, ?e = F_1(6) is the 6th quantile ofthe error distribution, /(?e/) is the error density for the 6th quantile, and lim T~lX'X = Q, which is positive definite. Thus, implementing inference procedures using regression quantiles requires es? timating the error density for each quantile. Under the best of circumstances (i.e., a large sample), density estimation is difficult. For some quantiles, how? ever, the number of observations may be very small (see Bassett and Koenker (1982) for an example).

Another major issue in the use of quantiles regression is how to aggregate the m vectors of estimates.22 Linear functions of regression quantiles may be written as

(12) poo = 2ir(e/)p*(e.), where ir is a symmetric weighting scheme. Koenker and Bassett show the sym? metry of ir, and e means p(ir) is an estimate of p, rather than an estimate of a particular p*(6). Their conjecture that

(13) ^(pc^-p^^o^'Httg-1) was proven by Ruppert and Carroll (1980). This is important because if F is non- Gaussian, it is straightforward to find tt(6) that make tt'Utt smaller than the variance of F. Thus, regression quantiles-based estimates of p have superior asymptotic efficiency relative to least squares. Judge, et al. (1985) discuss ap? proaches to getting consistent estimates ofthe elements of ft.

21 See Bassett and Koenker (1982) for a more intuitive discussion of this with examples. 22 Reporting estimates for all quantiles may not be practical since the number of quantiles can

exceed the number of observations easily, even for small samples.



LAD solves this aggregation problem by putting all weight on the 0 = 0.5 quantile. Bassett and Koenker examine LAD in some detail and prove that

(14) /r[p*(0.5) - p]-WV(0, [2/(0)]-2e_1 ,

where f(0) is the asymptotic variance of the sample median from samples with distribution function F. Bassett and Koenker (1978) show that whenever the median is a better measure of central tendency than the mean, LAD has smaller confidence ellipsoids than least squares. The superiority of LAD holds for error distributions with peaked density at the median and/or long tails.

Trimmed least squares (TRLS) is another approach to the aggregation prob? lem. Assuming a symmetric trimming process with trimming proportion a, cal? culate the regression quantiles p*(a) and p*(l - a). By Theorem 3.4 of Koenker and Bassett (1978), there will be Ta observations below p*(a) and 7a observa? tions above P*(l - a). Dropping these observations and performing least squares on whatever remains on the sample yields the TRLS estimates. Of course, this is essentially least squares on a censored sample. It involves sampling from F with F truncated at F~ l(a) and F~ l(l - a).

Ruppert and Carroll find that TRLS based on preliminary least-squares esti? mates is inferior to a procedure in which sample trimming is based on regression quantiles estimates. The resulting estimates, pa, are location and scale invariant and Ruppert and Carroll show that

(15) /j(Pa-p)^iV(0/(a,F)G-1) . Judge, et al. and Ruppert and Carroll discuss methods of consistent estimation of s2(a,F). One advantage of TRLS over the regression quantiles approach is that to find s2(a,F) requires estimating only one error density.

The Ruppert and Carroll analysis does not require symmetry of F, but if F is symmetric, (3a is unbiased. Nonsymmetrieal error distributions lead to bias ofthe intercept estimate, but the slope coefficient estimates are unbiased.

C. L-Estimates of the Weekend Effect

Due to program limitations, the regression quantiles, LAD, and TRLS pro? cedures were applied only to the three-year subsamples.23 Table 4 reports LAD estimates of both models along with asymptotic ^-scores. The major result is that a Monday return anomaly exists through the 1972-1974 sample. It disappears afterward except for the VW returns in the 1978-1980 and 1981-1983 samples. Using the size-adjusted t-values, however, the weekend effect persists only in the VW returns for the 1978-1980 sample. This curious difference across return measures does not seem to have attracted much attention in earlier studies.

Sometimes, the size of the negative Monday differential is slightly smaller compared to the OLS estimates in Table 1. On average, the Monday return dif-

23 A FORTRAN program to calculate RQ, LAD, and TRLS estimates is available upon request to the author. The program currently runs on a VAX Cluster and makes several calls to the IMSL double precision library. To install the program on non-VAX machines should require only minor modifications.


Connolly 149

ferential for Model 1 is 18 percent smaller using the LAD approach and exclud? ing the 1975-1977 sample. As expected, r-scores also are smaller. The average reduction for the Monday coefficient is 5.2 percent, excluding the 1975-1977 sample. The reductions are very large for the 1969-1971, 1978-1980, and 1981-1983 samples.

Table 5 reports TRLS estimates using an a = 0.05 and a = 0.20 trim. Given the evidence of nonnormality presented in Section III, a trim of 0.05 is almost certainly necessary. A trim of 0.20 is almost certainly as large as the data require. Comparing the inferences from each is a useful form of sensitivity analy? sis. While the LAD and OLS coefficient estimates are about the same, for exam?

ple, TRLS confidence intervals are much larger and grow as the size of the trim increases. The average reduction in the t-statistic on ax for TRLS estimates (a = 0.05) compared to OLS is 27.8 percent. Thus, relying on the TRLS estimates (a = 0.20) would probably have led the data analyst to conclude there was no weekend effect in any sample.24 If the trim is only 0.05, we again find the earlier pattern of a weekend effect that disappears after 1974.

If the data analyst believes the data have a substantial number of outliers (heavy contamination), a larger trimming proportion is probably appropriate, thus leading to a finding of no weekend effect.25 By contrast, a lower trimming proportion is approriate if the researcher believed the data were of reasonably high quality (i.e., fairly normal). Since there is no analysis of how to determine the appropriate trimming proportion, it is important for researchers to communi- cate the sensitivity of results to alternative trimming proportions.

V. Distribution-Free Estimation: M-Estimators

A. Theory of /W-Estimators

Another set of robust estimation and testing methods is based on M-estimators. Like L-estimators, the advantage of M-estimators is that outlying observa? tions exert a smaller influence on coefficient estimates and confidence intervals.

Ordinary least squares solves for P to minimize %(yt ? xfi)2. Equivalently, OLS solves for p so that Xxt(yt ? xfo) = 0. In a maximum-likelihood (ML) frame? work where the errors et have a density function/(jr ? xfi), ML solves for the p that minimizes -2,\n[f(yt-xfo)]. Equivalently, ML solves for P so that ^xt\f'(?)//(')] = 0. If the error distribution is normal with zero mean and constant variance, Xxt(yt-xfi) = 0 = 2*,[f (?)//(')]?

Huber's (1981) proposed M-estimator replaces/'(?)//(*) with another func? tion that will be robust to outliers. Define k = cct where c = 1.54 and a is a

dispersion measure. Huber's estimator finds the value of P corresponding to ?*r(e,*) = 0. The er* are Winsorized residuals calculated as

? or if e < ? k

(16) ?* = e, if |e,| < * cv if e > k .

24 This is particularly true when using the sample size-adjusted /-values computed using (7). 25 The box plots discussed in Section III strongly suggest the presence of outliers. Thus, a small

trim probably errs on the side of removing only the biggest outliers.



Note Absolute value of asymptotic f-statistics is in parentheses below coefficient estimates.

The iterative weighted least-squares approach to calculating Huber's M-estimates is as follows:

1) estimate the model using LAD and retrieve the residuals;

2) calculate a as the median of the absolute value of LAD residuals or the interquartile range of LAD residuals;

3) calculate weights for weighted least-squares using

1 if|?,|*t W' ?r/|e,| if |e,| > k;


Connolly 151

TABLE 5

Trimmed Least Squares Estimates of DOW and Weekend Effect Models

Model 1 Model 2

(continued on next page)



TABLE 5 (continued)

Model 1 Model 2

Note- Absolute value of asymptotic f-statistics is in parentheses below coefficient estimates.

4) calculate weighted least-squares (WLS) estimates using

p = (X'WXyXX'Wy W = dmg (w{,w2,...,wTy, and

5) repeat steps 2-4 using WLS outputs until convergence.

The asymptotic properties of Huber's M-estimator have been investigated by Huber (1973), (1981) and Yohai and Maronna (1979). Their work established that the M-estimator described above is consistent and that

(17) fT(p-fS)^N[0,<T2W,F)Q-1] .


Connolly 153

For the estimator used in this work, (17) specializes to

g2e*'e* v,v,_i n _ l + ((klT)(l-y))/v <18) 2p = *^-l-(X'X)~ P

where \x is the percentage of non-Winsorized residuals. Thus, asymptotically consistent estimates of standard errors are fairly easy to calculate.26

B. /W-Estimates of the Weekend Effect

Table 6 reports estimates of Model 1 and Model 2 using the weighted least- squares implementation of Huber's M-estimator just described. After eight itera? tions, only minor changes in the third or fourth decimal place were found. Con? sequently, all estimates in Table 6 are based on 12 iterations. In addition to the coefficient estimates and the associated t-values, Table 6 reports the percentage of non-Winsorized residuals for each case. This value ranges from 0.28 to 0.41 with the mean value of 0.33. This is additional evidence of the presence of outliers. If the error distribution were normal, the percentage of non-Winsorized residuals ought to exceed 90 percent.

The estimates of ax for Model 1 are negative in most periods but the t- values are much smaller. With only a few exceptions, the Monday coefficients are insignificantly different from zero after the 1969-1971 sample. Furthermore, only the Monday coefficients for the VW returns for 1963-1965 and 1966-1968 are significant using the sample size-adjusted t-values. Of course, this is pre? cisely what econometric theory suggests should happen.

The change in t-values is less dramatic in Model 2. Here, the $x coefficient estimates are significantly negative for most cases through the 1912-191A sam? ple even using the sample size-adjusted critical values. For the post-1975 period, only the VW returns are significantly negative after adjusting for the effects of sample size. This result is entirely consistent with the findings from the other robust estimators in Section IV. It supports the view that the weekend effect ceased to exist by the mid-1970s.

C. Sensitivity Analysis

The reader may object that the M-estimator results are due to a particular (and perhaps peculiar) reweighting of observations. This naturally creates an in? terest in measuring the sensitivity of coefficient and standard error estimates to more general reweighting schemes. In a generalized least-squares model, Leamer (1984) derives measures of coefficient sensitivity. The simplest of his results is that the r-statistic on a coefficient completely summarizes the sensitivity of a coefficient to reweighting. He also shows how to derive bounds on the coef? ficient estimates for a given class of reweighting procedures. These bounds and the r-statistic are natural sensitivity measures. They are useful in establishing whether the M-estimator results are due to a special weighting matrix or are ro? bust. More generally, Leamer's analysis is useful in understanding the sensitivity of inferences to arbitrary reweightings that occur when heteroskedasticity correc-

26 The M-estimates reported here were computed using RATS.



Note: Absolute value of asymptotic f-statistics is in parentheses below coefficient estimates.

tions such as White are used. This is important to do in view of Chesher and Jewitt's (1987) demonstration that the White consistent covariance matrix proce? dure is easily biased by outliers'.

More specifically, define / as the covariance matrix of "usual" errors, V as the covariance matrix of "unusual" errors, 2 = / +V, 2_1 = W, and P(2) = (X'2~1X)_1X'2~1y. Leamer's analysis derives bounds on p(2) as V changes. For example, if V is bounded 0 ^ V ^ V* where V =ss V* means V* - V is posi? tive definite, the set of estimates of p is elliptical.27 The condition 0 ^ V ^ V* implies the following bounds on the variance of a linear combination (selected by

27 Leamer (1978), (1982) provides fundamental analysis of this issue.


Connolly 155

the vector of constants d) of residuals (given by u): d'd ^ var(d'w) ^ d'd + d'V*d. If V* = /, the inequalities become d'd ^ var(d'w) ^ d'd + d'd. In this case, the weights used in the generalized least-squares problem vary between 1 and 2.

The basic result of most interest here gives bounds on linear combinations of coefficients, i|/p. Leamer shows that the extreme estimates of i|/p are given by

i|/p - ty(x'xyxx'(y*-x +m)~ my/2 (19) _j 0.5

?U'(x'xylxf(V*~* +m) x(x'x)~*i|ic) ,

where M = I + X(X,X)~XX' and c is a measure ofthe length of MY. If the upper bound V* = #/, the boundary of the set of estimates can be fully described in terms of traditional t- values

(20) ^'$[\?q(T -kf512t(\ + qf5] ,

where t is the usual r-statistic. One implication of (20) is that reweighting will have a smaller impact on the coefficient estimates as the r-statistic for that coeffi? cient is larger.

If q = 1, some observations have twice the weight of others.28 With q = 1, the sign of the coefficient estimate is insensitive to reweighting if the coefficient's r-statistic exceeds [(T-k)/S]?5. The maximum percentage change in the coefficient estimates is given by [(T ? &)/8]05/ \ t\ .

For Model 1, Table 7 reports coefficient estimates, r-scores, upper and lower bounds, and the minimum r-score for a coefficient's sign to be resistant to reweighting. These results are for two-year samples 1963-1984.29 The extreme bounds for every coefficient cross zero. In no case is the coefficient r-statistic large enough to imply that the sign is resistant to reweighting. Table 8 contains the same results for Model 2. Here, too, coefficient estimates display consider? able sensitivity to alternate weighting schemes. For both models and every coef? ficient, the interval reported in the table encompasses the M-estimator/weighted least-squares estimates reported in Table 6.

This demonstration shows that the estimates reported in Table 6 cannot rea? sonably be characterized as extreme or due to a peculiar weighting matrix. To the contrary, the M-estimator results lie very close to the midpoint of the intervals reported in Tables 7 and 8. They are also fairly close to the OLS estimates. As the theory suggests, the biggest impact is on confidence intervals. The minimum r-scores reported in Tables 7 and 8 also suggest that empirical work using data

28 As q ?> 0, P(S) ?> P(OLS). Note that as q increases, the influence of outliers declines since a large weight is applied to the outlying ("unusual") observations. Some ofthe robust estimators (see Huber, (1964)) use very large weights that effectively cause outlying observations to have no influ? ence on the coefficient estimate.

29 These estimates were made using the matrix program GAUSS. Program limitations blocked the matrix computations required for the three-year samples. The program could accommodate the two-year samples that have about 505 observations each; the 1967-1968 sample has only 477 obser? vations.



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VI. GARCH Model of Stock Returns

A. Econometric Theory

To this point, the robustness analysis has assumed that some form of outlier contamination accounts for the observed nonnormality. The empirical results are from econometric methods that are robust to contamination of an otherwise nor?

mal error distribution. Another hypothesis is that the errors are heteroskedastic and, accordingly, the nonnormality tests are really providing evidence that the error distribution does not have constant variance. A limitation of the foregoing analysis is that the M- and L-estimators cannot easily account for autocorrelation or heteroskedasticity.

This section applies a time-varying heteroskedasticity model based on En? gle's (1982) Autoregressive Conditional Heteroskedasticity (ARCH) model. This model encompasses an autocorrelation correction and is robust to underly? ing nonnormality. Bollerslev (1987) applied this model to stock return indices and exchange rates.30 This model is attractive for several reasons. First, it incor? porates heteroskedasticity in a sensible way (for a time series). Second, it can be expanded to include other effects on conditional variances. Third, it can distin- guish between nonnormal conditional and unconditional errors.31 Thus, this mo? del offers considerable flexibility in robust modelling of stock returns.

Engle's ARCH regression model is given by

(21) yt i^_! ~ N(xtb,h),

(22) ht = aQ + ?a.?,2_. ,

(23) ?, = yt - xtb .

Here, \\ft_{ summarizes the information set at time t ? 1, xt is a vector of indepen? dent variables included in \\it_,, and ht is the variance of the errors. In Engle's formulation, the conditional distribution of the yt is normal but the yt's are not jointly or marginally normal. For stock returns, this formulation neatly captures the stylized properties of returns: the return distribution is unimodal, symmetric, and leptokurtic but returns are not independent.

30 There are a growing number of applications of these models to speculative price data. Engle and Bollerslev (1986) and Connolly (1988b) contain partial listings. Connolly and McMillan (1988) find strong evidence of time-varying heteroskedasticity in a study of capital structure changes and propose a generalized ARCH model for event studies.

31 Connolly (1988b) applies this model to return series for a variety of equity and fixed income securities.


Connolly 161

Bollerslev (1986) generalized the conditional variance Expression (22) to incorporate moving average terms

(24) ht = a0 + fja^ + ^jht_jlt_l_j, i=\ j=\

where p > 0, q > 0, at ^ 0, and P, ^ 0. He shows that under suitable condi? tions, the resulting GARCH(p,q) model is essentially a stationary ARCH(^) process. The advantage ofthe GARCH(p,q) form is parsimony since the ARCH form with moving average terms is

l

*' = ao(i-|jp') + !><-<?' with suitable restrictions on the coefficients.

Bollerslev (1987) extended the GARCH(p,q) model to allow conditionally Student f-distributed errors. More specifically, he derives the GARCH (p,q) mo? del assuming

(25)

= r(z)r(v/2)-1((v-2)^u_1)"?'5(l+er/Ir|/_1(v-2)-l)"Z,

where T(*) is the gamma function, v > 2, z = (v+ 1)12, and/l/e/ | *\ft-X) is the conditional density function for et. As Bollerslev notes, the f-distribution ap?

proaches a normal distribution with variance ht | t_x as 1/v ?> 0.32 Thus, in the expanded model, the error distribution may be conditionally heteroskedastic and nonnormal. This is useful because the unconditional leptokurtosis may be traced to nonnormality in the conditional error distribution and/or to time-varying heteroskedasticity. If v > 30 but ai9 P, > 0, time-varying heteroskedasticity accounts for the fat-tailed error distribution. If v < 10 and ai9 $( > 0, both non? normality and time-varying heteroskedasticity produce the fat-tailed error distri? bution. Later in this section estimates of v are reported.33

B. Estimation and Testing with the GARCH Model

Bollerslev shows that apart from some constants, the log likelihood function for the sample y x,..., YT is given by

(26) Lj.Ce) = ?log/v(?, !*,_,),

32 Bollerslev also shows how the GARCH model can be derived as a subordinate stochastic

process from the conditional normal error model. 33 Fama (1965) and French and Roll (1986) noted that the variance of Monday's return appeared

to be higher than for other days. In the present model, this proposition can be tested by including a Monday dummy variable to capture any systematic increase in Monday variance. Specifically, the variance equation for a GARCH(1,1) process with the Monday variance shift is given by

ht = Oo + OLjC^, + 0,fc,_i +&,M, .



where 0 is the parameter vector including the degree of freedom parameter v. Maximum likelihood methods applied to (26) readily yield estimates of 0 and associated standard errors; inference procedures are straightforward. There is only one published study of the small sample distribution properties of ARCH models (see Engle, Hendry, and Trumble (1985)). It did not cover the model described by (21), (25), and (26), however. Given the relatively large sample sizes used here, the presumption is that asymptotic properties are approximated reasonably closely. Verification of this conjecture is obviously an important topic for future work.

C. GARCH Model Results

Table 9 reports tests for ARCH effects. To compute these test statistics, retrieve residuals from an OLS regression, square them, compute an autoregres- sion of lag length n, and calculate T-R2 where R2 comes from the autoregres- sion.34 Engle shows T-R2 has a x2 distribution with n degrees of freedom. There is strong evidence of ARCH effects for all sample periods and return measures, even using critical values for a 0.0001 test.

GARCH models were estimated for each of the seven subperiods and likeli? hood ratio tests were performed to find day-of-the-week and GARCH effects. Table 10 contains these test results. The first three columns list the likelihood

ratio test statistics from tests of the GARCH speeification against the constant variance model. For these tests, we used three models for returns: a constant mean model, Model 1, and Model 2. In all but one case, the test statistics reject the constant variance assumption in favor ofthe GARCH variance model.

The right-hand side of Table 10 contains results of likelihood ratio tests for weekend effects based on a GARCH (1,1) model ofthe regression variance. Col? umn IV compares the constant mean model to Model 2. Column V compares Model 2 to Model 1 and Column VI compares the constant mean model to Model 1. Through the 1963-1974 period, the likelihood ratio test results clearly favor Model 2 for the S&P and EW return measures. For the VW return measure, the tests select Model 1 for the 1963-1968 period and favor Model 2 for the 1969-1974 period. For the 1975-1983 period, test results show the constant mean model is best for the S&P and EW return measures and Model 2 is best for

the VW return measure. There is no obvious explanation for this divergence of results between the return measures.

To be certain that the coefficient estimates and their standard errors do not

depend on the normal error assumption, we used Bollerslev's modified likeli? hood function and estimated the degrees of freedom parameter on the error distri? bution. Table 11 reports complete results from reestimating the models selected by the likelihood ratio tests but using this modified likelihood function.35

There are several interesting findings. First, although the likelihood ratio tests selected Model 1 for the VW return measure for 1963-1965 and

1966-1968, estimated Monday returns are actually positive for both periods. The

34 Engle (1982) outlines this test. 35 The test results reported in Table 10 are computed assuming v = 33 but are unaffected by

allowing vto vary.


Connolly 163

same result obtains for the 1975-1977 and 1978-1980 samples: Monday returns are different from the rest of the days but the point estimates are positive. Only the 1969-1974 and 1981-1983 samples generate negative Monday returns for the VW measure. For the S&P and EW return measures, the negative Monday returns disappear after 1974.

A second aspect of the results in Table 11 concerns the degrees of freedom parameter, v, on the error distribution. The majority of estimates of v are less than 10 and only three are greater than 13. This means that the kurtosis in the unconditional error distribution cannot be attributed entirely to heteroskedas? ticity. Some portion of the kurtosis is due to fundamental nonnormality since the conditional error distribution is leptokurtic.

For some regressions, particularly those estimated on data from 1969 to 1983, the sum ofthe ax and a2 estimates is quite close to one. This is the princi? pal characteristic ofthe integrated-in-variance GARCH (IGARCH) model. Bol? lerslev (1988) and Engle and Bollerslev (1986) analyze this model. Its major implication is that

the multistep variance conditional on \\it will always depend onht+l (i.e., Et(ht+S) = ht+x), but for shorter forecast horizons additional information will also be important in forming optimal forecasts of the conditional variances. The integrated GARCU(p,q) models, both with or without trend, are therefore part of a wider class of models with a property called "persistent variance" in which the current informa? tion remains important for the forecasts of the conditional variances for all horizons (Engle and Bollerslev (1986), p. 27).

This contrasts with the usual idea that in efficient capital markets nothing about returns is forecastable.36

VII. Summary and Conclusions

A considerable share of empirical work in finance over the past decade has investigated apparent anomalies in stock return behavior. This paper has exam? ined the robustness of evidence on the DOW and the weekend effects. The work

reported here documents some of the same basic findings reported in earlier pa? pers.

After accounting for the impact of very large sample sizes, we have shown the sample evidence quite often favors the null hypothesis of equal returns across days of the week. We also have documented a number of important statistical flaws in the regressions used for weekend effect tests. Some of these flaws bias the hypothesis tests in the direction of finding an anomaly.

We considered the impact on inferences of four alternative estimation meth? ods, each of which displays some attractive robustness properties. Results from the L-estimators all suggest that the weekend effect is smaller than heretofore believed and that it ceased to exist by the mid- 1970s. The M-estimator results are even stronger. Estimates from the GARCH model with conditional Student-f errors strongly indicate GARCH effects in variance. Correcting for these effects and for autocorrelation generates weak evidence of a weekend effect until the

36 Connolly (1988b) reports tests for integrated-in-variance models using size-sorted portfolios and various bond return measures.



Note. The number of degrees of freedom in each test equals the lag length value. The critical x2 values are listed below each column.

mid-1970s. In addition, the size, strength, and stability ofthe weekend effect depends on the return measure. This result is particularly puzzling.

From a finance theory perspective, the evidence of a weekend anomaly is clearly dependent on the estimation method and the sample period. When trans? actions costs are taken into account, the probability that arbitrage profits are available from weekend-oriented trading strategies seems very small. This con? clusion is obviously consistent with an efficient markets approach.

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Connolly 165

TABLE 10

Likelihood Ratio Tests in GARCH DOW and Weekend Effect Models

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