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1 This article was downloaded by: [Columbia University] On: 10 December 2014, At: 05:45 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Applied Financial Economics Publication details, including instructions for authors and subscription information: Day-of-the-week effects in the pre-holiday returns of the Standard & Poor's 500 stock index Stephen P. Keef & Melvin L. Roush a Faculty of Commerce and Administration, Victoria University of Wellington, PO Box 600, Wellington, New Zealand b Faculty of Commerce and Administration, Victoria University of Wellington, PO Box 600, Wellington, New Zealand Published online: 02 Feb To cite this article: Stephen P. Keef & Melvin L. Roush (2005) Day-of-the-week effects in the pre-holiday returns of the Standard & Poor's 500 stock index, Applied Financial Economics, 15:2, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Applied Financial Economics, 2005, 15, Day-of-the-week effects in the pre-holiday returns of the Standard & Poor s 500 stock index Stephen P. Keef and Melvin L. Roush* Faculty of Commerce and Administration, Victoria University of Wellington, PO Box 600, Wellington, New Zealand This study investigates the day-of-the-week effects in the pre-holiday returns of the Standard & Poor s 500 stock index. The period investigated is The analysis is based on within-day contrasts and betweenday contrasts. There are three major findings. First, the results are consistent with prior research in that there is a strong pre-holiday effect up to 1987, but the pre-holiday effect is greatly diminished after Second, contrary to that frequently observed in the literature for typical days, there is no evidence of a weekend effect in pre-holiday returns. Third, a Labor Day effect is observed in the pre-1987 era. The return on the day before Labor Day is significantly greater than the return before other holidays that fall on a Monday. However, this effect is not observed after A number of other findings are discussed. I. Introduction Dimson and Marsh (1999, p. 53), in their review of the UK small-firm premium, say that Stock market anomalies generate strong and conflicting opinions from students of finance. This statement is not too surprising. Anomalies are, after all, abnormal. Anomalies represent deviations from an efficient market. They are patterns in prices that are not in accord with theoretical expectations. Research into stock market anomalies has a substantial history. Wilson et al. s (2001) review of the literature traces the occurrence of predictable regularities as far back as Cowles and Jones (1937). Anomalies in stock returns come in various forms; large firms versus small firms, long term compared to short term, over and under reactions to information, seasonal effects and others. Some say they ought not to occur. The conventional wisdom is that knowledgeable traders should eliminate them. Yet, market anomalies are frequently reported in the empirical finance and economic literature. Therefore, these potential departures from the efficient market hypothesis must be rigorously analysed and tested. Further analysis will lead to a greater understanding of the aberrant behaviour. This understanding, in turn, will provide insights toward an explanation of the anomaly. There can be little doubt that the day-of-the-week anomaly is one that has attracted considerable attention in the literature. Early research into the topic was conducted by Fama (1965), Cross (1973) and French (1980). The references cited by Agrawal and Tandon (1994), Chang et al. (1993) and Sias and Starks (1995) provide a comprehensive review of the literature. The reading of this literature is that the depressed Monday (or weekend) effect is the most commonly reported anomaly. Three hypotheses are offered to explain dayof-the-week effects in stock indices. The weekend effect is explained by the information release hypothesis *Corresponding author. Melvin.Roush@vuw.ac.nz. Applied Financial Economics ISSN print/issn online # 2005 Taylor & Francis Group Ltd DOI: /

3 108 S. P. Keef and M. L. Roush (French, 1980; Rogalski, 1984; DeFusco et al., 1993). This hypothesis argues that business leaders delay the release of negative information until after the stock exchange has closed on Friday (French, 1980, p. 66). Second, the settlement regime hypothesis (Gibbons and Hess, 1981; Lakonishok and Levi, 1982) argues that the delay in the cash payment for the security can lead to enhancements in the rates of return on specific days due to the extra credit occasioned by the two days of the weekend. Studies into the account effects of the London Stock Exchange are presented by Board and Sutcliffe (1988), Mills and Coutts (1995, pp ) and Arsad and Coutts (1997). There is little empirical support for this hypothesis with stock indices. A possible explanation may lie in the fact that the economic benefit associated with increased credit is small. Third, there is the information processing hypothesis (Lakonishok and Maberly, 1980; Miller, 1988; Damodaran, 1989; Chang et al., 1993; Abraham and Ikenberry, 1994; Sias and Starks, 1995). This hypothesis focuses on the asymmetry in costs faced by small versus large investors and the effects of firm size. A review of the literature reveals conflicting evidence on the validity of this hypothesis. The pre-holiday anomaly has also been extensively researched. In this paper, a pre-holiday is the normal trading day that precedes the holiday. The returns observed on pre-holidays are, on average, many times greater than the return on other trading days (Merrill, 1966; Ariel, 1990; Kim and Park, 1994; Vergin and McGinnis, 1999). A substantial amount of research effort has been devoted to verifying or refuting the existence of the pre-holiday anomaly. Although results are mixed, the phenomenon has been observed across time, in many countries, in auction and dealer markets, and over a range of high and low share prices (Pettengill, 1989; Kim and Park, 1994; Brockman and Michayluk, 1998). The impact of the pre-holiday return can be enormous. Ariel (1990) found that the pre-holiday effect contributed 34% to the total annual returns of the Dow Jones Industrial Average. The long-standing abnormal return on preholidays is currently not well understood. A number of explanations have been advanced. As detailed below, these are not consistently supported by the evidence. Clientele investment decisions have been offered as an explanation (Harris and Gurel, 1986; Ritter, 1988). Ariel s (1990) results do not support a specialist activity hypothesis. He discounts the possibility that the effect is a consequence of covering by short-sellers who seek to close out their risky positions over the holiday period. Because the preholiday effect is observed internationally, Kim and Park (1994, p. 156) argue that neither...trading methods, clearing mechanisms, settlement procedures [nor]...bid-ask spreads... can solve the riddle of abnormal pre-holiday returns. However, despite the lack of convincing explanations for the anomaly, it continues to be found; and as Brockman and Michayluk (1998, p. 205) argue,...further research in the area is both necessary and inevitable. This study extends the knowledge on the behaviour of the pre-holiday effect by integrating the dayof-the-week anomaly with the pre-holiday anomaly. The primary objective is to investigate whether preholiday returns are affected by the day-of-the-week anomaly, that is, to what degree can the frequently observed abnormal pre-holiday return be attributable to the also frequently observed day-of-of-theweek anomaly? This issue has not been previously addressed in the literature in a rigourous manner. The research question is examined by analysing the daily returns of the Standard & Poor s 500 stock index over the period 1 January 1930 to 31 December Ariel (1990) and Kim and Park (1994) present evidence relating to the interaction of the pre-holiday effect and day-of-the-week effects. Both studies rightly claim that the pre-holiday effect persists after controlling for day-of-the-week effects. However, as illustrated in the following section, existing research does not permit an unambiguous estimate of day-of-theweek effects in pre-holiday returns. Regression model of prior research The model employed by Ariel (1990, Table IV, Panel C, p. 1622) and Kim and Park (1994, Table 2, Panel B, p. 150) is r ¼ 0 þ 1 Mon þ 2 Wed þ 3 Thur þ 4 Fri þ 5 PreHol þ 6 NewYear þ error ð1þ where r is the observed rate of return for day t on the stock index and Mon, Wed, Thur and Fri represent the conventional day-of-the-week 0, 1 dummy variables. Liano et al. (1992) adopt a similar analysis. The Tuesday day-of-the-week effect is captured by the constant 0. PreHol and NewYear are dummy variables with a value of 1 if the day is a pre-holiday or New Year s Eve, respectively, otherwise zero. Regression model (Equation 1) tests for the preholiday effect captured by coefficient 5 after controlling for the day-of-the-week effects (coefficients ) and after controlling for the New Year s Eve effect (coefficient 6 ). Since the constant 0 captures the mean return on Tuesdays, coefficient

4 Pre-holiday returns of the S & P 500 stock index measures the pre-holiday effect referenced against the average Tuesday return. Such a design is sensible if Tuesday is an ordinary weekday in the context of day-of-the-week anomalies. There is little evidence to deny this, but whether Tuesday is, or is not, actually a typical trading day has not been specifically examined in the literature. The regression analyses in these prior studies include all the daily data in the time period investigated, that is, pre-holiday days together with trading days that are not pre-holidays. Here, the latter are called typical days. In both the Ariel (1990) and the Kim and Park (1994) studies, the time series data consists of 97% typical daily rates of return and 3% pre-holiday returns. There are approximately 260 weekdays per year and eight holidays. Hence, there are 252 trading days. Of these 252 trading days, there are eight pre-holidays and 244 typical trading days. Therefore, the day-of-the-week effects captured by coefficients represent a mixture of the effects for the typical days together with the effects for the pre-holidays. This aggregation leads to regression results that are difficult to interpret in the context of the presence of day-of-the-week effects in pre-holiday returns. A regression model that would, in the spirit of the Ariel (1990) and Kim and Park (1994) studies, provide unambiguous estimates of day-of-the-week effects in pre-holiday returns is r ¼ 0 þ 1 Mon þ 2 Wed þ 3 Thur þ 4 Fri þ 5 PreHol þ 1 PreHol Mon þ 2 PreHol Wed þ 3 PreHol Thur þ 4 PreHol Fri þ error ð2þ where, as explained later, it is assumed that the returns for New Year s Eve are deleted. To illustrate the new notation, variable PreHol Mon represents the interaction (product) of the pre-holiday dummy variable PreHol with the Monday dummy variable Mon. In this model, the new coefficients provide estimates of the differences in the day-ofthe-week effects for pre-holidays compared to typical days. The Monday effect for typical days is given by 1 and the Monday effect for pre-holidays is given by 1 þ 1. A similar interpretation applies to Wednesday... Friday. The Tuesday effect in pre-holiday returns is estimated by 0 þ 5. Equation 2 could be split into two regression equations. That is to say, one regression analysis based on the sample of typical returns (when PreHol ¼ 0) and the other regression based on the sample of pre-holiday returns (when PreHol ¼ 1). Each equation, of the form, r ¼ 0 þ 1 Mon þ 2 Wed þ 3 Thur þ 4 Fri þ error ð3þ could be examined separately. These two regression analyses would give, in the final analysis, the same estimates of the coefficients as could be obtained from regression Equation 2. In the present study day-of-the-week effects in typical returns are not examined since: (i) there is extensive evidence on this topic, (ii) pre-holiday returns are obviously different from typical returns; and (iii) the focus is on the day-of-the-week effect in pre-holiday returns. Furthermore, although Equation 3 is a conventional approach to day-of-theweek effects, a more suitable empirical model is employed, as discussed in the next section. The remainder of the paper is structured as follows. Section II outlines the methodology employed. Section III presents and discusses the results. Section IV provides brief conclusions. II. Methodology The model employed in this study is based on orthogonal contrasts. As explained below, the orthogonal contrasts consist of (i) within-day contrasts and (ii) between-day contrasts. First, however, the process for identifying the relevant holidays must be explained. Classifying holidays Brockman and Michayluk (1998) observe that the definition of a holiday varies between researchers. There are good reasons for this. Although it is an everyday word, a holiday is not easy to define in the context of stock exchanges. One definition contemplated was a day, other than Saturday or Sunday, upon which the exchange is closed (Lakonishok and Smidt, 1988). The problem with this definition is that it includes exceptional events. Examples are the funerals of Presidents, the joyous celebrations of VJ Day (15 and 16 August 1945) and the first landing on the moon (21 July 1969), catastrophes such as a snow storm (10 February 1969), a power blackout (14 July 1977) and Hurricane Gloria (27 September 1985). It is not obvious that these diverse events systematically share similar characteristics with regular holidays. Furthermore, national holidays change over time. For example, Martin Luther King Jr Day (third Monday in January) was not established until 1983.

5 110 S. P. Keef and M. L. Roush President s Day was introduced in It replaced the holidays of Lincoln s Birthday (12 February) and Washington s Birthday (22 February). Memorial Day was celebrated on 30 May up till In 1971 this holiday was moved to the last Monday in May. These changes pose difficulties for researchers. To circumvent these difficulties, the focus is on the seven national holidays that were normally observed each year throughout the period investigated. The holidays are President s Day (including Washington s Birthday and Lincoln s Birthday prior to 1971), Good Friday, Memorial Day, Independence Day (July Fourth), Labor Day, Thanksgiving Day and Christmas Day. It was consciously decided to omit the return on New Year s Eve. It is an anomaly in its own right. The general consensus is that the return on New Year s Eve is higher than the return on other pre-holidays (Ariel, 1990; Liano et al., 1992; Kim and Park, 1994). Christie-David and Chaudhry (2000) provide a comprehensive survey of the hypotheses that compete to explain the January (or year-end) effect. Further support for the decision to delete pre-holiday returns for New Year s Day is found in the empirical models employed by Ariel (1990), Kim and Park (1994) and Brockman and Michayluk (1997). They treat the return on New Year s Eve separately. Their use of the dummy NewYear effectively abstracts the effects of the excess return (compared to the constant) observed on New Year s Eve. Furthermore, this study is only focused on the corresponding days that the New York Stock Exchange was closed for trading on a weekday in celebration of these holidays. For example, Christmas Day fell on a weekend in 1937, 1943 and 1948, and the exchange did not close on either the prior Friday or the following Monday. The view is taken that it is not a true stock exchange holiday unless the market is closed on a weekday. Within-day contrasts The first part of this study focuses on the three holidays that fall on a fixed weekday throughout the entire period under investigation. Labor Day falls on the first Monday in September, Thanksgiving Day is the fourth Thursday in November, and Good Friday is the Friday before Easter Sunday. The three within-day contrasts are based on conventional 0, 1 dummy variables for each of the fixed day holidays. Consider, as an illustration, the holidays that fall on a Monday. Variable M Labor has a value of 1 if the holiday in question is Labor Day, otherwise zero. Variable M Other has a value of 1 if the holiday in question falls on a Monday that is not a Labor Day, otherwise zero. As a consequence, this contrast, and the other two that follow, take on a value of either þ1 or 1. This reading is necessary in the interpretation of the regression coefficients. The Labor Day Contrast, C Labor, compares the return on the day before the Labor Day holiday with the return on the day before other holidays that fall on a Monday. It is defined as C Labor, t ¼ M Labor, t M Other, t ð4þ Labor Day is a time of celebration of and for the worker. In most of the country, it is celebrated by engaging in pleasant outdoor activities, such as picnics and baseball games. It seems plausible to expect that there could be an exuberance effect associated with the holiday. Therefore, the working hypothesis is that the Labor Day pre-holiday return will be greater than the return associated with other holidays falling on a Monday. The Thanksgiving Day Contrast, C Thanksgiving, compares the return on the day before the Thanksgiving Day holiday with the return on the day before other holidays that fall on a Thursday. It is defined as C Thanksgiving, t ¼ Th Thanksgiving, t Th Other, t ð5þ Variable Th Thanksgiving has a value of 1 if the holiday in question is Thanksgiving Day, otherwise zero. Variable Th Other has a value of 1 if the holiday in question falls on a Thursday that is not a Thanksgiving Day, otherwise zero. Thanksgiving Day is a joyous celebration of the Pilgrim Fathers safe arrival in North America after their escape from religious persecution in England. Thus it is tentatively hypothesized that the pre-thanksgiving Day return will be greater than the return on the day before other holidays that fall on a Thursday. The Good Friday Contrast, C Good Friday, compares the return on the day before the Good Friday holiday with the return on the day before other holidays that fall on a Friday. The contrast is defined as C Good Friday, t ¼ F Good Friday, t F Other, t ð6þ Variable F Good Friday has a value of 1 if the holiday in question is Good Friday, otherwise zero. Variable F Other has a value of 1 if the holiday in question falls on a Friday that is not Good Friday, otherwise zero. Good Friday is a somber religious occasion. This provides support for the hypothesis that returns prior to Good Friday would be depressed compared to days before other holidays falling on a Friday. Between-day contrasts The conventional Mon, Tue, Wed, Thur and Fri dayof-the-week 0, 1 dummy variables are transformed

6 Pre-holiday returns of the S & P 500 stock index 111 into orthogonal contrasts (Keef and McGuinness, 2001). Contrast D 1 tests for a long weekend effect. Holidays falling on a Monday or on a Friday give rise to a long weekend of three days. For the other holidays the break is only one day. It is hypothesized that the anticipation of a longer holiday will enhance the return on the day before a long weekend. The contrast is defined as D 1, t ¼ 3 ðmon t þ Fri t Þ 2 ðtue t þ Wed t þ Thur t Þ ð7þ In the interpretation of the results it is necessary to note that this contrast takes on a value of þ3 if it is a three-day holiday and a value of 2 ifitis a one-day holiday. There were four occasions when a long holiday weekend was greater than three days. These occurred at Christmas 1945 (Monday 25 and Tuesday 26 December), Christmas 1956 (Monday 25 and Tuesday 26 December) and Christmas 1958 (Thursday 25 and Friday 26 December) as well as Independence Day 1968 (Thursday 4 and Friday 5 July). Each of these double holidays occurred prior to, or immediately after, a weekend. They represent breaks of four days. With these cases, the first day of the holiday is used to determine the day-of-the-week. The first two cases are treated as Monday holidays and the last two cases are treated as Thursday holidays. Thus the four day effect, if indeed it exists, is spread evenly across the two levels of Contrast D 1. Contrast D 2 compares the return on the day before Monday holidays with the return on the day before Friday holidays. The contrast tests for the consistency of the postulated long weekend effect. It is specified as D 2, t ¼ Mon t Fri t ð8þ A depressed Monday return is regularly reported with the stock index being studied. Thus, another hypothesis is that the return on the before a Tuesday holiday (this is a Monday return) will reflect this empirical regularity. Contrast D 3, which is defined as D 3, t ¼ 2 Tue t ðwed t þ Thur t Þ ð9þ tests for the often reported Monday effect. The days before holidays falling on a Wednesday or on a Thursday are used as the reference. Contrast D 4 is the residual contrast given the constraints of orthogonality and the three contrasts specified earlier. It is defined as D 4, t ¼ Wed t Thur t ð10þ This contrast can be interpreted as testing for a mid-week effect in the pre-holiday return. There is no obvious reason why the return on these pre-holidays would differ. Therefore, this contrast is not expected to be significant. These seven orthogonal contrasts are integrated into the regression model of the form r t ¼ 0 þ 1 C Labor, t þ 2 C Thanksgiving, t þ 3 C Good Friday, t þ 4 D 1, t þ 5 D 2, t þ 6 D 3, t þ 7 D 4, t þ error t ð11þ where, r t represents the observed rate of return for day t on the Standard & Poor s 500 stock index (I ). r t is calculated in the conventional fashion as r t ¼ log e ði t Þ log e ði t 1 Þ ð12þ In ideal circumstances, the error term error t is normally distributed with a mean of zero and finite variance. Coefficient 0 captures the mean preholiday rate of return. Recently, Vergin and McGinnis (1999) present results to show that the pre-holiday effect in the Standard & Poor s 500 stock index has disappeared since Thus the data is divided into two periods as at 1 January Each period is examined separately. The period 1 January December 1986 is called the pre-1987 era and the period 1 January December 1999 is called the post era. To distinguish between the two regressions, the superscripts Pre and Post are added, where necessary, to the coefficients of Equation 11, that is to say, Pre i refers to coefficient i in the pre-1987 era. Diagnostics: the pre-1987 era There are 403 pre-holiday returns in the period 1 January December Preliminary diagnostic tests indicate that the regression residuals for the pre-1987 era are not well behaved. Three problems are present in the errors of Equation 11. These are regularly reported in the literature analysing the daily rates of return of stock indices. The problems are heteroscedasticity, serial correlation and non-normality. The presence of heteroscedasticity is established using the battery of seven tests in White (1997). Using Engle s (1982) ARCH test as an illustration, the estimated chi-square on one degree of freedom is ( p<0.001). The residual correlogram reveals significant ( t >1.96) serial correlations at lag 9 (t ¼ 2.40) and lag 10 (t ¼ 2.82). The Jarque and Bera (1980) test statistic, with a chi-square distribution on 2 degrees of freedom, provides convincing evidence of non-normality. The critical p ¼ 0.05 chi-square value on 2 degrees of freedom is It is clear that fat tails (t Excess Kurtosis ¼ 49.57) dominate the positive skewness (t Skewness ¼ 8.33).

7 112 S. P. Keef and M. L. Roush The conventional approach to the serial correlation problem is to apply the Newey and West (1987) adjustment with a suitable lag. The Newey and West (1987) adjustment also provides control for the influences of heteroscedasticity. It is an enhancement on White s (1980) adjustment for unknown forms of heteroscedasticity. The complication of adopting suitable prophylactics for heteroscedasticity and serial correlation is that the effects of non-normality remain unknown. There exist a number of statistical procedures to accommodate non-normality; see, for example, the texts of Greene (1993) and Judge et al. (1985) or the empirical analyses of Connolly (1989). The difficulty with these approaches is that they cannot adequately cope with heteroscedasticity and serial correlation. The issue of non-normality is addressed by the use of the bootstrap method (see Noreen, 1989 or Mooney and Duval, 1993). An advantage of the bootstrap method is that it does not require the rejection of information contained within the data as is encountered with trimming procedures. A bootstrap process entails two decisions. The first decision concerns the resampling process. Here, the non-parametric method is adopted. The attraction of this approach, compared to the alternative parametric method, is that it is devoid of assumptions relating to the distribution of the data. The data in this study is an n k matrix where n is the sample size (403) and k is the number of variables in the regression model this being eight. The eight variables are the rate of return and the seven independent contrasts. A random sample, with replacement, of 403 full-rows is generated. Regression on this resample is used to estimate the parameters (coefficients) of interest. The resampling is repeated a sufficiently large number of times 9999 is used in this study. The second decision in a bootstrap analysis relates to the parameter used to characterize the effect and its confidence interval. The bootstrap-t (or percentile-t) method is used to estimate the coefficients at the conventional confidence level of p ¼ 0.05 (Mooney and Duval, 1993, p. 40). The resampling process and the associated regression (Equation 11) generate 9999 t-statistics per independent variable. When ranked in ascending order, the 250th and 9750th values form the basis of the lower and upper confidence intervals for a p ¼ 0.05 double tailed test, that is, 2.5% in each tail. Using ^ i to denote the estimate of coefficient i, with standard error given by ^ i obtained from the regression of the original raw data, the coefficient s p ¼ 0.05 confidence interval is given by ^ i ^ i t i, 250 to ^ i þ ^ i t i, 9750 ð13þ where t i, j ¼ð^ i, j ^ i Þ= ^ i, j ð14þ with the superscript * denoting the estimate is obtained from the resamples and the additional subscript j represents the number of the resample (that is, j ¼ 1, 2,..., 9999). If the confidence interval in Equation 13 does not contain zero, then it can be concluded that the effect is statistically non-zero at the chosen level of statistical significance. That is to say, the null hypothesis is rejected. The estimate of the standard error ^ i, j for each of the resamples is obtained after the application of the Newey and West (1987) adjustment with a lag of ten. The random sampling process, with replacement, shuffles the temporal order of the data. This aspect is addressed by sorting the data, by a temporal indicator, prior to running each of the resample regressions. For practical reasons it was not possible to determine the appropriate lag for each of the sorted resamples. Thus the use of a lag of 10 is a pragmatic compromise. Diagnostics: the post-1987 era There are 91 pre-holiday returns in the period 1 January December Preliminary diagnostic tests show that the regression residuals for the post 1987 era are reasonably well behaved. The Jarque and Bera (1980) test statistic 5.81 ( p>0.05) does not deny the assumption of normality. The largest serial correlation coefficient is for lag 3 (t ¼ 1.20). Six of the seven tests for heteroscedasticity fail to reject the null hypothesis. However the Glejser test ( 2 ¼ on 7 degrees of freedom, p<0.05) indicates the presence of modest heteroscedasticity. As a consequence an OLS regression of Equation 11 incorporating White s (1980) heteroscedasticityconsistent covariance matrix is used. This adjustment controls for all forms of heteroscedasticity. III. Results and Discussion The results of the bootstrapping exercise for the pre-1987 period are presented in Table 1. In order to provide the full picture, the results of the preliminary OLS regression for that period are also presented in the same table. The OLS regression of the data and the bootstrapping exercise generate consistent results. The implication is that the non-normality does not greatly influence the estimates of the coefficients. Such an outcome could not have been predicted

8 Pre-holiday returns of the S & P 500 stock index 113 Table 1. Day-of-the-week effects (a) r t ¼ 0 þ 1 C Labor, t þ 2 C Thanksgiving, t þ 3 C Good Friday, t þ 4 D 1, t þ 5 D 2, t þ 6 D 3, t þ 7 D 4, t þ error t Variable Coefficient Pre-1987 (b) OLS with Newey and West (1987) lag 10 Bootstrapped-t p ¼ 0.05 confidence interval Estimated coefficient t-statistic Lower median upper Post-1987 OLS with White s (1980) adjustment Estimated coefficient t-statistic Constant % % 0.282% 0.374% 0.123% 1.56 C Labor % % 0.180% 0.272% 0.098% 0.55 C Thanksgiving % % 0.062% 0.200% 0.396% 3.11 C Good Friday % % 0.117% 0.052% 0.132% 0.84 D % % 0.005% 0.046% 0.016% 0.49 D % % 0.041% 0.147% 0.023% 0.19 D % % 0.100% 0.251% 0.003% 0.03 D % % 0.105% 0.254% 0.356% 4.14 Notes: (a) Significant ( p<0.05) coefficients are in italics. (b) OLS and Bootstrapped-t methods generate similar results. ex ante. Nevertheless, the analysis focuses on the bootstrapped results for the pre-1987 period. For the post-1987 period, the diagnostics do not indicate that the assumptions of OLS regression are denied. Therefore, bootstrapping was not performed on this data, and only the OLS regression results are presented in Table 1. There is no support in the pre-1987 or post-1987 data for the hypothesis (Contrast C Good Friday ) that returns would be depressed on the Good Friday pre-holidays ( p>0.05), and the results presented in Table 1 do not support the hypotheses (Contrasts D 1 and D 2 ) that returns would be greater on the day before a long (three day) holiday weekend than on the day before a holiday break of only one day ( p>0.05). The results for the two separate periods highlight four statistically significant effects. Each significant effect is discussed separately under the subheadings of: (i) Pre-holiday effect (coefficient Pre 0 for the pre era), (ii) Labor Day Monday effect (Contrast C Labor captured by Pre 1 for the pre-1987 era), (iii) Within-Thursday effect (Contrast C Thanksgiving represented by coefficient Post 2 for the post-1987 era), and (iv) Wednesday versus Thursday effect (Contrast D 4 corresponding to coefficient Post 7 for the post-1987 era). Contrast D 4 and Contrast C Thanksgiving involve the holidays falling on either a Wednesday or on a Thursday. As shall be illustrated, the interpretation of the results for these contrasts requires a degree of judgment in the post-1987 era since the cell frequencies for that period are small. This is an unavoidable consequence of the data available. To aid the analysis, Table 2 presents the mean pre-holiday return of the raw data broken down by day of-the-week. The statistically significant results are highlighted by the use of boxes. Pre-holiday effect The median pre-holiday return in the pre-1987 era ( Pre 0 ¼ 0.282%) is statistically greater than zero. The bootstrap p ¼ 0.05 confidence limits of 0.182% and 0.374% do not contain zero. These findings are consistent with results of prior research (see Vergin and McGinnis, 1999, Table 1, p. 478 for a summary). After 1987, however, the mean pre-holiday return ( Post 0 ¼ 0.123%) is considerably smaller. It is not statistically significant (t ¼ 1.56). Thus, Vergin and McGinnis s (1999) finding that the pre-holiday effect has greatly diminished since 1987 is corroborated. A way of assessing the strength of the pre-holiday return is to decompose the annual rate of return into pre-holidays returns and standard returns (Lakonishok and Smidt, 1988; Brockman and Michayluk, 1998). Assuming eight pre-holidays per year and 244 standard trading days, the pre-1987 holiday effect represents 32.9% of the annual return. This figure is consistent with prior research (see Ariel, 1990 and Vergin and McGinnis, 1999). In the post-1987 era, the holiday effect accounts for only 6.4%. This represents a fivefold decrease. The apparent drop in the pre-holiday effect could be due to two separate influences. First, there is the drop in the size of the pre-holiday returns, per se. Second, there is the change in the mean typical daily rate of return. These are the one-day (threeday in the case of Mondays) rates of return, which are directly comparable with the estimated

9 114 S. P. Keef and M. L. Roush Table 2. Breakdown of observed pre-holiday returns by day-of-the-week (a) Mean daily rate of return on pre-holiday Pre-1987 Post-1987 Day of the holiday Within-day Between-days Within-day Between-days All Mondays 0.349% (n ¼ 155) 0.103% (n ¼ 47) Labor Day Mondays 0.579% (n ¼ 57) 0.244% (n ¼ 13) Other Mondays 0.216% (n ¼ 98) 0.050% (n ¼ 34) All Tuesdays 0.598% (n ¼ 27) 0.087% (n ¼ 3) All Wednesdays 0.391% (n ¼ 31) 0.451% (n ¼ 3) All Thursdays 0.143% (n ¼ 90) 0.013% (n ¼ 16) Thanksgiving Thursdays 0.056% (n ¼ 57) 0.135% (n ¼ 13) Other Thursdays 0.294% (n ¼ 33) 0.657% (n ¼ 3) All Fridays 0.313% (n ¼ 100) 0.169% (n ¼ 22) Good Friday Fridays 0.252% (n ¼ 57) 0.060% (n ¼ 13) Other Fridays 0.394% (n ¼ 43) 0.325% (n ¼ 9) Total 0.314% (n ¼ 403) 0.110% (n ¼ 91) Note: (a) Significant effects (see Table 1) are highlighted within boxes. pre-holiday rates of return. The mean return in the pre-1987 era is 0.021% and the mean return in the post-1987 era is 0.053%. These two dimensions are now investigated. Define the relative magnitude RM of the preholiday effect as the ratio mean pre-holiday daily return RM ¼ mean typical daily return which can be written as ð15þ RM ¼ mean pre-holiday daily return 1 ð16þ mean typical daily return The RM in Equation 16 can be represented as the area in a two-dimensional space with one axis described by mean pre-holiday return and the other axis described by the inverse of the mean typical daily return. Figure 1 presents a stylized representation of the pre-holiday effect for the two periods. The thick line defines the relative magnitude of the pre-holiday effect in the period before RM Pre is the area within the thick line. The area within the thin line represents the relative magnitude for the post-1987 era, RM Post. As illustrated by Fig. 1, the change in the relative magnitude of the pre-holiday effect defined as RM ¼ RM Pre RM Post is represented by three distinct influences which are denoted by areas A, B and C. Using RM Post as the reference, since it is the smaller value, these influences are: (i) the change in the mean pre holiday daily return (area A), (ii) the change in the mean typical daily return (area B) and (iii) the interaction term (area C). Mean pre-holiday daily return % A Post-1987 a Inverse of the mean typical daily return % Fig. 1. Decomposition of the relative magnitude ratio RM (Note: (a) The post-1987 area is superimposed on the pre-1987 area) The relative magnitude of the pre-holiday effect in the pre-1987 era is mean pre-holiday daily return RM Pre ¼ mean typical daily return ¼ 0:314% ¼ 15:0 ð17þ 0:021% The mean pre-holiday return is obtained from Table 2. The use of the bootstrapped estimate generates a relatively similar result. The mean typical daily return in Equation 17 represents the average daily rate of return of the index for the period 1 January December 1986 excluding all pre-holiday returns and excluding the returns on all New Year s Eves and by excluding returns before other closings. The relative measure of the pre-holiday effect in the pre-1987 era as shown in Equation 17 is That is to say, the average pre-holiday rate of return is equivalent to five C B

10 Pre-holiday returns of the S & P 500 stock index 115 weeks of return on typical trading days. As mentioned by others, this is a staggering figure. The relative magnitude of the pre-holiday effect in the post-1987 era RM Post is mean pre-holiday daily return RM Post ¼ mean typical daily return ¼ 0:110% ¼ 2:1 ð18þ 0:053% The data is obtained in the same way outlined immediately above with the exception that the period is 1 January December The relative measure of the pre-holiday effect in the post-1987 era as shown in Equation 18 is 2.1. Although in isolation this is an economically meaningful ratio, it is relatively small compared to the pre-1987 ratio. The 12.9 unit decrease in the relative measure of the pre-holiday effect, from 15.0 for the pre-1987 era to 2.1 for the post-1987 era, can be explained as follows. Approximately 3.8 units (29.9%) are attributed to the drop in the pre-holiday mean daily rate of return see area A in Fig. 1. Approximately 3.2 units (24.6%) can be attributed to the increase in the mean typical daily return see area B. Finally, approximately 5.9 units (45.5%) are attributed to the combination of these two effects (area C). These results could be interpreted as indicating that the strong pre-holiday effect in the pre-1987 era was caused by the pre-holiday return being higher than it should be and the returns on typical days as being lower than they should be. A new equilibrium may be in the process of being established that will be more in line with theoretical expectations. Labor Day Monday effect Contrast C Labor compares the pre-labor Day return with the return observed on the day before other holidays falling on a Monday. In the pre-1987 era, the pre-labor Day return is significantly larger than the return on the day before other holidays falling on a Monday. The p ¼ 0.05 bootstrapped confidence limits of Pre 1 are 0.048% to 0.272% with a median of 0.180%. The confidence interval does not include zero. Noting Contrast C Labor takes on a value of þ1 or 1, the within-monday difference is estimated as 0.360%, which, by any account, is an economically significant number. The mean return for the day before Labor Day is 0.579%; the mean return for the day before other holidays falling on a Monday is 0.216% (Table 2). This Labor Day effect has not been previously reported in the literature. In the post-1987 era, a similar effect is apparent (using a naı ve eye-ball test) in the raw data (Table 2). However, coefficient Post 1 is close to half the size of coefficient Pre 1 and its standard error is larger. In statistical terms, the Labor Day effect is absent in the post-1987 era (t ¼ 0.55). Therefore the hypothesis that the return prior to Labor Day would be larger than the return before other holidays falling on a Monday is supported before 1987 but not after that date. The hypothesis was based on the psychological notion of exuberance. Shiller s (2000) latest book, Irrational Exuberance, offers insights into anomalous behaviour by investors as reflected in stock prices. Notwithstanding, it is fair to comment that the psychological cum economic underpinnings of the pre-holiday effect are not well understood. The same can be said of the Labor Day effect observed in this study. Thus it comes as no surprise that it is not easy to explain why the Labor Day effect has declined since A conventional interpretation of the statistical results indicates that something about the social and commercial environment in the USA must have changed dramatically about that time, but no convincing economic explanations are immediately apparent. Furthermore, it should be noted that the hypothesized Labor Day effect may actually be present in the background in the post-1987 era. The raw numbers support this prospect. The effect simply might not be statistically significant due to the small sample size for that period. Within-Thursday effect Contrast C Thanksgiving compares the returns on the days before Thanksgiving Day with the returns on days before other holidays falling on Thursdays. The hypothesized Thanksgiving Day effect is absent in the pre-1987 era ( p>0.05) but is present in the post-1987 era (t ¼ 3.11). The mean return on the day before Thanksgiving Day (0.135%) is reasonably close to the average pre-holiday return (0.110%) in the post-1987 era. It can be viewed as a typical pre-holiday return. However, the mean return on the days before other holidays falling on a Thursday ( 0.657%) is large and negative. It is anomalous since it is dramatically different in terms of magnitude and sign from the average pre-holiday return. Thus the statistical analysis has faithfully captured the underlying structure of the data. However, the small sample size raises the specter of the results being a reflection of aberrant returns. That is to say, the sample may not be truly representative of the population. The small samples are discussed in a following section.

11 116 S. P. Keef and M. L. Roush Wednesday versus Thursday effect Contrast D 4 compares the returns on the day before holidays falling on a Wednesday with the returns on the day before a holiday falling on a Thursday. A Wednesday versus Thursday effect is absent in the pre-1987 era ( p>0.05) but is present in the post-1987 era (t ¼ 4.14). Given the results reported immediately above for Contrast C Thanksgiving in the post-1987 era, there is a degree of uncertainty as to the proper economic content of the mean return on the day before a Thursday holiday. The mean return on the days before holidays falling on a Wednesday (0.451%) is four times larger than the typical pre-holiday returns (0.110%) but is based on a relatively small sample size of three. Thus there is a question mark hanging over both elements of the Wednesday versus Thursday effect. Further considerations Another point to consider is the reason why the within-thursday effect and the Wednesday versus Thursday effect appear after These effects are clearly (given the more than adequate sample sizes) absent in the pre-1987 era. To further complicate the interpretation, these effects in pre-holiday returns apparently emerge when the pre-holiday return, per se, is greatly reduced. The importance of Contrast C Thanksgiving and Contrast D 4 in the post-1987 era can be questioned. Both contrasts involve the holidays falling on a Wednesday or on a Thursday. The corresponding trading days are Tuesday and Wednesday, respectively. The estimates of the statistical importance of both contrasts are based on a small sample size of n ¼ 3 (see the two boxes on the right hand side of Table 2). The mechanics of the statistical process can accommodate this aspect. However, the question of whether these cases are true representations of the underlying economic phenomena is more pressing. Thus a review of these cases has utility. The relevant details of the six cases are shown in Table 3. Independence Day and Christmas Day are the only holidays involved. This selection process is purely a consequence of the calendar nothing else. All bar the last case appear in the Vergin and McGinnis (1999, Table 3A at p. 480) data set. A concern with these small samples is that one result is unnaturally influencing the mean value. This issue was addressed by the use of boxplots see Fig. 2. Visual inspection indicates the degree to which an outlier influences the mean. For the three Thursday pre-holidays involved in Contrast C Thanksgiving, this analysis indicates the data is symmetrically distributed. Thus an outlier does not influence the statistical result for this contrast. For the three Wednesday pre-holidays involved in Contrast D 4, the return of 0.172% on the day before Independence Day 1990 is visually indicated as an outlier. This outlier is, however, strongly biased toward the null hypothesis. Thus the evidence points to the fact that the statistical significance of this contrast cannot be attributed to a deviant case. For these enigmatic reasons it would not be prudent to lay claims that these latter results are Thursday Wednesday Rate of Return % Fig. 2. Boxplot of returns on the day before holidays occurring on Wednesday and Thursday in the post-1987 era Table 3. Details of the six cases that occurred on Wednesday and Thursday after 1987 Contrast Day-of-the-week of the holiday Holiday Return on the day before the holiday C Thanksgiving Thursday Independence Day % Independence Day % Christmas Day % Mean 0.657% D 4 Wednesday Independence Day % Christmas Day % Christmas Day % Mean 0.451%

12 Pre-holiday returns of the S & P 500 stock index 117 a reflection of either: (i) a systematic economic effect or (ii) a reflection of aberrant results. There are arguments in favour of either interpretation. However, it is sufficient to infer that future research might consider these aspects when further data becomes available. Monday effect The most often reported day-of-the-week anomaly is the Monday or weekend effect. This effect is normally attributed to the information release hypothesis. Itis argued that the effect is a consequence of executives delaying the release of negative information until after the stock exchange has closed on a Friday (French, 1980). This negative information results in the depression of share prices on a Monday. Connolly (1989) reports that the Monday effect has disappeared since Thus, the observation that the hypothesized Monday effect is not present in pre-holiday returns in the post-1987 era is not unexpected. However, the absence of a Monday effect in the pre-1987 pre-holiday returns needs to be explained. The situation is made even more puzzling because, in the pre-1987 era, the Monday return (0.598%) on the day before a Tuesday holiday is, on average, the highest of the week (Table 2). Four possible explanations were considered for the absence of the Monday effect in pre-holiday returns in the pre-1987 era. A point to note is that the holiday is on Tuesday and here the focus is on the release of information on, or just prior to, the Friday before. First, negative information does not arise in the days immediately before a Tuesday holiday. This is an unlikely explanation since the general consensus is that information (negative or positive) arises randomly. Second, executives release the information immediately as it arises during the week preceding the holiday. That is to say, they refrain from the conjectured delaying tactic. The veracity of this explanation is not easy to test. Third, executives continue to delay the release of negative information, as is generally argued by the information release hypothesis, but the pre-holiday effect swamps it. The support for this interpretation is weak. The mean Monday effect for the period with the index we examine (using data reported in Connolly, 1989, Table 2, p. 138) was 0.306%. This is of the same magnitude, but with a reversed sign, of the mean pre-holiday effect (0.314%). The observed Monday return (0.598%) prior to a Tuesday holiday implies that the net holiday effect is thus estimated as 0.904% (¼0.598% þ 0.306%). This is a staggering figure that suggests this explanation is improbable. Finally, the data leads to the possibility that executives delay the release of good information during the week and release it on the Friday immediately before a Tuesday holiday. Essentially, this is the reverse of the conventional information release hypothesis. This idea could account for the fact that the Monday pre-holiday effect is larger than the average pre-holiday effect. However, the notion that executives delay the release of good information appears somewhat outlandish. Such a strategy has little economic or psychological sense to it. It is surmised that the reason that executives delay the release of bad information is to reduce or eliminate the negative overshoot effect. That is to say, they believe that the market will have correctly adjusted the share price after carefully assimilating the negative information over the weekend. The authors are not convinced that a similar psychology would apply to the treatment of good information; it is believed that executives would rejoice in a positive overshoot. It would put them in a good light. The following share price reversal could be attributed to day-to-day vagaries of the stock market. IV. Conclusions As reported elsewhere, the returns on the day prior to a holiday on the Standard & Poor s 500 stock index are considerably higher than the average rate of return on what can be called normal trading days in the pre-1987 era. The Vergin and McGinnis (1999) finding that the pre-holiday return has greatly diminished since 1987 is confirmed. Using the notion of the relative magnitude ratio, insights into the degree that the pre-holiday effect has declined are offered. The analysis shows that the decline in the pre-holiday rate of return, per se, accounts for about 30% of the decrease in the relative magnitude ratio. The remaining decline in the relative magnitude ratio is attributed to the increase in the typical daily return and to the interaction effect. Five new and important results are obtained in this study of the day-of-the-week effect in pre-holiday returns that have not been previously reported in the literature. First, two within-day effects are observed. They are the Labor Day effect in the pre-1987 era and the within-thursday effect in the post-1987 era. Both effects are absent in the complementary periods. Tentative explanations for the Labor Day effect are offered. The within-thursday effect has to be treated with caution due to the small sample sizes. Second, there is evidence to support the presence of a Wednesday versus Thursday between-day effect in the post 1987 era. The effect is absent before 1987.