The Halloween effect: Trick or treat?

International Review of Financial Analysis 19 (2010) 379–387

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International Review of Financial Analysis

The Halloween effect: Trick or treat? K. Stephen Haggard ⁎, H. Douglas Witte 1 Missouri State University, 901 S. National Ave., Springfield, MO 65897 United States

a r t i c l e

i n f o

Article history: Received 13 November 2009 Received in revised form 18 August 2010 Accepted 15 October 2010 Available online 23 October 2010 JEL classification: G10 G14

a b s t r a c t Research documents higher stock returns in November through April than for the rest of the year. This anomaly is known as the “Halloween effect” and results in the following trading rule: sell stocks in early May, invest in T-bills, and re-invest in stocks on Halloween. In contrast to recent studies, we show that the Halloween effect is robust to consideration of outliers and the “January effect.” Additionally, we show that investing in a “Halloween portfolio” provides risk-adjusted returns in excess of buy and hold equity returns even after consideration of transaction costs. © 2010 Elsevier Inc. All rights reserved.

Keywords: Anomalies Market efficiency Calendar

1. Introduction The “Halloween effect,” identified by Bouman and Jacobsen (2002), is an equity return anomaly in which the months of November through April provide higher returns than the remaining months of the year. This effect, if real, is perhaps of greater interest to investors than most other anomalies because the trading rule is simple to implement with low transactions costs, making exploitation of this anomaly potentially profitable. More recent studies posit that this anomaly might be driven by outliers or is simply the “January effect” in disguise. In this study, we examine the robustness of the Halloween effect to the consideration of outliers and the January effect. We also construct mean-variance efficient portfolios to determine whether investing in a Halloween portfolio can result in risk-adjusted returns superior to those of a buy-and-hold market portfolio. Finally, we examine the impact of transaction costs on the returns to investing in a Halloween portfolio. 2. Literature review In their seminal paper, Bouman and Jacobsen (2002) analyze stock returns across 37 countries from January 1970 through August 1998 and find a Halloween effect in 36 of these markets. This finding is remarkable in light of the adage “sell in May and go away” having appeared numerous times in the financial press before and during ⁎ Corresponding author. Tel.: + 1 417 836 5567; fax: +1 417 836 6224. E-mail addresses: [email protected] (K.S. Haggard), [email protected] (H.D. Witte). 1 Tel.: + 1 417 836 8478; fax: +1 417 836 6224. 1057-5219/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2010.10.001

their sample period. Most return anomalies disappear after discovery, presumably as opportunistic traders exploit them. The effect is particularly strong in European countries and is not the result of risk differences between the May–October and November–April timeframes that delineate the Halloween effect. Bouman and Jacobsen also demonstrate the economic significance of Halloween-based investment, even when transaction costs are considered. Bouman and Jacobsen's results for U.S. stock returns are more marginal. When the January effect is not considered, the Halloween effect attains statistical significance at the 5% level. After incorporating the January effect, significance falls just short of the 10% level they employ as a cutoff. However, the November–April period has a slightly smaller return standard deviation than the May–October period, adding to its attractiveness. Jacobsen and Visaltanachoti (2009) examine differences in the Halloween effect among U.S. stock market sectors and show that the effect is strongest for production sectors and weakest for defensive, consumer-oriented sectors. Maberly and Pierce (2004) examine monthly U.S. stock returns over the same 1970–1998 period as Bouman and Jacobsen. By treating the October 1987 (− 22.55%) and August 1998 (− 15.81%) returns as outliers, the authors purport to show the dependence of the Halloween effect on these two extreme returns. The effect is diminished and not statistically significant at any conventional level in an alternative specification that controls for these two observations. The authors do not provide an objective basis for identifying exactly two outliers and do not investigate the impact of considering additional outliers. Maberly and Pierce (2003) also examine the impact of outliers on the Halloween effect in Japanese equity markets. Galai, Kedar-Levy, and Schreiber (2008) also posit a relation between the Halloween effect and outliers. In contrast to the results of

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Maberly and Pierce (2004), Galai et al. (2008) find that, in daily S&P 500 returns, the Halloween effect is significant only after controlling for outliers. This difference in findings might be due to analyzing daily returns versus monthly, the different time period analyzed (1980– 2002 versus 1970–1998), or even the dropping of return observations from the sample. “Returns on non-consecutive days, other than weekend returns, are excluded, as they are not daily returns” (Galai et al., 2008, pp. 786–787). Outliers are an important aspect researchers have investigated as a possible source of the Halloween effect. In this study, we utilize a formal set of econometric techniques known as robust regression to determine the size and significance of the Halloween effect after controlling for extreme returns. The advantage of this approach is that it does not require any ad hoc specification of the number of outliers or the number of standard deviations from the mean an observation must be before it is considered an outlier. Rather, robust regression reduces the influence extreme returns have on the ordinary least squares (OLS) estimates in rough proportion to the departure of the observation from the regression fit. Our approach eliminates the possibility of data mining in the determination of the number of outliers for which to control. Lucey and Zhao (2008) examine U.S. stock data from 1926 to 2002 to determine the robustness of the Halloween effect to the consideration of the January effect, first identified by Wachtel (1942) and reinforced by Rozeff and Kinney (1976), in which equity returns are significantly higher in January than in other months. They find no evidence of a Halloween effect in their full sample. Using subperiod analysis, they show that neither the Halloween effect nor the January effect are consistently significant for value-weighted returns and that only the January effect is consistently significant for equal-weighted returns. The authors contend that the Halloween effect, when it does appear, might simply be the January effect in disguise. The subperiod findings of Lucey and Zhao (2008) are likely the result of the relatively short subperiods they examine, as we demonstrate in this study. Over three subperiods since 1946, their average estimate of the Halloween effect for the CRSP value-weighted index is a large 1.02% per month. However, because the sampling subperiods are small, the tests have reduced power and statistical significance is found only in the 1946–1965 period. We update the CRSP returns through 2008 and, with larger subperiods, find a Halloween effect over the most recent 55 years that it is significant and independent of the January effect. Like Lucey and Zhao, we find no evidence of a Halloween effect in the earliest subperiod we examine, 1926–1953. However, over both the 1954–1980 and the 1981–2008 subperiods, our evidence suggests a sizable Halloween effect that is independent of the January effect. This might explain the timing of the earliest reference to “sell in May and go away,” which appears in a 1964 issue of the Financial Times. In this study, we examine the robustness of the Halloween effect to outliers and the January effect over the period from 1926 to 2008. We also investigate how investment in a “Halloween portfolio,” which holds equities from November to April (excluding January) and Treasury bills the remainder of the year, might improve upon the Sharpe (1966) ratio attainable using a buy-and-hold investment strategy. 3. Sample and method We use monthly value-weighted and equal-weighted stock returns from the Center for Research in Security Prices (CRSP) over the period 1926–2008. We use the following regression model, identical to that of Lucey and Zhao (2008), in our examination: Rt = α + β1 Wt + β2 Jt +
t

ð1Þ

where Rt is the return on the index, Wt is the Halloween indicator, which has a value of “1” in the months from November to April and

“0” otherwise, and Jt is the January indicator, which has a value of “1” in January and “0” otherwise. In addition to using OLS regression, which is sensitive to outliers, we use the M-estimation techniques of Huber (1964) and Hampel (1974), which are more robust in the presence of outliers. OLS coefficient estimates are the solution to a sum of squared errors minimization problem. Each sample observation has an associated squared error, which receives the same weight, w(e) = 1, in the following minimization, min β ∑ Tt = 1w(et)e2t . The concept behind the M-estimators of Huber and Hampel is to dampen the influence of extreme errors by applying reduced weights to larger squared errors. Both estimators set w(e) = 1 for errors up to one or more threshold levels but reduce weights for errors beyond these levels. For example, the Huber estimator has a single threshold, kσ, and sets w(e) = 1 for |e| ≤ kσ and sets w(e) = kσ/|e| for |e| N kσ. The parameter k in the threshold level is referred to as a “tuning constant” and it is common practice in application to estimate the “scale factor” σ, ˆ = MAR = ð:6745Þ, where MAR is the median absolute error with σ from OLS. Aside from the different number of threshold levels, the primary differentiating point of the Hampel estimator is that it applies a finite rejection point, beyond which the observation is classified as an outlier and given zero weight. The estimation of the Huber and Hampel regressions can be viewed as a weighted least squares problem, minimizing the generalized sum of squared errors expression above. The solution to −1 this minimization is given by bˆ = ðX ′ WX Þ X ′ WY, where W is a T × T diagonal weight matrix with elements wtt = w(et). The weights, however, depend upon the errors, the errors depend upon the estimated coefficients, and the estimated coefficients depend upon the weights. An iterative solution, in this case iteratively reweighted least squares (IRLS), must be applied. To perform IRLS we start by first computing OLS estimates. Second, from the error terms, we calculate weights according to the weighting scheme specified by the particular M-estimator. Third, we solve for the new weighted least squares coefficient estimates. We repeat the second and third steps, using the error terms from the previous iteration, until the estimated coefficients converge. Upon convergence, we calculate an estimated asymptotic covariance matrix for the coefficients to determine their statistical significance (see Fox, 1997, for details). Our selection of tuning constants is based on the work of Hoaglin, Mosteller, and Tukey (1983), who study the effect of varying these constants for many different robust estimators. In general, a smaller tuning constant provides more resistance to outliers at the expense of lower efficiency in the case of normally distributed errors. We select mid-range values for the tuning constants in both the Huber and Hampel regressions. As noted above, we use the normalized median absolute deviation as the scale factor in the Huber regressions. As is common practice, we use the median absolute deviation as the scale factor in the Hampel regressions. We perform the OLS, Huber, and Hampel regressions for the entire U.S. sample period and for the subperiods of 1926–1953, 1954–1980, and 1981–2008. We also perform these regressions using a global sample of the same 37 countries examined by Bouman and Jacobsen (2002). To further demonstrate the impact of outliers on the Halloween effect, we use a deletion diagnostic method (Belsley, Kuh, & Welsch, 1980) to calculate the sensitivity of the estimated regression coefficient for the Halloween indicator to extreme observations. Last, in order to assess the investment significance of the Halloween effect, we form mean-variance efficient portfolios using the method of Britten-Jones (1999). Britten-Jones (1999) demonstrates that OLS regression can be applied to determine optimal portfolio weights. The Britten-Jones framework regresses a T-vector of ones on K independent variables representing the excess returns on K risky assets (or portfolios) considered for investment from time 1 to T.

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The vector of ones is representative of a portfolio with an infinite Sharpe ratio—returns have a mean of one and a variance of zero. The regression does not include an intercept and finds the combination of risky assets that comes closest to the Sharpe ratio of this ideal portfolio. The regression coefficient estimates represent optimal portfolio weights for the K risky assets. The novelty of this approach is that estimation and standard inference procedures on optimal portfolio weights can be conducted in the same way as for the coefficients in an OLS regression, allowing us to easily determine whether a particular asset is a statistically significant component of a mean-variance efficient portfolio constructed from a given set of assets. In the mean-variance analysis, we select from three possible funds to construct optimal portfolios. Our regression equation is as follows: 1 = β1 rm;t + β2 rh;t + β3 rj;t + ε t

ð2Þ

where r m,t is the return to the “buy-and-hold fund,” which continuously invests in the market portfolio, rh,t is the return to the “Halloween fund,” which invests in the market portfolio in the months November through April (excluding January), T-bills otherwise, and rh,t is the return to the “January fund,” which invests in the market portfolio in January and T-bills otherwise. Given this regression framework, the sample mean-variance efficient portfolio will call for βˆ 1 to be invested in the market portfolio during May through October, βˆ 1 + βˆ 2 to be invested in the market portfolio during November through April (excluding January), and βˆ 1 + βˆ 3 to be invested in the market portfolio during January. All remaining funds are invested in T-bills. We scale the coefficient estimates such that their sum is equal to unity and report them as portfolio weights. However, since the Halloween fund and the January fund are never invested in the market portfolio at the same time, the maximum proportion of the portfolio that will ever be invested in the market portfolio will be the greater of ˆ 2 or β ˆ1+β ˆ 3 (assuming that one or both βˆ 2 and βˆ 3 are positive). ˆ1+β β If the larger sum is less than 1, the returns to the portfolio can be increased to the maximum achievable without leverage by taking the portion of the portfolio that is continuously invested in T-bills and, instead, investing that portion in the market portfolio. For example, suppose βˆ 1 = 0.5, βˆ 2 = 0.3, and βˆ 3 = 0.2. The portfolio invests 50% in the market during May through October, 80% in the market during November through April (excluding January), and 70% in the market during January. In reporting the performance statistics of the portfolio, we would multiply the portfolio weights by 1/0.8 = 1.25. This is equivalent to moving “farther out” along the capital allocation line afforded by the three funds. This step proportionally increases both the portfolio return and standard deviation but keeps the relative weightings constant and gives the maximum portfolio return achievable without leverage. The increased returns that result from this rescaling have no impact on the Sharpe ratio but do impact the manipulationproof performance measure (MPPM) of Goetzmann, Ingersoll, Spiegel, and Welch (2007). The manipulation-proof performance measure, which typically gives less consideration to risk than the Sharpe ratio, can be materially different for two portfolios with the same Sharpe ratio. We perform these portfolio exercises to examine the economic importance of the Halloween effect relative to the January effect and to demonstrate any potential improvements in risk-adjusted returns that might be achieved by holding a Halloween portfolio. To explore the potential profitability of the Halloween strategy, we repeat this exercise including hypothetical transaction costs.

Fig. 1. Value-weighted and equal-weighted CRSP returns in percent by month. Monthly returns data are for 1954–2008.

alongside the CRSP equal-weighted returns for the period 1954–2008. If the Halloween effect is real, monthly returns for November through April should be higher than monthly returns for May through October. The value-weighted returns displayed in Fig. 1 are consistent with the Halloween effect return pattern. The equal-weighted returns are also consistent with this pattern, but the return for January towers above the returns for other months in this series. Keim (1983) documents higher returns for small firms than for large firms and reports that approximately 50% of this size effect occurs in January. Given that small firms receive greater weighting under an equal weighting scheme than under a value weighting scheme, it is not surprising to see a strong January effect in the equal-weighted returns. However, the same effect is not apparent in the value-weighted returns. Next, we examine monthly returns by subperiod. Fig. 2 displays value-weighted returns for the 1954–2008 subperiod, as well as the subperiods of 1954–1980 and 1981–2008. The Halloween effect is visible in all three periods, but the return patterns are not entirely consistent between subperiods. For example, July returns are positive

4. Results We begin our investigation of the Halloween effect by examining the monthly returns on the CRSP value-weighted portfolio, which we use as a proxy for the market portfolio. Fig. 1 shows these returns

Fig. 2. Value-weighted CRSP returns in percent by month. Monthly returns data for are 1954–2008, 1954–1980, and 1981–2008.

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and large in magnitude in the 1954–1980 subperiod but are negative in the 1981–2008 subperiod. Note that January value-weighted returns are similar in magnitude to November and December returns in all three periods. We use OLS regression for a formal statistical examination of returns over the larger 1926–2008 period. Table 1 presents the results of regressions performed using model (1). β1 is the estimated coefficient on the Halloween indicator and β2 is the estimated coefficient on the January indicator. In column 1, we exclude the January indicator from the model. In column 2, we include both the Halloween indicator and the January indicator. We perform these regressions using both value-weighted and equal-weighted CRSP returns. In column 1 for value-weighted returns, the coefficient on the Halloween indicator is positive and significant at the 10% level. Returns are 58.7 basis points higher in the months of November–April than over the rest of the year, consistent with a Halloween effect. However, once we include the January indicator in column 2, the coefficient on the Halloween indicator remains positive but loses significance. The coefficient on the January indicator is not significant for this regression. For equal-weighted returns, we find a positive and significant coefficient on the Halloween indicator in column 1, which loses significance after the introduction of the January indicator. In the case of equal-weighted returns, the January indicator coefficient is positive and significant in column 2. The results in Table 1 are, at first glance, consistent with the notion that the Halloween effect in the U.S. is driven by the January effect. Bouman and Jacobsen (2002) first identify the Halloween effect in the period 1970–1998. Lucey and Zhao (2008) study the persistence of the Halloween effect over time and find the effect is independently significant of the January effect in only the 1946–1965 subperiod. The differences between the findings of these two studies are suggestive of time variation in the Halloween effect, as well as the reduced power of statistical tests in smaller subperiods. As Sauer, Brajer, Ferris, and Marr (1988) point out, “splitting one large sample into many small samples makes for a less powerful test” (p. 207). To study these issues, we break our sample into three subperiods: 1926–1953, 1954–1980, and 1981–2008. The fact that we have six more years of return data and break the whole sample into three subperiods rather than the four in Lucey and Zhao (2008) both positively impact the power of our tests. Table 2 presents OLS regressions of model (1) over these three subperiods. Columns 1 and 2 in this table are identical to columns 1 and 2 in Table 1, with the January indicator included only in column 2. For the subperiod 1926–1953, we find no significant Halloween effect or January effect in the value-weighted returns. Using equalweighted returns, we find no Halloween effect for this subperiod, but we do document a positive and significant January effect, consistent

Table 1 Regressions of monthly returns on indicators for the Halloween effect (β1) and the January effect (β2). U.S. monthly returns data are for 1926–2008. (t-statistics in parentheses, returns in basis points)

Intercept β1

Value-weighted index

Equal-weighted index

1

2

1

2

60.1 (2.466)** 58.7 (1.702)*

60.1 (2.466)** 49.0 (1.356) 57.8 (0.885) 0.0037 0.0017

141.7 (4.111)*** 122.6 (2.516)**

141.7 (4.161)*** 45.8 (0.908) 460.6 (5.042)*** 0.0311 0.0292

β2 R2 Adjusted R2

0.0029 0.0019

0.0063 0.0053

β1 is the estimated coefficient on the Halloween indicator, which has a value of “1” in the months of November–April, “0” otherwise. β2 is the estimated coefficient on the January indicator, which has a value of “1” in January, “0” otherwise. The symbols ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.

Table 2 Regressions of monthly returns on indicators for the Halloween effect (β1) and the January effect (β2). U.S. monthly returns data are for 1926–1953, 1954–1980, and 1981–2008. (t-statistics in parentheses, returns in basis points) Value-weighted index

Equal-weighted index

1

1

2

289.9 (3.457)*** − 27.5 (− 0.232)

289.9 (3.482)*** − 117.1 (− 0.948) 537.3 (2.405)** 0.0172 0.0113

2

A. Regressions for the subperiod 1926–1953 Intercept 106.0 106.0 (1.918)* (1.917)* − 39.1 − 61.7 β1 (−0.501) (− 0.754) 135.4 β2 (0.913) 2 0.0007 0.0032 R − 0.0022 − 0.0027 Adjusted R2 B. Regressions for the subperiod 1954–1980 Intercept 34.1 34.1 (1.049) (1.047) 120.9 117.6 β1 (2.629)*** (2.434)** 20.0 β2 (0.229) R2 0.0210 0.0212 0.0180 0.0151 Adjusted R2 C. Regressions for the subperiod 1981–2008 Intercept 39.3 39.3 (1.144) (1.047) 96.4 93.6 β1 (1.986)** (1.836)* β2 16.8 (0.182) 0.0117 0.0118 R2 2 0.0087 0.0058 Adjusted R

0.0002 − 0.0028

54.0 (1.261) 205.2 (3.391)***

0.0345 0.0315

78.0 (1.926)* 193.1 (3.372)***

0.0329 0.0300

54.0 (1.294) 125.5 (2.030)** 478.2 (4.273)*** 0.0864 0.0807

78.0 (1.957)* 131.9 (2.232)** 367.1 (3.433)*** 0.0656 0.0604

β1 is the estimated coefficient on the Halloween indicator, which has a value of “1” in the months of November–April, “0” otherwise. β2 is the estimated coefficient on the January indicator, which has a value of “1” in January, “0” otherwise. The symbols ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.

with the role that small stocks play in the January effect. For the subperiod 1954–1980, we find a positive and significant Halloween effect for value-weighted returns, which does not lose significance once the January indicator is included. For these returns, the Halloween effect is significant but the January effect is not, in contrast to the findings of Lucey and Zhao (2008). In the equal-weighted returns regressions, the Halloween effect is positive and significant, remaining so after the inclusion of the January indicator. As expected for equal-weighted returns, the coefficient on the January indicator is also positive and significant for this subperiod. For the subperiod 1981–2008, we find a similar pattern to the 1954–1980 subperiod. The Halloween effect is positive and significant in both valueweighted return regressions, but the January effect does not attain significance. When we examine this period using equal-weighted returns, we obtain positive and significant results for the Halloween effect in both models, indicative of a significant January effect that does not subsume the Halloween effect. Thus, the Halloween effect is present in both the 1954–1980 and 1981–2008 subperiods and is not subsumed by the January effect as posited by Lucey and Zhao (2008). The results in Table 2 suggest that a significant Halloween effect has persisted in the U.S. over the past 55 years independent of the January effect. Next, we turn our attention to the impact of outliers on the Halloween effect. Maberly and Pierce (2004) contend that the Halloween effect disappears after controlling for the two biggest outliers in value-weighted returns. For both value- and equalweighted returns over the 1954–2008 period, we compute an OLS influence vector for the observations. An influence vector measures the influence of an observation by calculating OLS coefficient

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estimates with that observation omitted. The observation omissions that most impact the estimates (relative to the estimates with the observation included) are deemed “influential observations” or outliers. The influence vector we compute measures the change in the coefficient estimate on the Halloween indicator in regressions of model (1). In Table 3 we report the ten most influential return outliers in descending order of their influence on the coefficient estimate on the Halloween indicator in regressions of model (1). At first glance, it is apparent that nearly all of these outliers are negative returns. For both value-weighted and equal-weighted returns, 9 of the 10 biggest outliers are negative. This result is not unusual. Cutler, Poterba, and Summers (1989) document that 8 of the 10 biggest one-day movements in the S&P 500 index from 1946 to 1987 have been negative. Campbell and Hentschel (1992) provide a theoretical explanation for this phenomenon. The arrival of news creates volatility, which depresses stock prices. When good news arrives, its positive effect on returns is partially offset by this volatility effect. However, when bad news arrives, its negative effect on returns is magnified by the volatility effect. As such, most outliers are negative. When evaluating the impact of outliers on the Halloween effect, it is important to recognize that deleting a negative (or positive) outlier can either increase or decrease the magnitude of the effect, depending on which month contains the outlier. Removing a negative October outlier reduces the magnitude of the Halloween effect, but removing a negative April outlier increases the magnitude of the Halloween effect. Furthermore, positive and negative outliers occurring in any given period, excluding January, have an equivalent, albeit opposite, impact on the Halloween effect estimate as long as the outliers represent an equivalent departure from average returns. For value-weighted returns we identify the biggest outlier, like Maberly and Pierce (2004), as October 1987 at − 22.55%. Omitting this observation would reduce the Halloween coefficient by 6.96 basis points from its original value of 105.39 basis points for this sample period. The second greatest outlier is October 2008, which has a return of −18.42%. Omitting this outlier reduces the Halloween coefficient by 5.71 basis points. The final column in Table 3 shows the

Table 3 Impact of outliers on the Halloween effect. Change in β1 resulting from omitting outliers in descending order of magnitude. U.S. monthly returns data are for 1954–2008. Month

Year

Outlier return

Outlier omitted

Cumulative

A. Value-weighted index returns. β1 = 105.39 basis points in the original regression over 1954–2008 1 October 1987 −22.55% −6.96 − 6.96 2 October 2008 − 18.42% − 5.71 −12.72 3 November 1973 − 12.13% 4.94 −7.77 4 October 1974 16.58% 4.93 − 2.85 5 August 1998 − 15.81% − 4.92 − 7.84 6 March 1980 − 12.04% 4.91 − 2.89 7 April 1970 − 10.54% 4.37 1.55 8 November 2000 − 10.32% 4.28 5.93 9 February 2001 − 9.95% 4.15 10.21 10 November 2008 − 8.40% 3.58 13.95 B. Equal-weighted index returns. β1 = 128.78 basis 1954−2008 1 October 1987 −25.20% 2 November 1973 − 17.45% 3 March 1980 − 16.14% 4 April 1970 −16.00% 5 August 1998 − 18.99% 6 October 2008 − 17.94% 7 October 1978 − 17.87% 8 November 2000 − 11.46% 9 February 1991 15.05% 10 November 2008 −11.04%

points in original regression over −7.86 7.08 6.60 6.55 − 5.97 − 5.65 − 5.63 4.89 −4.78 4.74

− 7.86 − 0.78 5.87 12.52 6.50 0.77 − 4.97 0.05 − 4.70 0.20

β1 is the estimated coefficient on the Halloween indicator from model (1), which has a value of “1” in the months of November–April, “0” otherwise.

383

cumulative effect of omitting the outliers. Omitting the first two outliers reduces the Halloween coefficient by 12.72 basis points. However, the next two largest outliers impact the Halloween coefficient in a positive way. If we discard the return of − 12.13% in November 1973, the Halloween coefficient increases by 4.94 basis points. If we also discard the return of 16.58% in October 1974, the Halloween coefficient increases by another 4.93 basis points, bringing the cumulative change in the Halloween coefficient to just −2.85 basis points. The impact of omitting outliers 5–10 actually causes the Halloween coefficient to become greater than its original value. A similar pattern emerges for the equal-weighted returns, where the cumulative effect of omitting outliers bounces back and forth from being negative to being positive. The alternating results of omitting outliers demonstrated in Table 3 raise a question: How many outliers are appropriate to omit? Surely any answer to this question would be subjective. A more objective analytical approach would include all of the data but limit the ability of outliers to unduly influence the results. In Table 4, we use two such robust regression methods to investigate the Halloween effect. First, we use the Huber (1964) M-estimation technique, the most common general method of robust regression, and then apply the redescending estimator of Hampel (1974). Both of these techniques limit the influence of outliers on regression results without arbitrarily selecting outliers ex ante for omission. Table 4 presents the results of OLS, Huber, and Hampel regressions of model (1) for both value- and equal-weighted returns over the 1954–2008 period, as well as the subperiods of 1954–1980 and 1981– 2008. For value-weighted returns over 1954–2008, the Halloween effect is positive and significant in all three types of regression. Using the coefficient from the Huber regression, returns in the months November through April (excluding January) are, on the average, 96.7 basis points higher than the returns of from May to October. Over this period, the Halloween effect is robust to the consideration of outliers and to the January effect, which does not achieve significance in any of these regressions. Equal-weighted returns over the same period display a significant Halloween effect but also display a significant January effect. Once again, this is due to the greater weight placed on small stocks in equal-weighted returns. An identical pattern of significance occurs for the subperiod 1954–1980, and a similar pattern exists over the subperiod 1981–2008. The only difference between the 1954–1980 results and the 1981–2008 results is the lack of significance of the Halloween coefficient in the Hampel regression of equal-weighted returns. The results presented in Table 4 reinforce the robustness of the Halloween effect to both outliers and the January effect over the 1954–2008 time period. We perform, but do not report in a table, robust regressions of model (1) over the entire 1926–2008 sample period. Using the Huber regression, the Halloween effect is significant at the 10% level for value-weighted returns and significant at the 5% level for equalweighted returns. The Hampel regression results are qualitatively similar. These results are the robust regression analogue of the OLS full sample results reported in column 2 of Table 1. There, when we control for the January effect but not for outliers, the Halloween effect is not significant. This contrast between the OLS and robust regression results suggests that, over the entire 1926–2008 sample period, outliers have the net effect of obscuring the Halloween effect in OLS regressions. Outliers appear to be the primary reason for the lack of significance in our full sample OLS results as well as those reported by Lucey and Zhao (2008) for their full 1926–2002 sample period. This does not imply, however, the Halloween effect is pervasive over the entire sample. Our subperiod analysis, as well as that of Lucey and Zhao, strongly indicates that the Halloween effect is not present in the earliest part of the sample. To this point, our analysis has focused solely on the U.S. stock market. However, Bouman and Jacobsen (2002) document the Halloween effect using MSCI data from around the world. In the

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Table 4 Robust regressions of U.S. monthly returns on indicators for the Halloween effect (β1) and the January effect (β2). Monthly returns data are for 1954–2008, 1954–1980, and 1981– 2008. (t-statistics in parentheses, returns in basis points) Value-weighted index OLS

Equal-weighted index Huber

Hampel

OLS

Huber

Hampel

A. Regressions for the subperiod 1954–2008 Intercept 36.7 (1.553) 105.4 β1 (3.003)*** 18.4 β2 (0.289)

57.2 (2.590)*** 96.7 (2.950)*** − 10.5 (−0.177)

70.2 (3.225)*** 90.9 (2.817)*** − 10.6 (− 0.181)

66.2 (2.301)** 128.8 (3.018)*** 421.6 (5.461)***

92.6 (3.465)*** 117.2 (2.957)*** 349.1 (4.868)***

108.3 (4.089)*** 105.3 (2.681)*** 318.4 (4.482)***

B. Regressions for the subperiod 1954–1980 Intercept 34.1 (1.047) 117.6 β1 (2.434)** 20.0 β2 (0.229)

46.2 (1.507) 115.8 (2.548)** − 15.5 (−0.189)

58.3 (1.835)* 112.6 (2.388)** − 15.1 (− 0.177)

54.0 (1.294) 125.6 (2.030)** 478.2 (4.273)***

69.3 (1.759)* 136.8 (2.340)** 359.5 (3.400)***

76.1 (2.025)** 143.3 (2.573)** 311.9 (3.095)***

C. Regressions for the subperiod 1981–2008 Intercept 39.3 (1.143) 93.6 β1 (1.836)* 16.8 β2 (0.182)

67.5 (2.135)*** 77.5 (1.651)* − 42.8 (−0.050)

80.9 (2.689)*** 69.3 (1.554) − 4.6 (− 0.056)

78.0 (1.957)* 131.9 (2.232)** 367.1 (3.433)***

113.5 (3.109)*** 100.5 (1.857)* 338.3 (3.453)***

138.0 (3.704)*** 69.8 (1.263) 324.9 (3.250)***

β1 is the estimated coefficient on the Halloween indicator, which has a value of “1” in the months from November to April, “0” otherwise. β2 is the estimated coefficient on the January indicator, which has a value of “1” in January, “0” otherwise. Huber (1964) M-Estimation is performed with k = 1.5. Hampel (1974) estimation is performed with a = 1.5, b = 3.6 and c = 8. The symbols ***, **, and * denote significance at the 1, 5, and 10 percent levels respectively.

interest of completeness, we now analyze the impact of outliers on the Halloween effect in stock markets around the world. Table 5 presents the β1 coefficient estimates from the OLS, Huber, and Hampel regressions of model (1) for MSCI equity returns (including dividends) from 37 countries over the 1970–2008 period. The earliest data available in the MSCI set is from 1970, and 18 of the countries (including the U.S.) have data for the entire 39-year sample period, 17 have data for the last 21 years of the sample period, while South Africa and Russia have data for the last 16 and 14 years, respectively. The countries in Table 5 are arranged in descending order of t-statistic for the β1 coefficient estimated using OLS. Using the OLS estimate, the Halloween effect is significant in 22 of 37 countries. Note that the United States, the primary focus of our paper, is solidly in the middle of the pack by statistical significance of the Halloween effect coefficient estimate. The average estimate of the Halloween effect for these 37 countries using OLS is 174.5 basis points. Using the Huber estimates results in a small reduction to 18 countries that display a statistically significant Halloween effect. However, the impact of outliers on statistical significance is not uniform across countries. For example, Malaysia and Indonesia lose significance altogether, even though their Halloween effects were significant at the 5% level using OLS. Finland's Halloween effect, which did not attain statistical significance with OLS, attains it at the 10% level using the Huber estimates. The Halloween effect in the United States maintains significance using the Huber estimate, but the estimate is reduced from 85.1 basis points to 72.5 basis points, a 14.8% decrease versus OLS. The average estimate of the global Halloween effect falls to 129.3 basis points, a 25.9% decrease versus OLS. Using the Hampel estimates, we find a significant Halloween effect in 15 of 37 countries. The pattern of significance for the Hampel estimates is more predictable than the pattern for the Huber estimates. No country without a significant OLS Halloween effect estimate has a significant Hampel estimate. The Halloween effect in the United States maintains significance using the Hampel estimate, but the estimate is further reduced to 63.5 basis points, a 25.4%

decrease versus OLS. The average estimate of the global Halloween effect falls to 104.9 basis points, a 39.9% decrease versus OLS. We reach the same conclusion from the Huber and Hampel results. Outliers impact the Halloween effect more globally than they do in the United States, but the Halloween effect is still significant in many parts of the world after controlling for outliers. In our final exercise of the study, we examine whether the Halloween effect can be exploited for profit and whether the January effect might subsume such profit. We form mean-variance efficient portfolios using the regression-based method of Britten-Jones (1999). The attractive features of the Britten–Jones framework are that the scaled coefficient estimates represent the portfolio weightings, which create the most efficient (highest Sharpe ratio) portfolios possible and the t-statistics for the coefficient estimates allow us to infer whether the weightings are significantly different from zero. We consider three investment options. One option is a buy-andhold fund that continuously holds all CRSP stocks. The second option is a Halloween fund that holds equities during November through April (except January) and Treasury bills otherwise. The third option is a January fund that holds equities during January and Treasury bills otherwise. We perform this exercise to examine the economic significance of the Halloween effect relative to the January effect and to demonstrate any potential improvements in risk-adjusted returns that might be earned by investing in the Halloween fund. Table 6 presents the results. Column 1 represents investment of the entire portfolio in the buy-and-hold fund (represented by the value-weighted returns on all CRSP stocks). Column 2 represents potential investment in two funds, the buy-and-hold fund and the Halloween fund. Column 3 represents potential investment in two funds, the buy-and-hold fund and the January fund. Column 4 represents potential investment in all three funds: the buy-and-hold fund, the Halloween fund, and the January fund. To explore the potential profitability of the Halloween strategy, we repeat the column 4 exercise, including transaction costs of 1% per year (spread evenly over each month of the year) for the Halloween fund and

K.S. Haggard, H.D. Witte / International Review of Financial Analysis 19 (2010) 379–387 Table 5 Robust Regressions of Monthly Returns on Indicators for the Halloween Effect (β1). Data for N most recent Monthly Returns ending December 2008.

385

Table 6 Maximum Sharpe ratio portfolios composed of the buy-and-hold fund, the Halloween fund, and the January fund. U.S. monthly returns data are for 1954–2008.

(t-statistics in italics, returns in local currencies in percentages)

(t-statistics in parentheses, returns and MPPMs in basis points)

Countries

N

OLS

OLS t

Huber

Huber t

Hampel

Hampel t

Belgium Austria Netherlands France Sweden Italy Japan United Kingdom Germany Spain Taiwan Ireland South Africa Switzerland Canada Malaysia Indonesia United States Portugal Russia Brazil Singapore Greece Finland Australia Norway Thailand Mexico Korea Turkey Chile Phillipines Argentina New Zealand Hong Kong Jordan Denmark Average Estimate

468 468 468 468 468 468 468 468 468 468 252 252 192 468 468 252 252 468 252 168 252 468 252 252 468 468 252 252 252 252 252 252 252 252 468 252 468

196.4 214.2 186.5 200.9 216.9 211.5 163.0 169.7 158.3 163.5 352.4 198.0 211.0 97.0 97.2 209.9 307.3 85.1 153.0 485.5 396.1 128.7 207.6 199.3 81.3 101.8 187.0 134.6 141.3 221.3 89.5 105.2 223.7 54.6 56.7 37.3 12.7 174.5

3.83*** 3.83*** 3.69*** 3.56*** 3.45*** 3.31*** 3.14*** 3.11*** 2.86*** 2.85*** 2.61*** 2.40** 2.29** 2.06** 2.03** 2.00** 1.98** 1.97** 1.91* 1.71* 1.69* 1.66* 1.64 1.62 1.42 1.41 1.31 1.25 1.16 1.10 1.07 0.94 0.80 0.73 0.58 0.52 0.26

137.2 129.4 161.8 159.7 167.2 187.0 143.8 146.7 114.0 115.2 244.3 160.4 158.5 70.9 92.0 120.1 180.7 72.5 140.8 365.6 306.3 62.3 136.2 196.0 47.6 68.2 63.9 129.2 126.4 171.8 47.1 62.8 201.0 26.1 52.7 38.2 − 17.8 129.3

3.41*** 3.25*** 3.75*** 2.98*** 2.92*** 3.08*** 3.00*** 3.19*** 2.37** 2.22** 1.99** 2.23** 1.89* 1.69* 2.17** 1.34 1.48 1.84* 2.01** 1.58 1.56 1.05 1.27 1.80* 1.00 1.03 0.52 1.21 1.14 0.92 0.61 0.60 1.25 0.37 0.66 0.60 − 0.38

107.3 92.3 156.8 143.5 143.1 167.7 130.4 132.2 94.0 86.7 212.4 156.7 124.8 55.0 85.7 75.6 139.9 63.5 136.8 288.7 236.4 41.3 100.6 173.0 34.3 54.4 4.6 126.4 125.4 118.0 28.8 34.9 147.7 3.0 40.4 49.0 −28.7 104.9

2.83*** 2.67*** 3.72*** 2.57** 2.59*** 2.72*** 2.83*** 3.05*** 1.94* 1.72* 1.84* 2.21** 1.46 1.33 2.05** 0.94 1.24 1.65* 2.08** 1.35 1.36 0.81 1.07 1.63 0.73 0.80 0.04 1.24 1.13 0.61 0.37 0.34 1.00 0.04 0.55 0.79 − 0.65

1

2

3

A. Portfolio weights. Value-weighted index returns Market 1.000 0.107 0.505 (0.624) (2.562)** Halloween 0.893 (3.081)*** January 0.495 (0.838) Return 51.0 43.4 30.8 Std. Dev. 434.0 257.9 252.5 Sharpe Ratio 0.117 0.168 0.122

MPPM (ρ = 3)

22.2

33.4 p = 0.406 (2 vs. 1)

21.2

B. Portfolio weights. Equal-weighted index returns Market 1.000 0.486 0.448 (3.557)*** (4.148)*** Halloween 0.514 (2.193)** January 0.552 (2.385)** Return 125.7 94.2 82.9 Std. Dev. 544.9 382.1 332.3 Sharpe ratio 0.231 0.247 0.249

MPPM (ρ = 4)

65.8

64.7 p = 0.910 (2 vs. 1)

64.0

4

5

− 0.015 (− 0.139) 0.578 (3.423)*** 0.437 (1.707)* 50.2 276.8 0.181

0.101 (0.686) 0.532 (2.254)** 0.367 (1.032) 34.0 281.0 0.148 F = 2.61 p = 0.074 (5 vs. 1) 29.7

38.6 p = 0.036 (4 vs. 3)

0.077 (0.932) 0.421 (3.368)*** 0.502 (3.497)*** 105.4 371.9 0.284

78.8 p = 0.027 (4 vs. 3)

0.156 (1.627) 0.354 (2.421)** 0.490 (2.927)*** 96.1 363.4 0.264 F = 5.20 p = 0.057 (5 vs. 1) 70.8

β1 is the estimated coefficient on the Halloween indicator, which has a value of “1” in the months from November to April, “0” otherwise. Huber (1964) M-Estimation is performed with k = 1.5. Hampel (1974) estimation is performed with a = 1.5, b = 3.6 and c = 8. The symbols ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively. Significance Level: 1% 5% 10% OLS: 11 7 4 Huber: 8 6 4 Hampel: 7 4 4

The buy-and-hold fund is composed of all stocks in CRSP. The Halloween fund holds the buy-and-hold fund from November to April (excluding January) and 3-month T-bills at all other times. The January fund holds the buy-and-hold fund during January and 3-month T-bills at all other times. Column 1 represents investing in the buy-and-hold fund continually. Column 2 represents the portfolio weights for the maximum Sharpe ratio portfolio when only the buy-and-hold fund and the Halloween fund are available. Column 3 represents the portfolio weights for the maximum Sharpe ratio portfolio when only the buy-and-hold fund and the January fund are available. Column 4 represents the portfolio weights for the maximum Sharpe ratio portfolio when the buy-and-hold fund, the Halloween fund, and the January fund are available. Column 5 is identical to column 4, except for the consideration transaction costs of one percent per year (spread evenly over each month of the year) for the Halloween portfolio and transaction costs of 0.25% per year for the January portfolio. Calculations follow Britten-Jones (1999). MPPM is the manipulation-proof performance measure of Geotzmann et al. (2007). The symbols ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.

transaction costs of 0.25% per year for the January fund and present the results in column 5. For each column of Table 6, given the fund choices available, we calculate the optimal portfolio weights, the excess return of the portfolio, the standard deviation of the return, and the Sharpe ratio. Using value-weighted returns, the buy-and-hold fund has excess returns of 51 basis points per month and standard deviation of 434 basis points, resulting in a Sharpe ratio of 0.117. The mean-variance efficient portfolio composed of the buy-and-hold fund and the Halloween fund would invest 10.7% of the portfolio in the buy-andhold fund and 89.3% in the Halloween fund. The resulting portfolio would have an excess return of 43.4 basis points per month, which is less than the return on the market portfolio. However, the standard deviation of such a portfolio would drop to 257.9 basis points, increasing the Sharpe ratio to 0.168—a material improvement over holding only the buy-and-hold fund. Britten-Jones (1999) demonstrates that the p-value for an F-test of the differences between the Sharpe ratios of two portfolios, which differ by only one portfolio component (in this case, the Halloween fund) is the same as the p-value for the t-statistic of the

regression coefficient representing the portfolio weight of the additional component. Since the t-value of 3.081 on the Halloween fund weight in column 2 is significant at the 1% level, we can also state that the Sharpe ratio for the column 2 portfolio is significantly greater than the Sharpe ratio for the column 1 portfolio at the 1% level. In column 3, we restrict the fund selections to the buy-and-hold fund and the January fund. Such a portfolio would be roughly evenly split between the two funds, although the weight invested in the January fund is not significantly different from zero (t = 0.838). The return on this portfolio is 30.8 basis points with a standard deviation of 252.5 basis points, resulting in a Sharpe ratio of 0.122. Based on the t-statistic for the January fund portfolio weight, this Sharpe ratio is not significantly different than that of the buy-and-hold fund by itself. If we include both the Halloween fund and the January fund as investment options in addition to the buy-and-hold fund, the optimal portfolio would actually short the buy-and-hold fund by holding −1.5% and hold 57.8% of the portfolio in the Halloween fund and 43.7% of the portfolio in the January fund. Note that, if the Halloween effect were merely the January effect in disguise, this portfolio would

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invest primarily in the January fund and not in the Halloween fund. As discussed in the Sample and Methods section, we scale up the returns and standard deviations on portfolios containing both the Halloween and January fund to account for the fact that such a portfolio is never completely invested in the market portfolio, as the Halloween and January funds are never invested in the market portfolio at the same time. Similar to the market portfolio, the excess return on this portfolio is 50.2 basis points per month, but the standard deviation is far lower at 276.8 basis points per month, resulting in a Sharpe ratio of 0.181. Once again, we find that the Sharpe ratio for the portfolio containing the Halloween fund (column 4) is significantly higher than for the portfolio excluding this fund (column 3), given that the t-value for the portfolio weight placed on the Halloween fund in column 4 is significant at the 1% level. Finally, we include the transaction costs of 1% per year (spread evenly over each month of the year) for the Halloween fund and transaction costs of 0.25% per year for the January fund to determine the robustness of our results to transaction costs. Doing so increases the attractiveness of the buy-and-hold fund, which bears no transaction costs since it is continuously invested in the market. Such a portfolio would invest 10.1% in the buy-and-hold fund, 53.2% in the Halloween fund, and 34.0% in the January fund. Excess returns of the portfolio are 34 basis points per month with a standard deviation of 281 basis points. The resulting Sharpe ratio is 0.148, which is 21% greater than the Sharpe ratio for the buy-and-hold fund. Note that, for all portfolios that allow for investment in the Halloween fund, the positive portfolio weightings on the Halloween fund are highly significant. We determine the statistical significance of the difference between the Sharpe ratio of this portfolio and the Sharpe ratio of the buy-and-hold fund by using the F-test specified by Britten-Jones (1999). Doing so results in an F2,658 of 2.61, which has a p-value of 0.074. Thus, using value-weighted returns, the Sharpe ratio for a mean-variance efficient portfolio composed of all three funds is significantly greater than the Sharpe ratio for the buy-and-hold fund at the 10% level even after consideration of transaction costs. The results for the equal-weighted returns are similar. As one would expect, the January fund is more heavily weighted under equalweighted returns due to the inordinate contribution of small stocks to the January effect. Small stocks receive greater weight under an equal weighting scheme. However, the January fund does not totally dominate the Halloween fund, which still accounts for 35.4% of the portfolio after consideration of transaction costs. The Sharpe ratio of the column 5 portfolio exceeds the Sharpe ratio of the buy-and-hold fund by 14.29% using equal-weighted returns. As is the case for valueweighted returns, Halloween fund weightings are highly significant for equal-weighted returns. The F-test for the significance of the difference in Sharpe ratios between column 5 and column 1 has a p-value of 0.057. Thus, using equal-weighted returns, the Sharpe ratio for a mean-variance efficient portfolio composed of all three funds is significantly greater than the Sharpe ratio for the buy-and-hold fund at the 10% level even after consideration of transaction costs. To further demonstrate the beneficial impact of investing in the Halloween fund over and above investment in the buy-and-hold fund and the January fund, we calculate the manipulation-proof performance measure (MPPM) of Goetzmann, Ingersoll, Spiegel, and Welch (2007) for all the portfolios considered in Table 6. The primary reason for this analysis is the possibility that investment in the Halloween fund, which stipulates investment in T-bills 7 months out of the year, distorts the underlying portfolio return distribution in such a way that it deviates from normality. If this is the case, the Sharpe ratio can conceivably be manipulated upward in a manner which does not add value to a fund's investor. The manipulation-proof performance measure estimate for portfolio i is given by ˆi = Θ

 
i1−ρ  1 1 T h
ln ∑t = 1 1 + ri;t = 1 + rf ;t ð1−ρÞ T

ð3Þ

where ri,t is the return to portfolio i during month t and rf,t is the riskˆ i statistic represents the free return over the same period. The Θ monthly excess return certainty equivalent provided by the portfolio to an investor with constant relative risk aversion equal to ρ. Guided by Goetzmann et al. (2007), we use a value of 3 for the parameter ρ when examining value-weighted index returns and a value of 4 when examining equal-weighted index returns. To derive a consistent estimator for the standard error of the difference in MPPMs, we apply the delta method and a heteroskedasticity and autocorrelation robust (HAC) kernel estimator of the covariance matrix. The specific kernel we use is a quadratic-spectral kernel with automatic bandwidth selection (see Andrews, 1991, for details). Ledoit and Wolf (2008) show that this HAC estimator is more robust to common features of real-world data than are methods similar to that of Jobson and Korkie (1981), which assume the data are normally distributed. We calculate p-values for a test of the difference in MPPMs for columns 1 and 2 and a test of the difference in MPPMs between columns 3 and 4. We find that, for both value-weighted and equalweighted index returns, the addition of the Halloween fund to a portfolio containing the buy-and-hold fund and the January fund results in a MPPM that is significantly larger at the 5% level. From an investment strategy perspective, the value-weighted index results presented in Table 6 are more realistic for actual investors. Small firm stocks are not as liquid and might not be as available for immediate purchase or sale as compared to large firm stocks. Therefore, constructing an equal-weighted portfolio might prove to be far more difficult than constructing a value-weighted portfolio. 5. Conclusion In this study, we show that the Halloween effect in U.S. returns is significant in the period 1954–2008, but not before. Anomalies usually are present only in older data, given that they can be exploited for profit by savvy investors once they are identified. This does not appear to be the case with the Halloween effect. We also show that the Halloween effect is robust to consideration of outliers, the January effect, and transactions costs. Some anomalies, such as those related to weather, would require many transactions per year and a touch of clairvoyance to exploit, making profitability from exploitation of these anomalies questionable. By contrast, the Halloween effect is an especially attractive anomaly for investors, given the low number of transactions required and the easily predictable dates of those transactions. The greater risk-adjusted returns available by investing in a Halloween portfolio are a challenge to the efficient markets hypothesis. Further research is needed to reconcile these results with rational human behavior. References Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817−858. Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: Wiley. Bouman, S., & Jacobsen, B. (2002). The Halloween indicator: Sell in May and go away. The American Economic Review, 92, 1618−1635. Britten-Jones, M. (1999). The sampling error in estimates of mean-variance efficient portfolio weights. Journal of Finance, 54, 655−671. Campbell, J. Y., & Hentschel, L. (1992). No news is good news. Journal of Financial Economics, 31, 281−318. Cutler, D. M., Poterba, J. M., & Summers, L. H. (1989). What moves stock prices? Journal of Portfolio Management, 15, 4−12. Fox, J. (1997). Applied Regression Analysis, Linear Models, and Related Methods. Thousand Oaks, California: Sage Publications. Galai, D., Kedar-Levy, H., & Schreiber, B. Z. (2008). Seasonality in outliers of daily stock returns: A tail that wags the dog? International Review of Financial Analysis, 17, 784−792. Goetzmann, W., Ingersoll, J., Spiegel, M., & Welch, I. (2007). Portfolio performance manipulation and manipulation-proof performance measures. Review of Financial Studies, 20, 1503−1546. Hampel, F. R. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383−393.

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