Abstract
Regression analysis is often used to estimate a linear relationship between security abnormal returns and firm-specific variables. If the abnormal returns are caused by a common event (i.e., there is “event clustering”) the error term of the cross-sectional regression will be heteroskedastic and correlated across observations. The size and power of alternative test statistics for the event clustering case has been evaluated under ideal conditions (Monte Carlo experiments using normally distributed synthetic security returns) by Chandra and Balachandran (J Finance 47:2055–2070, 1992) and Karafiath (J Financ Quant Anal 29(2):279–300, 1994). Harrington and Shrider (J Financ Quant Anal 42(1):229–256, 2007) evaluate cross-sectional regressions using actual (not simulated) stock returns only for the case of cross-sectional independence, i.e., in the absence of clustering. In order to evaluate the event clustering case, random samples of security returns are drawn from the data set provided by the Center for Research in Security Prices (CRSP) and the empirical distributions of alternative test statistics compared. These simulations include a comparison of OLS, WLS, GLS, two heteroskedastic-consistent estimators, and a bootstrap test for GLS. In addition, the Sefcik and Thompson (J Accounting Res 24(2):316–334, 1986) portfolio counterparts to OLS, WLS, and GLS, are evaluated. The main result from these simulations is none of the other estimator shows clear advantages over OLS or WLS. Researchers should be aware, however, that in these simulations the variance of the error term in the cross-sectional regression is unrelated to the explanatory variable.
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Notes
Portions of the exposition rely heavily on Karafiath (1994).
Harrington and Shrider (2007) were published while this paper was under second revision.
Harrington and Shrider (2007) evaluate this estimator using simulations that have cross-sectional independence.
I.e., when the null hypothesis is true the empirical rejection rate conforms to type I error.
I.e., under the null hypothesis the empirical rejection rate is inconsistent with type I error alone.
The goal of the experimental design is to replicate case 1 from Karafiath (1994) with actual stock returns in place of simulated returns.
I.e., the minimum correlation criterion applies to the first column (or row) of the portfolio correlation matrix.
This exposition is based on Sefcik and Thompson (1986).
See Sefcik and Thompson (1986, p. 324) especially the second paragraph.
For each of the Sefcik–Thompson portfolios, one of the cross-sectional regression coefficients is obtained directly as the \(\hat{\gamma}_{pn} \) estimate from the following regression:
$$ R_{pt} =\alpha _p +\beta _p R_{mt} +\sum_{n=1}^{T_2 } {\gamma _{pn} D_{pt} } +\varepsilon _{pt,}\quad t=1,\ldots,T_1, T_1 +1,\ldots,T_1 +T_2 $$where: R pt = return to a Sefcik–Thompson portfolio on day t; R mt = return to the market index on day t;
$$ D_{nt} =\left\{ {{\begin{array}{l} {1\,\,{\rm if}\,\,t=T_1 +n} \\ {0\,\,{\rm otherwise}\,\,\,\,} \\ \end{array} }} \right. $$T 1 = number of days in the “estimation” window; T 2 = number of days in the “event” window.
See Sefcik and Thompson (1986, pp. 332–333).
I thank an anonymous review for suggesting the bootstrap.
Chou (2004) and Hein and Westfall (2004) implement a variant of this bootstrap. These authors calculate the test statistic for each iteration of the bootstrap and identify the 1% and 5% critical values from the empirical distribution. Obtaining a reliable value for the 1% tails of the distribution requires far more than the 500 iterations used here. Since each iteration requires inverting a covariance matrix for up to 75 securities in a portfolio; the computation becomes prohibitively time-consuming for 3,000 runs. The method of calculating the empirical variance represented by Eq. 4 is suggested by Greene (2003).
Nonetheless, there are some interesting similarities between the results in Table 6 and the results for regression slope coefficients presented by Harrington and Shrider (2007, p. 252) in their Table 8. Their Table 8 reports the power of several tests for non-zero regression slope coefficients when there is an increase in error variance. When the increase in error variance is unrelated to the regressor, and the regression has high explanatory power, the heteroskedastic-consistent estimators offer little or no gain in power relative to OLS or WLS—very similar to the result reported in Table 6 of this paper. Their Table 8 shows that heteroskedastic-consistent estimators offer significant increase in power only when the regression has very low explanatory power i.e., R 2 = 10%. A complete examination of the event clustering case with increases in error variances is beyond the scope of this paper.
Thus, the conditions for asymptotically unbiased standard errors described by Greenwald (1983) are satisfied by construction. The Greenwald condition is discussed in Bernard (1987) and Karafiath (1994). Harrington and Shrider (2007) document the bias in regression based tests when the Greenwald condition is violated for the case of cross-sectional independence, ie., the absence of event clustering.
See Sefcik and Thompson (1986, pp. 326–328) for details.
See Binder (1998) for a recent discussion of event study methods.
A final caveat is in order: as with all simulations, the results may be specific to the experimental design, and these simulations do not include increases in error term variances due to the event. In particular researchers should be aware that OLS and WLS test statistics will be biased if the variance of the error term is related to one or more of the explanatory variables in the regression.
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Acknowledgments
An earlier version of this paper was prepared while the author was a Visiting Scholar in the Private Enterprise Research Center at Texas A&M University. I thank the Center's director, Thomas R. Saving, for valuable insight. I also thank John Glascock, Allan Zebedee, the participants at the Texas A&M finance seminar series, and two anonymous reviewers for their comments.
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Karafiath, I. Is there a viable alternative to ordinary least squares regression when security abnormal returns are the dependent variable?. Rev Quant Finan Acc 32, 17–31 (2009). https://doi.org/10.1007/s11156-007-0079-y
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DOI: https://doi.org/10.1007/s11156-007-0079-y