The Statistical Significance of Excess Dollar Returns

The following post comes to us from Tiago Duarte-Silva and Maria Tripolski-Kimel, both of Charles River Associates.

The literature on event studies has long established the properties of excess returns and tests of their statistical significance. However, it is useful in certain settings to examine excess dollar returns. For example, mergers and acquisitions often require the examination of dollar returns to assess the impact on the wealth of securities’ holders. Other examples include the analysis of managerial skill on actively managed funds, of the magnitude of price manipulation, or of the impact of disclosure events on prices in securities litigation.

A common practitioner use of event studies involving excess dollar returns is in analyzing the impact of disclosure events on prices in the context of securities litigation (see, e.g., Francis et al., 1994). In typical Section 10(b) cases in securities litigation, plaintiffs allege that a company misrepresented or failed to disclose material information and therefore the prices of the company’s common stock at the time of plaintiffs’ transactions were inflated. The per-share damages to plaintiffs are computed based on inflation in the company’s common stock at the time of transactions. The stock price inflation is often measured on an alleged disclosure day by comparing the decline in the stock price from the value it could have had absent the disclosure event according to a regression model. The statistical question of interest, therefore, is whether this decline is due to random chance or to the materiality of the information disclosed.

In June 2011, a post on this Forum covered one of the few papers examining the properties of excess dollar returns: Event Study Analysis: Correctly Measuring the Dollar Impact of an Event by Saha and Ferrell. The authors observed through an approximation how a statistic of excess dollar returns differs from the more commonly used statistic of excess log returns. Further, Saha and Ferrell demonstrated through a numerical example that the statistical significance of a day’s excess log return does not necessarily imply statistical significance of the same day’s excess dollar return. This would have serious implications for securities litigation and other domains: the effects of news on securities prices were presented as different depending on whether returns were measured in logs or dollars.

We examine this claim in our paper, Testing Excess Returns on Event Days: Log Returns vs. Dollar Returns, which was recently accepted for publication in Finance Research Letters, and find that the results of testing excess dollar returns should not be different than the results of testing excess log returns under the standard linear regression model.

To obtain this result, we start by investigating how the hypothesis of non-materiality of disclosure could be written through a statistical model of log returns and equivalently through the definition of ex post excess log returns as used in the financial literature. We derive a similar hypothesis by looking at ex post excess dollar returns. Additionally, we state that excess dollar returns should be defined as conditional on the prior day’s price as it is part of the information set for the returns being measured. This change in setup leads to the more intuitive result that—in practice—statistically significant returns imply statistically significant dollar returns.

We also investigate why Saha and Ferrell’s (2011) test led to different results. We find that they use a biased estimator of the mean of excess dollar returns and misapply the delta method to approximate excess dollar returns’ distribution. Moreover, the delta method may not be suitable for deriving the standard error of the excess dollar returns as it assumes an approximately normal distribution of the excess dollar returns. We also show by simulation that the distribution of excess dollar returns is skewed.

In summary, our paper returns the literature to the intuitive result that the effect of news on the value of a security or a firm is of the same statistical significance regardless of whether returns are measured in logs or in dollars.

The full paper is available for download here.

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