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Bias-Corrected Estimation of Price Impact in Securities Litigation

Bias-Corrected Estimation of Price Impact in Securities Litigation Abstract The single-firm event studies that securities litigants use to detect the impact of a corrective disclosure on a firm’s stock price have low statistical power. As a result, observed price impacts are biased against defendants and systematically overestimate the effect on firm value. We use the empirical distribution of daily stock returns to analyze the bias and develop bias-corrected estimators of price impact in securities litigation. Because of low statistical power, the ex ante incentives against committing securities fraud are also too low. We analyze the adjustment for optimal deterrence and find that it is material, but is nowhere equal to the opposing truncation bias. 1. Introduction A small but growing literature1 examines problems with the single-firm event studies that became ubiquitous in federal securities litigation following the decision of the U.S. Supreme Court in Basic v. Levinson.2 In Basic, a case under Section 10(b) of the Securities Exchange Act of 1934, the Court endorsed the “fraud-on-the-market” doctrine. The fraud-on-the-market doctrine is a presumption that “the market price of shares traded on well-developed markets reflects all publicly available information, and, hence, any material misrepresentations.”3 The major implication of this doctrine is that a false disclosure or omission in well-developed securities markets will be reflected in the security’s price regardless whether every individual trader is aware of the misrepresentation. A later “corrective disclosure”—often an admission by the security issuer—then reveals that previous statement or omission was false or misleading.4 Investors usually sue the maker of the false statement or omission, alleging that the misstatement or omission was the result of an intent to deceive investors. Litigants in such cases nearly always offer the opinion evidence of expert witnesses who conduct event studies on the security price at issue. An event study is a statistical methodology used by academic researchers to determine whether corporate events such as stock splits, merger announcements, or dividend changes are associated with a statistically significant change in the stock prices of companies subject to the event.5 After Basic, securities litigants use event studies to answer two critical questions: First, did the corrective disclosure cause a price impact in the security at issue?6 Second, if there was a price impact, how much of it was due to the corrective disclosure as opposed to other causes such as broad market movements—a fact relevant to loss causation and damages, among other things?7 Event studies have become a key part of almost every securities case.8 Event studies in securities litigation differ importantly from event studies in academic work. When researchers apply the event study methodology in academic work, they examine the price impact of some set of events on a sample of many firms subject to that type of event. When expert witnesses apply the event study methodology in litigation, they examine the price impact of a single event—the corrective disclosure—at a single firm. Recently, Brav and Heaton (2015) argued that greater attention should be given to three methodological problems with these “single-firm event studies” as used in securities litigation.9 The first problem is low statistical power: It is difficult for single-firm event studies to detect price impacts that actually exist, because price impacts must be quite large to be detected when only one event is in the sample. The second issue is confounding effects: Without many firms in the sample to average away price moves that were unrelated to the event at issue, the single-firm event study is a noisy measure of the price impact. The third problem that Brav and Heaton identified was new in the literature: low power and confounding effects combine to generate biased measures of price impact. That is, the price impact estimates from single-firm event studies are systematically too large. The reason is straightforward. When statistical power is low, a small price impact will be detected only when confounding effects—price changes unrelated to the corrective disclosure—combine with the disclosure’s true price impact to push the total price change (the sum of confounding effects plus the true price impact) past the threshold of statistical significance. Suppose, for example, that the true price impact is |$-$|2.0%, but only returns larger in magnitude than |$-$|2.5% are statistically significant. On its own the true price impact would not be statistically significant. But suppose there are |$-$|0.6% of additional negative confounding effects. This pushes the total observed price impact past the threshold to |$-$|2.6%. In effect, the event returns that are statistically significant are drawn from a distribution that is truncated above at the threshold of statistical significance. The expected value of a draw from that truncated distribution is not equal to the true price impact, which comes from the untruncated distribution. Thus, conditional on statistical significance, the stock return on the event date overestimates the event’s price impact. The practical effect of this bias is to drive up compensatory damages calculations and settlements. This article expands on Brav and Heaton (2015)’s insight in two ways. First, Brav and Heaton illustrate their argument using only simulations that assume normally distributed returns.10 We estimate and study the bias in a more realistic fashion using the empirical distribution of actual daily stock returns, which are non-normally distributed. We find that the bias is often material, especially for small price impacts and high volatility stocks, and that the bias is larger the lower (i.e., more stringent) the threshold of statistical significance. Second, we present methods to correct for the bias. We develop and validate bias-corrected estimators of the true price impact. We analyze their performance and find that while all five improve on the uncorrected event date return, a median bias-corrected (MBC) estimator performs best. We provide procedures and make code available that enables the production and validation of bias-corrected estimates of price impact in single-firm event studies. While our article is in one sense narrowly methodological—proposing a method of correcting one statistical bias that is present in securities litigation event studies—the correction we propose raises difficult questions. After all, a consequence of the low statistical power of single-firm event studies is that many smaller (but still economically important) securities frauds will go undetected because they do not reach statistical significance. This means that the ex ante incentives against committing securities fraud are already too low. Here, we propose a method to ensure the frauds that are detected are not overestimated. This arguably further reduces the ex ante incentives against committing securities fraud. Thus, the methods we develop in this article make price impact estimates in securities litigation more accurate, even as they may reduce the deterrent effect of securities litigation. A precedent for this discussion is the problem of unlawful search and seizure under the Fourth Amendment. Scholars have documented an increase in crime after the U.S. Supreme Court greatly limited the use of evidence from unlawful search and seizure in Mapp v. Ohio.11|$^{,}$|12 Nevertheless, many would argue the answer is better enforcement that does not rely on unlawful search and seizure, not the continued admissibility of illegally obtained evidence. Here as well, we argue that the answer is explicit punitive damages13 in securities litigation, not back-door additional damages in the form of badly applied methodology. Punitive damages are, however, impermissible under current federal securities law.14 We show that our methods also yield an accurate estimate of the optimal deterrence factor as developed by Polinsky and Shavell (1998), and our estimates yield two conclusions. First, the need for additional damages to deter securities fraud is widespread. Second, the upward adjustment for optimal deterrence is nowhere equal to the downward adjustment for truncation bias. If lack of deterrence is a problem—and the data suggest that it is—the answer is to properly correct for truncation bias, which reduces compensatory damages, and then to optimize deterrence by awarding additional punitive damages. If securities law should change to make explicit punitive damages permissible, the methods developed in this article yield accurate estimates of both the bias correction and the optimal deterrence adjustment. 2. Statistical Bias in Single-Firm Event Studies Suppose a company makes a corrective disclosure, say an announcement of a restatement of earnings, that reduces the market’s valuation of the firm’s equity by the fraction |$S$|⁠. We cannot observe the true price impact |$S$| because price movement on the event date also occurs for reasons unrelated to the corrective disclosure including the impact of other information relevant to firm value, trading by investors seeking to invest or divest the stock for other reasons, and trading by noise traders for unexplained reasons. As a result, we observe a stock return on the event date that includes both the price impact of the corrective disclosure and the net effect of unrelated price movements: \begin{equation} r^{EVENT} = S + r_t. \end{equation} (1) We assume that the distribution of the noise term |$r_t$| is the distribution of daily returns on nonevent dates (i.e., days without corrective disclosures). Thus, on the day of the corrective disclosure (the “event date”) we observe the price impact |$S$| plus a draw from the nonevent return distribution.15 Courts generally require that plaintiffs in a securities case demonstrate that the event-date return is statistically significant.16 As a result, the set of litigated negative event-date returns is truncated above at the threshold of statistical significance. Thus a litigated event-date return represents a draw from the distribution: \begin{equation} r^{OBSERVED} \in \{ r^{EVENT} \, | \, r^{EVENT}<T_{p} \}, \end{equation} (2) where |$T_p$| is the truncation threshold at significance level |$p$|⁠, i.e., the return threshold that the corrective disclosure return must exceed (in absolute value) to be statistically significant. Because the observed distribution is truncated above, the expected value of |$r^{OBSERVED}$| is lower (more negative) than the true price impact |$S$|⁠. On average, the corrective disclosure return overestimates the negative magnitude of the price impact with the following bias: \begin{align} bias(\tilde{r}, T_p, S) &= E[r^{EVENT} \, | \, r^{EVENT} < T_{p}] -S\nonumber\\ &= E[S + r_t \, | \, S + r_t<T_{p}] -S\nonumber\\ &= E[r_t \, | \, r_t<T_{p}-S]. \end{align} (3) Without any distributional assumptions, we can make three predictions: Prediction 1: The bias is larger the larger is the dispersion or volatility of the stock’s returns. (This follows because higher dispersion increases the probability of truncation but does not change the unconditional expectation.) Prediction 2: The bias is smaller the larger is the true price impact |$S$|⁠. (This follows because the partial derivative with respect to |$S$| is negative.) Prediction 3: The bias is larger the lower is the threshold |$T_p$|⁠, i.e., the smaller is |$p$|⁠. (This follows because smaller |$p$| increases the probability of truncation.)17 We now turn to examining the potential magnitude of this bias using the empirical distribution of actual daily market-adjusted stock returns. 2.1. Data Our data are from the Center for Research in Security Prices (“CRSP”). We extract daily returns for all U.S. common stocks in CRSP from 2010 to 2015 and adjust the returns for each stock using the standard market model: \begin{equation} R^i_t = \alpha^i + \beta^i R^{Mkt}_t + r^i_t, \end{equation} (4) where |$R^i_t$| is the daily return to stock |$i$| on date |$t$| |$R^{Mkt}_t$| is the daily return to the CRSP value-weighted market portfolio on date |$t$| |$r^i_t$| is the daily market-adjusted return to stock |$i$| on date |$t$| To test Prediction 1, we divide the sample stocks into low volatility |$(\sigma < 4\%)$| and high volatility |$(\sigma > 4\%)$| groups. All our results are similar if we use a different breakpoint such as 3 or 5%.18 From each of the two groups we randomly sample, without replacement, 10,000 blocks where each block consists of 100 consecutive market-adjusted daily returns for a single stock. 2.2. Estimating the Bias For each block, we compute significance thresholds using the 100 daily returns as the distribution of nonevent daily returns. Next, we add a simulated price impact |$S$| to all 100 returns to simulate the distribution of event-date returns. To mimic the selection effects of requiring litigated cases to have a statistically significant event-date return, we then drop simulated event-date returns that are not significant according to a two-sided |$t$|-test with |$P<0.05$|⁠, which produces a truncated distribution of event-date returns. We estimate the bias by comparing the mean of 100 draws with replacement from the truncated distribution to the mean of the untruncated distribution. Finally, we take the mean of the estimated bias across all 10,000 blocks, which yields the mean expected bias across one million simulated single-firm event studies. Figure 1 plots the mean event-date return, conditional on a significant |$t$|-test, across a range of simulated price impacts |$S$| within the high volatility (Figure 1a) and low volatility (Figure 1b) groups. The vertical distance from the 45-degree line equals the expected bias. For low volatility stocks, there is essentially no bias for price impacts larger than |$-$|8%. This is intuitive: for low volatility stocks a large event-date return is almost surely due to a large price impact and not due to nonevent noise. For low volatility stocks, there is still potentially material bias for price impacts of |$-$|8% or smaller. Figure 1. View largeDownload slide Truncation Bias in Simulated Event Studies. Each figure plots the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$|⁠, over a range of simulated true price impacts. The figures show the average across 1 million simulated single-firm event studies using CRSP stocks with low volatility stocks (⁠|$\sigma < 4\%$|⁠) (a) and high volatility stocks (⁠|$\sigma > 4\%$|⁠) (b) of daily market-adjusted returns. The vertical distance from the 45-degree line equals the mean bias. Figure 1. View largeDownload slide Truncation Bias in Simulated Event Studies. Each figure plots the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$|⁠, over a range of simulated true price impacts. The figures show the average across 1 million simulated single-firm event studies using CRSP stocks with low volatility stocks (⁠|$\sigma < 4\%$|⁠) (a) and high volatility stocks (⁠|$\sigma > 4\%$|⁠) (b) of daily market-adjusted returns. The vertical distance from the 45-degree line equals the mean bias. For high volatility stocks, the bias is material across the full range of simulated price impacts. Even with a large price impact of |$-$|25% the mean event-date return is |$-$|26.03%, reflecting a mean bias of more than 1% of market capitalization. The larger bias in higher volatility stocks agrees with our Prediction 1, and in all cases the bias is larger for a smaller price impact, which agrees with Prediction 2. 2.3. Alternative Significance Levels Figure 2 shows the bias in the group of high volatility stocks when we use different thresholds of statistical significance. Relative to our first threshold of |$P<0.05$|⁠, a less strict threshold of |$P<0.10$| generates a smaller bias (dashed line).19 However, the bias is still present and material across the full range of simulated price impacts. Figure 2. View largeDownload slide Bias Using Different Significance Thresholds. The figure graphs the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$| and |$P<0.10$|⁠, as well as when the truncating threshold is set at zero (negative returns only). The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. Figure 2. View largeDownload slide Bias Using Different Significance Thresholds. The figure graphs the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$| and |$P<0.10$|⁠, as well as when the truncating threshold is set at zero (negative returns only). The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. Figure 2 further shows that a bias is present when we impose the mere requirement that the event-date return be negative. Even ignoring the typical requirement to demonstrate statistical significance at the 5% level, it is unlikely that a securities case would be brought if the event-date return were positive, and this much weaker condition also causes truncation of the observed event-date returns. The resulting bias is smaller but still clearly present across the full range of simulated price impacts. We conclude that, first, the pattern of statistical bias is consistent with our Prediction 3 above, and second, that even the weakest assumption on the extent of truncation results in material bias. 2.4. One-Tailed |$t$|-Test Our results above use a standard two-tailed |$t$|-test to evaluate statistical significance. A one-tailed test is unquestionably more appropriate in securities litigation,20 as it improves statistical power and reflects that the null hypothesis being tested is usually one-tailed, e.g., that the event return is nonnegative. However, the use of one-tailed tests is often contested.21 What effect does the use of a one-tailed test have on the statistical bias? Table 1 Column 1 shows the mean bias in our baseline specification, using a two-tailed |$t$|-test with |$P<0.05$|⁠. This is equivalent to a one-tailed |$t$|-test with |$P<0.025$|⁠. Column 2 shows the mean bias when we use a one-tailed |$t$|-test with |$P<0.05$| (equivalent to a two-tailed |$t$|-test with |$P<0.10$|⁠) instead. As in Figure 2, we see that the bias is smaller in all cases. Thus, our results highlight an additional benefit of using a one-tailed test: in addition to its being the clear correct choice in securities litigation, a one-tailed test makes the truncation less severe, which reduces the statistical bias when measuring price impact. Table 1. Bias Using Alternative Significance Tests |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% The table compares the mean statistical bias across a range of price impacts when we evaluate statistical significance with a two-tailed |$t$|-test (baseline specification) versus a one-tailed |$t$|-test or the SQ test of Gelbach et al. (2013). The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. View Large Table 1. Bias Using Alternative Significance Tests |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% The table compares the mean statistical bias across a range of price impacts when we evaluate statistical significance with a two-tailed |$t$|-test (baseline specification) versus a one-tailed |$t$|-test or the SQ test of Gelbach et al. (2013). The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. View Large 2.5. The SQ Test The |$t$|-test is widely used in securities-litigation event studies.22 However, Gelbach et al. (2013) find that the |$t$|-test can be poorly suited to statistical testing using daily stock returns because daily stock returns are skewed and fat-tailed, contradicting the distributional assumptions for a valid |$t$|-test. They present an alternative test based on sample quantiles, the SQ test, that does not rely on distributional assumptions. Table 1 Columns 3 and 4 show the mean bias using the SQ test with |$P<0.025$| and |$P<0.05$| to evaluate statistical significance. Comparing to Columns 1 and 2, we see that in all cases using the SQ test the bias is reduced relative to the |$t$|-test. This result is intuitive. As Gelbach et al. (2013) document, the non-normal distribution of daily stock returns leads the |$t$|-test to systematically underreject the null hypothesis, that is, to classify too many event-date returns as nonsignificant. This worsens the bias when the |$t$|-test is used. Thus, our results highlight an additional benefit of Gelbach et al. (2013)’s SQ test for single-firm event studies: the SQ test is both more accurate and reduces the statistical bias when measuring price impact. 2.6. Deterrence and Punitive Damages Next we use our analysis and data to examine the effect of low statistical power on deterrence. Polinsky and Shavell (1998) argue that when the injurer has a significant chance of escaping liability then punitive (additional, noncompensatory) damages are needed to increase deterrence. Thus, our results suggest that it would be socially optimal to increase the damages awarded in securities litigation above the bias-corrected best estimate. To do so is not currently permissible because punitive damages are unavailable to private securities litigants. Nevertheless, we can examine how material the deterrence effects and the appropriate punitive damages are in securities litigation. According to the formula derived in Polinsky and Shavell (1998), the total damages imposed “...should equal the harm multiplied by the reciprocal of the probability that the injurer will be found liable when he ought to be.” Denote the appropriate punitive damages by |$\mathcal{P}$|⁠: \begin{equation} \text{Total damages} = S + \mathcal{P} = \frac{S}{Prob[foundliable]} \end{equation} Rearranging, \begin{equation} \mathcal{P} = S \, \frac{1-Prob[foundliable]}{Prob[foundliable]} = S \, \frac{Prob[S + r_t \geq T_{p}]}{Prob[S + r_t<T_{p}]} \end{equation} (5) Figure 3 plots the statistically appropriate adjustment for deterrence across a range of simulated true price impacts, using our data drawn from CRSP daily stock returns. There are two main conclusions we draw from these results. First, the appropriate deterrence adjustment is material across the entire range of price impacts. Thus, the data suggest that the need for additional punitive damages to deter securities fraud is widespread. Second, the deterrence adjustment is nowhere equal to the truncation bias. Thus, these two countervailing motives—to correct for the downward truncation bias, which reduces the damages, and to optimize deterrence by awarding additional damages—do not cancel each other out, and must be estimated and incorporated separately into the final damages awarded. Figure 3. View largeDownload slide Statistically Appropriate Punitive Damages. The figure graphs the statistically appropriate punitive damages according to the formula of Polinsky and Shavell (1998), expressed as a fraction of the firm’s equity value, across a range of true price impacts. The bias due to statistical truncation is also presented for comparison. The distribution in each case is based on 1 million simulated single-firm event studies across the sample of all CRSP stocks. Figure 3. View largeDownload slide Statistically Appropriate Punitive Damages. The figure graphs the statistically appropriate punitive damages according to the formula of Polinsky and Shavell (1998), expressed as a fraction of the firm’s equity value, across a range of true price impacts. The bias due to statistical truncation is also presented for comparison. The distribution in each case is based on 1 million simulated single-firm event studies across the sample of all CRSP stocks. 3. Bias-Corrected Estimators We next examine how to correct for the bias in estimated price impacts in single-firm event studies. The basic task is to correct a consistent estimator for finite sample bias. In the original setting of event-study methodology—academic studies of the impact of a particular class of events such as stock splits—there may be several hundred firms that experience the event in question. In that setting, we observe hundreds of draws from the event-date distribution, so as long as the estimator is consistent—i.e., converges to an unbiased estimate as the number of observations grows—the bias in the estimate due to truncation of individual draws is negligible. The situation is very different in a securities litigation setting. We are typically examining a single event at a single firm. It is easy to see that finite sample bias is a major concern, because the “treated” set consists of a single daily return. It is simple to compute the bias for a known price impact. That is, given a value of the true price impact |$S$|⁠, we simply evaluate the expectation (3) as in Figure 1. The challenge is that in practice we do not know the true price impact and must estimate |$S$| given the event-date return that we observe. Fortunately, correcting for finite sample bias is an empirical question with a long history in econometrics, and many approaches have been proposed.23 Different approaches yield different estimates of |$S$| and the bias. We develop five bias-corrected estimators, and then examine how they perform in practice. 3.1. First-Order Corrections The first two bias correction methods, analytical bias corrected (ABC) and constant bias corrected (CBC), set the true price impact equal to the observed value |$r^{EVENT}$| (which is of course not the case). They then compute the bias and use it as an approximation to the bias at |$S$|⁠. In effect, this is equivalent to assuming that the bias is constant for values of |$S$| that are close to the observed value |$r^{EVENT}$|⁠. 3.1.1. Analytical Bias Correction We first compute an analytical bias correction assuming that returns are normally distributed. The bias in the mean of a truncated normal distribution is: \begin{equation} bias = E[x | x < T_p] - \mu \, , \, x \sim N(\mu,\sigma) = - \phi \biggl(\cfrac{T_p - \mu}{\sigma}\biggr) / \Phi \biggl(\cfrac{T_p - \mu}{\sigma}\biggr) \end{equation} and the ABC estimate of the bias is: \begin{equation} = - \phi \biggl(\cfrac{T_p - r^{EVENT}}{\sigma}\biggr) / \Phi \biggl(\cfrac{T_p - r^{EVENT}}{\sigma}\biggr) \end{equation} (6) Thus, the ABC estimator assumes that returns are normally distributed, and uses the bias evaluated at |$r^{EVENT}$| as an approximation of the bias at |$S$|⁠. 3.1.2. Constant Bias Correction We next examine the CBC estimator of MacKinnon and Smith (1998). For each simulated event study, we follow the steps: (1) Compute the threshold of statistical significance |$T_p$| based on the set of nonevent returns. (2) Shift the set of nonevent returns downward by the event-date return |$r^{EVENT}$|⁠, yielding distribution |$D_{all}$|⁠. (3) Drop any observations in |$D_{all}$| that are above |$T_p$|⁠, yielding distribution |$D_{truncated}$|⁠. (4) Compute the means of |$D_{all}$| and |$D_{truncated}$|⁠. The difference between the two means is the CBC estimate of the bias. Thus, the CBC estimator computes the bias as in Figure 1, evaluated at the observed event-date return |$r^{EVENT}$|⁠. This will yield an accurate estimate of the bias and the true price impact |$S$| as long as the bias evaluated at the true |$S$| is the same as the bias evaluated at |$r^{EVENT}$|⁠. 3.2. Higher Order Corrections The first two bias corrections, ABC and CBC, remove first-order bias in the measurement. However, if their implicit assumption that the bias is constant is inaccurate then they will not remove all bias in expectation. We explore higher order estimators, which can remove all bias in expectation. However, because they rely on more assumptions and computations, they can be less robust in practice. 3.2.1. Median Bias Correction We next examine a median bias-corrected estimator (MBC) following Andrews (1993). For each simulated event study we examine a grid of true price impacts |$S^*$| around the observed event-date return |$r^{EVENT}$|⁠. For each value of |$S^*$| we shift the nonevent distribution to the left by |$S^*$|⁠, truncate it above the threshold, and compute the median of the truncated distribution. We then pick the most negative (i.e., most conservative) value of |$S^*$| with median equal to |$r^{EVENT}$|⁠. 3.2.2. Linear and Nonlinear Bias Correction Finally, we examine the linear bias-corrected (LBC) and nonlinear bias-corrected (NBC) estimators of MacKinnon and Smith (1998). In both cases we begin with the CBC estimate of the bias, denoted |$S_1$|⁠. We then “move up” and compute a second bias estimate in the same fashion but evaluated at |$S_1$| instead of at |$r^{EVENT}$|⁠. Denote the second bias-corrected estimate of the price impact |$S_2$|⁠. We take the difference between the two estimates and extrapolate linearly to arrive at the LBC estimate. We continue iterating until successive estimates converge to arrive at the NBC estimate. There is one issue which makes the application of the LBC and NBC challenging: for small price impacts, the bias increases more than proportionately as the true price impact shrinks (see Figure 2). That is, the inverse of the bias function may not have a unique root. This issue does not affect the ABC or CBC estimators as the bias is evaluated at a single point, the observed |$r^{EVENT}$|⁠. Likewise, the MBC appears to be quite numerically stable in our setting, because we have a simple heuristic in place that avoids the case of multiple roots. However, the LBC and NBC estimators look for roots of the inverse bias function in a general way and so numerical convergence can be an issue with small price impacts. We deal with this issue by identifying individual cases where the LBC and NBC are converging poorly and in these cases we default to the CBC estimate. 4. Performance of the Bias-Corrected Estimators As the bias is most common and most material in high volatility stocks, we compare the estimators in that group of stocks. We evaluate the performance of the estimators as follows. We first specify a true price impact |$S$|⁠. Within each block of 100 returns, we shift the returns downward by |$S$| and truncate above at |$T_p$|⁠. We draw 100 simulated event-date returns from the truncated distribution. We bias-correct each of the simulated event-date returns, then compute the difference between the bias-corrected estimate and the true price impact |$S$|⁠. Across our 10,000 blocks this generates 1 million measurements of (1). the true price impact, (2) the uncorrected event-date return, and (3) the bias-corrected estimate. Across a range of simulated true price impacts, we then characterize the distribution of the bias-corrected estimates. Figure 4 plots the mean and median bias-corrected estimates for our five estimators across a range of simulated price impacts. The 45 degree line corresponds to an entirely unbiased estimate. For price impacts of |$-$|15% or more, all the estimators recover the true price impact quite accurately as measured both by means and medians. In contrast, for price impacts smaller than |$-$|15%, all five estimators fail to eliminate the bias, although all reduce the bias significantly relative to the uncorrected event-date return. Overall the median bias-corrected (MBC) estimator performs best in terms of the mean bias, while the LBC estimator performs slightly better in terms of the median bias. Figure 4. View largeDownload slide Performance of Bias-Corrected Estimators The figure plots mean and median estimates of the true price impacts for a variety of bias correction regimes, across a range of simulated true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. (a) Mean estimates for ABC and CBC. (b) Mean estimates for MBC, LBC and NBC. (c) Median estimates for ABC and CBC. (d) Median estimates for MBC, LBC and NBC. Figure 4. View largeDownload slide Performance of Bias-Corrected Estimators The figure plots mean and median estimates of the true price impacts for a variety of bias correction regimes, across a range of simulated true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. (a) Mean estimates for ABC and CBC. (b) Mean estimates for MBC, LBC and NBC. (c) Median estimates for ABC and CBC. (d) Median estimates for MBC, LBC and NBC. Table 2 compares the mean and median residual bias for our five bias-corrected estimators across a range of simulated true price impacts. The first column (“Uncorrected”) shows the statistical bias when we simply use the event-date return as our estimate of the price impact. The next five columns show the performance of the ABC, CBC, MBC, LBC, and NBC estimators. In each row the first- and second-best performing estimators are denoted by superscripts. Table 2. Performance of Bias-Corrected Estimators Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% The table presents means and medians of the bias induced by requiring statistical significance via two-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table 2. Performance of Bias-Corrected Estimators Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% The table presents means and medians of the bias induced by requiring statistical significance via two-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large As in Figure 4, all five estimators improve on the raw event-date return in terms of both mean and median bias. Evaluated in terms of the mean bias (Figure 4a), for large price impacts all three higher-order estimators (MBC, LBC, and NBC) perform best about equally well. For small price impacts of |$-$|10% or |$-$|5%, again as in Figure 4, the MBC estimator outperforms the rest. This pattern appears to arise because the MBC estimator is (1) a higher-order consistent estimator, meaning that it removes all bias in expectation and (2) numerically stable, even though it does not target the mean bias directly. Evaluated in terms of the median bias (Table 2b), the MBC estimator performs marginally better than the LBC and NBC for large price impacts and significantly better for small price impacts, which is not surprising since the MBC is the only estimator that targets the median bias directly while the other five all attempt to reduce the mean bias. In the Appendix, we compare the performance of the estimators when statistical significance is evaluated via one-tailed |$t$|-test or an SQ test instead. In both cases, the results are very similar to those in Table 2. In both alternative cases, there is less bias to correct, and all the estimators perform better. These results again support the use of one-tailed tests and the SQ test in practice. Whether an estimator’s performance should be evaluated in terms of mean or median bias or some other metric will depend on the user’s objective (or loss function, in statistical terms) as well as the specific data and significance criteria, a potential topic for future research. We further note that in practice there is no need to choose only one estimator. A researcher or expert witness carrying out a single-firm event study could compute bias-corrected estimates, for the case-specific data and significance criteria, using multiple estimators and then compare the resulting estimates.24 5. Conclusion Brav and Heaton (2015) raise the issue of potential bias in price impact estimates in securities litigation. This bias arises because price impacts must be statistically significant to be actionable, but low power and confounding effects in single-firm event studies make it likely that statistically significant event-date returns reflect more than the true price impact of the event under examination. We run simulations using the empirical distribution of daily stock returns and show that under very general conditions there is often material bias relative to the true price impact. We develop and evaluate bias-corrected estimators for single-firm event studies. All five improve on the uncorrected event-date return as an estimate of the true price impact. Which estimator performs best will depend on the objective function, the specific sample, and the significance criterion used, but we conclude that the MBC based on Andrews (1993) performs the best and most consistently. This article addresses one specific source of bias which is readily quantifiable. Future research could address an additional bias that we have not addressed here: The price impact of a corrective disclosure, in an efficient market, includes a component that reflects the burden of the anticipated litigation that corrective disclosure will generate. For corrective disclosures that cause price declines, even the true price impact overstates the harm. This occurs because the costs to the issuer of any anticipated litigation fall on the issuer, but the benefits accrue only to those who bought or sold prior to the corrective disclosure. That is, the prices paid after a corrective disclosure reflect the new valuation of the security in light of corrected information and the expected costs of litigation, settlements, and fines, but do not reflect any benefits of litigation (such as expected settlements) because the right to compensation on a securities fraud claim does not pass to the new buyer. We are aware of no reported case that adjusts for this bias and scholarship on securities fraud price impact and damages calculations largely ignores it. While quantification of this effect will be difficult, it remains a necessary step in increasing the reliability of financial-market evidence in securities litigation.25 Further, there is a clear argument that the low statistical power of single-firm event studies means that additional punitive damages are justified to deter securities fraud. Our data and methods allow us to put numbers to this intuition. Our results suggest that the need for deterrence adjustments in securities litigation is widespread. We show that our methods yield a simple calculation of the statistically appropriate deterrence adjustment in securities litigation—should explicit punitive damages ever become legally permissible. In sum, our paper provides both new insights and new tools for litigants, expert witnesses, and other users of single-firm event studies. A. Appendix This appendix provides additional results and extensions to the main text. A.1. Performance of Bias-Corrected Estimators Using Different Statistical Tests This section compares the performance of our five bias-corrected estimators using different criteria of statistical significance than in the main text. In the main text we evaluate statistical significance using a two-tailed |$t$|-test with |$P<0.05$|⁠. The results are all consistent with our main findings. The main difference to our main findings is that when the SQ test is used instead of a |$t$|-test, the first-order bias corrections (ABC and CBC) perform relatively well especially for large price impacts. Table A.1 presents results when we use a one-tailed |$t$|-test with |$P<0.05$|⁠. Evaluated in terms of the mean bias (Table A.1a), for large price impacts all three higher-order estimators (MBC, LBC, and NBC) perform best about equally well while for small price impacts of |$-$|10% or |$-$|5% the MBC estimator significantly outperforms the others. Evaluated in terms of the median bias (Table 2b), the MBC estimator performs marginally better than the LBC and NBC for large price impacts and significantly better for small price impacts. Table A.1. Performance of Bias-Corrected Estimators Using a One-Tailed |$t$|-Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% The table presents means and medians of the bias induced by requiring statistical significance via one-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table A.1. Performance of Bias-Corrected Estimators Using a One-Tailed |$t$|-Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% The table presents means and medians of the bias induced by requiring statistical significance via one-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table A.2 presents results when we use an SQ test with |$P<0.05$|⁠. Gelbach et al. (2013) propose the SQ test and show it has superior properties to the |$t$|-test for accurately evaluating statistical significance in daily returns data. In terms of mean bias, all the bias-corrected estimators perform well for large price impacts; the ABC and CBC estimators perform best; for small price impacts the MBC and LBC estimator perform best. In terms of median bias, the MBC estimator performs best across the board. Table A.2. Performance of Bias-Corrected Estimators Using an SQ Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% The table presents means and medians of the bias induced by requiring statistical significance via SQ test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table A.2. Performance of Bias-Corrected Estimators Using an SQ Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% The table presents means and medians of the bias induced by requiring statistical significance via SQ test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large We thank the editor and two anonymous referees for comments which substantially improved the paper. Any errors are our own. Footnotes 1. In addition to Brav and Heaton (2015), discussed below, the literature includes Dybvig et al. (2000), Gelbach et al. (2013), Baker (2016), and Fisch et al. (2018). 2. 485 U.S. 224 (1988). 3. Id. at 246. 4. See, e.g., Arkansas Teachers Ret. Sys. v. Goldman Sachs Grp., Inc., 879 F.3d 474, 480, n3 (2d Cir. 2018) (“A ‘corrective disclosure’ is an announcement or series of announcements that reveals to the market the falsity of a prior statement.”) (citation omitted). 5. A recent book covering event study methodology is Kliger and Gurevich (2014). Other sources are MacKinlay (2007) and Kothari and Warner (2007). 6. See, e.g., Willis v. Big Lots, Inc., 242 F. Supp.3d 634 (S.D. Ohio 2017) (“[P]rice impact is demonstrated either through evidence that a stock’s price rose in a statistically significant manner after a misrepresentation or that it declined in a statistically significant manner after a corrective disclosure.”); see also Hatamian v. Advanced Micro Devices, Inc., No. 14-CV-00226 YGR, 2016 WL 1042502, at *7 (N.D. Cal. Mar. 16, 2016) (“Price impact in securities fraud cases is not measured solely by price increase on the date of a misstatement; it can be quantified by decline in price when the truth is revealed.”). 7. See, e.g., Bricklayers and Trowel Trades Int’l Pension Fund v. Credit Suisse First Boston, 853 F. Supp. 2d 181, 190 (D. Mass. 2012), aff’d sub nom. Bricklayers & Trowel Trades Int’l Pension Fund v. Credit Suisse Sec. (USA) LLC, 752 F.3d 82 (1st Cir. 2014) (“An event study that fails to disaggregate the effects of confounding factors must be excluded because it misleadingly suggests to the jury that a sophisticated statistical analysis proves the impact of defendants’ alleged fraud on a stock’s price when, in fact, the movement could very well have been caused by other information released to the market on the same date.”). 8. See, e.g., Bricklayers & Trowel Trades Int’l Pension Fund v. Credit Suisse Secs. LLC, 752 F.3d 82, 86 (1st Cir. 2014) (“The usual—it is fair to say ‘preferred’—method of proving loss causation in a securities fraud case is through an event study, in which an expert determines the extent to which the changes in the price of a security result from events such as disclosure of negative information about a company, and the extent to which those changes result from other factors.”). 9. The United States Court of Appeals for the Second Circuit recently discussed and approved of the Brav and Heaton (2015) analysis. See In re Petrobras Sec., 862 F.3d 250, 279 (2d Cir. 2017) (“These methodological challenges counsel against imposing a blanket rule requiring district courts to, at the class certification stage, rely on directional event studies and directional event studies alone.”). 10. Brav and Heaton (p. 591, n17) “focus on the normal case because standard practice still rests heavily on the normality assumption, despite strong evidence that daily abnormal returns are non-normal[,]” citing Gelbach et al. (2013). 11. Atkins and Rubin (2003). 12. 367 U.S. 643 (1961). The Court held that evidence obtained by searches and seizures violating the Constitution is constitutionally inadmissible in state court proceedings. 13. Polinsky and Shavell (1998) show that when the injurer has a significant chance of escaping liability then punitive (additional, noncompensatory) damages are needed to increase deterrence. 14. 15 U.S.C.A. 78bb (West) (“No person permitted to maintain a suit for damages under the provisions of this chapter shall recover, through satisfaction of judgment in 1 or more actions, a total amount in excess of the actual damages to that person on account of the act complained of.”) (emphasis added); Bernard v. Lombardo, No. 16 CV. 863 (RMB), 2016 WL 7377240, at *4 (S.D.N.Y. November 23, 2016) (“Plaintiffs will not receive punitive damages for their securities fraud claims because it is well established that an award for punitive damages is not permissible for violations of Section 10(b) of the 1934 Act and Rule 10b-5 claims.”) (internal quotations and citations omitted). 15. We abstract here from another potentially important confounding effect, one that is positively correlated with the true impact. Specifically, price impacts of corrective disclosures—whether negative (the corrective disclosure of bad news) or positive (the corrective disclosure of good news)—are biased because the price impact includes expected costs of litigation to the issuer from the circumstances surrounding the corrective disclosure. That is, suppose |$S=F+E(L)$|⁠, where |$S$| is the total reduction following the corrective disclosure, |$F$| is the decline due to the revaluation of the business’s fundamental valuation in light of the corrected information, and |$E(L)$| is the expected total cost of litigation at the time of the corrective disclosure. While we assume here that |$E(L)=0$|⁠, |$E(L)$| may be large in particular applications, a fact that has been ignored in the literature and case law. 16. Consider, e.g., Goldkrantz v. Griffin, 1999 WL 191540, No. 97 Civ. 9075 (DLC) (S.D.N.Y. April 6, 1999), where plaintiff shareholders brought a securities class action against defendants for an alleged misrepresentation. When the defendant company cured the alleged misrepresentation by corrective disclosure in a 10K filing, the stock price fell |$-$|2.64%. Defendants’ expert submitted an event study where the critical return for statistical significance was |$-$|4.41%, so the price fall of |$-$|2.64% was rejected as statistically insignificant. See also Willis v. Big Lots, Inc., 242 F. Supp. 3d 634 (S.D. Ohio 2017) (“Defendants failed to show that there was no statistically significant price impact following the corrective disclosures in this case.”) 17. Prediction 3 reflects a fundamental tension in statistical inference. Making the statistical test more stringent results in fewer false positives—in this case, spurious securities litigation—but increases the severity of the bias in observed price impacts. 18. It is important to note that both firm and market volatility vary over time and that time-varying volatility can also affect inference in single-firm event studies. Baker (2016) documents and explores this topic in detail. Fisch et al. (2018) develop a generalized way to adjust for time-varying volatility that is compatible with the SQ test. 19. Interestingly, courts have rejected event-date returns that were significant at the 10% level without any apparent analysis of the benefits and tradeoffs of a higher level than 5%. See, e.g., In re Intuitive Surgical Sec. Litig., No. 5:13-CV-01920-EJD, 2016 WL 7425926, at *15 (N.D. Cal. December 22, 2016) (“The court finds a lack of price impact in connection with the release of Intuitive’s financial results between April 18 and April 19, 2013. First, neither Lehn nor Coffman found a statistically significant price impact at the 95% confidence level for this date. Rather, Coffman’s analysis resulted in a price impact at a 90% confidence level. Although Plaintiffs argue that price impact at a 90% confidence level is statistically significant, the district court in Halliburton Tex adopted 95% confidence level as the threshold requirement and this court finds no reason to deviate here.”) (citations omitted). 20. See Brav and Heaton (2015) p. 614, n21) (“[O]ne-tailed tests may be more appropriate in testing for the alternative of a price impact that is less than zero (the usual case for a corrective disclosure) or greater than zero (the usual case at the time of a misrepresentation that allegedly inflates the security price); Fisch et al. (2018, p. 589) (“In event studies used in securities fraud litigation, by contrast, price must move in a specific direction to support the plaintiffs’ case. For example, an unexpected corrective disclosure should cause the stock price to fall. Thus, tests of statistical significance based on event study results should be conducted in a ‘one-sided’ way so that an estimated excess return is considered statistically significant only if it moves in the direction consistent with the allegations of the party using the study. The one-sided–two-sided distinction is one that courts and expert witnesses regularly miss, and it is an important one.”) 21. See, e.g., Premium Plus Partners, L.P. v. Davis, 653 F. Supp.2d 855, 867 (N.D. Ill. 2009), aff’d sub nom. Premium Plus Partners, L.P. v. Goldman, Sachs & Co., 648 F.3d 533 (7th Cir. 2011) (“Goldman contends that Donaldson’s comparative analyses are unreliable. Goldman argues that Donaldson should have used a two-tail test rather than a one-tail test. However, Premium has offered sufficient justifications for the use of a one-tail test.) 22. See, e.g., Declaration of Steven P. Feinstein, Ph.D., CFA in Support of Plaintiffs’ Motion for Class Certification, January 28, 2016, ¶128, In re Genworth Financial Inc. Sec. Litig., 2016 WL 4495143 (S.D.N.Y.) (“For each event, a statistical test called a |$t$|-test was conducted to determine whether the residual return of Genworth common stock was statistically significant.”) 23. For a complete technical discussion of the topic see MacKinnon and Smith (1998). 24. Code for the estimators and replicating our results can be found at http://www.davidsonheath.com. 25. To understand the problem, suppose that a stock is trading at $\$$100 due to on-going but unknown misrepresentations. A corrective disclosure is made, and the price falls to $\$$92. If the expected cost of private securities litigation (assume no regulatory action is expected) is $\$$2 per share and the expected decline in the value of the business from the now-disclosed truth is $\$$6/share, then only $\$$6 should be attributed to any alleged fraud; $\$$2 of the price move reflects expected corporate losses from compensating the alleged fraud. To count that as fraud overcompensates damages for those who purchased stock at inflated prices (or bought calls at inflated prices or wrote puts at deflated prices). Moreover, the overall price impact must then be the solution to an iterative problem, where the final price impact is the point of convergence to the problem: “If the actual price impact is $\$$6, but is overestimated as $\$$8 for litigation purposes, then the expected cost of the litigation will be $\$$2,” that is, the overestimation of damages itself raises the cost of litigation to the issuer. The dynamic plays out differently when the price impact is positive. Suppose that a stock is trading at $\$$100 due to on-going but unknown misrepresentations. A corrective disclosure is made, and the price increases to $\$$108. If the expected cost of private securities litigation (assume no regulatory action is expected) is $\$$2 per share and the expected increase in the value of the business from the now-disclosed truth is $\$$10/share. Since the plaintiffs in this case will be those that sold their stock early (or, who purchased put options at inflated prices or wrote calls at deflated prices) then $\$$10 should be attributed to any alleged fraud; $\$$2 of the price move reflects expected corporate losses from compensating the alleged fraud. To count only $\$$8 as the price impact will under-compensate plaintiffs. References Andrews, D. W. K. 1993 . “ Exactly Median-Unbiased Estimation of First Order Autoregressive/Unit Root Models, ” 61 Econometrica 139 – 165 . http://www.jstor.org/stable/2951781. Google Scholar Crossref Search ADS Atkins, R. , and Rubin P. . 2003 . “ Effects of Criminal Procedure on Crime Rates: Mapping Out the Consequences of the Exclusionary Rule, ” 46 The Journal of Law and Economics 157 – 79 . Google Scholar Crossref Search ADS Baker, A. C. 2016 . “ Single-firm Event Studies, Securities Fraud, and Financial Crisis: Problems of Inference, ” 68 Stanford Law Review 1207 – 61 . Brav, A. , and Heaton J. B. . 2015 . “ Event Studies in Securities Litigation: Low Power, Confounding Effects, and Bias, ” 93 Washington University Law Review 583 – 614 . http://openscholarship.wustl.edu/law_lawreview/vol93/is s2/15. Dybvig, P. H. , Gong N. , and Schwartz R. . 2000 . “ Bias of Damage Awards and Free Options in Securities Litigation, ” 9 Journal of Financial Intermediation 149 – 68 . Google Scholar Crossref Search ADS Fisch, J. E. , Gelbach J. B. , and Klick J. . 2018 . “ The Logic and Limits of Event Studies in Securities Fraud Litigation, ” 96 Texas Law Review 495 – 541 . Gelbach, J. B. , Helland E. , and Klick J. . 2013 . “ Valid Inference in Single-Firm, Single-Event Studies, ” 15 American Law and Economics Review 553 – 621 . Google Scholar Crossref Search ADS Kliger, D. , and Gurevich G. . 2014 . Event Studies for Financial Research: A Comprehensive Guide . New York : Palgrave Macmillan . https://www.palgrave.com/us/book/9781137435385. Kothari, S. P. , and Warner J. P. . 2007 . “ Econometrics of Event Studies ” in Espen Eckbo, B. ed., Handbook of Corporate Finance: Empirical Corporate Finance . North-Holland : Elsevier , 3 – 36 . http://www.sciencedirect.com/science/article/pii/B97804 44532657500159. MacKinlay, A. C. 2007 . “ Event Studies in Economics and Finance, ” 35 Journal of Economic Literature 13 – 39 . MacKinnon, J. G. , and Smith Jr A. A. . 1998 . “ Approximate Bias Correction in Econometrics, ” 85 Journal of Econometrics 205 – 30 . Google Scholar Crossref Search ADS Polinsky, A. M. , and Shavell S. . 1998 . “ Punitive Damages: An Economic Analysis, ” 111 Harvard Law Review 869 – 962 . Google Scholar Crossref Search ADS © The Author(s) 2019. Published by Oxford University Press on behalf of the American Law and Economics Association. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png American Law and Economics Review Oxford University Press

Bias-Corrected Estimation of Price Impact in Securities Litigation

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Abstract

Abstract The single-firm event studies that securities litigants use to detect the impact of a corrective disclosure on a firm’s stock price have low statistical power. As a result, observed price impacts are biased against defendants and systematically overestimate the effect on firm value. We use the empirical distribution of daily stock returns to analyze the bias and develop bias-corrected estimators of price impact in securities litigation. Because of low statistical power, the ex ante incentives against committing securities fraud are also too low. We analyze the adjustment for optimal deterrence and find that it is material, but is nowhere equal to the opposing truncation bias. 1. Introduction A small but growing literature1 examines problems with the single-firm event studies that became ubiquitous in federal securities litigation following the decision of the U.S. Supreme Court in Basic v. Levinson.2 In Basic, a case under Section 10(b) of the Securities Exchange Act of 1934, the Court endorsed the “fraud-on-the-market” doctrine. The fraud-on-the-market doctrine is a presumption that “the market price of shares traded on well-developed markets reflects all publicly available information, and, hence, any material misrepresentations.”3 The major implication of this doctrine is that a false disclosure or omission in well-developed securities markets will be reflected in the security’s price regardless whether every individual trader is aware of the misrepresentation. A later “corrective disclosure”—often an admission by the security issuer—then reveals that previous statement or omission was false or misleading.4 Investors usually sue the maker of the false statement or omission, alleging that the misstatement or omission was the result of an intent to deceive investors. Litigants in such cases nearly always offer the opinion evidence of expert witnesses who conduct event studies on the security price at issue. An event study is a statistical methodology used by academic researchers to determine whether corporate events such as stock splits, merger announcements, or dividend changes are associated with a statistically significant change in the stock prices of companies subject to the event.5 After Basic, securities litigants use event studies to answer two critical questions: First, did the corrective disclosure cause a price impact in the security at issue?6 Second, if there was a price impact, how much of it was due to the corrective disclosure as opposed to other causes such as broad market movements—a fact relevant to loss causation and damages, among other things?7 Event studies have become a key part of almost every securities case.8 Event studies in securities litigation differ importantly from event studies in academic work. When researchers apply the event study methodology in academic work, they examine the price impact of some set of events on a sample of many firms subject to that type of event. When expert witnesses apply the event study methodology in litigation, they examine the price impact of a single event—the corrective disclosure—at a single firm. Recently, Brav and Heaton (2015) argued that greater attention should be given to three methodological problems with these “single-firm event studies” as used in securities litigation.9 The first problem is low statistical power: It is difficult for single-firm event studies to detect price impacts that actually exist, because price impacts must be quite large to be detected when only one event is in the sample. The second issue is confounding effects: Without many firms in the sample to average away price moves that were unrelated to the event at issue, the single-firm event study is a noisy measure of the price impact. The third problem that Brav and Heaton identified was new in the literature: low power and confounding effects combine to generate biased measures of price impact. That is, the price impact estimates from single-firm event studies are systematically too large. The reason is straightforward. When statistical power is low, a small price impact will be detected only when confounding effects—price changes unrelated to the corrective disclosure—combine with the disclosure’s true price impact to push the total price change (the sum of confounding effects plus the true price impact) past the threshold of statistical significance. Suppose, for example, that the true price impact is |$-$|2.0%, but only returns larger in magnitude than |$-$|2.5% are statistically significant. On its own the true price impact would not be statistically significant. But suppose there are |$-$|0.6% of additional negative confounding effects. This pushes the total observed price impact past the threshold to |$-$|2.6%. In effect, the event returns that are statistically significant are drawn from a distribution that is truncated above at the threshold of statistical significance. The expected value of a draw from that truncated distribution is not equal to the true price impact, which comes from the untruncated distribution. Thus, conditional on statistical significance, the stock return on the event date overestimates the event’s price impact. The practical effect of this bias is to drive up compensatory damages calculations and settlements. This article expands on Brav and Heaton (2015)’s insight in two ways. First, Brav and Heaton illustrate their argument using only simulations that assume normally distributed returns.10 We estimate and study the bias in a more realistic fashion using the empirical distribution of actual daily stock returns, which are non-normally distributed. We find that the bias is often material, especially for small price impacts and high volatility stocks, and that the bias is larger the lower (i.e., more stringent) the threshold of statistical significance. Second, we present methods to correct for the bias. We develop and validate bias-corrected estimators of the true price impact. We analyze their performance and find that while all five improve on the uncorrected event date return, a median bias-corrected (MBC) estimator performs best. We provide procedures and make code available that enables the production and validation of bias-corrected estimates of price impact in single-firm event studies. While our article is in one sense narrowly methodological—proposing a method of correcting one statistical bias that is present in securities litigation event studies—the correction we propose raises difficult questions. After all, a consequence of the low statistical power of single-firm event studies is that many smaller (but still economically important) securities frauds will go undetected because they do not reach statistical significance. This means that the ex ante incentives against committing securities fraud are already too low. Here, we propose a method to ensure the frauds that are detected are not overestimated. This arguably further reduces the ex ante incentives against committing securities fraud. Thus, the methods we develop in this article make price impact estimates in securities litigation more accurate, even as they may reduce the deterrent effect of securities litigation. A precedent for this discussion is the problem of unlawful search and seizure under the Fourth Amendment. Scholars have documented an increase in crime after the U.S. Supreme Court greatly limited the use of evidence from unlawful search and seizure in Mapp v. Ohio.11|$^{,}$|12 Nevertheless, many would argue the answer is better enforcement that does not rely on unlawful search and seizure, not the continued admissibility of illegally obtained evidence. Here as well, we argue that the answer is explicit punitive damages13 in securities litigation, not back-door additional damages in the form of badly applied methodology. Punitive damages are, however, impermissible under current federal securities law.14 We show that our methods also yield an accurate estimate of the optimal deterrence factor as developed by Polinsky and Shavell (1998), and our estimates yield two conclusions. First, the need for additional damages to deter securities fraud is widespread. Second, the upward adjustment for optimal deterrence is nowhere equal to the downward adjustment for truncation bias. If lack of deterrence is a problem—and the data suggest that it is—the answer is to properly correct for truncation bias, which reduces compensatory damages, and then to optimize deterrence by awarding additional punitive damages. If securities law should change to make explicit punitive damages permissible, the methods developed in this article yield accurate estimates of both the bias correction and the optimal deterrence adjustment. 2. Statistical Bias in Single-Firm Event Studies Suppose a company makes a corrective disclosure, say an announcement of a restatement of earnings, that reduces the market’s valuation of the firm’s equity by the fraction |$S$|⁠. We cannot observe the true price impact |$S$| because price movement on the event date also occurs for reasons unrelated to the corrective disclosure including the impact of other information relevant to firm value, trading by investors seeking to invest or divest the stock for other reasons, and trading by noise traders for unexplained reasons. As a result, we observe a stock return on the event date that includes both the price impact of the corrective disclosure and the net effect of unrelated price movements: \begin{equation} r^{EVENT} = S + r_t. \end{equation} (1) We assume that the distribution of the noise term |$r_t$| is the distribution of daily returns on nonevent dates (i.e., days without corrective disclosures). Thus, on the day of the corrective disclosure (the “event date”) we observe the price impact |$S$| plus a draw from the nonevent return distribution.15 Courts generally require that plaintiffs in a securities case demonstrate that the event-date return is statistically significant.16 As a result, the set of litigated negative event-date returns is truncated above at the threshold of statistical significance. Thus a litigated event-date return represents a draw from the distribution: \begin{equation} r^{OBSERVED} \in \{ r^{EVENT} \, | \, r^{EVENT}<T_{p} \}, \end{equation} (2) where |$T_p$| is the truncation threshold at significance level |$p$|⁠, i.e., the return threshold that the corrective disclosure return must exceed (in absolute value) to be statistically significant. Because the observed distribution is truncated above, the expected value of |$r^{OBSERVED}$| is lower (more negative) than the true price impact |$S$|⁠. On average, the corrective disclosure return overestimates the negative magnitude of the price impact with the following bias: \begin{align} bias(\tilde{r}, T_p, S) &= E[r^{EVENT} \, | \, r^{EVENT} < T_{p}] -S\nonumber\\ &= E[S + r_t \, | \, S + r_t<T_{p}] -S\nonumber\\ &= E[r_t \, | \, r_t<T_{p}-S]. \end{align} (3) Without any distributional assumptions, we can make three predictions: Prediction 1: The bias is larger the larger is the dispersion or volatility of the stock’s returns. (This follows because higher dispersion increases the probability of truncation but does not change the unconditional expectation.) Prediction 2: The bias is smaller the larger is the true price impact |$S$|⁠. (This follows because the partial derivative with respect to |$S$| is negative.) Prediction 3: The bias is larger the lower is the threshold |$T_p$|⁠, i.e., the smaller is |$p$|⁠. (This follows because smaller |$p$| increases the probability of truncation.)17 We now turn to examining the potential magnitude of this bias using the empirical distribution of actual daily market-adjusted stock returns. 2.1. Data Our data are from the Center for Research in Security Prices (“CRSP”). We extract daily returns for all U.S. common stocks in CRSP from 2010 to 2015 and adjust the returns for each stock using the standard market model: \begin{equation} R^i_t = \alpha^i + \beta^i R^{Mkt}_t + r^i_t, \end{equation} (4) where |$R^i_t$| is the daily return to stock |$i$| on date |$t$| |$R^{Mkt}_t$| is the daily return to the CRSP value-weighted market portfolio on date |$t$| |$r^i_t$| is the daily market-adjusted return to stock |$i$| on date |$t$| To test Prediction 1, we divide the sample stocks into low volatility |$(\sigma < 4\%)$| and high volatility |$(\sigma > 4\%)$| groups. All our results are similar if we use a different breakpoint such as 3 or 5%.18 From each of the two groups we randomly sample, without replacement, 10,000 blocks where each block consists of 100 consecutive market-adjusted daily returns for a single stock. 2.2. Estimating the Bias For each block, we compute significance thresholds using the 100 daily returns as the distribution of nonevent daily returns. Next, we add a simulated price impact |$S$| to all 100 returns to simulate the distribution of event-date returns. To mimic the selection effects of requiring litigated cases to have a statistically significant event-date return, we then drop simulated event-date returns that are not significant according to a two-sided |$t$|-test with |$P<0.05$|⁠, which produces a truncated distribution of event-date returns. We estimate the bias by comparing the mean of 100 draws with replacement from the truncated distribution to the mean of the untruncated distribution. Finally, we take the mean of the estimated bias across all 10,000 blocks, which yields the mean expected bias across one million simulated single-firm event studies. Figure 1 plots the mean event-date return, conditional on a significant |$t$|-test, across a range of simulated price impacts |$S$| within the high volatility (Figure 1a) and low volatility (Figure 1b) groups. The vertical distance from the 45-degree line equals the expected bias. For low volatility stocks, there is essentially no bias for price impacts larger than |$-$|8%. This is intuitive: for low volatility stocks a large event-date return is almost surely due to a large price impact and not due to nonevent noise. For low volatility stocks, there is still potentially material bias for price impacts of |$-$|8% or smaller. Figure 1. View largeDownload slide Truncation Bias in Simulated Event Studies. Each figure plots the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$|⁠, over a range of simulated true price impacts. The figures show the average across 1 million simulated single-firm event studies using CRSP stocks with low volatility stocks (⁠|$\sigma < 4\%$|⁠) (a) and high volatility stocks (⁠|$\sigma > 4\%$|⁠) (b) of daily market-adjusted returns. The vertical distance from the 45-degree line equals the mean bias. Figure 1. View largeDownload slide Truncation Bias in Simulated Event Studies. Each figure plots the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$|⁠, over a range of simulated true price impacts. The figures show the average across 1 million simulated single-firm event studies using CRSP stocks with low volatility stocks (⁠|$\sigma < 4\%$|⁠) (a) and high volatility stocks (⁠|$\sigma > 4\%$|⁠) (b) of daily market-adjusted returns. The vertical distance from the 45-degree line equals the mean bias. For high volatility stocks, the bias is material across the full range of simulated price impacts. Even with a large price impact of |$-$|25% the mean event-date return is |$-$|26.03%, reflecting a mean bias of more than 1% of market capitalization. The larger bias in higher volatility stocks agrees with our Prediction 1, and in all cases the bias is larger for a smaller price impact, which agrees with Prediction 2. 2.3. Alternative Significance Levels Figure 2 shows the bias in the group of high volatility stocks when we use different thresholds of statistical significance. Relative to our first threshold of |$P<0.05$|⁠, a less strict threshold of |$P<0.10$| generates a smaller bias (dashed line).19 However, the bias is still present and material across the full range of simulated price impacts. Figure 2. View largeDownload slide Bias Using Different Significance Thresholds. The figure graphs the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$| and |$P<0.10$|⁠, as well as when the truncating threshold is set at zero (negative returns only). The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. Figure 2. View largeDownload slide Bias Using Different Significance Thresholds. The figure graphs the mean event-date return conditional on a significant two-tailed |$t$|-test with |$P<0.05$| and |$P<0.10$|⁠, as well as when the truncating threshold is set at zero (negative returns only). The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. Figure 2 further shows that a bias is present when we impose the mere requirement that the event-date return be negative. Even ignoring the typical requirement to demonstrate statistical significance at the 5% level, it is unlikely that a securities case would be brought if the event-date return were positive, and this much weaker condition also causes truncation of the observed event-date returns. The resulting bias is smaller but still clearly present across the full range of simulated price impacts. We conclude that, first, the pattern of statistical bias is consistent with our Prediction 3 above, and second, that even the weakest assumption on the extent of truncation results in material bias. 2.4. One-Tailed |$t$|-Test Our results above use a standard two-tailed |$t$|-test to evaluate statistical significance. A one-tailed test is unquestionably more appropriate in securities litigation,20 as it improves statistical power and reflects that the null hypothesis being tested is usually one-tailed, e.g., that the event return is nonnegative. However, the use of one-tailed tests is often contested.21 What effect does the use of a one-tailed test have on the statistical bias? Table 1 Column 1 shows the mean bias in our baseline specification, using a two-tailed |$t$|-test with |$P<0.05$|⁠. This is equivalent to a one-tailed |$t$|-test with |$P<0.025$|⁠. Column 2 shows the mean bias when we use a one-tailed |$t$|-test with |$P<0.05$| (equivalent to a two-tailed |$t$|-test with |$P<0.10$|⁠) instead. As in Figure 2, we see that the bias is smaller in all cases. Thus, our results highlight an additional benefit of using a one-tailed test: in addition to its being the clear correct choice in securities litigation, a one-tailed test makes the truncation less severe, which reduces the statistical bias when measuring price impact. Table 1. Bias Using Alternative Significance Tests |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% The table compares the mean statistical bias across a range of price impacts when we evaluate statistical significance with a two-tailed |$t$|-test (baseline specification) versus a one-tailed |$t$|-test or the SQ test of Gelbach et al. (2013). The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. View Large Table 1. Bias Using Alternative Significance Tests |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% |$t$|-Test |$t$|-Test SQ Test SQ Test Two-tailed: |$P<0.05$| |$P<0.10$| |$P<0.05$| |$P<0.10$| One-tailed: |$P<0.025$| |$P<0.05$| |$P<0.025$| |$P<0.05$| Price Impact Mean Statistical Bias |$-$|30% |$-$|0.87% |$-$|0.81% |$-$|0.41% |$-$|0.27% |$-$|25% |$-$|1.03% |$-$|0.97% |$-$|0.68% |$-$|0.38% |$-$|20% |$-$|1.49% |$-$|1.27% |$-$|1.15% |$-$|0.63% |$-$|15% |$-$|2.55% |$-$|1.98% |$-$|2.21% |$-$|1.15% |$-$|10% |$-$|5.23% |$-$|3.85% |$-$|4.60% |$-$|2.47% |$-$|5% |$-$|10.37% |$-$|8.31% |$-$|9.25% |$-$|5.64% The table compares the mean statistical bias across a range of price impacts when we evaluate statistical significance with a two-tailed |$t$|-test (baseline specification) versus a one-tailed |$t$|-test or the SQ test of Gelbach et al. (2013). The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. View Large 2.5. The SQ Test The |$t$|-test is widely used in securities-litigation event studies.22 However, Gelbach et al. (2013) find that the |$t$|-test can be poorly suited to statistical testing using daily stock returns because daily stock returns are skewed and fat-tailed, contradicting the distributional assumptions for a valid |$t$|-test. They present an alternative test based on sample quantiles, the SQ test, that does not rely on distributional assumptions. Table 1 Columns 3 and 4 show the mean bias using the SQ test with |$P<0.025$| and |$P<0.05$| to evaluate statistical significance. Comparing to Columns 1 and 2, we see that in all cases using the SQ test the bias is reduced relative to the |$t$|-test. This result is intuitive. As Gelbach et al. (2013) document, the non-normal distribution of daily stock returns leads the |$t$|-test to systematically underreject the null hypothesis, that is, to classify too many event-date returns as nonsignificant. This worsens the bias when the |$t$|-test is used. Thus, our results highlight an additional benefit of Gelbach et al. (2013)’s SQ test for single-firm event studies: the SQ test is both more accurate and reduces the statistical bias when measuring price impact. 2.6. Deterrence and Punitive Damages Next we use our analysis and data to examine the effect of low statistical power on deterrence. Polinsky and Shavell (1998) argue that when the injurer has a significant chance of escaping liability then punitive (additional, noncompensatory) damages are needed to increase deterrence. Thus, our results suggest that it would be socially optimal to increase the damages awarded in securities litigation above the bias-corrected best estimate. To do so is not currently permissible because punitive damages are unavailable to private securities litigants. Nevertheless, we can examine how material the deterrence effects and the appropriate punitive damages are in securities litigation. According to the formula derived in Polinsky and Shavell (1998), the total damages imposed “...should equal the harm multiplied by the reciprocal of the probability that the injurer will be found liable when he ought to be.” Denote the appropriate punitive damages by |$\mathcal{P}$|⁠: \begin{equation} \text{Total damages} = S + \mathcal{P} = \frac{S}{Prob[foundliable]} \end{equation} Rearranging, \begin{equation} \mathcal{P} = S \, \frac{1-Prob[foundliable]}{Prob[foundliable]} = S \, \frac{Prob[S + r_t \geq T_{p}]}{Prob[S + r_t<T_{p}]} \end{equation} (5) Figure 3 plots the statistically appropriate adjustment for deterrence across a range of simulated true price impacts, using our data drawn from CRSP daily stock returns. There are two main conclusions we draw from these results. First, the appropriate deterrence adjustment is material across the entire range of price impacts. Thus, the data suggest that the need for additional punitive damages to deter securities fraud is widespread. Second, the deterrence adjustment is nowhere equal to the truncation bias. Thus, these two countervailing motives—to correct for the downward truncation bias, which reduces the damages, and to optimize deterrence by awarding additional damages—do not cancel each other out, and must be estimated and incorporated separately into the final damages awarded. Figure 3. View largeDownload slide Statistically Appropriate Punitive Damages. The figure graphs the statistically appropriate punitive damages according to the formula of Polinsky and Shavell (1998), expressed as a fraction of the firm’s equity value, across a range of true price impacts. The bias due to statistical truncation is also presented for comparison. The distribution in each case is based on 1 million simulated single-firm event studies across the sample of all CRSP stocks. Figure 3. View largeDownload slide Statistically Appropriate Punitive Damages. The figure graphs the statistically appropriate punitive damages according to the formula of Polinsky and Shavell (1998), expressed as a fraction of the firm’s equity value, across a range of true price impacts. The bias due to statistical truncation is also presented for comparison. The distribution in each case is based on 1 million simulated single-firm event studies across the sample of all CRSP stocks. 3. Bias-Corrected Estimators We next examine how to correct for the bias in estimated price impacts in single-firm event studies. The basic task is to correct a consistent estimator for finite sample bias. In the original setting of event-study methodology—academic studies of the impact of a particular class of events such as stock splits—there may be several hundred firms that experience the event in question. In that setting, we observe hundreds of draws from the event-date distribution, so as long as the estimator is consistent—i.e., converges to an unbiased estimate as the number of observations grows—the bias in the estimate due to truncation of individual draws is negligible. The situation is very different in a securities litigation setting. We are typically examining a single event at a single firm. It is easy to see that finite sample bias is a major concern, because the “treated” set consists of a single daily return. It is simple to compute the bias for a known price impact. That is, given a value of the true price impact |$S$|⁠, we simply evaluate the expectation (3) as in Figure 1. The challenge is that in practice we do not know the true price impact and must estimate |$S$| given the event-date return that we observe. Fortunately, correcting for finite sample bias is an empirical question with a long history in econometrics, and many approaches have been proposed.23 Different approaches yield different estimates of |$S$| and the bias. We develop five bias-corrected estimators, and then examine how they perform in practice. 3.1. First-Order Corrections The first two bias correction methods, analytical bias corrected (ABC) and constant bias corrected (CBC), set the true price impact equal to the observed value |$r^{EVENT}$| (which is of course not the case). They then compute the bias and use it as an approximation to the bias at |$S$|⁠. In effect, this is equivalent to assuming that the bias is constant for values of |$S$| that are close to the observed value |$r^{EVENT}$|⁠. 3.1.1. Analytical Bias Correction We first compute an analytical bias correction assuming that returns are normally distributed. The bias in the mean of a truncated normal distribution is: \begin{equation} bias = E[x | x < T_p] - \mu \, , \, x \sim N(\mu,\sigma) = - \phi \biggl(\cfrac{T_p - \mu}{\sigma}\biggr) / \Phi \biggl(\cfrac{T_p - \mu}{\sigma}\biggr) \end{equation} and the ABC estimate of the bias is: \begin{equation} = - \phi \biggl(\cfrac{T_p - r^{EVENT}}{\sigma}\biggr) / \Phi \biggl(\cfrac{T_p - r^{EVENT}}{\sigma}\biggr) \end{equation} (6) Thus, the ABC estimator assumes that returns are normally distributed, and uses the bias evaluated at |$r^{EVENT}$| as an approximation of the bias at |$S$|⁠. 3.1.2. Constant Bias Correction We next examine the CBC estimator of MacKinnon and Smith (1998). For each simulated event study, we follow the steps: (1) Compute the threshold of statistical significance |$T_p$| based on the set of nonevent returns. (2) Shift the set of nonevent returns downward by the event-date return |$r^{EVENT}$|⁠, yielding distribution |$D_{all}$|⁠. (3) Drop any observations in |$D_{all}$| that are above |$T_p$|⁠, yielding distribution |$D_{truncated}$|⁠. (4) Compute the means of |$D_{all}$| and |$D_{truncated}$|⁠. The difference between the two means is the CBC estimate of the bias. Thus, the CBC estimator computes the bias as in Figure 1, evaluated at the observed event-date return |$r^{EVENT}$|⁠. This will yield an accurate estimate of the bias and the true price impact |$S$| as long as the bias evaluated at the true |$S$| is the same as the bias evaluated at |$r^{EVENT}$|⁠. 3.2. Higher Order Corrections The first two bias corrections, ABC and CBC, remove first-order bias in the measurement. However, if their implicit assumption that the bias is constant is inaccurate then they will not remove all bias in expectation. We explore higher order estimators, which can remove all bias in expectation. However, because they rely on more assumptions and computations, they can be less robust in practice. 3.2.1. Median Bias Correction We next examine a median bias-corrected estimator (MBC) following Andrews (1993). For each simulated event study we examine a grid of true price impacts |$S^*$| around the observed event-date return |$r^{EVENT}$|⁠. For each value of |$S^*$| we shift the nonevent distribution to the left by |$S^*$|⁠, truncate it above the threshold, and compute the median of the truncated distribution. We then pick the most negative (i.e., most conservative) value of |$S^*$| with median equal to |$r^{EVENT}$|⁠. 3.2.2. Linear and Nonlinear Bias Correction Finally, we examine the linear bias-corrected (LBC) and nonlinear bias-corrected (NBC) estimators of MacKinnon and Smith (1998). In both cases we begin with the CBC estimate of the bias, denoted |$S_1$|⁠. We then “move up” and compute a second bias estimate in the same fashion but evaluated at |$S_1$| instead of at |$r^{EVENT}$|⁠. Denote the second bias-corrected estimate of the price impact |$S_2$|⁠. We take the difference between the two estimates and extrapolate linearly to arrive at the LBC estimate. We continue iterating until successive estimates converge to arrive at the NBC estimate. There is one issue which makes the application of the LBC and NBC challenging: for small price impacts, the bias increases more than proportionately as the true price impact shrinks (see Figure 2). That is, the inverse of the bias function may not have a unique root. This issue does not affect the ABC or CBC estimators as the bias is evaluated at a single point, the observed |$r^{EVENT}$|⁠. Likewise, the MBC appears to be quite numerically stable in our setting, because we have a simple heuristic in place that avoids the case of multiple roots. However, the LBC and NBC estimators look for roots of the inverse bias function in a general way and so numerical convergence can be an issue with small price impacts. We deal with this issue by identifying individual cases where the LBC and NBC are converging poorly and in these cases we default to the CBC estimate. 4. Performance of the Bias-Corrected Estimators As the bias is most common and most material in high volatility stocks, we compare the estimators in that group of stocks. We evaluate the performance of the estimators as follows. We first specify a true price impact |$S$|⁠. Within each block of 100 returns, we shift the returns downward by |$S$| and truncate above at |$T_p$|⁠. We draw 100 simulated event-date returns from the truncated distribution. We bias-correct each of the simulated event-date returns, then compute the difference between the bias-corrected estimate and the true price impact |$S$|⁠. Across our 10,000 blocks this generates 1 million measurements of (1). the true price impact, (2) the uncorrected event-date return, and (3) the bias-corrected estimate. Across a range of simulated true price impacts, we then characterize the distribution of the bias-corrected estimates. Figure 4 plots the mean and median bias-corrected estimates for our five estimators across a range of simulated price impacts. The 45 degree line corresponds to an entirely unbiased estimate. For price impacts of |$-$|15% or more, all the estimators recover the true price impact quite accurately as measured both by means and medians. In contrast, for price impacts smaller than |$-$|15%, all five estimators fail to eliminate the bias, although all reduce the bias significantly relative to the uncorrected event-date return. Overall the median bias-corrected (MBC) estimator performs best in terms of the mean bias, while the LBC estimator performs slightly better in terms of the median bias. Figure 4. View largeDownload slide Performance of Bias-Corrected Estimators The figure plots mean and median estimates of the true price impacts for a variety of bias correction regimes, across a range of simulated true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. (a) Mean estimates for ABC and CBC. (b) Mean estimates for MBC, LBC and NBC. (c) Median estimates for ABC and CBC. (d) Median estimates for MBC, LBC and NBC. Figure 4. View largeDownload slide Performance of Bias-Corrected Estimators The figure plots mean and median estimates of the true price impacts for a variety of bias correction regimes, across a range of simulated true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for the group of high volatility CRSP stocks. The vertical distance from the 45-degree line equals the expected bias. (a) Mean estimates for ABC and CBC. (b) Mean estimates for MBC, LBC and NBC. (c) Median estimates for ABC and CBC. (d) Median estimates for MBC, LBC and NBC. Table 2 compares the mean and median residual bias for our five bias-corrected estimators across a range of simulated true price impacts. The first column (“Uncorrected”) shows the statistical bias when we simply use the event-date return as our estimate of the price impact. The next five columns show the performance of the ABC, CBC, MBC, LBC, and NBC estimators. In each row the first- and second-best performing estimators are denoted by superscripts. Table 2. Performance of Bias-Corrected Estimators Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% The table presents means and medians of the bias induced by requiring statistical significance via two-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table 2. Performance of Bias-Corrected Estimators Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.61% |$-$|0.12% |$-$|0.20% 0.03%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$|  |$-$|25% |$-$|0.84% |$-$|0.15% |$-$|0.28% |$-$|0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|1.31% |$-$|0.21% |$-$|0.48% |$-$|0.08%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.11%  |$-$|15% |$-$|2.28% |$-$|0.52% |$-$|1.03% |$-$|0.16%|$^{\rm a}$| |$-$|0.17%|$^{\rm b}$| |$-$|0.45%  |$-$|10% |$-$|4.63% |$-$|2.12% |$-$|2.84% |$-$|0.98%|$^{\rm a}$| |$-$|1.41%|$^{\rm b}$| |$-$|2.17%  |$-$|5% |$-$|9.45% |$-$|6.81% |$-$|7.49% |$-$|5.03%|$^{\rm a}$| |$-$|5.84%|$^{\rm b}$| |$-$|6.78% (b) Median bias  |$-$|30% |$-$|0.36% |$-$|0.15% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.42% |$-$|0.11% |$-$|0.05%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm a}$|  |$-$|20% |$-$|0.59% 0.00%|$^{\rm a}$| |$-$|0.02% |$-$|0.01%|$^{\rm b}$| 0.12% 0.04%  |$-$|15% |$-$|1.08% 0.17% |$-$|0.13% |$-$|0.04%|$^{\rm b}$| 0.25% |$-$|0.01%|$^{\rm a}$|  |$-$|10% |$-$|2.69% |$-$|0.42% |$-$|1.06% |$-$|0.37%|$^{\rm b}$| |$-$|0.03%|$^{\rm a}$| |$-$|1.01%  |$-$|5% |$-$|6.84% |$-$|4.03% |$-$|4.78% |$-$|3.11%|$^{\rm a}$| |$-$|3.22%|$^{\rm b}$| |$-$|4.92% The table presents means and medians of the bias induced by requiring statistical significance via two-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large As in Figure 4, all five estimators improve on the raw event-date return in terms of both mean and median bias. Evaluated in terms of the mean bias (Figure 4a), for large price impacts all three higher-order estimators (MBC, LBC, and NBC) perform best about equally well. For small price impacts of |$-$|10% or |$-$|5%, again as in Figure 4, the MBC estimator outperforms the rest. This pattern appears to arise because the MBC estimator is (1) a higher-order consistent estimator, meaning that it removes all bias in expectation and (2) numerically stable, even though it does not target the mean bias directly. Evaluated in terms of the median bias (Table 2b), the MBC estimator performs marginally better than the LBC and NBC for large price impacts and significantly better for small price impacts, which is not surprising since the MBC is the only estimator that targets the median bias directly while the other five all attempt to reduce the mean bias. In the Appendix, we compare the performance of the estimators when statistical significance is evaluated via one-tailed |$t$|-test or an SQ test instead. In both cases, the results are very similar to those in Table 2. In both alternative cases, there is less bias to correct, and all the estimators perform better. These results again support the use of one-tailed tests and the SQ test in practice. Whether an estimator’s performance should be evaluated in terms of mean or median bias or some other metric will depend on the user’s objective (or loss function, in statistical terms) as well as the specific data and significance criteria, a potential topic for future research. We further note that in practice there is no need to choose only one estimator. A researcher or expert witness carrying out a single-firm event study could compute bias-corrected estimates, for the case-specific data and significance criteria, using multiple estimators and then compare the resulting estimates.24 5. Conclusion Brav and Heaton (2015) raise the issue of potential bias in price impact estimates in securities litigation. This bias arises because price impacts must be statistically significant to be actionable, but low power and confounding effects in single-firm event studies make it likely that statistically significant event-date returns reflect more than the true price impact of the event under examination. We run simulations using the empirical distribution of daily stock returns and show that under very general conditions there is often material bias relative to the true price impact. We develop and evaluate bias-corrected estimators for single-firm event studies. All five improve on the uncorrected event-date return as an estimate of the true price impact. Which estimator performs best will depend on the objective function, the specific sample, and the significance criterion used, but we conclude that the MBC based on Andrews (1993) performs the best and most consistently. This article addresses one specific source of bias which is readily quantifiable. Future research could address an additional bias that we have not addressed here: The price impact of a corrective disclosure, in an efficient market, includes a component that reflects the burden of the anticipated litigation that corrective disclosure will generate. For corrective disclosures that cause price declines, even the true price impact overstates the harm. This occurs because the costs to the issuer of any anticipated litigation fall on the issuer, but the benefits accrue only to those who bought or sold prior to the corrective disclosure. That is, the prices paid after a corrective disclosure reflect the new valuation of the security in light of corrected information and the expected costs of litigation, settlements, and fines, but do not reflect any benefits of litigation (such as expected settlements) because the right to compensation on a securities fraud claim does not pass to the new buyer. We are aware of no reported case that adjusts for this bias and scholarship on securities fraud price impact and damages calculations largely ignores it. While quantification of this effect will be difficult, it remains a necessary step in increasing the reliability of financial-market evidence in securities litigation.25 Further, there is a clear argument that the low statistical power of single-firm event studies means that additional punitive damages are justified to deter securities fraud. Our data and methods allow us to put numbers to this intuition. Our results suggest that the need for deterrence adjustments in securities litigation is widespread. We show that our methods yield a simple calculation of the statistically appropriate deterrence adjustment in securities litigation—should explicit punitive damages ever become legally permissible. In sum, our paper provides both new insights and new tools for litigants, expert witnesses, and other users of single-firm event studies. A. Appendix This appendix provides additional results and extensions to the main text. A.1. Performance of Bias-Corrected Estimators Using Different Statistical Tests This section compares the performance of our five bias-corrected estimators using different criteria of statistical significance than in the main text. In the main text we evaluate statistical significance using a two-tailed |$t$|-test with |$P<0.05$|⁠. The results are all consistent with our main findings. The main difference to our main findings is that when the SQ test is used instead of a |$t$|-test, the first-order bias corrections (ABC and CBC) perform relatively well especially for large price impacts. Table A.1 presents results when we use a one-tailed |$t$|-test with |$P<0.05$|⁠. Evaluated in terms of the mean bias (Table A.1a), for large price impacts all three higher-order estimators (MBC, LBC, and NBC) perform best about equally well while for small price impacts of |$-$|10% or |$-$|5% the MBC estimator significantly outperforms the others. Evaluated in terms of the median bias (Table 2b), the MBC estimator performs marginally better than the LBC and NBC for large price impacts and significantly better for small price impacts. Table A.1. Performance of Bias-Corrected Estimators Using a One-Tailed |$t$|-Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% The table presents means and medians of the bias induced by requiring statistical significance via one-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table A.1. Performance of Bias-Corrected Estimators Using a One-Tailed |$t$|-Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.50% |$-$|0.12% |$-$|0.15% 0.04%|$^{\rm a}$| |$-$|0.05%|$^{\rm b}$| |$-$|0.06%  |$-$|25% |$-$|0.68% |$-$|0.11% |$-$|0.20% |$-$|0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| |$-$|0.06%  |$-$|20% |$-$|1.00% |$-$|0.10% |$-$|0.29% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| |$-$|0.08%  |$-$|15% |$-$|1.72% |$-$|0.21% |$-$|0.63% |$-$|0.11%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| |$-$|0.33%  |$-$|10% |$-$|3.42% |$-$|1.06% |$-$|1.77% |$-$|0.45%|$^{\rm a}$| |$-$|0.66%|$^{\rm b}$| |$-$|1.50%  |$-$|5% |$-$|7.51% |$-$|4.77% |$-$|5.53% |$-$|3.37%|$^{\rm a}$| |$-$|4.19%|$^{\rm b}$| |$-$|5.56% (b) Median bias  |$-$|30% |$-$|0.33% |$-$|0.19% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.08%  |$-$|25% |$-$|0.37% |$-$|0.13% |$-$|0.06% 0.01%|$^{\rm a}$| |$-$|0.01%|$^{\rm b}$| |$-$|0.03%  |$-$|20% |$-$|0.48% |$-$|0.03% |$-$|0.01%|$^{\rm b}$| 0.00%|$^{\rm a}$| 0.09% 0.04%  |$-$|15% |$-$|0.8% 0.20% 0.00%|$^{\rm a}$| |$-$|0.02%|$^{\rm b}$| 0.25% 0.05%  |$-$|10% |$-$|1.88% 0.16%|$^{\rm b}$| |$-$|0.46% |$-$|0.14%|$^{\rm a}$| 0.29% |$-$|0.50%  |$-$|5% |$-$|5.29% |$-$|2.46% |$-$|3.25% |$-$|1.80%|$^{\rm a}$| |$-$|1.92%|$^{\rm b}$| |$-$|3.80% The table presents means and medians of the bias induced by requiring statistical significance via one-tailed |$t$|-test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table A.2 presents results when we use an SQ test with |$P<0.05$|⁠. Gelbach et al. (2013) propose the SQ test and show it has superior properties to the |$t$|-test for accurately evaluating statistical significance in daily returns data. In terms of mean bias, all the bias-corrected estimators perform well for large price impacts; the ABC and CBC estimators perform best; for small price impacts the MBC and LBC estimator perform best. In terms of median bias, the MBC estimator performs best across the board. Table A.2. Performance of Bias-Corrected Estimators Using an SQ Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% The table presents means and medians of the bias induced by requiring statistical significance via SQ test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large Table A.2. Performance of Bias-Corrected Estimators Using an SQ Test Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% Price Impact Uncorrected ABC CBC MBC LBC NBC (a) Mean bias  |$-$|30% |$-$|0.28% |$-$|0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.16% 0.07% 0.04%  |$-$|25% |$-$|0.41% 0.01%|$^{\rm a}$| 0.02%|$^{\rm b}$| 0.13% 0.10% 0.06%  |$-$|20% |$-$|0.63% 0.08%|$^{\rm b}$| |$-$|0.01%|$^{\rm a}$| 0.13% 0.19% 0.09%  |$-$|15% |$-$|1.12% 0.16% |$-$|0.15%|$^{\rm b}$| 0.16% 0.28% 0.03%|$^{\rm a}$|  |$-$|10% |$-$|2.36% |$-$|0.20% |$-$|0.85% 0.06%|$^{\rm b}$| |$-$|0.02%|$^{\rm a}$| |$-$|0.70%  |$-$|5% |$-$|5.46% |$-$|2.61% |$-$|3.48% |$-$|1.76%|$^{\rm a}$| |$-$|2.38%|$^{\rm b}$| |$-$|3.71% (b) Median bias  |$-$|30% |$-$|0.30% |$-$|0.22% |$-$|0.09% 0.02%|$^{\rm a}$| |$-$|0.07%|$^{\rm b}$| |$-$|0.09%  |$-$|25% |$-$|0.33% |$-$|0.17% |$-$|0.04% 0.02%|$^{\rm a}$| |$-$|0.03%|$^{\rm b}$| |$-$|0.03%|$^{\rm b}$|  |$-$|20% |$-$|0.39% |$-$|0.08% 0.03%|$^{\rm b}$| 0.02%|$^{\rm a}$| 0.09% 0.05%  |$-$|15% |$-$|0.61% 0.14% 0.10%|$^{\rm b}$| 0.01%|$^{\rm a}$| 0.26% 0.12%  |$-$|10% |$-$|1.39% 0.40% |$-$|0.09%|$^{\rm b}$| |$-$|0.04%|$^{\rm a}$| 0.43% |$-$|0.17%  |$-$|5% |$-$|4.18% |$-$|1.28% |$-$|2.09% |$-$|0.81%|$^{\rm a}$| |$-$|0.98%|$^{\rm b}$| |$-$|2.80% The table presents means and medians of the bias induced by requiring statistical significance via SQ test, |$P<0.05$|⁠, for a variety of bias correction regimes across a range of true price impacts. The distribution in each case is based on 1 million simulated single-firm event studies for high volatility CRSP stocks. |$^{\rm a}$| and |$^{\rm b}$| denote the first- and second-best performing estimators in each row. View Large We thank the editor and two anonymous referees for comments which substantially improved the paper. Any errors are our own. Footnotes 1. In addition to Brav and Heaton (2015), discussed below, the literature includes Dybvig et al. (2000), Gelbach et al. (2013), Baker (2016), and Fisch et al. (2018). 2. 485 U.S. 224 (1988). 3. Id. at 246. 4. See, e.g., Arkansas Teachers Ret. Sys. v. Goldman Sachs Grp., Inc., 879 F.3d 474, 480, n3 (2d Cir. 2018) (“A ‘corrective disclosure’ is an announcement or series of announcements that reveals to the market the falsity of a prior statement.”) (citation omitted). 5. A recent book covering event study methodology is Kliger and Gurevich (2014). Other sources are MacKinlay (2007) and Kothari and Warner (2007). 6. See, e.g., Willis v. Big Lots, Inc., 242 F. Supp.3d 634 (S.D. Ohio 2017) (“[P]rice impact is demonstrated either through evidence that a stock’s price rose in a statistically significant manner after a misrepresentation or that it declined in a statistically significant manner after a corrective disclosure.”); see also Hatamian v. Advanced Micro Devices, Inc., No. 14-CV-00226 YGR, 2016 WL 1042502, at *7 (N.D. Cal. Mar. 16, 2016) (“Price impact in securities fraud cases is not measured solely by price increase on the date of a misstatement; it can be quantified by decline in price when the truth is revealed.”). 7. See, e.g., Bricklayers and Trowel Trades Int’l Pension Fund v. Credit Suisse First Boston, 853 F. Supp. 2d 181, 190 (D. Mass. 2012), aff’d sub nom. Bricklayers & Trowel Trades Int’l Pension Fund v. Credit Suisse Sec. (USA) LLC, 752 F.3d 82 (1st Cir. 2014) (“An event study that fails to disaggregate the effects of confounding factors must be excluded because it misleadingly suggests to the jury that a sophisticated statistical analysis proves the impact of defendants’ alleged fraud on a stock’s price when, in fact, the movement could very well have been caused by other information released to the market on the same date.”). 8. See, e.g., Bricklayers & Trowel Trades Int’l Pension Fund v. Credit Suisse Secs. LLC, 752 F.3d 82, 86 (1st Cir. 2014) (“The usual—it is fair to say ‘preferred’—method of proving loss causation in a securities fraud case is through an event study, in which an expert determines the extent to which the changes in the price of a security result from events such as disclosure of negative information about a company, and the extent to which those changes result from other factors.”). 9. The United States Court of Appeals for the Second Circuit recently discussed and approved of the Brav and Heaton (2015) analysis. See In re Petrobras Sec., 862 F.3d 250, 279 (2d Cir. 2017) (“These methodological challenges counsel against imposing a blanket rule requiring district courts to, at the class certification stage, rely on directional event studies and directional event studies alone.”). 10. Brav and Heaton (p. 591, n17) “focus on the normal case because standard practice still rests heavily on the normality assumption, despite strong evidence that daily abnormal returns are non-normal[,]” citing Gelbach et al. (2013). 11. Atkins and Rubin (2003). 12. 367 U.S. 643 (1961). The Court held that evidence obtained by searches and seizures violating the Constitution is constitutionally inadmissible in state court proceedings. 13. Polinsky and Shavell (1998) show that when the injurer has a significant chance of escaping liability then punitive (additional, noncompensatory) damages are needed to increase deterrence. 14. 15 U.S.C.A. 78bb (West) (“No person permitted to maintain a suit for damages under the provisions of this chapter shall recover, through satisfaction of judgment in 1 or more actions, a total amount in excess of the actual damages to that person on account of the act complained of.”) (emphasis added); Bernard v. Lombardo, No. 16 CV. 863 (RMB), 2016 WL 7377240, at *4 (S.D.N.Y. November 23, 2016) (“Plaintiffs will not receive punitive damages for their securities fraud claims because it is well established that an award for punitive damages is not permissible for violations of Section 10(b) of the 1934 Act and Rule 10b-5 claims.”) (internal quotations and citations omitted). 15. We abstract here from another potentially important confounding effect, one that is positively correlated with the true impact. Specifically, price impacts of corrective disclosures—whether negative (the corrective disclosure of bad news) or positive (the corrective disclosure of good news)—are biased because the price impact includes expected costs of litigation to the issuer from the circumstances surrounding the corrective disclosure. That is, suppose |$S=F+E(L)$|⁠, where |$S$| is the total reduction following the corrective disclosure, |$F$| is the decline due to the revaluation of the business’s fundamental valuation in light of the corrected information, and |$E(L)$| is the expected total cost of litigation at the time of the corrective disclosure. While we assume here that |$E(L)=0$|⁠, |$E(L)$| may be large in particular applications, a fact that has been ignored in the literature and case law. 16. Consider, e.g., Goldkrantz v. Griffin, 1999 WL 191540, No. 97 Civ. 9075 (DLC) (S.D.N.Y. April 6, 1999), where plaintiff shareholders brought a securities class action against defendants for an alleged misrepresentation. When the defendant company cured the alleged misrepresentation by corrective disclosure in a 10K filing, the stock price fell |$-$|2.64%. Defendants’ expert submitted an event study where the critical return for statistical significance was |$-$|4.41%, so the price fall of |$-$|2.64% was rejected as statistically insignificant. See also Willis v. Big Lots, Inc., 242 F. Supp. 3d 634 (S.D. Ohio 2017) (“Defendants failed to show that there was no statistically significant price impact following the corrective disclosures in this case.”) 17. Prediction 3 reflects a fundamental tension in statistical inference. Making the statistical test more stringent results in fewer false positives—in this case, spurious securities litigation—but increases the severity of the bias in observed price impacts. 18. It is important to note that both firm and market volatility vary over time and that time-varying volatility can also affect inference in single-firm event studies. Baker (2016) documents and explores this topic in detail. Fisch et al. (2018) develop a generalized way to adjust for time-varying volatility that is compatible with the SQ test. 19. Interestingly, courts have rejected event-date returns that were significant at the 10% level without any apparent analysis of the benefits and tradeoffs of a higher level than 5%. See, e.g., In re Intuitive Surgical Sec. Litig., No. 5:13-CV-01920-EJD, 2016 WL 7425926, at *15 (N.D. Cal. December 22, 2016) (“The court finds a lack of price impact in connection with the release of Intuitive’s financial results between April 18 and April 19, 2013. First, neither Lehn nor Coffman found a statistically significant price impact at the 95% confidence level for this date. Rather, Coffman’s analysis resulted in a price impact at a 90% confidence level. Although Plaintiffs argue that price impact at a 90% confidence level is statistically significant, the district court in Halliburton Tex adopted 95% confidence level as the threshold requirement and this court finds no reason to deviate here.”) (citations omitted). 20. See Brav and Heaton (2015) p. 614, n21) (“[O]ne-tailed tests may be more appropriate in testing for the alternative of a price impact that is less than zero (the usual case for a corrective disclosure) or greater than zero (the usual case at the time of a misrepresentation that allegedly inflates the security price); Fisch et al. (2018, p. 589) (“In event studies used in securities fraud litigation, by contrast, price must move in a specific direction to support the plaintiffs’ case. For example, an unexpected corrective disclosure should cause the stock price to fall. Thus, tests of statistical significance based on event study results should be conducted in a ‘one-sided’ way so that an estimated excess return is considered statistically significant only if it moves in the direction consistent with the allegations of the party using the study. The one-sided–two-sided distinction is one that courts and expert witnesses regularly miss, and it is an important one.”) 21. See, e.g., Premium Plus Partners, L.P. v. Davis, 653 F. Supp.2d 855, 867 (N.D. Ill. 2009), aff’d sub nom. Premium Plus Partners, L.P. v. Goldman, Sachs & Co., 648 F.3d 533 (7th Cir. 2011) (“Goldman contends that Donaldson’s comparative analyses are unreliable. Goldman argues that Donaldson should have used a two-tail test rather than a one-tail test. However, Premium has offered sufficient justifications for the use of a one-tail test.) 22. See, e.g., Declaration of Steven P. Feinstein, Ph.D., CFA in Support of Plaintiffs’ Motion for Class Certification, January 28, 2016, ¶128, In re Genworth Financial Inc. Sec. Litig., 2016 WL 4495143 (S.D.N.Y.) (“For each event, a statistical test called a |$t$|-test was conducted to determine whether the residual return of Genworth common stock was statistically significant.”) 23. For a complete technical discussion of the topic see MacKinnon and Smith (1998). 24. Code for the estimators and replicating our results can be found at http://www.davidsonheath.com. 25. To understand the problem, suppose that a stock is trading at $\$$100 due to on-going but unknown misrepresentations. A corrective disclosure is made, and the price falls to $\$$92. If the expected cost of private securities litigation (assume no regulatory action is expected) is $\$$2 per share and the expected decline in the value of the business from the now-disclosed truth is $\$$6/share, then only $\$$6 should be attributed to any alleged fraud; $\$$2 of the price move reflects expected corporate losses from compensating the alleged fraud. To count that as fraud overcompensates damages for those who purchased stock at inflated prices (or bought calls at inflated prices or wrote puts at deflated prices). Moreover, the overall price impact must then be the solution to an iterative problem, where the final price impact is the point of convergence to the problem: “If the actual price impact is $\$$6, but is overestimated as $\$$8 for litigation purposes, then the expected cost of the litigation will be $\$$2,” that is, the overestimation of damages itself raises the cost of litigation to the issuer. The dynamic plays out differently when the price impact is positive. Suppose that a stock is trading at $\$$100 due to on-going but unknown misrepresentations. A corrective disclosure is made, and the price increases to $\$$108. If the expected cost of private securities litigation (assume no regulatory action is expected) is $\$$2 per share and the expected increase in the value of the business from the now-disclosed truth is $\$$10/share. Since the plaintiffs in this case will be those that sold their stock early (or, who purchased put options at inflated prices or wrote calls at deflated prices) then $\$$10 should be attributed to any alleged fraud; $\$$2 of the price move reflects expected corporate losses from compensating the alleged fraud. To count only $\$$8 as the price impact will under-compensate plaintiffs. References Andrews, D. W. 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A. . 1998 . “ Approximate Bias Correction in Econometrics, ” 85 Journal of Econometrics 205 – 30 . Google Scholar Crossref Search ADS Polinsky, A. M. , and Shavell S. . 1998 . “ Punitive Damages: An Economic Analysis, ” 111 Harvard Law Review 869 – 962 . Google Scholar Crossref Search ADS © The Author(s) 2019. Published by Oxford University Press on behalf of the American Law and Economics Association. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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American Law and Economics ReviewOxford University Press

Published: May 1, 2019

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