Econometrics of Event Studies Part 2

9 3. Characterizing Event Study Methods 3.1 An event study: the model An event study typically tries to examine return behavior for a sample of firms experiencing a common type of event (e.g., a stock split). The event might take place at different points in calendar time or it might be clustered at a particular date (e.g., a regulatory event affecting an industry or a subset of the population of firms). Let t = 0 represent the time of the event. For each sample security i, the return on the security for time period t relative to the event, Rit, is: Rit = Kit + eit (1) where Kit is the “normal” (i.e., expected or predicted return given a particular model of expected returns), and eit is the component of returns which is abnormal or unexpected.2 Given this return decomposition, the abnormal return, eit, is the difference between the observed return and the predicted return: eit = Rit - Kit (2) Equivalently, eit is the difference between the return conditional on the event and the expected return unconditional on the event. 2 Thus, the abnormal return is a direct measure of the This framework is from Brown and Warner (1980) and Campbell, Lo, and MacKinlay (1997). 10 (unexpected) change in securityholder wealth associated with the event. The security is typically a common stock, although some event studies look at wealth changes for firms’ preferred or debt claims. A model of normal returns (i.e., expected returns unconditional on the event but conditional on other information) must be specified before an abnormal return can be defined. A variety of expected return models (e.g., market model, constant expected returns model, capital asset pricing model) have been used in event studies.3 Across alternative methods, both the bias and precision of the expected return measure can differ, affecting the properties of the abnormal return measures. Properties of different methods have been studied extensively, and are discussed later. 3.2 Statistical and economic hypotheses Cross-sectional aggregation. An event study seeks to establish whether the cross- sectional distribution of returns at the time of an event is abnormal (i.e., systematically different from predicted). Such an exercise can be conducted in many ways. One could, for example, examine the entire distribution of abnormal returns. This is equivalent comparing the distributions of actual with the distribution of predicted returns and asking whether the distributions are the same. In the event study literature, the focus almost always is on the mean of the distribution of abnormal returns. Typically, the specific null hypothesis to be tested is whether the mean abnormal return (sometimes referred to as the average residual, AR) at time t is equal to zero. Other parameters of the cross-sectional distribution (e.g., median, variance) and determinants of the cross-sectional variation in abnormal returns are sometimes studied as well. The focus on mean effects, i.e., the first moment of the return distribution, makes sense if one 3 For descriptions of each of these models, see Brown and Warner (1985) or Campbell, Lo, and MacKinlay (1997). 11 wants to understand whether the event is, on average, associated with a change in security holder wealth, and if one is testing economic models and alternative hypotheses that predict the sign of the average effect. For a sample of N securities, the cross-sectional mean abnormal return for any period t is: ARt = N 1 N ∑e i =1 it . (3) Time-series aggregation. It is also of interest to examine whether mean abnormal returns for periods around the event are equal to zero. First, if the event is partially anticipated, some of the abnormal return behavior related to the event should show up in the pre-event period. Second, in testing market efficiency, the speed of adjustment to the information revealed at the time of the event is an empirical question. Thus, examination of post-event returns provides information on market efficiency. In estimating the performance measure over any multi-period interval (e.g., time 0 through +6), there are a number of methods for time-series aggregation over the period of interest. The cumulative average residual method (CAR) uses as the abnormal performance measure the sum of each month’s average abnormal performance. Later, we also consider the buy-and-hold method, which first compounds each security’s abnormal returns and then uses the mean compounded abnormal return as the performance measure. The CAR starting at time t1 through time t2 (i.e., horizon length L = t2 - t1 + 1) is defined as: CAR (t1 , t 2 ) = t2 ∑ AR t = t1 t . (4) Both CAR and buy-and-hold methods test the null hypothesis that mean abnormal performance is equal to zero. Under each method, the abnormal return measured is the same as 12 the returns to a trading rule which buys sample securities at the beginning of the first period, and holds through the end of the last period. CARs and buy-and-hold abnormal returns correspond to security holder wealth changes around an event. Further, when applied to post-event periods, tests using these measures provide information about market efficiency, since systematically nonzero abnormal returns following an event are inconsistent with efficiency and imply a profitable trading rule (ignoring trading costs). 3.3 Sampling distributions of test statistics For a given performance measure, such as the CAR, a test statistic is typically computed and compared to its assumed distribution under the null hypothesis that mean abnormal performance equals zero.4 The null hypothesis is rejected if the test statistic exceeds a critical value, typically corresponding to the 5% or 1% tail region (i.e., the test level or size of the test is 0.05 or 0.01). The test statistic is a random variable because abnormal returns are measured with error. Two factors contribute to this error. First, predictions about securities’ unconditional expected returns are imprecise. Second, individual firms’ realized returns at the time of an event are affected for reasons unrelated to the event, and this component of the abnormal return does not average to literally zero in the cross-section. For the CAR shown in equation (4), a standard test statistic is the CAR divided by an estimate of its standard deviation.5 Many alternative ways to estimate this standard deviation have been examined in the literature (see, for example, Campbell, Lo, and MacKinlay, 1997). The test statistic is given by: 4 Standard tests are “classical” rather than “Bayesian.” A Bayesian treatment of event studies is beyond the scope of this chapter. 5 An alternative would be a test statistic that aggregates standardized abnormal returns, which means each observation is weighted in inverse proportion of the standard deviation of the estimated abnormal return. The standard deviation of abnormal returns is estimated using time-series return data on each firm. While a test using standardized abnormal returns is in principle superior under certain conditions, empirically in short-horizon event studies it typically makes little difference (see Brown and Warner, 1980, and 1985). 13 CAR (t1 , t 2 ) [σ 2 (t1 , t 2 )] ½ , (5) (t1 , t 2 ) = L σ 2 ( AR t ) (6) where σ 2 and σ 2 ( ARt ) is the variance of the one-period mean abnormal return.. Equation (6) simply says that the CAR has a higher variance the longer is L, and assumes time-series independence of the one-period mean abnormal return. The test statistic is typically assumed unit normal in the absence of abnormal performance. This is only an approximation, however, since estimates of the standard deviation are used. The test statistic in eq. (5) is well-specified provided the variance of one-period mean abnormal return is estimated correctly. Event-time clustering renders the independence assumption for the abnormal returns in the cross-section incorrect (see Collins and Dent, 1984, Bernard, 1987, and Petersen, 2005, and more detailed discussion in section 4 below). This would bias the estimated standard deviation estimate downward and the test statistic given in eq. (5) upward. To address the bias, the significance of the event-period average abnormal return can be and often is gauged using the variability of the time series of event portfolio returns in the period preceding or after the event date. For example, the researcher can construct a portfolio of event firms and obtains a time series of daily abnormal returns on the portfolio for a number of days (e.g., 180 days) around the event date. The standard deviation of the portfolio returns can be used to assess the significance of the event-window average abnormal return. The crosssectional dependence is accounted for because the variability of the portfolio returns through 14 time incorporates whatever cross-dependence that exists among the returns on individual event securities. The portfolio return approach has a drawback, however. To the extent the event period is associated with increased uncertainty, i.e., greater return variability, the use of historical or postevent time-series variability might understate the true variability of the event-period abnormal performance. An increase in event-period return variability is economically intuitive. The event might have been triggered by uncertainty-increasing factors and/or the event itself causes uncertainty in the economic environment for the firm. In either case, the event-period return variability is likely to exceed that during other time periods for the event firms. Therefore, the statistical significance of the event-window abnormal performance would be overstated if it is evaluated on the basis of historical variability of the event-firm portfolio returns (see Brown and Warner, 1980, and 1985, and Collins and Dent, 1984). One means of estimating the likely increase in the variability of event-period returns is to estimate the cross-sectional variability of returns during the event and non-event periods. The ratio of the variances during the event period and non-event periods might serve as an estimate of the degree of increase in the variability of returns during the event period, which can be used to adjust for the bias in the test statistic calculated ignoring the increased event-period uncertainty.6 3.4 Criteria for “reliable” event study tests Using the test statistics, errors of inference are of two types. A Type I error occurs when the null hypothesis is falsely rejected. A Type II error occurs when the null is falsely accepted. Accordingly, two key properties of event study tests have been investigated. The first is whether the test statistic is correctly specified. A correctly-specified test statistic yields a Type I error 6 Use of non-parametric tests of significance, as suggested in Corrado (1989), might also be effective in performing well-specified tests in the presence of increased event-period uncertainty. 15 probability equal to the assumed size of the test. The second concern is power, i.e., a test’s ability to detect abnormal performance when it is present. Power can be measured as one minus the probability of a Type II error. Alternatively, it can be measured as the probability that the null hypothesis will be rejected given a level of Type I error and level of abnormal performance. When comparing tests that are well-specified, those with higher power are preferred. 3.5 Determining specification and power The joint-test problem. While the specification and power of a test can be statistically determined, economic interpretation is not straightforward because all tests are joint tests. That is, event study tests are well-specified only to the extent that the assumptions underlying their estimation are correct. This poses a significant challenge because event study tests are joint tests of whether abnormal returns are zero and of whether the assumed model of expected returns (i.e. the CAPM, market model, etc.) is correct. Moreover, an additional set of assumptions concerning the statistical properties of the abnormal return measures must also be correct. For example, a standard t-test for mean abnormal performance assumes, among other things, that the mean abnormal performance for the cross-section of securities is normally distributed. Depending on the specific t-test, there may be additional assumptions that the abnormal return data are independent in time-series or cross-section. The validity of these assumptions is often an empirical question. This is particularly true for small samples, where one cannot rely on asymptotic results or the central limit theorem. Brown-Warner simulation. To directly address the issue of event study properties, the standard tool in event study methodology research is simulation procedures that use actual security return data. The motivation and specific research design is initially laid out in Brown and Warner (1980, 1985), and has been followed in almost all subsequent methodology research. 16 Much of what is known about general properties of event study tests comes from such large-scale simulations. The basic idea behind the event study simulations is simple and intuitive.7 Different event study methods are simulated by repeated application of each method to samples that have been constructed through a random (or stratified random) selection of securities and random selection of an event date to each. If performance is measured correctly, these samples should show no abnormal performance, on average. This makes it possible to study test statistic specification, that is, the probability of rejecting the null hypothesis when it is known to be true. Further, various levels of abnormal performance can be artificially introduced into the samples. This permits direct study of the power of event study tests, that is, the ability to detect a given level of abnormal performance. Analytical methods. Simulation methods seem both natural and necessary to determine whether event study test statistics are well-specified. Once it has been established using simulation methods that a particular test statistic is well-specified, analytical procedures have also been used to complement simulation procedures. Although deriving a power function analytically for different levels of abnormal performance requires additional distributional assumptions, the evidence in Brown and Warner (1985, p. 13) is that analytical and simulation methods yield similar power functions for a well-specified test statistic. As illustrated below, these analytical procedures provide a quick and simple way to study power. 3.6 A quick summary of our knowledge Qualitative properties. Table 2 highlights, in qualitative terms, what is known about the properties of event study tests. The table shows the characteristics of event study methods along three dimensions: specification, power against specific types of alternative hypotheses, and the 7 This characterization of simulation is from Brown and Warner (1985, p. 4). 17 sensitivity of specification to assumptions about the return generating process. The table also shows how these properties can differ sharply for short and long horizon studies. Much of the remainder of the chapter deals with the full details of this table. From Table 2, horizon length has a big impact on event study test properties. First, shorthorizon event study methods are generally well-specified, but long-horizon methods are sometimes very poorly specified. While much is understood about how to reduce misspecification in long horizon studies (see section 4), no procedure in whose specification researchers can have complete confidence has yet been developed. Second, short-horizon methods are quite powerful if (but only if) the abnormal performance is concentrated in the event window. For example, a precise event date is known for earnings announcements, but insider trading events might be known to have occurred only sometime during a one-month window. In contrast to the short-horizon tests, long-horizon event studies (even when they are wellspecified) generally have low power to detect abnormal performance, both when it is concentrated in the event window and when it is not. That power to detect a given level of abnormal performance is decreasing in horizon length is not surprising, but the empirical magnitudes are dramatic (see below). Third, with short-horizon methods the test statistic specification is not highly sensitive to the benchmark model of normal returns or assumptions about the cross-sectional or time-series dependence of abnormal returns. This contrasts with long-horizon methods, where specification is quite sensitive to assumptions about the return generating process. Along several lines, however, short- and long-horizon tests show similarities, and these results are easy to show using either simulation or analytical procedures. First, a common problem shared by both short- and long-horizon studies is that when the variance of a security’s 18 abnormal returns conditional on the event increases, test statistics can easily be misspecified, and reject the null hypothesis too often. This problem was first brought to light and has been studied mainly in the context of short-horizon studies (Brown and Warner, 1985, and Corrado, 1989). A variance increase is indistinguishable from abnormal returns differing across sample securities at the time of an event, and would be expected for an event. Thus, this issue is likely to be empirically relevant both in a short- and long-horizon context as well. Second, power is higher with increasing sample size, regardless of horizon length. Third, power depends on the characteristics of firms in the event study sample. In particular, firms experiencing a particular event can have nonrandom size and industry characteristics. This is relevant because individual security variances (and abnormal return variances) exhibit an inverse relation to firm size and can vary systematically by industry. Power is inversely related to sample security variance: the noisier the returns, the harder to extract a given signal. As shown below, differences in power by sample type can be dramatic. Quantitative results. To provide additional texture on Table 2, below we show specific quantitative estimates of power. We do so using the test statistic shown previously in equations (5) and (6), using two-tailed tests at the 0.05 significance level.8 Since this test statistic is wellspecified, at least at short horizons, the power functions are generated using analytic (rather than simulation) procedures. The estimates are for illustrative purposes only, however, and only represent “back of the envelope” estimates. The figures and the test statistic on which they are based assume independence of the returns (both through time and in the cross-section), and that all securities within a sample have the same standard deviation. The power functions also 8 This format for displaying power functions is similar to Campbell, Lo, and MacKinlay (1997, pp. 168-172). Our test statistic and procedures are the same as for their test statistic J1, but as discussed below we use updated variance inputs. 19 assume that return and abnormal return variances are the same (i.e., the model of abnormal returns is the “mean-adjusted returns” model of Brown and Warner, 1980). Volatility. In calculating the test statistic in an event study, a key input required here is the individual security return (or abnormal return) variance (or standard deviation). To determine a reasonable range of standard deviations, we estimate daily standard deviations for all CRSP listed firms from 1990 to 2002. Specifically, for each year, we: i) calculate each stock’s standard deviation, and ii) assign firms to deciles ranked by standard deviation. From each decile, the averages of each year’s mean and median values are reported in Table 3. The mean daily standard deviation for all firms is 0.053. This is somewhat higher than the value of 0.026 reported by Brown and Warner (1985, p. 9) for NYSE/AMEX firms and the value of 0.035 reported by Campbell and Wasley (1993, p. 79) for NASDAQ firms. The differences reflect that individual stocks have become more volatile over time (Campbell, Lettau, Malkiel, Xu, 2001)). This is highly relevant because it suggests that the power to detect abnormal performance for events over 1990-2002 is lower than for earlier periods. From Table 3, there is wide variation across the deciles. Firms in decile 1 have a mean daily standard deviation of 0.014, compared to 0.118 for decile 10. The figure of 0.118 for decile 10 seems very high, although this is likely to represent both very small firms and those with low stock prices. Further, there is a strong negative empirical relation between volatility and size. Our qualitative results apply if ranking is by firm size, so table 3 is not simply picking up measurement error in volatility. Results. Figure 1 shows how, for a sample comprised of securities of average risk and 10% abnormal performance, the power to detect abnormal performance falls with horizon length. This level of abnormal performance seems economically highly significant. If the abnormal performance is concentrated entirely in one day (and the day in known with certainty), a sample 20 of only six stocks detects this level of abnormal performance 100% of the time. In contrast, if the same abnormal performance occurs over six months, a sample size of 200 is required to detect the abnormal performance even 65% of the time. These various rejection frequencies are lower than those using pre-1990 volatilities (not reported), although this is not surprising. Figure 2a through 2c show related results using a one-day horizon for samples whose individual security standard deviations correspond to the average standard deviation for: the lowest decile (Figure 2a); all firms (Figure 2b); and the highest decile (Figure 2c). For decile 1 firms, with 1% abnormal performance a 90% rejection rate requires only 21 stocks. For firms in decile 10, even with 5% abnormal performance a 90% rejection rate requires 60 stocks. These comparisons may distort the differences in actual power if high variance firms are less closely followed and events are bigger surprises. When the effect of events differs cross-sectionally, analysis of test properties (i.e., power and specification) is more complicated. Collectively, our results illustrate that power against alternative hypotheses can be sensitive to calendar time period and sample firm characteristics, and highlight the importance already recognized in the profession of studying test statistic properties for samples stratified by firm characteristics. A complete analysis of these issues would focus on abnormal return (rather than return) volatility, and study how specification (and abnormal return distributional properties such as skewness) varies across time and firm characteristics. 3.7 Cross-Sectional Tests This section’s focus thus far has been event study tests for mean stock price effects. These tests represent the best understood class of event study tests. To provide a more complete the picture of event-related tests, we briefly call attention to cross-sectional tests. These tests 21 examine how the stock price effects of an event are related to firm characteristics. For a crosssection of firms, abnormal returns are compared to (e.g., regressed against) firm characteristics. This provides evidence to discriminate among various economic hypotheses. Cross-sectional tests are a standard part of almost every event study. They are relevant even when the mean stock price effect of an event is zero. In addition, they are applicable regardless of horizon length. They are simple to do, but as discussed below, “one must be careful in interpreting the results” (Campbell, Lo, and Mackinlay, 1997, p.174). One reason that abnormal returns vary cross-sectionally is that the economic effect of the event differs by firm. For such a situation, Sefcik and Thompson (1986) examine the statistical properties of cross-sectional regressions. They are concerned with the effects of cross- sectionally correlated abnormal returns and heteroscedasticity in the abnormal returns. They argue that accounting for each appears to be potentially important for inferences, and they suggest procedures to deal with these issues. Abnormal returns also vary cross-sectionally because the degree to which the event is anticipated differs by firm. For example, for firms which are more closely followed (e.g., more analysts), events should more predictable, all else equal. Further, events are endogenous, reflecting a firm’s self selection to choose the event, which in turn reflects insiders’ information. Recognizing these factors, and recognizing that it is the unexpected information provided by an event that determines the stock price effect, has numerous consequences. For example, standard estimates of cross-sectional coefficients can be biased (Eckbo, Maksimovic, and Williams (1990)). Appropriate procedures for treating self-selection and partial anticipation issues is the subject of an entire chapter by Li and Prabhala in this volume (see Ch 2). 22 Quite apart from the issues discussed in the context of Li and Prabhala, there are several additional dimensions where our understanding of cross-sectional tests is incomplete, and where additional work is potentially fruitful. One area concerns the power of cross-sectional procedures. While specification of cross-sectional regression methods (i.e., biases in regression coefficients) has received much attention, the power of alternative procedures to detect underlying cross-sectional effects has received less study. A related point is that a simple type of cross-sectional procedure is to form portfolios based on firm characteristics, and compare portfolio abnormal returns. Such procedures are common, but methodological comparisons to cross-sectional regressions would prove useful. Portfolio procedures seem less amenable to multivariate comparisons than do regression procedures, but the relative empirical merits of each in an event-study context have not been investigated. We also note that some studies focus not on the stock price effect of an event, but on predicting a corporate event (e.g., management turnover, or a security issue of a particular type), sometimes using past stock prices as one explanatory variable. These tests use cross-sectional methods in the sense that the cross-section includes both event and non-event firms. Typically, discrete choice models (e.g., probit or logit model) relate whether or not the event occurred to firm-specific characteristics. This seems intuitive, since we would like to know what factors led the firm to have the event. These methods complement standard event study methods. Methodological work on prediction models could enhance our understanding of how to best to use information about events to test economic hypotheses about firm behavior. Finally, additional important issues to consider in an event study are: (i) whether the event was partially anticipated by market participants (e.g., a governance-related regulation 23 might be anticipated following corporate scandals or CEO turnover is likely in the case of a firm experiencing steep stock-price decline and poor accounting performance), and (ii) whether the partial anticipation is expected to vary cross-sectionally in a predictable fashion (e.g., market participants might anticipate that managers of firms experiencing high price run-ups are likely to make value-destroying stock acquisitions, but the negative announcement effect of an actual merger announcement might have been largely anticipated for the firms who have experienced relatively high prior price run up). These issues arising from the nature of information arrival, partial anticipation of events, and cross-sectional variation in the degree of anticipation are also beyond the scope of this chapter. Interested readers will find treatments in Malatesta and Thompson (1985), Eckbo, Maksimovic, and Williams (1990), and, especially, Thompson (1995) of considerable interest.