Industry Indices in Event Studies. Joseph M. Marks Bentley University. Jim Musumeci* Bentley University. Aimee Hoffmann Smith Bentley University

Transcription

1 Industry Indices in Event Studies Joseph M. Marks Bentley University Jim Musumeci* Bentley University Aimee Hoffmann Smith Bentley University Current Draft: March, 2018 * Corresponding author. The authors are grateful to Atul Gupta and Richard Sansing for their helpful comments. The usual disclaimer applies.

2 Industry Indices in Event Studies Abstract Event studies compare a sample of stock returns relative to their benchmark returns at the time of an event and test whether deviations from the benchmarks are significantly different from zero. There are two desirable characteristics of these benchmarks: (1) that they be unbiased, i.e., absent an event, the average deviation from the benchmark is zero, and (2) that the prediction error has as small a variance as possible. King (1966) found that a firm s industry explains about 10% of its variance of returns. Despite this, event-study benchmarks typically ignore this industry effect. We consider several common factor models and examine the results when an industry factor is used to replace or supplement a market factor. We find that inclusion of an industry factor increases event study test power by approximately 10% on average and up to 18.4% in samples of 500 events. JEL classification: G14, C10, C15

3 Industry Indices in Event Studies In general, there are two desirable requirements for a benchmark to be used in an event study. First, it should be unbiased; biasedness will necessarily produce Type I error that increases in sample size. Second, it should have as low a variance of prediction error as possible. Ceteris paribus, a lower prediction-error variance will improve test power. Consider a perhaps not-too-distant future in which worldwide stock markets trade continuously around the clock, and both global and national stock market indices are readily available. If you were examining an event s effect on a sample of U.S. stocks only, you could use a global market index to find benchmark returns. However, while international events affect the entire world s economy, certainly some countries will be affected to a greater extent than others will. For example, while political upheaval in, say, China, would necessarily have implications for worldwide markets, we would expect it to affect Asian stocks substantially more than U.S. stocks. Similarly, while any disruption of NAFTA would have global implications, we would expect it to have a larger impact on U.S., Canadian, and Mexican stocks than on South African stocks. Thus, if an entire sample of stocks is from the same country, we would expect a global index to overweight events that have minor implications for that country s stocks and underweight events that have major implications. For this reason, a U.S. market index seems a better choice than a global index for a sample composed of U.S. companies. In general, a market index that includes the stocks of interest but does not include too much extraneous noise seems a wiser choice than an index that is too broad. 1 King (1966) found that the market accounts for about half the total variance of a U.S. stock, and that the stock s industry explains about an additional 10%. This suggests use of an industry factor is likely to produce residuals with a smaller variance and thus lead to more powerful tests. Consider, for example, an increase in the value of the dollar. Ceteris paribus, we would expect this to have a positive effect on industries that are net importers and a negative effect on industries that are net exporters. A national index would necessarily reflect only the average effect on all firms and would be less informative than an industry index that is more likely to capture the net exporter vs. net importer effect. If all the stocks in an event-study sample were from the same industry, then the same industry index could be used for each stock and there would be no problem with inconsistency. However, it might seem that some comparability problems could occur if we used different indices for different stocks in the same sample. For example, if two stocks in different industries had the same total variance of returns, but one industry contained fairly homogenous firms while the other did not, we would expect the estimation-period residuals and event-day abnormal return for the stock in the homogenous industry to have a smaller 1 However, an index cannot be too narrow, either. For example, an equally weighted index that includes only the stocks in the sample would necessarily produce an average residual of zero on the event day or any other day.

4 variance than those for the stock in the heterogeneous industry. Because the main two eventstudy methods, Patell (1976) and Boehmer et al. (1991), normalize the event-period abnormal returns by the standard deviation of the estimation-period residuals, we expect this difference in the variances of the raw (unnormalized) abnormal returns to be inconsequential because the point of normalization is to create standardized abnormal returns that have approximately equal variances. Consistent with the findings of King (1966), our results confirm that use of an industry index produces more powerful tests. Specifically, the performance of four standard event-study benchmarks (i.e., the CRSP equally weighted index, the CRSP value-weighted index, the Fama- French Three-Factor Model, and the Carhart Four-Factor Model) can be improved substantially by simply replacing or supplementing the market index with an industry index. This is the case even in the presence of event-induced variance. We demonstrate that inclusion of an industry factor increases event study test power by roughly 10% on average and up to 18.4% in samples of 500 events when = 1% and an equally weighted industry index is used to supplement the Fama-French Three-Factor Model. These findings suggest that industry indices should be more frequently incorporated into event study methodologies. In the following section, we discuss use of equally weighted versus value-weighted indices in event studies. Next, we describe how we use (1) common methods employing a market index, (2) an industry index replacing the market index, and (3) an industry index supplementing a market index to form 16 different benchmarks. We then proceed to describe how we simulate events and abnormal returns, after which we compare the specification and power of tests using the various benchmarks. Of particular interest will be comparisons of commonly used methods with benchmarks that either replace or supplement a market index with an industry index. The final section concludes. I. Equally Weighted vs. Value-Weighted Indices For many years the market model using the CRSP equally weighted index was the main event-study benchmark, and indeed it is still the default in Eventus. Although the CRSP valueweighted index is occasionally used, it can lead to an interesting paradox. Consider an economy with 9 small firms, each constituting 5% of the total market cap, and one large firm constituting the remaining 55%. Without loss of generality, assume that each of the 10 companies has a β of 1 and an α of 0. Suppose also that on some event date the stock price of the large firm increased by 0.9%, while the stock prices of the nine small firms declined by 1.1% each. The value-weighted index didn t change because.45(-.011) +.55(.009) = 0, while an equally weighted index would produce a market return of 9(.011)+1(.009) = -0.9%. 10 An event study testing whether the abnormal return of some sample of stocks (which, unbeknownst to you, happens to be all the stocks in this market) is different from zero will 4

5 produce an odd result. If you use the value-weighted index but weight the residuals equally, you find that nine stocks underperformed the 0% return of the value-weighted index by 1.1% and one outperformed it by.9%, yielding an average abnormal return of 9(.011)+1(.009) = -0.9%. 10 The conclusion is that the abnormal performance of your sample (i.e., the market) was negative, oddly suggesting that the market performed worse than itself. This problem is resolved if you use the equally weighted index. Since the nine small stocks underperformed the equally weighted market return of -0.9% by 0.2% each while the single large firm outperformed it by 1.8%, the average abnormal return was 9(.002)+ 1(.018) 10 = 0. Similarly, the value-weighted index avoids the paradox if you value-weight the residuals (but no one does that). Brown and Warner (1980) also make this observation on pp , and they go on to find use of the value-weighted index produces less powerful tests than when an equally weighted index is used. This paradox notwithstanding, there is a powerful intuitive motivation for using a valueweighted index. Specifically, if your sample consists primarily of large stocks, then those stocks are likely to share a higher correlation with a value-weighted index than with an equally weighted one. Accordingly, a single-factor model using the value-weighted index may have lower prediction error and so be preferable. Another possibility is to use a benchmark with only large stocks (to keep the high correlation), but to use an equally weighted index of these stocks to avoid the paradox discussed above. To the best of our knowledge, no one has tried this. More recently, some have used the Fama-French Three-Factor Model or the Carhart Four- Factor Model to compute benchmark returns, and these models use a value-weighted market index. They also produce the interesting result observed by Cremers et al. (2012) that common market benchmarks have alphas that are consistently non-zero. The reason for this unusual result is a mirror image of the paradox just discussed. In our example, the residuals are equally weighted, therefore an equally weighted index is consistent and avoids the problem. Cremers et al., on the other hand, examine value-weighted benchmarks and find their paradox is mitigated when, consistent with their benchmarks examined and with the Fama-French market factor, they value-weight the HML factor as well. Our study compares the use of industry indices with commonly applied methods, some of which include an equally weighted market index (e.g., the single-factor CRSP equally weighted index that is the default in Eventus) and others of which use a value-weighted market index (e.g., the Fama-French and Carhart models). To maintain consistency with our use of both equally and value-weighted market portfolios, we examine both equally weighted and valueweighted industry indices. These indices were constructed by calculating the daily return on an appropriate portfolio of all stocks in the same industry as per the Fama and French (1997) 48 industry classifications. 5

6 II. Various Plausible Benchmarks In the following sections, we compare simulated event-study results using the 16 benchmarks described below: I. A single-factor model using the CRSP equally weighted index (CRSP EW, the default in Eventus) II. A single-factor model using an equally weighted industry index III. A two-factor model using the CRSP and industry equally weighted indices IV. A single-factor model using the CRSP value-weighted index (CRSP VW) V. A single-factor model using a value-weighted industry index VI. A two-factor model using the CRSP and industry value-weighted indices VII. The Fama-French Three-Factor Model (FF) VIII. A Three-Factor Model with an equally weighted industry index as well as the Fama- French SMB and HML factors (i.e., the Fama-French Three-Factor Model with an equally weighted industry index replacing the market index) IX. A Three-Factor Model with a value-weighted industry index as well as the Fama-French SMB and HML factors (i.e., the Fama-French Three-Factor Model with a value-weighted industry index replacing the market index) X. The Fama-French Three-Factor Model with a fourth factor equal to the industry equally weighted index XI. The Fama-French Three-Factor Model with a fourth factor equal to the industry valueweighted index XII. The Carhart Four-Factor Model (Carhart) XIII. A Four-Factor Model with an equally weighted industry index as well as the Fama- French SMB and HML factors and the Carhart momentum factor (i.e., the Carhart Four- Factor Model with an equally weighted industry index replacing the market index) XIV. A Four-Factor Model with a value-weighted industry index as well as the Fama-French SMB and HML factors and the Carhart momentum factor (i.e., the Carhart Four-Factor Model with a value-weighted industry index replacing the market index) XV. The Carhart Four-Factor Model with a fifth factor equal to the industry equally weighted index XVI. The Carhart Four-Factor Model with a fifth factor equal to the industry value-weighted index Consistent with common practice, in the first six (single- and two-factor) benchmarks, the market and industry indices are raw values, i.e., not in risk-premium form. In the remaining ten 6

7 benchmarks, all market and industry indices are in risk-premium form, i.e., Rmarket RF or Rindustry index RF. While it is possible to simply look at the results of all 16 benchmarks and choose the one that is well specified and most powerful, we believe a better method of analyzing the results in the following sections is to consider subsets of the 16 benchmarks. We begin with a simple comparison of the four models that are currently available in Eventus, specifically I (CRSP EW), IV (CRSP VW), VII (FF), and XII (Carhart). We then proceed to examine whether each of these four models produces better tests when the market index is replaced by or supplemented with an industry index. For example, a researcher who plans to use the default in Eventus (benchmark I, the CRSP equally weighted index) will find a comparison with the equally weighted industry index (benchmark II) or the CRSP and industry equally weighted indices (benchmark III) most similar and useful. We note that while adding an independent variable will necessarily produce a higher R 2, or equivalently a lower variance of the estimation-period residuals, event-studies generally make out-of-sample predictions. The adjustment for out-of-sample prediction variance is given for a single-factor model in Patell (1976) to be C = (R M,E R M) 2 [1] T t ( R M,t R M) 2 for bivariate models. In the multivariate case, the analogous adjustment is given in Kmenta (1971, p. 375) as C = T + (X 0 X) (X X) 1 (X 0 X) [2] In this latter case, multicollinearity will increase the variance of the prediction error. Thus, while adding a factor (e.g., changing from benchmark II to benchmark III) will necessarily produce a lower in-sample variance, it may well produce inferior results because the independent variables are correlated, leading to a higher value of C and thus a larger denominator for the t-statistic and, consequently, weaker tests. Panel A of Table I reports correlations between the main indices used (value-weighted or equally weighted market indices, SMB, HML, and MOM) for the entire sample period from November 3, 1926 through December 30, The large correlation between the valueweighted and equally weighted indices (0.899) is not surprising and is a good reason not to include both in any benchmark model. Because no current research of which we are aware mixes an equally weighted market index with any of SMB, HML, or MOM, we confine our attention to the correlations between all pairs of the value-weighted index, SMB, HML, and MOM. Of these, the largest magnitude of correlation is between the CRSP value-weighted 7

8 index and HML at 0.181, with that between the CRSP value-weighted index and SMB a close second at Whether these are sufficiently large to cause the value of C from eq. [2] to increase so that it causes a decrease in test power is an empirical question that is addressed in Table VII-XII. Panels B and C of Table I report the and results after dividing our sample period into 198 non-overlapping windows, each consisting of 120 days; the results are similar except that the magnitudes of the correlations are generally smaller. Table II reports the correlation between the 48 equally weighted industry indices and each of the other variables (CRSP EW, CRSP VW, SMB, HML, and MOM). As in Panels A and B from Table I, the correlations for the full sample and the average correlations for the 120-day windows are typically similar. The largest average population correlation of is between the equally weighted industry indices and the equally weighted CRSP index. This is relevant for benchmark III, which combines the equally weighted CRSP index with an equally weighted industry index to form a two-factor model. The individual industry indices correlations with CRSP EW range from (Industry 35, Computers) to (Industry 17, Construction Materials). The second largest average industry-index correlation with another factor is and is with the value-weighted CRSP index. This correlation is relevant for benchmarks X and XV (FF and Carhart, each supplemented with the equally weighted industry indices). The correlations between CRSP VW and the individual industry indices range from a low of (Industry 28, Non-Metallic and Industrial Metal Mining) to a high of (Industry 17, Construction Materials). Whether these correlations are sufficiently large to impair test power is an empirical issue addressed later. Finally, Table III reports the correlation between the 48 value-weighted industry indices and each of the other variables. As in Tables I and II, the correlations for the full sample and the average correlations for the 120-day windows are typically similar. The value-weighted industry indices have the largest average correlation of with the CRSP VW Index. This creates potential multicollinearity problems for benchmark VI (the two-factor model consisting of the value-weighted CRSP index and a value-weighted industry index) as well as benchmarks XI and XVI (FF and Carhart, each supplemented with a value-weighted industry index). The correlations with the individual industry indices range from a low of for Industry 35, Computers, to a high of for Industry 21, Machinery. For the value-weighted industry indices in Table 3, the second largest average correlation (0.633) is with the CRSP EW index, with individual correlations ranging from a low of for Industry 35, Computers, to a high of for Industry 21, Machinery. III. Generation of Simulated Abnormal Returns and Results Our initial candidates for simulated events consisted of the entire population of stocks listed in the daily CRSP database. For every observation in CRSP, we estimated parameters for 8

9 the 16 benchmarks introduced in Section II using an estimation period of the 120 days preceding the event, provided the stock traded for at least $5 on the event date and there were no missing observations during the 120 days preceding the event. This generated 37,587,725 parameter estimates and event-day abnormal returns. As expected, the average event-day abnormal return was essentially zero for all 16 benchmarks. A summary of the regression results is reported in Table IV. In general, adding more factors increases R 2 and decreases in-sample residual standard deviation, but also produces larger forecast-error adjustments (C from eq. [2]) because of multicollinearity. For example, the commonly used CRSP equally weighted index (benchmark I) produces a residual standard deviation of and a forecast error adjustment, C, of , while the Carhart model (benchmark XII) has a slightly lower in-sample residual standard deviation of , but a higher forecast error adjustment of We then proceeded to generate 10,000 random pseudo-portfolios, each consisting of 5,000 hypothetical events. For each portfolio, we constructed nested subsets with sizes of 2,500, 1,000, 500, 250, and 100 events. The sampling was without replacement within each portfolio so that the same firm-day event cannot appear twice within a portfolio. Additionally, the sampling was performed without imposing controls on firm or temporal distribution (i.e., the same firm or the same date but not both may appear multiple times within a given portfolio). For each event date for each stock we then simulated an abnormal return with a of equal to either 0% or 0.125%, and a variance equal to (0 or 1) times the estimation-period variance. Thus, for example, = 0% and = 0 simulates no abnormal return at all (and provides evidence regarding test specification), while = 0.125% and = 1 simulates an event that causes share price to increase by an average of an eighth of a percent, with a variance that is equal to the stock s residual variance during the estimation period. To ensure that we did not obtain an aberrant simulation, we repeated the pseudo-portfolio generation process four additional times and selected the set with the difference between the average BMP test statistics for benchmarks I (CRSP EW) and II (equally weighted industry indices). The results were reasonably consistent across all five simulations. 2 A: = 0% Tables V and VI report the results for = 0%, = 0 and = 0%, = 1. Thus, they provide evidence regarding whether the various benchmarks produce tests that are well-specified, ing they reject a null hypothesis of no abnormal performance with a frequency equal to the purported value of. 2 For example, across the five simulations, the average BMP test statistic for benchmark II (equally weighted industry index) ranged from a low of to a high of when N = 500, = 0.125%, and = 0. The simulation chosen featured an average BMP test statistic of in this case. 9

10 Consistent with Marks and Musumeci (2017), who tested only benchmark I (CRSP equally weighted index), we find the Patell test is misspecified and rejects H0: SAR = 0 too frequently across all 16 models, even absent any event-induced variance (Table V). We find little evidence that the BMP test is misspecified for any of the 16 models except occasionally when N = 5,000. The results are even more dramatic when the event creates an increase in variance as in Table VI ( = 0%, = 1). Here, the Patell test rejects a true null between three and four times as often as it should when the significance level is 5% (Panel A), and over seven times as often as it should when the significance level is 1% (Panel B). This is consistent with previous research [Boehmer et al. (1991), Harrington and Shrider (2007), and Marks and Musumeci (2017)], except we extend the analysis beyond only benchmark I and find the problem occurs for any of the 16 benchmarks. As was the case when = 0, we find little evidence that the BMP test is misspecified when = 1. Because the Patell test is misspecified both in the absence or presence of event-induced variance, we consider only the BMP test in our examination of test power. B: = 0.125% Panels A C of Table VII report the results for = 0.125%, = 0. Not surprisingly, for a given sample size, we did not find a dramatic absolute difference between the average BMP test statistics or average rejection rates for the various benchmarks. For example, absent any event-induced variance ( = 0) and when N = 500 events, the benchmark with the largest average BMP t-statistic was III (CRSP equally weighted index plus an equally weighted industry index). The average t-statistic for this benchmark was (Panel A) and the rejection rate was 33.30% for α=5% (Panel B). On the other extreme, the benchmark with the lowest average t-value was IV (CRSP value-weighted index) with an average BMP t-statistic of and a rejection rate of 29.51% for α=5%. While the absolute difference of between these two average BMP t-statistics does not appear large, it does represent a proportional increase of = 7.83%. This is roughly equivalent to the increase in power that would be expected by increasing sample size by 16% from 500 to A simple test of the difference in BMP t-statistics for these two rejects H0: BMP III BMP IV = 0 with a t-statistic of The results were fairly similar in the presence of event-induced variance (Table VII, Panels D F). Here, the average t-values were unsurprisingly lower than those of Panel A, as more noise invariably produces less powerful tests, and event-induced variance is essentially a type of noise. Specifically, the average rejection rate across all 16 benchmarks for N = 500 and α=5% 3 Estimated as the value of N satisfying SAR I = s 2 I /(N 2) SAR IV. 2 /(5000 2) s IV 10

11 was 32.16% when = 0 (Panel B), but only 19.42% when = 1 (Panel E). The best and worst performers were again benchmark III (average BMP t-statistic = and rejection rate = 19.83%) and benchmark IV (average BMP t-statistic = and rejection rate = 18.16%). Once more, the absolute difference in rejection rates appears small, but the proportional increase from least to most powerful is = 7.90%. A simple test of the difference in BMP t-statistics for these two rejects H0: BMP III BMP IV = 0 with a t-statistic of Before comparing the results when industry indices replace or supplement the market return as an additional independent variable, we first compare the commonly used benchmarks that are currently options in Eventus, specifically, benchmarks I (CRSP EW), IV (CRSP VW), VII (Fama-French Three-Factor), and XII (Carhart Four-Factor). As we found in the regression results from Table IV, the additional independent variables of benchmarks VII and XII will result in lower standard deviations of the estimation-period residuals, but also larger values of C because multicollinearity adversely affects out-of-sample predictions. Which of these two factors will dominate is primarily an empirical question. In Table VIII, we find that there is a good reason the CRSP equally weighted index (benchmark I) is the default in Eventus. When compared with the other three options using N = 500 in the absence of event-induced variance, it produces t-statistics that exceed those of the competing benchmarks by anywhere from for benchmark XII (Carhart Four-Factor) to for benchmark IV (CRSP VW). In all cases, its t-statistics are significantly larger, both with and without event-induced variance. For example, comparing the CRSP equally weighted benchmark (I) with the Fama-French benchmark (VII) for portfolios of 500 events absent eventinduced variance, we find that the difference in average t-values is and that we can reject H0: BMP I BMP VII = 0 with a t-statistic of The results are only slightly closer in the presence of event-induced variance, with a difference in average t-values of and a t- statistic of for H0: BMP I BMP VII = 0. The results in Table VIII also indicate that the weakest of the set is the CRSP VW index (benchmark IV), which is not surprising given the discussion in section I, where we compared equally weighted and value-weighted indices. It is interesting to note that the comparison of best vs. worst benchmarks from Table VII found that benchmark III beat benchmark IV with t-values in the ballpark of 7 to 8 when N = 500, and yet Table VIII finds that benchmark I is better than benchmark IV with t-values close to 40. This may seem contradictory as one might expect the best to beat the worst with the greatest t-values, but this is neither a contradiction nor an error. Benchmarks I (CRSP EW) and IV (CRSP VW) have an average correlation with each other of around 0.9 and so they track each other very closely. This necessarily implies that the difference between the t-statistics obtained from either method will be fairly stable and have a low standard deviation (around 0.19 when = 0 and 0.14 when = 1). In contrast, benchmark III (CRSP EW with an additional factor equal to the equally weighted industry index) does not track benchmark IV (CRSP VW) as closely because of the additional industry factor. The difference between these two benchmarks 11

12 average t-statistics is larger than the difference between the average t-statistics for benchmarks I (CRSP EW) and IV (CRSP VW). Specifically, the results in Panels A and D of Table VII indicate that when N = 500, the difference between the average t-statistics for benchmarks III and IV is ( for benchmark III for benchmark IV) when = 0 and ( for benchmark III for benchmark IV) when = 1. The corresponding differences for benchmarks I and IV are ( for benchmark I for benchmark IV) when = 0 and ( for benchmark I for benchmark IV) = 1. However, the standard deviation of the difference between the average test statistics for benchmarks III and IV is so much larger than that of the difference between the test statistics for I and IV that, despite the larger numerator, it generates lower t-statistics. Specifically, the standard deviation of BMPIII - BMPIV is approximately when = 0 and when = 1, which is about seven times as great as that of BMPI BMPIV in both scenarios. Thus, even though the of BMPIII - BMPIV is over 40% larger than that of BMPI BMPIV, its standard deviation is so much larger that it has a substantially smaller t-statistic for the test BMP = 0. 4 This emphasizes the limitations of t- statistics, a point made by Ziliak and McCloskey (2008) and related to the perils of relying too heavily on p-values (e.g., see Nuzzo (2014)). However, the primary purpose of this paper is not just to compare currently used methods, but to determine whether an industry index improves the performance of commonly used models when it is used instead of or in addition to a market index. Thus, we find it natural not to compare all the models with each other, but to compare them within similar groups. We now examine what happens when we take a commonly used model and replace or supplement the market index with an industry index. For example, the default in Eventus is a single-factor model using the CRSP equally weighted index (benchmark I). The two most natural comparisons involving an industry index are benchmark II (an equally weighted industry index is used instead of the market index) and benchmark III (an equally weighted industry index is used in addition to the market index). The results are reported in Table IX. With the sole exceptions of N = 500 and N = 1,000 when = 0, benchmark II slightly outperforms the more common benchmark I. However, regardless of sample size or the presence of eventinduced variance, benchmark III consistently outperforms benchmarks I and II. This was not a foregone conclusion, as while it is true that an additional independent variable will necessarily increase in-sample R 2, it will also increase out-of-sample forecast error because of the rather large correlation between industry and market indices. Nevertheless, the simulations suggest the benefit of adding an equally weighted industry index to the CRSP equally weighted market index outweighs the costs created by multicollinearity. Returning once more to the results of 4 For example, consider three sequences of observed t-values, TA = {xi}, TB = {xi (-1) n }, and TC {xi (- 1) n }. The sequences of differences will be TB-A = { (-1) n } and TC-A = {.2 +.2(-1) n }. The and standard deviation of TB-A are.1 and.01, while those of TC-A are.2 and.2. Thus while the of TC-A is twice that of TB-A, its standard deviation is 20 times as large and so it will have t-values that are only one-tenth those of TB-A. 12

13 Panels A and D in Table VII, when N = 500 events, the increase in the average t-value from for benchmark I to for the two-factor benchmark III when = 0 (or to when = 1). As might be expected, these increases in average BMP test statistics are also associated with an increase in test power. Next, we will see that this pattern is repeated for variations of the commonly used benchmarks CRSP EW, CRSP VW, Fama-French, and Carhart, although the improvements in test power are generally greater in those cases. Table X reports similar results when we use value-weighted indices (benchmarks IV, V, and VI). Here, when N = 500 and absent any event-induced variance, replacing the market index with an industry index increases the average BMP test statistic by , which is significantly greater than zero (t = 3.935). Once again, supplementing CRSP VW with a value-weighted industry factor increases the BMP test statistic to an even greater extent, by (t = 5.856). In the presence of event-induced variance, the improvements are a bit more muted, but are still substantial. In this case, replacing CRSP VW with a value-weighted industry index increases the average test statistic by (t = ), while supplementing CRSP VW with that valueweighted industry index increases the average BMP statistic by (t = ). We next consider natural peers of the Fama-French Three-Factor Model (benchmark VII). Specifically, we examine what happens if the market index is replaced with an equally weighted industry index (benchmark VIII) or a value-weighted industry index (benchmark IX), or if it is supplemented by these industry indices (benchmarks X and XI). The comparisons are summarized in Table XI. Once again there is a familiar theme: use of an industry index improves test power. For example, in the absence of event-induced variance when N = 500, the FF model s (benchmark VII) average BMP test statistic is increased by (t = ) when its market index is replaced with an equally weighted industry index (benchmark VIII), and by (t = ) when it is replaced with a value-weighted industry index (benchmark IX). There are stronger improvements when the industry indices supplement the market index; for example, adding an equally weighted industry index to the FF model to construct benchmark X increases the BMP test statistic by (t = ), and adding a value-weighted industry index to create benchmark XI increases it by (t = ). In terms of test power, = 5% rejection rates increase by 14.2% and = 1% rejection rates increase by 18.4%. 5 The results are similar but slightly less dramatic in the presence of event-induced variance. Finally, we consider similar comparisons for the Carhart Four-Factor Model (benchmark XII), the results of which are summarized in Table XII. The improvement in test power from use of industry indices is substantial, but a bit more muted than when the FF Model is altered. For example, absent event-induced variance and when N = 500, the average BMP test statistic is increased by (t = ) when the market risk premium is replaced by an equally weighted industry risk premium to form benchmark XIII and by (t = ) when it is 5 Calculated as 33.96% 29.74% 14.81% 1 and 1 based on the results of Panels B and C in Table VII % 13

14 replaced by a value-weighted industry risk premium to form benchmark XIV. There is greater improvement when the industry index supplements the market index. For instance, when N = 500 and θ = 0, the average BMP statistic rises by (t = ) when the Carhart Four- Factor Model is supplemented with an equally weighted industry index to create benchmark XV and it increases by (t = ) when a value-weighted industry index is added instead to construct benchmark XVI. The results are similar in the presence of event-induced variance. IV. Conclusions We compare a number of benchmarks for event studies and find several that improve test power compared with the current Eventus options. Of the available choices in Eventus, we find the default (the CRSP equally weighted index) outperforms the CRSP value-weighted index, Fama-French Three-Factor, and Carhart Four-Factor benchmarks. However, the performance of all four standard benchmarks can be improved by either replacing or supplementing the market index with an industry index. King (1966) found that an industry index explains about 10% of a stock s variance of returns, but for some reason, subsequent event-study methods have bypassed the use of industry indices. We confirm that use of an industry index produces even more powerful tests. Replacing the market index with a stock s industry index improves test power substantially except in the case of the CRSP equally weighted index, when it leaves the average BMP test statistic essentially unchanged. Supplementing the market index with an industry index improves test power to an even greater extent, even after making the standard adjustment that multicollinearity creates for out-of-sample predictions (C of eq. [2]). We find that absent eventinduced variance in samples of 500 events, consideration of industry indices improves test power by up to 18.4% (when = 1% and an equally weighted industry index is used to supplement the Fama-French Three-Factor Model), and also improves test power (albeit by a slightly smaller amount) when it is used to replace the market index. Consideration of industry indices also increases test power, albeit by slightly smaller amounts, when event-induced variance is present. Given that more powerful tests are superior, we recommend that industry indices be more widely used in event studies. 14

15 References Boehmer, E., J. Musumeci, and A. Poulsen, 1991, Event Study Methodology under Conditions of Event-Induced Variance, Journal of Financial Economics, 30: Brown, S.J., and J.B. Warner, 1980, Measuring Security Price Performance, Journal of Financial Economics, 8(3): Brown, S.J., and J.B. Warner, 1985, Using Daily Stock Returns: The Case of Event Studies, Journal of Financial Economics, 14(1): Carhart, M., 1997, On Persistence in Mutual Fund Performance, The Journal of Finance, 52(1): Cowan, A.R., 2007, Eventus, Eventus User s Guide: Software Version 8.0, Standard Edition 2.1, Cowan Research L.C. Cremers, M., A. Petajisto, and E. Zitzewitz, 2012, Should Benchmark Indices Have Alpha? Revisiting Performance Evaluation, Critical Finance Review 2: Fama, E., and K. French, 1992, The Cross-Section of Expected Stock Returns, The Journal of Finance, 47(2): Fama, E., and K. French, 1997, Industry Costs of Equity, Journal of Financial Economics, 43(2): Harrington, S., and D. Shrider, 2007, All Events Induce Variance: Analyzing Abnormal Returns When Effects Vary across Firms, Journal of Financial and Quantitative Analysis, 42(1): King, B., 1966, Market and Industry Factors in Stock Price Behavior, Journal of Business 39(1): Kmenta, J., 1971, Elements of Econometrics, (MacMillan, New York, NY). Marks, J., and J. Musumeci, 2017, Misspecification in Event Studies, Journal of Corporate Finance, 45: Nuzzo, R., 2014, Scientific Method: Statistical Errors. Nature 506:

16 Patell, J., 1976, Corporate Forecasts of Earnings Per Share and Stock Price Behavior: Empirical Test, Journal of Accounting Research, 14(2): Ziliak, S.T., and D. N. McCloskey, 2008, The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice, and Lives, Ann Arbor, MI: University of Michigan Press. 16

17 Table I: Correlations between Factor Returns Panel A: Correlations between Index Returns for the Full Sample (11/03/ /30/2016) CRSPEW CRSPVW SMB HML MOM CRSPEW CRSPVW SMB HML MOM Panel B: Mean Correlations between Index Returns for the 198 Non-overlapping 120-day Windows CRSPEW CRSPVW SMB HML MOM CRSPEW CRSPVW SMB HML MOM Panel C: Median Correlations between Index Returns for the 198 Non-overlapping 120-day Windows CRSPEW CRSPVW SMB HML MOM CRSPEW CRSPVW SMB HML MOM This table displays correlations between various common factor returns. The CRSP Equally Weighted Index never appears in a regression with any of the other indices in the table, so its correlations with those indices do not concern us and are shown only for completeness. Of the remaining indices, the largest full-sample (Panel A) correlation of is between the CRSP value-weighted index and the HML factor. Within 120-day windows, however, the magnitudes of the correlations generally have smaller s and s. The exception is MOM, which not only changes sign, but also has a larger correlation with the CRSP VW index. Whether this creates an increase in the out-of-sample prediction error (C in eq. [2]) that reduces test power is an empirical question addressed in Tables VII XII. 17

18 Table II: Correlations between Equally Weighted Industry Indices and Other Common Indices Industry CRSP EW CRSP VW SMB HML MOM full sample avg 120- day windows full sample avg 120- day windows full sample avg 120- day windows full sample avg 120- day windows full sample avg 120- day windows average This table reports correlations between the 48 equally weighted industry indices and five other common indices. Full sample refers to the entire set of observations, while avg. 120-day windows is the average correlation across 198 disjoint intervals of 120 trading days each. The industry indices share the largest correlations with the market index, but whether this reduces test power when both are included in a regression (as in benchmarks III, X, and XV) is an empirical question that is answered in Tables IX, XI, and XII.

19 Table III: Correlations between Value-Weighted Industry Indices and Other Common Indices Industry CRSP EW CRSP VW SMB HML MOM full sample avg 120- avg 120- avg 120- avg 120- avg 120-day full day full sample day full sample day full sample day windows sample windows windows windows windows average This table presents correlations between the 48 value-weighted industry indices and five other common indices. Full sample refers to the entire set of observations, while avg. 120-day windows is the average correlation across 198 disjoint intervals of 120 trading days each. As in Table II, the industry indices share the largest correlations with the market index, but whether this reduces test power when both are included in a regression (as in benchmarks VI, XI, and XVI) is an empirical question that is addressed in Tables X, XI, and XII. 19

20 Table IV: Summary Statistics for Regressions Benchmark: I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI Alpha CRSP EW CRSP VW SMB HML MOM Industry EW Industry VW R Residual C (forecast error adjustment) This table summarizes the regression results for our simulated events. The parameters are based on a 120-day estimation period preceding each event candidate in the CRSP database. There were 37,587,725 observations satisfying the $5 price filter. Not surprisingly, benchmarks with more factors generally produced higher values for R 2 and smaller in-sample standard deviations of the residual, but larger values of the forecast error adjustment, C, because of correlation between the factors.