Generalized Rank Test for Testing Cumulative Abnormal Returns in Event Studies Professor James Kolari, Chase Professor of Finance, Texas A&M Univer- sity, TAMU - 4218, Finance Dept., College Station, TX 77843-4218, Email: [email protected], Office phone: 979-845-4803, Fax: 979-845-3884 Professor Seppo Pynonnen, 1 Department of Mathematics and Statistics, Uni- versity of Vaasa, P.O.Box 700, FI-65101, Vaasa, Finland, Email address: sjp@uwasa.fi, Office phone: +358-6-3248259, Fax: +358-6-3248557 Abstract. Corrado’s (1989) rank test and its modification in Corrado and Zivney (1992) that accounts for possible volatility changes due to the event effect appear to have good (empirical) power properties against the para- metric tests of Patell (1976) and Boehmer, Musumeci and Poulsen (BMP) (1991). However, the Corrado-Zivney test is derived for an one-day event window. The ranks of abnormal returns are dependent by construction, which introduces incremental bias in the standard error in the denomina- tor of the simple CAR t-statistic of ranks as the accumulation period grows. This paper proposes a generalized rank test that can be used both for testing cumulative abnormal returns as well as one-day abnormal returns. Empiri- cal properties of the test statistics are studied with simulations using CRSP returns. The results show that the some popular test statistics, including the ordinary t-test and adjusted Corrado-Zivney test with cumulated ranks tend to under-reject the null hypothesis as the CAR period increases. In ad- dition, the power of the cumulated ranks Corrado-Zivney test seems to suffer when the abnormal return is randomly assigned to a single day within the event window. The proposed generalized rank test is robust against these problems. Furthermore, it is robust to abnormal return serial correlation, event-induced volatility, and cross-correlation due to event day clustering, with competitive (empirical) power relative to the standard parametric tests of Patell and BMP. JEL Classification: G14; C10; C15 EFM Classification: 350; 760 1 Presenting author
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Generalized Rank Test for Testing
Cumulative Abnormal Returns in Event
Studies
Professor James Kolari, Chase Professor of Finance, Texas A&M Univer-sity, TAMU - 4218, Finance Dept., College Station, TX 77843-4218, Email:[email protected], Office phone: 979-845-4803, Fax: 979-845-3884
Professor Seppo Pynonnen,1 Department of Mathematics and Statistics, Uni-versity of Vaasa, P.O.Box 700, FI-65101, Vaasa, Finland, Email address:[email protected], Office phone: +358-6-3248259, Fax: +358-6-3248557
Abstract. Corrado’s (1989) rank test and its modification in Corrado andZivney (1992) that accounts for possible volatility changes due to the eventeffect appear to have good (empirical) power properties against the para-metric tests of Patell (1976) and Boehmer, Musumeci and Poulsen (BMP)(1991). However, the Corrado-Zivney test is derived for an one-day eventwindow. The ranks of abnormal returns are dependent by construction,which introduces incremental bias in the standard error in the denomina-tor of the simple CAR t-statistic of ranks as the accumulation period grows.This paper proposes a generalized rank test that can be used both for testingcumulative abnormal returns as well as one-day abnormal returns. Empiri-cal properties of the test statistics are studied with simulations using CRSPreturns. The results show that the some popular test statistics, includingthe ordinary t-test and adjusted Corrado-Zivney test with cumulated rankstend to under-reject the null hypothesis as the CAR period increases. In ad-dition, the power of the cumulated ranks Corrado-Zivney test seems to sufferwhen the abnormal return is randomly assigned to a single day within theevent window. The proposed generalized rank test is robust against theseproblems. Furthermore, it is robust to abnormal return serial correlation,event-induced volatility, and cross-correlation due to event day clustering,with competitive (empirical) power relative to the standard parametric testsof Patell and BMP.
Abstract. Corrado’s (1989) rank test and its modification in Corrado andZivney (1992) that accounts for possible volatility changes due to the eventeffect appear to have good (empirical) power properties against the para-metric tests of Patell (1976) and Boehmer, Musumeci and Poulsen (BMP)(1991). However, the Corrado-Zivney test is derived for an one-day eventwindow. The ranks of abnormal returns are dependent by construction,which introduces incremental bias in the standard error in the denomina-tor of the simple CAR t-statistic of ranks as the accumulation period grows.This paper proposes a generalized rank test that can be used both for testingcumulative abnormal returns as well as one-day abnormal returns. Empiri-cal properties of the test statistics are studied with simulations using CRSPreturns. The results show that the some popular test statistics, includingthe ordinary t-test and adjusted Corrado-Zivney test with cumulated rankstend to under-reject the null hypothesis as the CAR period increases. In ad-dition, the power of the cumulated ranks Corrado-Zivney test seems to sufferwhen the abnormal return is randomly assigned to a single day within theevent window. The proposed generalized rank test is robust against theseproblems. Furthermore, it is robust to abnormal return serial correlation,event-induced volatility, and cross-correlation due to event day clustering,with competitive (empirical) power relative to the standard parametric testsof Patell and BMP.
Rejection rates based on 1,000 simulations for portfolios of size 50 securities with estimation period of 239 days and event
period 21 days. The event day is the day 250 denoted as t = 0. Cumulative abnormal returns, CAR(−d, +d), with
d = 0, 1, 5, and 10 are cumulated around the event day. Securities from CRSP and event dates from period 1990 2005
are randomly selected with replacement. Ordinary t-test, tpatell, and tbmp are parametric tests, tcumrankr, and tgrank are
nonparametric tests. a = 10%, b = 5%, and c = 1% significant. GRANK and CUM RANK coincide for the single event
day abnormal return AR(0).
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Table 2. Rejection rates in two-tailed test at the 5 % level of the null hypoth-esis of no mean event effects with different event windows and with differentlevels of event induced volatility.
Market model abnormal returns.
Event induced volatility,√
c σ
Test statistics c = 1.0 c = 1.5 c = 2.0 c = 3.0
Panel A: AR(0)
ORDIN tordin [Eq. (18)] 0.043 0.094 0.145 0.220
PATELL tpatell [Eq. (20)] 0.051 0.116 0.178 0.268
BMP tbmp [Eq. (21)] 0.045 0.042 0.043 0.043
CUM RANK tcumrank [Eq. (28)] 0.045 0.047 0.051 0.042
GRANK tgrank [Eq. (12)] 0.045 0.047 0.051 0.042
Panel B: CAR(−1, +1)
ORDIN tordin [Eq. (18)] 0.037 0.084 0.128 0.208
PATELL tpatell [Eq. (20)] 0.069 0.115 0.161 0.261
BMP tbmp [Eq. (21)] 0.054 0.051 0.056 0.056
CUM RANK tcumrank [Eq. (25)] 0.037 0.038 0.036 0.038
GRANK tgrank [Eq. (12)] 0.048 0.047 0.046 0.049
Panel C: CAR(−5, +5)
ORDIN tordin [Eq. (18)] 0.029 0.063 0.105 0.195
PATELL tpatell [Eq. (20)] 0.057 0.105 0.154 0.250
BMP tbmp [Eq. (21)] 0.057 0.061 0.059 0.062
CUM RANK tcumrank [Eq. (28)] 0.028 0.027 0.028 0.028
GRANK tgrank [Eq. (12)] 0.059 0.058 0.055 0.056
Panel D: CAR(−10, +10)
ORDIN tordin [Eq. (18)] 0.026 0.076 0.121 0.212
PATELL tpatell [Eq. (20)] 0.045 0.103 0.151 0.246
BMP tbmp [Eq. (21)] 0.066 0.071 0.064 0.063
CUM RANK tcumrank [Eq. (28)] 0.022 0.023 0.022 0.022
GRANK tgrank [Eq. (12)] 0.067 0.066 0.066 0.069
Rejection rates based on 1,000 simulations for portfolios of size 50 securities with estimation period of 239 days and
event period 21 days. The event day is day 250 denoted as t = 0. Cumulative abnormal returns, CAR(−d, +d), with
d = 0, 1, 5, and 10 are cumulated around the event day. Securities from CRSP and event dates from period 1990 to
2005 are randomly selected with replacement. The ordinary t-test tordin, the PATELL test tpatell, and the BMP test tbmp
are parametric tests, the CUM RANK test, tcumrank, and the GRANK test tgrank are nonparametric tests. GRANK and
CUM RANK coincide for the single event day abnormal return AR(0). The 95 percent confidence interval around the 0.05
rejection rate is [0.036, 0.064] and the respective 99 percent confidence interval is [0.032, 0.068]
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Table 3. Two-tailed average rejection rates at the 0.05 significance levelfor selected test statistics sampled from 1,000 random portfolios of n = 50securities with randomly assigned (cumulative) abnormal return on one eventday within the cumulated window.
Test statistic
Panel A: AR(0) ORDIN PATELL BMP CUM RANK GRANK
−3.0 0.998 1.000 0.999 1.000 1.000
−2.0 0.939 1.000 0.995 0.999 0.999
−1.0 0.433 0.855 0.822 0.876 0.876
−0.5 0.129 0.324 0.348 0.378 0.378
0.0 0.043 0.051 0.045 0.045 0.045
0.5 0.120 0.324 0.314 0.384 0.384
1.0 0.405 0.852 0.824 0.900 0.900
2.0 0.935 0.997 0.992 0.998 0.998
3.0 0.995 1.000 0.999 1.000 1.000
Panel B: CAR(−1, +1) ORDIN PATELL BMP CUM RANK GRANK
−4.2 0.989 1.000 0.998 0.998 0.999
−2.1 0.599 0.956 0.936 0.909 0.967
−1.0 0.173 0.498 0.514 0.475 0.544
0.0 0.037 0.069 0.054 0.037 0.048
1.0 0.161 0.405 0.439 0.471 0.506
2.1 0.552 0.935 0.910 0.929 0.968
4.2 0.986 1.000 0.996 1.000 1.000
Panel C: CAR(−5, +5) ORDIN PATELL BMP CUM RANK GRANK
−8.5 0.993 1.000 1.000 0.846 1.000
−5.1 0.793 0.996 0.995 0.742 0.995
−2.5 0.266 0.684 0.699 0.463 0.716
−0.8 0.056 0.155 0.182 0.119 0.169
0.0 0.029 0.057 0.057 0.028 0.059
0.8 0.030 0.078 0.105 0.083 0.138
2.5 0.206 0.566 0.605 0.457 0.686
5.1 0.731 0.990 0.970 0.833 0.994
8.5 0.992 1.000 0.998 0.967 0.999
Panel D: CAR(−10, +10) ORDIN PATELL BMP CUM RANK GRANK
−10.1 0.967 1.000 0.999 0.505 0.999
−6.1 0.659 0.980 0.982 0.482 0.976
−4.0 0.362 0.795 0.819 0.390 0.811
−2.0 0.114 0.329 0.376 0.205 0.347
−1.0 0.058 0.132 0.163 0.094 0.146
0.0 0.026 0.045 0.066 0.022 0.067
1.0 0.024 0.047 0.070 0.040 0.096
2.0 0.059 0.172 0.213 0.133 0.291
4.0 0.247 0.647 0.667 0.352 0.759
6.1 0.527 0.953 0.946 0.524 0.977
10.1 0.958 1.000 1.000 0.728 1.000
GRANK and CUM RANK coincide for the single event day abnormal return AR(0).
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Figure 1. Estimated power functions with different CAR-windows for
ORDIN, PATELL, BMP, CUM RANK, and GEN RANK tests based on 1,000
samples of n = 50 security portfolios from the CRSP database: Significance
level is 0.05, two-sided tests, and no event-induced variance.
GRANK and CUM RANK coincide for the single event day abnormal return AR(0).