The practice of quantitative research not only involves statistical calculations and formulas but also involves the understanding of statistical techniques related to real world applications. You might not become a quantitative researcher nor use statistical methods in your profession, but as a consumer, citizen, and scholar-practitioner, it will be important for you to become a critical consumer of research, which will empower you to read, interpret, and evaluate the strength of claims made in scholarly material and daily news.
For this Assignment, you will critically evaluate a scholarly article related to t tests.
To prepare for this Assignment:
Search for and select a quantitative article specific to your discipline and related to t tests. Help with this task may be found in the Course guide and assignment help linked in this week’s Learning Resources.
For this Assignment:
Write a 2- to 3-page critique of the article. In your critique, include responses to the following:
Why did the authors use this t test?
Do you think it’s the most appropriate choice? Why or why not?
Did the authors display the data?
Do the results stand alone? Why or why not?JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 27, NO. 3, SEPTEMBER 1992
The Specification and Power of the Sign Test in
Event Study Hypothesis Tests Using Daily Stock
Returns
Charles J. Corrado and Terry L. Zivney*
Abstract
This paper evaluates a notiparametric sign test for abnormal security price performance
in event studies. The sign test statistic examined here does not require a symmetrical
distribution of security excess returns for correct specification. Sign test performance is
compared to a parametric /-test and a nonparametric rank test. Simulations with daily
security return data show that the sign test is better specified under the null hypothesis
and often more powerful under the alternative hypothesis than a /-test. The performance
of the sign test is dominated by the performance of a rank test, however, indicating
that the rank test is preferable to the sign test in obtaining nonparametric inferences
concerning abnormal security price performance in event studies.
I. Introduction
Iti financial event studies, a sign test is commonly used to specify statistical
significance independently of an assumption concerning the distribution of the
excess return population from which data are collected. Seemingly a completely
nonparametric procedure, nevertheless, a sign test can be misspecified if an incorrect
assumption about the data is imposed. For example, Jain ((1986), p. 88)
reports that 53 percent of a sample of five million excess returns is negative.
Consequently, Brown and Warner (1980), (1985) and Berry, Gallinger, and Henderson
(1990) demonstrate that a sign test assuming an excess return median of
zero is misspecified.
This paper studies the specification and power of a sign test that does not
assume a median of zero, but instead uses a sample excess return median to
calculate the sign of an event date excess return. This version of the sign test
is expected to be correctly specified no matter how skewed the distribution of
security excess returns and may be efficient compared to a f-test for distributions
‘College of Business and Public Administration, University of Missouri, 214 Middlebush
Hall, Columbia, MO 65211, and School of Business Administration, University of Tennessee, 615
McCallie Avenue, Chattanooga, TN 37403, respectively. The authors thank JFQA referee Robert
Connolly for helpful comments.
465
466 Journal of Financial and Quantitative Analysis
with heavier tailweights than the normal distribution.’ Sign test performance
is compared with two other event study procedures: a parametric f-test and a
nonparametric rank test. These provide benchmarks against which the sign test
is compared.
In addition to a comparative evaluation of the three test procedures discussed
above, this paper examines two relevant and related issues in event
study methodology. The first issue is the effect of short estimation periods on
the performance of test statistics commonly used in event studies. This may be
important when market model parameter instability is suspected and, as a result,
a researcher prefers to use a relatively short estimation period. The second, and
related issue, is the effect of a prediction error variance correction on test statistic
performance. A prediction error correction may become important as the
estimation period shortens; we examine how important the correction procedure
might become.^
The organization of the paper is as follows. Section II presents the test
procedures and test statistics used in this study. Sample construction is described
in Section III. Empirical results are presented in Section IV. Section V provides
a summary and conclusion.
II. Test Procedures and Test Statistics
We examine tests of the null hypothesis that the shift in the distribution
of event date excess returns is zero. Simulation experiments, as in Brown and
Warner (1985), are used where the excess return measure is the residual from
the standard market model.^ In each experiment, securities and event dates are
randomly selected and a portfolio is formed. A 250-day sample period surrounds
each event date. An event date is defined as day 0, and days -244 through – 6
comprise a 239-day estimation period from which market model parameters are
obtained.
In addition to simulations using 250-day sample periods, we also report
the effect on test statistic performance from using shorter sample periods of 100
days and 50 days, where days -94 through – 6 and days -44 through – 6 form
89-day and 39-day estimation periods, respectively.
We compare the performance of a sign test with two alternative tests: a ttest
and a rank test. These two benchmark statistics are T2 and Tj, respectively,
as specified in Corrado (1989). We first review test statistic construction for
T2 and Tj and, immediately following, we specify construction of the sign test
statistic, denoted by T^.
‘The efficiency of the sign test in the presence of normal and nonnormal distributions is discussed
in Hettmansperger (1984) and Lehmann (1975), (1986). Zivney and Thompson (1989) suggest
that a properly specified sign test may provide a more powerful test for abnormal security price
performance in event studies than the f-test.
The prediction error correction procedure is discussed in many statistics and econometrics texts.
A suggested reference is Maddala (1977).
‘Brown and Warner ((1985), pp. 210-211) discuss the importance of simulations in evaluating
event study methods.
Corrado and Zivney 467
A. Mest (72)
Let Ai, represent the excess return of security i on day t. Each excess
return is divided by its estimated standard deviation to yield a standardized
excess return,
(1) K = Ai,/SiAi),
where the standard deviation is calculated as ^
(2) SiAi) = .
The day 0 test statistic is given by
1 -^
V A^
238 ^ “•
‘^^° f=-244
1 ‘^
(3) T2 = ^Z^’o ‘
where A^ is the number of securities in the sample portfolio.
B. Rank Test (73)
Let Ki, denote the rank of the excess retum A,, in security /’s 250-day time
series of excess retums,
(4) Ki, = rankiAi,), t =-244,…,+5.
To allow for missing retums, ranks are standardized by dividing by one plus the
number of nonmissing retums in each firm’s excess retums time series,
(5) Ui, = Ki,/i\+Mi),
where M, is the number of nonmissing retums for security i. This yields order
statistics for the uniform distribution with an expected value of one-half.^ The
rank test statistic substitutes (f/» – 1/2) for the excess retum Ai,, yielding this
day 0 test statistic,*
(6)
1 ‘^
7-3 = ^y(f/,o-l/2)/S(t/).
“”For estimation periods of 89 days and 39 days, the standard deviation S(Ai) is calculated as
88 2.^^’ –^
(=-94
1 ^ 2
— 2 _ ‘*,-,. respectively,
‘ A discussion of order statistics is contained in Lehmann (1986).
*If there are no missing returns, i.e.. Mi the same for all i, then this statistic is identical to T-i as
specified in Corrado ((1989), p. 388), Without an adjustment for missing retums, the rank test may
be misspecified. Corrado ((1989), p. 389) avoids this problem by restricting event date selection to
allow a 250-day sample for each security.
468 Journal of Financial and Quantitative Analysis
The standard deviation SiU) is calculated using the entire sample period.
(7) SiU) =
+5
where N, represents the number of nonmissing retums in the cross-section of
N-firms on day t in event time.
C. Sign Test (74)
Let the median excess retum in security i’s time series of excess retums
be denoted by median (A,). For each day in the sample period, the sign of each
excess retum is calculated as
(8) Gi, = signiAi, – median (A,)) t = -244,…, -i-5,
where signix) is equal to -l-l, -1 , or 0 as x is positive, negative, or zero,
respectively. From the signs G,,, this day 0 test statistic is constructed
1 ‘^
(9) 74 = -j=yG
The standard deviation 5(G) is calculated using the entire sample period,^
(10) S(G) =
where A^, is the number of nonmissing retums in the cross-section of A’-finns
on day t in event time.
The sign test procedure specified above transforms security excess retums
into sign values where the probability of a value of +1 is equal to the probability
of a value of – 1 regardless of any asymmetry in the original distribution. This
procedure precludes the misspecification documented by Brown and Wamer
(1980), (1985) and Berry, Gallinger, and Henderson (1990) of a sign test that
assumes an excess retum median of zero.
III. Sample Construction
From the Center for Research in Security Prices (CRSP) Daily Retum Files,
we obtain daily retum data for 600 firms. All firms are listed over the entire
period from July 1962 through December 1986 and none have more than ten
missing retums. From this data base, we construct 1,200 portfolios each of 10
and 50 securities. Each time a security is selected for inclusion in a portfolio,
a hypothetical event date is randomly generated. Securities and event dates are
randomly selected with replacement.
‘For sample periods of 100 days and 50 days, the standard deviations 5(f,) and 5(G,) are
calculated by summation from day -94 through day +5, and day -44 through day +5, respectively.
Corrado and Zivney 469
IV. Empirical Results
A. Test Statistics with No Abnormal Performance
1. Test Statistics without a Variance Correction for Prediction Error
For sufficiently large sample sizes, the Central Limit Theorem implies that
the distribution of each test statistic will converge to normality. We examine
the completeness of this convergence for portfolio sizes of 10 and 50 securities.
Table 1 summarizes the empirical distributions of the three test statistics without
a variance correction for prediction error by reporting the first four sample
moments and the studentized range of each statistic for portfolio sizes of 10 and
50 securities and sample periods of 50, 100, and 250 days.
Summary
Test
Statistic
Measures of
Mean
250-Day Sample Period,
72
73
To’h£ 74
-0.022
-0.028
-0.014
-0.013
-0.037
-0.049
100-Day Sample Period
72
73
74
72
73
74
-0.035
-0.046
-0.051
-0.028
-0.042
-0.052
50-Day Sample Period,
72
73
72
73
7-4
-0.029
-0.033
-0.016
-0.039
-0.052
-0.066
TABLE 1
the Distribution of Test Statistics with No Abnormal
OLS Market Model Adjusted
Standard
Deviation
, Portfolio Size = 50
1.077
1.015
1.008
Portfolio Size = 10
1.053
1.007
1.012
, Portfolio Size = 50
1.098
1.009
1.008
Portfolio Size = 10
1.060
0.996
1.008
Portfolio Size = 50
1.136
1.015
0.992
Portfolio Size = 10
1.091
0.992
1.004
Skewness
0.133
-0.016
-0.064
0.261*
-0.006
-0.049
0.226*
-0.014
-0.074
0.231*
0.008
-0.034
0.276*
-0.016
-0.079
0.305*
-0.004
-0.039
Excess Returns
Kurtosis
3.634*
2.979
2.887
3.732*
2.917
2.776
3.936*
2.931
2.852
3.701*
2.852
2.688
4.441*
2.847
2.866
4.152*
2.736
2.590
Performance
Studentized
Range
7.375
6.488
6.259
8.129*
6.341
6.167
8.949*
6.540
6.531
7.919*
6.671
5.800
9.949*
6.199
6.464
8.991*
5.844
5.703
Test statistic distribution measures based on 1,200 simulation experiments: portfolio sizes
= 10, 50 securities, sample periods = 50, 100, 250 days. Randomly selected daily stock
returns and event dates over the period 1962 through 1986. T2 is a parametric f-test statistic,
73 is a nonparametric rank test statistic, and T4 is a nonparametrio sign test statistic.
‘Significant at 99-percent confidence level.
The distributions of the f-test statistic and the rank test statistic reported in
Table 1 for a 250-day sample period are similar to those reported in Brown and
470 Journal of Financial and Quantitative Analysis
Wamer (1985) and Corrado (1989). Compared to a standard normal distribution,
the f-test statistic distribution is significantly positively skewed with a coefficient
of skewness of 0.261 for portfolio sizes of 10 securities, but somewhat less
skewed with portfolio sizes of 50 securities where the coefficient of skewness is
0.133. The distribution of the f-test statistic is significantly kurtotic relative to a
normal distribution for all portfolio sizes and sample periods where the smallest
reported coefficient of kurtosis is 3.634.^ By contrast, the rank test statistic and
the sign test statistic sample moments are all close to those expected from a
standard normal population.
The significant kurtosis reported for the f-test suggests that the distribution
of the f-test statistic deviates from normality in the tails of the distribution.
Since statistical inferences are based on tailweight probabilities for a normal
distribution, the f-test might yield biased inferences in event studies using daily
stock retums. To assess the potential severity of this bias, Table 2 summarizes
the tailweight specification of the three test statistics by reporting rejection rates
in nominal 5-percent level and 1-percent level, upper-tail and lower-tail tests for
portfolio sizes of 10 and 50 securities and sample periods of 50, 100, and 250
days.
TABLE 2
Rejection Rates of Null Hypothesis with
Portfolio Size = 50
Lower Tail
5% 1%
250-Day Sample Period
T?
73
TA
5.6% 1.3%
5.8 1.3
6.4 1.3
100-Day Sample Period
T?
T3
TA
6.5% 1.3%
5.7 1.0
6.3 1.3
50-Day Sample Period
T?
73
TA
7.2%* 1.6%
6.1 1.2
5.0 1.1
Nominal Test
Upper Tail
5%
6.1%
4.8
5.3
6.1%
4.4
3.8
6.8%*
4.5
4.2
1%
2.1%*
1.2
0.7
2.3%*
1.1
0.6
2.3%*
1.1
0.6
No Abnormal Performance
Levels
Portfolio Size = 10
Lower Tail
5%
5.4%
5.8
6.6
5.3%
5.3
6.3
6.5%
5.9
6.2
1%
1.4%
1.0
1.6
1.7%
0.9
1.0
1.8%*
1.1
1.1
Upper Tail
5%
6.5%
4.3
4.8
6.6%
4.0
5.0
6.3%
4.3
4.5
1%
2.0%*
1.2
0.9
1.8%*
• 1.1
0.8
2.1%*
0.8
0.3
Rejection rates based on 1,200 simulation experiments: portfolio sizes = 10, 50 securities,
sample periods = 50, 100, 250 days. Randomly selected securities and event dates over
the period 1962 through 1986. T2 is a parametric f-test statistic, 73 is a nonparametric rank
test statistic, and TA is a nonparametric sign test statistic.
* Significant at 99-percent confidence level.
‘Normal population skewness and kurtosis coefficients are 0 and 3, respectively. Pearson and
Hartley ((1966), pp. 207-208) provide 99-percent critical values of 0.165 and 3.37 for skewness
and kurtosis coefficients, respectively, for samples of size 1200 from a normal population. The
studentized range 99-percent critical value of 7.80 is obtained from Fama ((1976), p. 40).
Corrado and Zivney 471
The results reported in Table 2 indicate that the f-test is well specified in
upper-tail and lower-tail 5-percent level tests with sample periods of 100 days
and 250 days; the largest deviation from a correct 5-percent rejection rate is a
rate of 6.6 percent obtained with a portfolio size of 10 and a 100-day sample
period. With a short 50-day sample period (39-day estimation period), f-test
specification deteriorates slightly with a tendency to reject the null hypothesis
too often.’ Since sample periods in event studies are usually longer than 100
days, the parametric f-test provides reliable test specification in 5-percent level
tests even with portfolios of as few as 10 securities.'”
The adverse effects of the significant skewness and kurtosis (reported in
Table 1) on f-test specification only became apparent in 1-percent level tests.
The f-test yields biased inferences in upper-tail 1-percent level tests where it
typically rejects the null hypothesis at rates of 2 percent or more.” In lower-tail
1-percent level tests, the f-test is somewhat better specified where the largest
rejection rate is 1.8 percent. By contrast, both the rank test and the sign test
are well specified in upper-tail and lower-tail 5-percent level and 1-percent level
tests for all portfolio sizes and sample periods examined here.
2. Test Statistics witii a Variance Correction for Prediction Error
In event studies using market model excess retums, the day 0 excess retums
variance is sometimes corrected for prediction error. Although theoretically correct,
the importance of a prediction correction declines as the estimation period
lengthens. Using 239-day estimation periods. Brown and Wamer ((1985), p. 8)
and Corrado ((1989), p. 387) both report that a variance correction for prediction
error did not discemibly affect the results of their simulation experiments. To
assess the impact of a prediction correction with shorter estimation periods, in
Table 3, we report test statistic specification with no abnormal performance for
portfolios of 10 securities using 89-day and 39-day estimation periods, corresponding
to 100-day and 50-day sample periods, respectively.’^
Comparing test specification with a prediction error variance correction reported
in Table 3 with test specification without a prediction correction reported
in Tables 1 and 2, we see that f-test specification is improved slightly by the
correction procedure; the coefficients of skewness and kurtosis are smaller, although
they still deviate significantly from the expected coefficients for a normal
distribution. Also, rejection rates for the f-test are closer to correct levels with
a prediction correction. The improvement is discemible in experiments using a
‘Significant deviations from correct specification can be ascertained by the critical values
c = a±2.515^Ja{\-a)/n.
where a is the correctly specified rejection rate, and 2.575 is the 99.5th percentile of the standard
normal distribution. With a sample size of n = 1200, for a = 5 percent, these critical values are 5
percent ±1.62 percent, and, for a = 1 percent, they are 1 percent ±0.74 percent.
‘”Using 239-day estimation periods. Brown and Wamer ((1985), p. 14) report reliable r-test
specification in 5-percent level tests for portfolios containing as few as 5 securities.
“Brown and Wamer ((1985), p. 14) caution that because the empirical distribution of the Mest
statistic is kurtotic relative to a normal distribution, “. . . significance levels should not be taken
literally.”
‘^The effect of a prediction error correction for portfolios of 50 securities was also examined;
we found similar, but less noticeable effects.
472 Journal of Financial and Quantitative Analysis
TABLE 3
Summary Measures of Test Statistic Performance with
Variance Correction for Prediction Error; No Abnormal Performance
OLS Market-Model Adjusted Excess Returns
Test
Statistic Mean
100-Day Sample Period, 1
72
73
74
-0.026
-0.042
-0.052
Standard
Deviation
Portfolio Size
1.045
0.996
1.006
50-Day Sample Period, Portfolio Size =
72
T3
TA
-0.036
-0.049
-0.067
1.056
0.991
1.007
Skewness
= 10
0.221*
0.009
-0.038
10
0.229*
-0.050
-0.006
Kurtosis
3.664*
2.844
2.677
3.717*
2.735
2.575
Studentized
Range
8.019*
5.811
6.635
8.054*
5.926
5.685
Nominal Test Levels: Portfolio Size = 10
Lower Tail Upper Tail Lower Tail Upper Tail
5% 1% 5% 1% 5% 1% 5% 1%
50-Day Sample Period 100-Day Sample Period
72
73
TA
5.2%
5.3
6.3
1.3%
0.9
1.0
6.0%
4.3
4.3
1.8%*
0.3
0.8
5.2%
5.3
6.3
1.3%
0.9
1.0
6.3%
4.1
5.0
1.8%*
1.1
0.8
Test statistic distribution measures based on 1,200 simulation experiments: portfolio size =
10 securities; sample periods = 50, 100 days. Randomly selected daily stock returns and
event dates over the period 1962 through 1986. 72 is a parametric f-test statistic, 73 is a
nonparametric rank test statistic, and TA is a nonparametric sign test statistic.
*Significant at 99-percent confidence level.
50-day sample period, but quite small in experiments using a 100-day sample
period. The specification of the rank test statistic and the sign test statistic are
not noticeably affected by the prediction error correction procedure.
In simulation results not reported here, the power of the parametric f-test
in detecting abnormal security price performance was typically slightly reduced
by a prediction error variance correction, but only by trivial magnitudes. We
conclude that for sample periods longer than 100 days, the presence or absence
of a variance correction for prediction error does not materially affect statistical
inferences in event studies.
B. Test Statistics with Abnormal Performance
We now assess the ability of the three test statistics to detect abnormal
performance in the day 0 excess retums distribution. As in Brown and Wamer
(1985), abnormal security price performance is simulated by adding a constant
to the day 0 retum of each security.’^
‘^Brown and Wamer ((1980), p. 212) argue that detecting mean shifts is the relevant phenomenon
when comparing event study test procedures.
Corrado and Zivney 473
Table 4 reports rejection rates of the null hypothesis at the 5-percent and
1-percent test levels for abnormal performance levels of ±1/2 percent and ±1
percent, portfolio sizes of 10 and 50 securities, and sample periods of 50 days,
100 days, and 250 days.
TABLE 4
Rejection Rates of Null Hypothesis with
Abnormal Performance of ± 1/2 percent and ± 1 percent
Test
Levels
– 1 %
5% 1%
^bno^mal
-1/2%
5%
250-Day Sample Period, Portfolio Size —
T2
74
TZ
V3
98.3%
99.3
98.3
57.4%
64.1
56.1
94.1%
97.2
90.3
65.0%
76.0
72.3
Portfolio Size -•
30.4%
36.6
25.3
23.3%
29.3
29.4
lob-Day Sample Period, Portfolio Size •
TA
72
T3
TA
98.5%
“”y9.6
96.3
59.6%
63.7
55.5
94.9%
97.2
89.6
66.8%
75.9
72.3
Portfolio Size ••
32.3%
35.3
22.4
24.9%
30.3
28.8
50-Day Sample. Period, Portfolio Size =
72
73
74
T2
T3
TA
98.7%
99.4 >
98.0
61.5%
63.4
53.1
95.0%
96.6
87.9
68.9%
75.1
69.3
Portfolio Size =
34.8%
•32.6
i9.8
27.0%
28.8
27.4
1%
= 50
39.0%
50.3
43.9
= 10
8.2%
9.6
8.8
= 50
41.8%
50.8
43.8
= 10
8.3%
9.4
7.9
50
43.1%
48.3
39.6
10
9.2%
9.7
6.6
Performance
+1/2%
5%
63.5%
75.6
68.2
21.6%
25.9
25.8
63.0%
76.3
67.9
22.3%
27.3
25.1
65.7%
76.9
67.2
23.2%
27.5
24.9
1%
36.4%
48.7
39.9
8.8%
7.8
8.1
37.8%
49.2
38.7
9.0%
8.0
6.8
40.6%
47.0
37.4
9.7%
7.0
4.7
+ 1%
5%
98.6%
99.8
98.5
53.5%
63.3
53.2
98.5%
99.8
98.8
53.7%
63.2
52.8
98.9%
99.5
97.6
55.0%
61.0
51.2
1%
93.8%
97.7
91.7
28.9%
33.3
24.7
94.8%
97.8
91.6
29.5%
32.5
21.3
94.8%
96.6
89.3
31.5%
30.8
17.2
Rejection rates’ based oh^1,200 simulation experiments: portfolio sizes = 10, 50 securities;
sample periods = 50, 100′; 250 days. Randomly selected securities and event dates over
the period 1962 through 198,6. T2 is a parametric f-test statistic, T3 is a nonparametric rank
test statistic, and TA is a noni^arametric sign test statistic.
\
\
1, ±1-Percent Abnormal Rer^formance, 5-Percent Level Tests
With ±1-percent ab.normalV performance introduced, the rank test is more
powerful than the f-test,^and the iXtest is more powerful than the sign test. With
-Hi-percent abnormal performance, aqrioss 50-day, 100-day, and 250-day sample
periods, in upper-tail tests for portfolios,of size 10, rejection rate averages are
54.1 percent, 62.5 percent, and 52.4 percx^.nt for the r-test, rank test, and sign
474 Journal of Financial and Quantitative Analysis
test, respectively.”” With -1-percent abnormal performance, in lower-tail tests
across the three sample periods for portfolios of size 10, rejection rate averages
are 59.5 percent, 63.7 percent, and 54.9 percent for the r-test, rank test, and sign
test, respectively.
2. ±1/2-Percent Abnormal Performance, 5-Percent Level Tests
With ± 1/2-percent abnormal performance introduced, the rank test dominates
the sign test, and the sign test dominates the r-test. With -(-1/2-percent
abnormal performance, across 50-day, 100-day, and 250-day sample periods, in
upper-tail tests for portfolios of size 50, the rejection rate averages are 64.1
percent, 76.3 percent, and 67.8 percent for the r-test, rank test, and sign test,
respectively. With -1/2-percent abnormal performance introduced, across the
three sample periods, in lower-tail tests for portfolios of size 50, rejection rate
averages are 66.9 percent, 75.7 percent, and 71.3 percent for the r-test, rank test,
and sign test, respectively.
C. Test Statistics with a Day 0 Variance Increase
Brown and Wamer (1985) and Corrado (1989) show that the parametric
r-test statistic is vulnerable to misspecification caused by an increase in th;,e
variance of the distribution of event date excess retums. To compare the spe.iification
and power of the three test statistics in the presence of a day 0 varVahce
increase, we follow Brown and Wamer’s (1985) procedure of transforPwing each
security’s day 0 excess retum by summing a day 0 excess retum and an excess
retum from outside the sample period, /
\
(11) Al = Ao+A,6.
The transformed excess retum A*o replaces the original excess retum Ajo when
computing test statistics with a simulated variance increase.
1. Test Statistics with a Cross-Sectional Variance Adjustment
Brown and Wamer (1985) and Corrado (1989) show that a r-test procedure
that solely relies on a cross-sectional standard deviation is nnt very powerful in
detecting abnormal security price performance.’^ Recently^nowever, Boehmer,
Musumeci, and Poulsen (1991) and Sanders and Robins,(1991) independently
demonstrate that a simple cross-sectional variance adjust’phent, applied after controlling
for cross-sectional heteroskedasticity, yields a^i^^ell-specified r-test when
there actually is an event date variance increase, and importantly, does not
noticeably affect r-test power when there is no vajriance increase. The crosssectionally
adjusted r-test statistic has the following form,
1 ‘^•’
(12) 7-2 (adjusted) = -=Y A
”’For example, the 54.1-percent average for the f-test is obtained as the average of 53.5 percent,
53.7 percent, and 55.0 percent, for sample peric’ds of 250, 100, and 50 days, respectively.
“This statistic is described ih Brown ,an\d Wamer ((1985), pp. 7-8) and Corrado ((1989),
pp. 386—387), and is T\ in Corrado’s nof.’ation.
Corrado and Zivney 475
where the standardized retums A’, are defined in Equation (1) and the day 0
cross-sectional standard deviation is calculated as
(13)
We shall here examine the impact of a cross-sectional variance adjustment on
the specification and power of the sign test and the rank test, with the adjusted
r-test as a benchmark. For the sign test and the rank test, a cross-sectional
variance adjustment may be implemented as described immediately below.
Based on the standardized excess retums A;,, we define the following standardized
excess retums series.
r = 6,
where the day 0 cross-sectional standard deviation S(Ao) is defined in Equation
(13).
A cross-sectional variance-adjusted rank test is obtained by first dividmg
the ranks of the excess retums defined in Equation (14) by one plus the number
of nonmissing retums (as specified in Equation (5)),
(15) Ui, = rank{Xi,)/{\+Mi\
and then proceeding to calculate the rank test statistic as specified in Equation
(6), which is reproduced here for convenience.
(6) T, = -^Y{Uio-l/2)/SiU).
A cross-sectional variance-adjusted sign test is obtained by defining the
signs of the excess retums in Equation (14) as follows,
(16) Gi, = sign (Xi,-median (Xi)) r =-244,…,-t-5,
and then proceeding to calculate a sign test statistic as specified in Equation (9),
which is reproduced here.
(9) , 7-4 = – ^
To assess the relative performance of the three test statistics, both before
and after a cross-sectional variance adjustment, the day 0 variance is doubled by
transforming the day 0 excess retums as specified in Equation (11) above. Table
5 reports the results of simulation experiments for portfolios of 50 securities with
abnormal performance levels of 0 percent, ±1/2 percent, and ±1 percent with
100-day and 250-day sample periods.
476 Journal of Financial and Quantitative Analysis
TABLE 5
Rejection Rates with a Day 0 Variance Doubling,
with and without a Cross-Sectional Variance Adjustment
Test
Statistic
Abnormal
Performance
250-Day Sample Period
7″2
73
TA
72
73
TA
72
73
TA
0%
-1/2% / +1/2%
– 1 % /+1 %
100-Day Sample Period
Tz
73
7″4
72
73
7″4′
Tz
7″3
74
0%
-1/2% / +1/2%
– 1 % /+1 %
With Adjustment
Lower
Tail
6.4%
6.8
6.8
40.9%
46.0
40.5
85.1%
89.1
81.8
6.7%
6.5
6.5
43.2%
46.8
39.5
85.7%
88.8
81.3
Upper
Tail
3.9%
4.5
4.3
36.4%
47.5
39.9
83.1%
90.9
83.9
3.5%
4.5
4.0
37.3%
47.2
38.8
83.1%
90.6
84.3
Without Adjustment
Lower
Tail
14.7%
12.0
7.8
60.9%
62.2
44.1
94.8%
95.8
84.3
15.4%
12.7
8.0
62.4%
61.8
41.6
95.2%
95.3
83.4
Upper
Tail
12.1%
5.4
3.3
57.8%
51.1
35.6
94.4%
92.5
81.3
11.8%
4.6
2.5
57.8%
51.4
35.9
93.8%
92.4
81.3
Test statistic performance results based on 1,200 simulation experiments with abnormal
performance of 0 percent, ±1/2 percent, ±1 percent; portfolio size = 50 securities; sample
periods = 100, 250 days. T2 is a parametric Mest statistic, T3 is a nonparametric rank
test statistic, and TA is a nonparametric sign test statistic. Rejection rates with negative
abnormal performance are reported for lower-tail 5-percent tests and rejection rates with
positive abnormal performance are reported for upper-tail 5-percent level tests.
As shown in Table 5, all three test statistics display some misspecification
without a cross-sectional variance adjustment, but the r-test is the most misspecified.
The specification of all three test statistics is improved by a day 0
cross-sectional variance adjustment in the presence of a day 0 variance increase.
Since the r-test was the most misspecified as a result of an event date variance
increase, the improvement is most remarkable for the r-test. For example, with
no abnormal performance and a 250-day sample period, the lower-tail rejection
rate of 14.7 percent for the r-test without a cross-sectional variance adjustment
is reduced to 6.4 percent with the variance adjustment.
With abnormal performance and a doubled day 0 variance, among the
three test statistics, the r-test is most affected by a cross-sectional variance
adjustment. For example, with -1-1/2-percent abnormal performance and a 250-
day sample period, the r-test rejection rate is 57.8 percent without a crosssectional
variance adjustment and 36.4 percent with the variance adjustment.
With the same abnormal performance and sample period, the rank test rejection
rate is 51.1 percent without, and 47.5 percent with the variance adjustment. The
Corrado and Zivney 477
corresponding sign test rejection rates are 35.6 percent, and 39.9 percent with
the variance adjustment.
When an event date variance increase is likely, correct specification for the
r-test requires that a cross-sectional variance adjustment be implemented.’^ For
the rank test, in contrast, a variance adjustment appears to be unimportant in tests
for positive abnormal performance, but necessary in tests for negative abnormal
performance. Sign test specification and power is only slightly improved by
a cross-sectional variance adjustment. After the variance correction is applied,
sign test and r-test power are comparable, but both are dominated by the rank
test.
D. Summary and Conclusions
We study the specification and power of a nonparametric sign test for
abnormal security price performance in event studies that does not require a
median of zero in the distribution of security excess retums for correct specification.
The performance of the sign test is compared with a parametric r-test
and a nonparametric rank test. The sign test is shown to be better specified
than the r-test under a complete null hypothesis of no abnormal performance
and no variance increase. Both the sign test and the rank test are equally well
specified under this complete null hypothesis. In the presence of an event date
variance increase, all three test statistics display some misspecification, but the
misspecification is most severe for the r-test.
When abnormal performance is present, sign test power is greater than that
of a r-test in detecting ± 1/2-percent abnormal performance, but less than that of
a r-test in detecting ±1-percent abnormal performance. The rank test dominates
both the sign test and the r-test in detecting both ± 1/2-percent and ±1-percent
abnormal performance.
The effect on test statistic performance from using short estimation periods
to obtain market model parameters was examined. With estimation periods
as short as 89 days, the performance of all three test statistics was virtually
unaffected. With 39-day estimation periods, only a slight deterioration in test
performance was noticed. Similarly, virtually no improvement in test statistic
performance resulted from the use of an event date excess retum correction for
prediction error.
The results of simulation experiments presented here indicate that a sign
test based on sample excess retum medians provides reliable, well-specified
inferences in event studies. This version of the sign test is better specified
than the r-test and has a power advantage over the r-test in detecting small
(± 1/2-percent) levels of abnormal performance. However, both the sign test
and the r-test are dominated by the rank test. This suggests that if a researcher
wishes to assess statistical significance independently of a parametric assumption
conceming the distribution of the data, the rank test is preferred to the sign test.
‘^Rohrbach and Chandra (1989) and Sanders and Robins (1991) provide tests for an event date
variance increase in market model residuals.
478 Journal of Financial and Quantitative Analysis
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