Improving Tests of Abnormal Returns by Bootstrapping the Multivariate Regression Model with
Event Parameters
Scott E. Hein
Texas Tech University
Peter Westfall
Texas Tech University
April 2004
We would like to thank the editors, Rene Garcia and Eric Renault, and two anonymous referees
and an associate editor, as well as Will Ashman, Keldon Bauer, Naomi Boyd, Phil English, Lisa
Kramer, Henry Oppenheimer, Bill Maxwell, Ted Moore, Ramesh Rao, Jon Scott, Shawn
Strothers, Kate Wilkerson, Eric Walden and Zhaohui Zhang for helpful comments. We would
also like to thank Jonathan Stewart and Brad Ewing for providing data used in this analysis.
Corresponding author and requests for reprints: Scott Hein, Jerry S. Rawls College of Business,
Texas Tech University, Lubbock, TX 79409-2101, (806) 742-3433, odhen@ba.ttu.edu.
Improving Tests of Abnormal Returns by Bootstrapping the Multivariate Regression Model with
Event Parameters
Abstract
Parametric dummy variable-based tests for event studies using multivariate regression are not
robust to nonnormality of the residual, even for arbitrarily large sample sizes. Bootstrap
alternatives are described, investigated, and compared for cases where there are nonnormalities,
cross-sectional and time series dependencies. Independent bootstrapping of residual vectors
from the multivariate regression model controls type I error rates in the presence of crosssectional correlation; and, surprisingly, even in the presence of time-series dependence
structures. The proposed methods not only improve upon parametric methods, but also allow
development of new and powerful event study tests for which there is no parametric counterpart.
Key words: Event study, Event parameter estimation, Cross-sectional correlation, Bootstrap,
Significance level, Simulation.
I. Introduction
Researchers have recently extensively applied the multivariate regression model
(MVRM), using a dummy variable representing a significant event date, to test the significance
of many different events on both financial asset prices and interest rates. Binder (1985a, b) and
MacKinlay (1997) provide surveys of event studies in finance and economics. Binder (1985a, b)
argues that a main advantage of employing an MVRM over examining cumulative abnormal
returns (CAR) lies in the fact that joint linear hypothesis tests can be carried out easily under the
former. This has been especially useful in testing whether a regulatory event has a significant
effect on a sample of firms.1 The MVRM approach relies on traditional t-statistics and F-tests to
test statistical significance, especially joint hypotheses, whereas testing joint linear hypotheses is
not typically done under the CAR approach. Rather the focus under the CAR approach has been
on the impact of events on individual firms.
Much research has been devoted to distributional concerns in the CAR analysis, since it
is widely recognized that the excess returns are generally not normally distributed. This
violation typically is generally associated with fat tails, but could also be attributed to other nonnormal characteristics such as skewness. It is understood that such problems will cause
significant statistical inference problems (Brown & Warner, 1980, 1985). Corrado (1989)
suggests a non-parametric rank test for event studies in the face of distributional problems. More
recently Lyon, Barber and Tsai (1999) suggest a bootstrap version of a skewness-adjusted tstatistic to control for the skewness bias in their tests of long-run abnormal returns in a CAR
setting.
There has been less analysis of distributional violations in the dummy variable/event
parameter estimation setting. The fact that tests on event parameter coefficients in the MVRM
can be seriously biased has gone largely unappreciated in the literature. Three exceptions to this
are Chou (2001), Kramer (2001) and Hein, Westfall and Zhang (2001, HWZ hereafter). These
papers find that violations of normality assumptions in the MVRM setting with event parameter
estimation do indeed have a significant influence on the statistical analysis of event significance.
As an example to show the need for bootstrapping, we consider the event tests of Stewart
and Hein (2002), who examined the December 4, 1990 announcement that the Federal Reserve
was eliminating the reserve requirement on non-personal time deposits. There was concern as to
whether the shock continued for some days after the announcement; hence event analysis was
performed for the days after the announcement. Stewart and Hein only show the univariate tests
of the events, but it is also desirable to find a multivariate overall summary to simplify
interpretation. We used the same data to test the hypothesis that the reserve requirement
announcement significantly affected the stocks of the largest banks in the country. Using the
largest five banks (Citicorp, Bank America, Nations, Chemical, and First Chicago), the
significance level of the multivariate normality-assuming test was p=0.039, while the HWZ
bootstrap shows p=0.078 (95% Confidence interval: 0.073-0.083, based on 10,000 samples). A
researcher using traditional MVRM procedures would have concluded that the event was
significant for the largest banks in the country at the 5% level, whereas the bootstrap indicates
that this significance is exaggerated.
Chou, Kramer, and HWZ provide different bootstrapping approaches aimed at rectifying
the bias in the event study tests: Chou and HWZ are similar in that they bootstrap the raw data to
estimate the distribution of the test statistics, hence such methods are called “data-based
bootstrap methods,” hereafter. On the other hand, Kramer bootstraps the test statistics themselves
to estimate their distribution, hereafter called a “test statistic-based bootstrap method”. The
current paper continues these research efforts by: (i) offering further analytical justifications for
these bootstrap methods, (ii) comparing different these different bootstrap methods in cases of
not only non-normality, but also time series and cross-sectional dependence structures, and (iii)
developing alternative, new methods that combine elements of the various proposals. We show
that the two basic types of bootstraps work well in the absence of cross-sectional correlation,
even though the data may exhibit time-series dependencies such as AR, ARCH and GARCH
effects. However, the test-statistic based bootstrap procedure results in grossly inflated type I
error rates in the presence of cross-sectional correlation (see also Bernard, 1987, for other
problems caused by cross-sectional correlation). The point is subtle but important: test statisticbased bootstrap methods are reasonable for event studies with multiple events at independent
time points, but should not be used for clustered event studies. Kramer noted this concern in her
dissertation, (Kramer, 1998); however, this caution was inadvertently left out of the published
article (Kramer, 2001).
Another serious concern with test statistic-based bootstraps is that it cannot be used at all
when the number of firms is small. In some cases, event tests on single firms are desirable.
Studies of dividend initiation or resumption like Boehme and Sorescu (2002) would illustrate
such an event study. While Boehme and Sorescu examine CARs, the data-based bootstrap not
only offers the flexibility to accommodate cross-sectional correlation, but also the ability to test
for events when there are a small number of firms, or just a single firm, as in this case.
On the other hand, the Kramer procedure of summing t-statistics has potentially more
power than the HWZ procedure that uses the less focused multivariate test of Binder (1985b) as
its base. In the final analysis, we recommend a new procedure, where the summed t-statistics are
used in conjunction with the data-based bootstrap. This new method has both good power and
control of type I error rates under the presence of cross-sectional correlation, as well as under
time series dependence structures, and is therefore our recommended procedure.
This paper is organized as follows: in section II we show that the classical parametric test
is inconsistent, with asymptotic type I error levels depending on kurtosis of the excess returns.
In section III motivate and present the data-based and test statistic-based bootstrap tests.
Theoretical results concerning effects of cross-sectional correlation are given in section IV, and
the various methods are compared theoretically and via simulation using a variety of models for
financial data in section V. Conclusions are given in section VI.
II. The multivariate regression model and asymptotic inconsistency
a. The model and event tests
Consider the traditional single market factor model
Rt = β0 + β1Rmt + εt, t = 1, … ,T;
(1)
where Rt is the return on a specific firm or portfolio of stocks, Rmt is the return on the market,
and εt represents the excess return. Additional predictors may be included in (1), these are
excluded to simplify the exposition. In matrix form,
R = Xβ + ε.
(2)
If the event time is t0, define the indicator vector D, having all elements 0's except for time t0,
where the value is 1. The event test is based on the model
R = Xβ + Dγ + ε,
(3)
and the event test is a test of H0: γ = 0, tested using the simple t-test, as reported in any standard
software that performs OLS analysis. The numerator of the t-statistic is the OLS estimate of γ,
which is simply the deleted residual of model (1) for time t0.
The MVRM models may be expressed as the simultaneous equations,
Ri = Xβi + Dγi + εi,
for i= 1,…,g (firms or portfolios).
(4)
Observations within row t of ε = [ε1 | … | εg] are cross-sectionally correlated. Row vectors of
ε are assumed to be independent and identically distributed (i.i.d.) in the classical MVRM model,
and this is also a major assumption of the resampling method. However, we find that i.i.d.assuming bootstrap method is marginally robust to non-i.i.d. data.
The clustered event null hypothesis is the multivariate (composite) test
H0: [γ1 | … | γg] = [0 | … | 0]
(5)
Traditional methodologies for testing (5) with MVRM and SUR models are discussed by Binder
(1985a,b), Schipper and Thompson (1985), and Karafiath (1988). Cross-sectional correlations
are allowed for elements within a given row of the residual matrix; a key feature of MVRM is
that such cross-section correlations are incorporated in the test of (5). The test of (5) is computed
easily and automatically with standard statistical software packages, using exact (under
normality) F-tests.
The hypothesis (5) tested for a given event may be restated in a more general sense
amenable to semi-parametric bootstrap-based testing as follows:
H0: The distribution of the abnormal return (or vector of abnormal returns in case of
MVRM) for the event time t0 is identical to the distribution of the abnormal returns (or
vectors of abnormal returns) for a given set of times.
The researcher determines the definition of the "given set of times". It may include all
times other than the event, all times other than a collection of event times, or a collection of
times from another time horizon. Under the assumption of normally distributed abnormal
returns, Schipper and Thompson (1985) give an exact test of H0. However, as with all tests, the
true type I error level differs from the nominal (usually 0.05) level under non-normality.
b. Asymptotic inconsistency
It is a common conception that type I error rates approach the nominal α-levels (typically
0.05) with larger sample sizes, because of the central limit theorem. However, such is not the
case for dummy variable based event study tests. To see why, consider model (3). The event
test is a test of H0: γ = 0, tested using the simple t-test. The numerator of the t-statistic is the
OLS estimate of γ, which is simply the deleted residual of model (2) for time t0. The argument
for large-sample validity of a regression test, despite nonnormally distributed residuals, requires
large-sample normality of the parameter estimate. Such a statement is made using the central
limit theorem, provided that the parameter estimate can, in some sense, be viewed as an
"average," or at least as a weighted average that is not dominated by one or a few weights. In the
case of event tests under our concern, the estimated event parameter does not become normally
distributed as the sample size (T) increases, since it is just the deleted residual itself, and not an
(weighted) average of residuals, as required by the central limit theorem. The distribution of the
estimated event parameter comes closer to the distribution of the true residual at time t0, but
unless this distribution is truly normal, the distribution of the estimated event parameter will not
converge to normal.
For a more formal view of this problem, consider the development of the asymptotic
normality of the OLS vector presented in Greene (1990, 295-296). The requirement that the
parameter estimate be an appropriate type of weighted average is equivalently formulated by the
requirement that (1/T)(G'G) converges (as T tends to infinity) to a non-singular matrix, where G
= [X | D]. However, in the case of event studies, the convergence requirement fails: Since
(1/T)(D'X) = (1/T)[1 | Rmt0] tends to [0 | 0 ] and (1/T)(D'D) = (1/T)(1) tends to (0) as T tends to
infinity, the matrix (1/T)(G'G) tends to a matrix whose third row and third column (assuming a
single-factor market model as in (1)) are composed entirely of zeros, and is therefore not
invertible. Thus, the conditions needed for the asymptotic normality of the estimates do not hold
for this type of event model test, and convergence of the type I errors to their nominal levels
cannot be assumed. This failure to converge to normality greatly affects the probability of
finding a "significant" event, even for large T, as shown below.
c. Kurtosis and convergence
Suppose that the data consist of known true residuals [εt1,…,εtg], as would be the case
when T is so large that the parameters can be estimated with essentially no error. Suppose also,
without loss of generality, that the variances are identically 1.0. If they are not, simply divide
each εtj by its standard deviation, whose values can be assumed to be known for large T.
Assume also that the variables are i.i.d. Suppose t0=1 is the event day in question. In this case,
g
the parametric test reduces to the statistic S = ∑ ε1i2 , with H0 rejected when S ≥ χ g2,1−α , where
i =1
χ g2,1−α denotes the (1-α) quantile of the chi-squared distribution with g degrees of freedom. This
test is exact (i.e., the type I error probability is exactly α) when ε1i has a normal distribution.
Now, suppose we apply the normality-assuming test when ε1i has a non-normal
distribution with kurtosis κ. Then by the central limit theorem,
g −1/ 2 ( S − g ) /(κ + 2)1/ 2 →d N (0,1) and g −1/ 2 ( χ g2,1−α − g ) / 21/ 2 → Z1−α , where Z1−α = Φ −1 (1 − α )
denotes the 1-α quantile of the standard normal distribution. Then
P ( S ≥ χ g2,1−α ) = P( g −1/ 2 ( S − g ) /(κ + 2)1/ 2 ≥ g −1/ 2 ( χ g2,1−α − g ) /(κ + 2)1/ 2 )
→ P( Z ≥ Z1−α (2 /(κ + 2))1/ 2 ) = 1 − Φ ( Z1−α (2 /(κ + 2))1/ 2 )
For outlier-prone data, as are common in financial markets, κ>0, and hence
Z1-α(2/(κ+2))1/2 < Z1-α, so the true type I error level exceeds α, with greater excess for larger κ, in
our asymptotic framework. Conversely, if the error distribution is less outlier-prone than normal
so that κ<0, the usual parametric test is too conservative.
III. Alternative Bootstrap Solutions
One of the major benefits of using the bootstrap is that the researcher need not specify
any distributions at all. It would be presumptuous to suppose that a single distribution applies
equally across all securities, as minor shocks are amplified to a greater extent for some market
sectors than others, resulting in differential kurtosis across sectors. It would be equally
presumptuous to assume that there is a single distribution that applies across all historic time
regimes.
First, the data-based bootstrap of HWZ and Chou is described, and then the test statisticbased bootstrap of Kramer is described.
a. The Data-based Bootstrap
We wish to estimate quantities such as p = P( S ≥ s | FS 0 ) for event test statistics S used to
test H0, where the probability is calculated under the true null distribution function FS0 of S.
When s is an observed (fixed) value of the random variable S, p is the “true p-value” of the test.
Under the multivariate normal assumption, the distribution FS0 is simply related to the ordinary
ANOVA F distribution and is known exactly as discussed by Schipper and Thompson (1985).
However, when the distributions are non-normal, the distribution FS0 depends on non-normal
characteristics such as kurtosis, and is unknown. We follow the motivation of the bootstrap
using the “plug-in principle” (Efron and Tibshirani, 1998, p. 35), estimating p = P( S ≥ s | FS 0 ) by
pˆ = P( S ≥ s | FˆS 0 ) , for a suitable estimate of FˆS 0 of the null distribution of S.
Using notation of Schipper and Thompson (1985, equation 3), suppose the event test
statistic is given by S = Γˆ ' A '[ A{ X '(Σˆ −1 ⊗ I ) X }−1 A ']−1 AΓ̂ with covariance matrix Σ̂ estimated as
in their equation (4). Then S depends on the data Ri in (4, above) only through the least squares
estimates of the event parameters and the sample residual covariance matrix. Noting that the
sample residual covariance matrix is a function of the sample OLS residuals ei, the sample
covariance matrix is identical, no matter whether it is calculated from the data Ri or the true
residuals εi: ei = (I-G(G’G)-1G’)Ri = (I-G(G’G)-1G’)εi. Similarly, under the null hypothesis, γˆ i
is identical, no matter whether it is calculated from the data Ri or the true residuals εi:
γˆ i =(0 0 1) (G’G)-1G’Ri = γi+(0 0 1) (G’G)-1G’εi =(0 0 1) (G’G)-1G’εi under the null hypothesis.
Thus, under H0: γi=0, i=1,…,g, the test statistic is identical, no matter whether it is computed
using the Ri or the εi: S(R1,…, Rg) = S(ε1,…,εg) under H0, and the null distribution of S is
completely determined by the multivariate distribution of the row vector ε=[ε1:…: εg]. Hence
p = P( S (R 1 ,..., R g ) ≥ s | FS 0 ) = P ( S (ε 1 ,..., ε g ) ≥ s | Fε ). Applying the plug-in principle, we
estimate p = P ( S ≥ s | Fε ) as pˆ = P( S ≥ s | Fˆε ) . 2
The plug-in estimate is conveniently evaluated by using the bootstrap to estimate the
distribution Fε . Since we do not know the true residuals vectors, we estimate Fε using the
empirical distribution of the sample residual vectors (see Freedman and Peters, 1984 for the
univariate regression case; see Rocke, 1989 for the multivariate case). Below is the bootstrap
algorithm as presented in HWZ:
1. Fit model (4). Obtain the usual statistic S for testing H0 using the traditional method
(assuming normality). Obtain also the T x g sample residual matrix e = [e1| … | eg].
2. Exclude the row corresponding to D = 1 from e, leaving the (T-1) x g matrix e-.
3. Sample T row vectors, one at a time and with replacement, from e-. This gives a
T x g matrix [ R1* | … | Rg* ].
4. Fit the MVRM model Ri* = Xβi + Dγi + εi, i = 1, …,g, and obtain the test statistic S*
using the same technique used to obtain the statistic S from the original sample.
5. Repeat 3 and 4 NBOOT times. The bootstrap p-value of the test is the proportion of the
NBOOT samples yielding an S* statistic that is greater than or equal to the original S
statistic from step 1.
In steps 1 and 2, the researcher finds residual vectors that are used to estimate the
(multivariate) distribution of the abnormal returns. The residual for the event day must be
excluded because its residual is identically zero, and residuals of exactly zero are not anticipated
under ordinary circumstances.3 In steps 3 and 4 the researcher creates the bootstrap response
variable. Because the null distribution of the test statistic depends only upon the residuals, as
described above, the bootstrap response variable need not contain the Xbi component.
The bootstrap estimate of p̂ found in step 5 has a Monte Carlo standard error equal to
pˆ (1 − pˆ ) / NBOOT. Thus, choosing a sufficiently large value of NBOOT can control the Monte
Carlo error; we recommend choosing NBOOT as large as computing resources will comfortably
allow to estimate the p-values with sufficient precision. While it is possible to reduce the
number of bootstraps if a simple "reject/accept" decision is all that is required (Davidson and
MacKinnon, 2000), ultimately one should report p-values in scientific reports. So that other
researchers may verify results, it is best that reported statistics, including p-values, not contain
random noise; we therefore recommend reducing such unnecessary noise in published reports by
employing as large a number of resampled data sets, as time constraints allow. With the easy
availability of modern high-speed computing, one can easily avoid unnecessary Monte Carlo
error (see also Andrews and Buchinsky, 1998).
b. The Test Statistic-Based Bootstrap
Kramer (2001) recently proposed a different bootstrap approach for event studies,
g
considering the test statistic Z =
∑
ti/(g1/2st) as a measure of the effect of the event, where ti is
i =1
the t-statistic from the univariate dummy-variable-based regression model for firm i, and st is the
sample standard deviation of the g t-statistics t1,…,tg.. The p-value of the test is obtained by
bootstrapping as follows: (i) create a pseudo-population of t-statistics ti* = ti - t (where t = Σ
ti/g), reflecting the null hypothesis case where the true mean of the t-statistics is zero; (ii) sample
g values with replacement from the pseudo-population and compute Z* from these pseudovalues; (iii) repeat (ii) NBOOT times, obtaining Z1*, …, ZNBOOT*. The p-value for the test is
then 2×min(pU, pL), where pL is the proportion of the NBOOT bootstrap samples yielding Zi* ≤ Z
and pU is the proportion yielding Zi* ≥ Z. In other words, Kramer suggests bootstrapping the test
statistic Z under the assumption that the statistics are independent. Lyon, Barber and Tsai (1999)
also suggest bootstrapping the test statistic, but they do so in a cumulative abnormal return
setting, as opposed to the dummy variable setting.
c. Type I error rates of bootstrap procedures under independence
Since both test-statistic based and data based bootstraps are justified only asymptotically,
simulation is required to evaluate their finite-sample operating characteristics. Figure 1 shows
the results of a simulation to compare traditional normality-assuming F-tests, the data-based
bootstrap, and the test-statistic based bootstrap. Data were simulated from the market model,
where the distribution of the market return was chosen as normal4, and the abnormal returns were
generated independently from distributions varying from extremely heavy-tailed to normal. To
model heaviness of the tail, we used t-distributions with 1, 2, 4, and 8 degrees of freedom (T1,
T2, T4, and T8)5, to simulate extremely heavy to lighter tails; and we included the normal
distribution for completeness. The bootstrapped p-values are calculated using 999 samples, and
the number of replications of the bootstrap is 10,000 in all cases6. The simulated type I error
rates for nominal α=0.05 level tests are shown in Figure 1.
We note the following from these simulations: First, as the number of firms (portfolios),
g, increases, actual type I error rate for the traditional method becomes larger (see panels (a) and
(d)). In contrast, both the data-based bootstrap and the test-statistic based bootstrap maintain the
type I error rate better for larger g. This finding is important, as there are many examples in the
finance literature where the event is supposed to affect a large number (g) of firms. (See, for
example, Stewart and Hein (2002).) Second, the closer the underlying distribution of the
abnormal returns is to the normal distribution (i.e., the less heavy-tailed it is), the closer the
actual type I error rate is to 0.05 for the traditional method, as expected. Third, even with larger
sample sizes, the true level of the traditional test does not converge to 0.05, (compare (a) with
(d)), confirming the argument that the central limit theorem does not apply in this test setting; on
the other hand, the data-based bootstrap does appear to converge asymptotically (compare (b)
with (e)). Finally, we note that the test-statistic based bootstrap cannot be used at all when g=1,
because it is undefined. When g=2, the algorithm produces bootstrap t-statistics that are
uniformly equal to 0, hence the bootstrap critical values also are zero, meaning that the event
hypothesis is always rejected. Thus, the test-statistic based bootstrap cannot be used for g=1 or
g=2. While unusual, the case g=1 still may be desired to assess event effects on individual firms
such as dividend announcements. On the other hand, the test statistic-based bootstrap controls
type I errors even better with g ≥ 4 than does the data-based bootstrap. It should be noted that
this simulation study considers independent returns only, and it also does not consider power of
the tests, we next consider robustness to dependence structures and power.
IV. The Effect of Cross-Sectional Correlation
While the data-based bootstrap is valid under cross-sectional correlation, as occurs in
clustered studies, the test statistic-based bootstrap is not. A simple structural model will shed
light on the problem. Suppose again that the data consist only of true residuals vectors, again
appealing to the large T case, and that the residuals are dependent with εti = ξt + δij, where (ξt,
δt1,...,δtg) are independent, mean zero, normal random variables with Var(ξt) ≡ σ ξ2 and Var(δti)
≡ σ δ2 . This model is sensible if the g multivariate responses at time t share an “industry effect” ξt,
in which case there will be a commonality to their excess returns. In this model, the excess
returns {εt1,…,εtg} at time t are dependent, with variance Var (ε ti ) = σ ξ2 + σ δ2 , and with common
cross-sectional correlation ρ = Corr (ε ti , ε tj ) = σ ξ2 /(σ ξ2 + σ δ2 ) (ρ is also known as the intraclass
correlation coefficient). Assuming large T, the variances Var (ε ti ) = σ ξ2 + σ δ2 are estimated
precisely, so the t-statistics for testing the null hypothesis that day t=1 is a non-event day are ti ≅
ε1i /(σ ξ2 + σ δ2 )1/ 2 . The sample variance of the t-statistics is
g
σ δ2
1 g
1
2
2
= 1 − ρ for large g. Thus, for large
s =
(δ1i − δ1 ) ≅ 2
∑ (ti − t ) ≅ ( g − 1)(σ 2 + σ 2 ) ∑
g − 1 i =1
σ ξ + σ δ2
i =1
ξ
δ
2
t
g
g and T, Kramer’s test-statistic is approximated by Z ≅ g −1/ 2 ∑ ε1i /{(σ ξ2 + σ δ2 )(1 − ρ )}1/ 2 . Now,
i =1
the “pseudo-population” of centered statistics {ti - t }≅ { (δ1i − δ1 ) /(σ ξ2 + σ δ2 )1/ 2 } approximates a
normal distribution with mean zero and variance (1-ρ) for large g; hence, the Z* statistics
obtained by sampling from this pseudo-population will approximate a normal distribution with
mean zero and variance one (the scale factor (1-ρ) vanishes because of the standardization by st).
This implies that the critical values of the bootstrap procedure are approximately ± Z1-α / 2 , and
that one rejects the null hypothesis that day t=1 is a non-event day approximately when
Z ≥ Z1-α / 2 .
Noting that Var(Z) ≅
g
⎛
⎞ 1+(g-1)ρ
1
(eg, Johnson and
Var ⎜ g -1/2 ∑ ε1i /(σ ξ2 +σ δ2 )1/2 ⎟ =
(1 − ρ )
1− ρ
i =1
⎝
⎠
Wichern, 1998, p. 470), the method has type I error level approximately equal to α when ρ=0
since the variance of the Z score is unity in that case. However, in general the true type I error
level is approximately equal to 2(1 − Φ ( Z1−α / 2 /((1 + ( g − 1) ρ ) /(1 − ρ ))1/ 2 )) . Thus, under the
assumed model with ρ ≠0, the true type I error level of the Kramer test approaches 1.0 as g
increases. In a nutshell, the problem is that natural structural effects ξt resulting in cross-sectional
correlation are incorrectly determined to be effects of an event, which causes type I error rates to
approach 1.0 for large g.
V. Comparison of Methods
In this section we begin by developing new alternative test procedures resulting from
combining aspects of the alternative bootstrap approaches. We then report simulation results that
compare these approaches in a number of different dimensions.
a. Old and new bootstrap methods
Kramer's summed Z statistic is more powerful than the general multivariate test statistic
when the event effects are all in the same direction. However, use of the standard deviation of
the t-statistics for normalization of Z can decrease power in cases where g is small, as random
noise incurred by estimating the standard deviation is included in the test statistic. While use of
such standardization to create pivotal statistics are known to improve the convergence rate of the
type I error in i.i.d. cases (Babu and Singh, 1983), the present case differs somewhat in that (a)
the statistics ti are already standardized and hence at least partially pivotal (free of mean and
scale factors), and (b) we consider that the ti are non-i.i.d. Pesarin (2001, p.148) considers using
the statistic Σti (the Liptak test) in related multivariate contexts, and uses resampling to estimate
its distribution.
To correct for lack of type I error control, we apply the data-based bootstrap algorithm of
section III.a. to the statistic S = Z, and call this the “BK” test. We also consider the same
bootstrapping of the statistic S = Σti test suggested by Pesarin, calling this the "BT" test. It
should be noted that, unlike the HWZ test, there is no parametric equivalent to the BT and BK
tests; bootstrapping is essential. We show below that BT tends to have higher power under
consistent shifts, and that it maintains the type I error rate. In summary, we compare the
following four bootstrap methods:
•
Method HWZ: The data-based bootstrap of the traditional parametric F test (the HWZ
proposal).
•
Method BT: The data-based bootstrap of the Σti statistic (a new proposal).
•
Method BK: The data-based bootstrap of Kramer's Z statistic (a new proposal).
•
Method K: The test statistic-based bootstrap of Kramer’s Z statistic (Kramer’s proposal).
b. Simulation study: Cross-sectional correlation effects
The simulation study summarized by Figure 1 assumes zero cross-sectional correlation and
does not consider power, only level. Our purpose now is to compare the various bootstrap
methods in terms of level and power, for non-independent data. Since the methods are supposed
to work under a variety of distributions, they should work in particular for the normal
distribution, so we initially study normal models, using a variety of different correlation
structures. We consider a normal MVRM with T=100 and equicorrelated cross-sectional errors
(as would be implied, for example, by the structural model of section IV) with no event effects.
We let g = 5 and g = 30 in two separate cases. Further, we allow the cross-sectional correlation
parameter to vary as ρ=0.0, 0.1, …,0.9. Table I displays simulated type I error rates using b=999
bootstrap samples and NSIM=1000 simulations. All entries should be close to the nominal
α=0.05 level.
[ Insert Table I Here ]
It is clear from Table I that the Kramer method does not control the type I error rate when
there is cross-sectional correlation. For g=5, with ρ>0.2, the rejection rate is consistently over
twice the nominal α = 0.05 level, and the problem is worse with larger g as shown in Panel B.
The values in Panel B for the Kramer method closely agree with the theoretical
values 2(1 − Φ ( Z1−α / 2 /((1 + ( g − 1) ρ ) /(1 − ρ ))1/ 2 )) developed in section IV.
We also note that the bootstrapped HWZ method is too conservative for g = 30. The reason
for this is that g is large (30) relative to T (100). In such cases, the covariance matrix of the
bootstrapped residual vectors tends toward singularity because of the repeated vectors in the
bootstrapped samples. This creates larger-than-expected bootstrapped F statistics, which in turn
creates larger-than-expected bootstrap p-values. This problem disappears when g is small
relative to T; we found in unreported simulations that the type I error level of HWZ is
approximately correct when g=10 and T=100, (estimated value of 0.053), and only slightly
conservative when g=15 when T=100 (estimated value of 0.041). Thus, we cannot recommend
the HWZ method where the number of firms or portfolios (g) examined is large relative to the
number of time points (T) because the researcher will too frequently fail to reject the null.
On the other hand, the BT and BK methods perform well in terms of preserving type I error,
showing again that bootstrapping of the residuals is superior to bootstrapping the test statistic in
the face of cross sectional correlation in returns. There is a slight tendency toward excess type I
errors in the independence case for BT and BK. Westfall and Young (1993, p. 127) document a
similar phenomenon in a related application, and note that such effects diminish with increasing
T.
c. Simulation study: Power
To compare the various bootstrap methods in terms of power, we assume that the first
observation is the event. We model this with a common mean shift γ, ranging from 0.0 to 4.0 for
all g variables, chosen to reflect a range of power values between 0 and 1. Since the variances
are unity in this study, the mean shifts are equivalently 0 to 4 standard deviations. Again, we
consider g = 5 and g = 30, with T = 100 in both cases. To simplify matters, and to make a fair
comparison with the Kramer method, we assume zero cross sectional correlation. Table II
shows simulated power in this case.
[Insert Table II Here]
Based on the simulations above, we can recommend the data-based bootstrapped,
modified Kramer test, BT, because the BT procedure (i) maintains the type I error level and (ii)
has higher power. As noted above, the HWZ approach suffers in power when g is large relative
to T, and it can be too conservative when g is large relative to T. We should also note in fairness
that the HWZ approach could detect cases where the event effects go in opposite directions (for
example if γ increased for some firms and decreased for others), whereas the BT, BK, and K
methods are virtually powerless in such instances. The HWZ approach would be more powerful
in cases such as examining dividend announcements -- initiations and omissions -simultaneously, say. Such cases may be somewhat rare, but HWZ has greater power in such
cases.
d. The effect of time-series dependence: Analytic and simulation results
Excess returns can be serially correlated, as well as cross-correlated. Interestingly, the
data-based bootstrap we have described is robust to autocorrelation despite its use of independent
sampling. The reason for this is precisely the same reason that it is non-robust to non-normality.
Namely, the test is based on a single random observation rather than a sum. If the single
suspected residual is large relative to the estimated distribution, then the event is called
significant. All that is required is that the marginal distribution be estimated consistently in order
for marginal type I error control to be maintained asymptotically. It is well known that OLS
estimates and error variances are consistent under AR disturbances (Greene, 1993, p. 422),
suggesting that the normality-assuming dummy variable tests should become essentially a
function of the true residuals and their variances for large T. Further, the empirical distribution
of the residuals converges almost surely to the true distribution under stationarity conditions
given by Yu (1993) and Zhengyan and Chuanrong (1996, Chapter 12), implying that the
bootstrap test procedures should maintain type I error rates in large samples under these
conditions as well.
Specifically, consider again the case where the test is based on known residuals and
known variances (taken to be 1.0 without loss of generality) and consider the BT method in the
case with g=1. In this case, the test statistic is simply ε1, assuming again that t=1 denotes the
suspected event day, and the critical values for the bootstrap test procedures are the α/2 and 1-
α/2 quantiles of the distribution of ε*, where ε* is sampled with replacement from the pseudopopulation {ε2,…,εT}. In this case, the true bootstrap critical values are evaluated without
resorting to Monte Carlo sampling as F̂ε ,α / 2 and F̂ε ,1−α / 2 , respectively the (suitably interpolated)
α/2 and 1-α/2 empirical quantiles of the data {ε2,…,εT}. The empirical distribution of data from
stationary processes converges to the marginal distribution under conditions given by Yu (1993)
and Zhengyan and Chuanrong (1996, Chapter 12), in which case P(reject H0 | H0 true) = P(ε1
≤ F̂ε ,α / 2 or ε1 ≥ F̂ε ,1−α / 2 | H0 true) → P(ε1 ≤ Fε ,α / 2 or ε1 ≥ Fε ,1−α / 2 | H0 true) = α, for continuous
Fε. This result may seem surprising because it is generally thought that independence-based
bootstrapping fails in the case of non-i.i.d. data (Politis, 2003); however, the difference in the
present case is that we are bootstrapping the distribution of a single observation at a single point
in time, not of an average or some other combined statistic over the entire history.
We note that this asymptotic control of type I error rates is only in the marginal sense;
that is, the control is not conditional upon the recent history but rather unconditional, averaging
over all histories. Conditioning upon the recent histories, the independence-assuming bootstrap
test will have type I error rates that are sometimes greater, sometimes less, than the nominal
level. On average, over all histories, the type I error level is controlled. Marginal control of type
I error rates is a good property, and one that any procedure should possess. In particular, it is
instructive to compare marginal type I error rates of different procedures; those that do not
control marginal type I error rates are clearly inferior. On the other hand, the practical
implementation of a procedure that controls the type I error only marginally requires that recent
history be ignored. This is clearly an undesirable feature; therefore, despite the robustness of the
independence-assuming bootstrap to time series dependencies, it can only be recommended
when the data series is “reasonably close” to i.i.d. (although arbitrary cross-sectional correlation
is allowable in any case, even if there are perfect cross-sectional dependencies in the case of BT
and BK). Further research is needed to determine how significant the time series dependence
must be for the conditionality issue to become problematic for the i.i.d. bootstrap.
Table III shows results of simulations to estimate marginal type I error rates when T=100;
g=5 as a function of the serial autocorrelation parameter φ when there is no cross-sectional
correlation.
[Insert Table III Here]
Table III shows that all bootstrap tests control the type I error rates reasonably well, even
under substantial autocorrelation. However, the Kramer method remains not valid when there is
cross-sectional correlation (simulation results for cross-sectional correlation and time-series
dependence are not shown but available from the authors). We conclude that autocorrelation
itself is not a serious problem for marginal type I error rates of bootstrap-based dummy-variable
event tests, despite the fact that the model assumes i.i.d. residuals.
e. Simulation from observed populations
So far, the simulations have been somewhat contrived, and the connection between the
simulation models and real return data are not well established. It is well known that observed
excess returns frequently exhibit cross-sectional correlations, non-normalities, autoregressive
effects, and conditional heteroscedasticity. Rather than develop a simulation model that
incorporates all of these characteristics, we instead utilize existing financial data for our
simulation model. We used a data set consisting of daily returns on the S&P 500 as well as the
five insurance sub-indexes created by Standard and Poors, from January 3, 1990 to November 6,
2001 (3086 consecutive trading days). The excess returns are fairly highly correlated cross-
sectionally, with (min, ave, max) = (.41, .56, .79). There are also significant ARCH(1),
GARCH(1), and AR(1) effects in the univariate models.
In our simulation model, we randomly sampled the 6-dimensional vectors in 10 blocks of
20 consecutive time points to create simulated samples of length T=200 that preserve time-series
dependencies, cross-sectional dependencies, and non-normalities of the original data. The
starting point for the 20 consecutive time points was chosen at random from the possible starting
points 1,…,(3086-20+1), thus the selected series possibly overlap, and data occurring earlier in
the original “population” can occur later in the simulated sample. Künsch (1989), pioneered this
type of block resampling for time series data, and Hall et al. (1995) provide recommendations for
block sizes. Samples from the population exhibited similar time-series (autoregressive and
conditional heteroscedastic) characteristics, cross-sectional dependencies, and distributional
characteristics as found in the population, albeit with generally less statistical significance due to
the smaller sample size, and with somewhat smaller time series effect sizes due to local
independence induced by block resampling.
The event day was chosen at random from the simulated series of 200 observations. We
would expect that the various procedures should control the type I error rates in this analysis,
since there is nothing special about the chosen day. Results are shown in Table IV, panel A,
comparing the methods HWZ, BT, BK, and K described above, in addition to the classical
parametric F test with no bootstrapping (called BINDER in the Table). The simulation from
observed financial data confirms the essential results from the previous simulations. Namely, the
simulation confirms that: (1) the normality-assuming test fails to control the type I error rate, as
the Binder statistic too frequently rejects the null hypothesis, (2) the Kramer test statistic-based
method fails to control the type I error rate in the presence of cross-sectional correlation, and (3)
the data-bootstrap methods do control the type I error rate. Again, as in the previous section, we
reiterate that these are unconditional type I error levels, and that, while we would rather control
conditional type I error levels, unconditional control remains desirable as a basis for comparison
of the various procedures.
For all simulations, there were 10,000 samples of size T=200 as described above, and for
all bootstrap methods, B=999 simulations were used.
[ Insert Table IV Here ]
To evaluate power, we added common effects γ=.01,…,.05 to the excess returns for all
five indexes, allowing for a range of power from 0 to 1. The results are shown in Table IV,
panel B. Among the bootstrap tests that control the type I error levels, the BT test is preferred
for γ>.01, presumably because the normalization in the BK procedure adds excess variability.
On the other hand, the common shift alternative is meant to favor both BT and BK over HWZ; if
the alternative were that the variance was increased and not the mean, resulting in large positive,
as well as negative, excess returns, then the HWZ test would be preferred. Since the type I errors
of the Binder and Kramer tests are uncontrolled, it is inappropriate to compare their power with
the other procedures without a size adjustment. However, since these tests are anti-conservative,
a size adjustment must make their power smaller, and we can therefore conclude that the sizeadjusted Kramer test must be less powerful than the BT test for γ>.03, and that the size-adjusted
binder test must be less powerful than the BT test for γ>0.
VI. Conclusion
In this paper, we examine the popular test statistics used for event studies in MVRM
models. We show that when excess returns come from non-normal distributions, these traditional
test statistics are misspecified. The traditionally calculated p-values are biased downward
dramatically when the number of firms (or portfolios) is large and the residual distribution is
heavy-tailed, causing the researcher to conclude an event is significant to financial markets too
frequently. Importantly, this bias does not diminish as the sample size increases. Bootstrap
methods correctly and automatically adapt to non-normal characteristics of the data, reducing
this misspecification significantly and providing more accurate p-values.
Further analysis shows that Kramer's (2001) bootstrap method is preferred in terms of the
form of summed-t test statistic, which has good power, but is not preferred when there is crosssectional correlation, in which case the test has grossly inflated type I error probability. We
recommend a modification of the Kramer method using data-based bootstrapping, which
provides good power and maintains the type I error control. We also note that this type of
bootstrap is marginally robust in the face of time series dependence, despite its resampling of
independent residual vectors.
It is standard practice to use the t-distribution rather than the z-distribution for analysis of
mean values when the variance is unknown, simply because the procedures based on the tdistribution are generally more accurate. We argue for general use of the bootstrap rather than
the traditional method of MVRM tests of key events, for precisely the same reasons. Since the
bootstrap method performs better over a range of possible cases, as we have shown in numerous
simulations, and since its performance is not noticeably inferior in the case of normal
distributions, we feel it is generally much more prudent to use the bootstrap p-values than to use
the traditional p-values, regardless of the form of the underlying distribution. At the very least,
traditional analysis of event studies should be supplemented with a bootstrap analysis, so
investigators can evaluate the robustness of their inferences.
Limitations of the research are the restriction to MVRM models with a common event
time, and the lack of control of conditional error rates under non-i.i.d. time series data. Since the
BT method works well under cross-correlated data, extensions to separate, independent event
times, as considered by Kramer, should also work well, but we leave this for future study.
Further research is also needed to evaluate the conditionality issues described in section V.
Software to perform these analyses is available freely from the authors.
Footnotes
1
Smirlock and Kaufold (1987), Karafiath and Glascock (1989), Cornett and Tehranian
(1989 and 1990), De Jong and Thompson (1990), Eyssell and Arshadi (1990), Demirguc-Kunt
and Huizinga (1993), Madura et al. (1993), Unal, Demirguc-Kunt and Leung (1993), Clark and
Perfect (1996), Cornett, Davidson, and Rangan (1996), Johnson and Sarkar (1996), Bin and Chen
(1998), Cosimano and McDonald (1998), Sinkey and Carter (1999), and Stewart and Hein
(2002) are examples of empirical studies that use such an approach in examining the significance
of a wide variety of events.
2
All calculations are conditional on the observed matrix G; see Westfall and Young, 1993,
p. 123, for further discussion; see also the description of the SAS/STAT software PROC
MULTTEST, SAS Institute (1999), which uses an identical method to estimate p-values from
tests in multivariate linear models.
3
A technical distinction between the bootstrap approach suggested by Chou and HWZ is
that Chou fits null restricted models and does not suggest removing the residual from the
bootstrap sample. In our formulation, fitting null restricted models is not needed, and the
residuals that are identically zero should be removed. In cases where multiple event parameters
are modeled, one should also exclude any other sample residuals that are forced to be zero; the
program that is freely available from the authors allows this.
4
The distribution of the test statistic depends mainly on the distribution of the residuals; the
distribution chosen for the market return has relatively little effect.
5
Student t distributions are used for fitting financial models with heavy tails; see e.g. the
SAS/ETS procedure PROC AUTOREG.
6
While a large bootstrap sample size is recommended for reporting significance from an analysis
of a data set to ensure replicability, it is not as essential that the bootstrap sample size be so large
for simulation studies, in which case the major contributor to variability is the outer loop; see
Westfall and Young (1993, pp. 38-41).
Table I. Simulated type I error rates as a function of cross-sectional correlation.
Panel A: g = 5
ρ=0 ρ=0.1 ρ=0.2 ρ=0.3 ρ=0.4 ρ=0.5 ρ=0.6 ρ=0.7 ρ=0.8 ρ=0.9
HWZ 0.059 0.059 0.059 0.059 0.059 0.059 0.059 0.059 0.059 0.059
BT 0.055 0.057 0.055 0.057 0.056 0.058 0.056 0.057 0.060 0.056
BK 0.053 0.056 0.049 0.053 0.050 0.042 0.041 0.045 0.050 0.049
K
0.057 0.078 0.113 0.168 0.220 0.275 0.335 0.418 0.500 0.624
Panel B: g = 30
ρ=0 ρ=0.1 ρ=0.2 ρ=0.3 ρ=0.4 ρ=0.5 ρ=0.6 ρ=0.7 ρ=0.8 ρ=0.9
HWZ 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002
BT 0.070 0.057 0.056 0.059 0.062 0.064 0.051 0.058 0.056 0.049
BK 0.073 0.057 0.051 0.055 0.056 0.053 0.055 0.058 0.053 0.050
K
0.056 0.366 0.516 0.590 0.652 0.702 0.718 0.813 0.851 0.885
Table II. Simulated power as a function of event effect γ.
Panel A: g = 5
γ=0 γ=0.2 γ=0.5 γ=0.75 γ=1.0 γ=1.5 γ=2.0 γ=3.0 γ=4.0
HWZ 0.05 0.04 0.09 0.19 0.28 0.74 0.95
1
1
BT 0.02 0.09 0.2 0.42 0.6 0.92 0.99
1
1
BK 0.05 0.07 0.17 0.34 0.58 0.78 0.92 0.97
1
K
0.06 0.08 0.16 0.27 0.42 0.65 0.75 0.88 0.93
Panel B: g = 30
γ=0 γ=0.1 γ=0.2 γ=0.3 γ=0.4 γ=0.5 γ=0.6 γ=0.7 γ=0.8 γ=0.9 γ=1.0
HWZ 0.002 0.002 0.003 0.007 0.011 0.013 0.042 0.072 0.130 0.1970.291
BT 0.070 0.092 0.206 0.363 0.573 0.751 0.871 0.949 0.988 0.9991.000
BK 0.073 0.090 0.193 0.335 0.539 0.697 0.844 0.930 0.975 0.9921.000
K
0.056 0.087 0.185 0.345 0.535 0.703 0.853 0.933 0.977 0.9960.999
Table III. Simulated type I error rates as a function of serial correlation φ.
φ=0 φ=0.1 φ=0.2 φ=0.3 φ=0.4 φ=0.5 φ=0.6 φ=0.7 φ=0.8 φ=0.9
HWZ 0.052 0.054 0.056 0.057 0.057 0.059 0.072 0.085 0.113 0.176
BT 0.062 0.062 0.067 0.066 0.068 0.07 0.067 0.071 0.079 0.093
BK 0.062 0.06 0.064 0.065 0.067 0.064 0.062 0.067 0.067 0.061
K
0.053 0.052 0.047 0.048 0.045 0.048 0.046 0.046 0.052 0.045
Table IV. Simulated type I error rates and power for the various procedures under block
sampling from a financial data series, where the samples exhibit ARCH, GARCH,
autoregressive, and cross-sectional correlation effects.
Panel A: Type I error rates
Binder
HWZ
BT
BK
K
Nominal type I error rate
α=0.01 α=0.05 α=0.10
0.058
0.105
0.137
0.014
0.055
0.104
0.015
0.054
0.103
0.014
0.053
0.105
0.097
0.275
0.358
Panel B: Power for the nominal α=0.05 tests.
Binder
HWZ
BT
BK
K
γ=0.01 γ=0.02 γ=0.03 γ=0.04 γ=0.05
0.139 0.346 0.712 0.935 0.987
0.071 0.183 0.465 0.784 0.933
0.148 0.543 0.858 0.970 0.993
0.193 0.463 0.674 0.820 0.885
0.542 0.776 0.880 0.925 0.946
0.2
0.15
T1
Type I error
T2
T4
0.1
T8
T2
T4
0.1
T8
Z
Z
0.05
2
4
8
1
Number of Firms (or Portfolios)
a)
2
4
1
0.2
Type I error
Type I error
T8
T2
T4
0.1
T8
T2
T4
0.1
T8
Z
0.05
0.05
0
0
4
T1
Z
0.05
2
8
0.15
T1
Z
1
4
0.2
0.15
T1
T4
2
(c)
0.2
0.1
Z
Number of Firms (or Portfolios)
Number of Firms (or Portfolios)
T2
T4
T8
8
(b)
0.15
T2
0.1
0
0
1
T1
0.05
0.05
0
0
1
8
2
4
1
8
(e)
2
4
8
Number of Firms (or Portfolios)
Number of Firms (or Portfolios)
Number of Firms (or Portfolios)
(d)
0.15
T1
Type I error
Type I error
0.15
0.2
Type I error
0.2
(f)
Figure 1: True type I error rates for nominal α=0.05 event tests with normal and nonnormal distributions. (a) traditional F test,
(b) data bootstrap F test (HWZ), (c) test statistic bootstrap (Kramer), all for T=200; (d), (e), (f) repeat (a), (b), (c) when T=50. True
type I error rates are 0% when g=1 for test statistic bootstrap, 100% when g=2.
Figure 1: True type I error rates for nominal α=0.05 event tests with normal and nonnormal distributions. (a) traditional F test,
(b) data bootstrap F test (HWZ), (c) test statistic bootstrap (Kramer), all for T=200; (d), (e), (f) repeat (a), (b), (c) when T=50.
True type I error rates are 0% when g=1 for test statistic bootstrap, 100% when g=2.
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