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LONG-RUN ABNORMAL STOCK PERFORMANCE: SOME ADDITIONAL EVIDENCE J.F. BACMANN
a AND M. DUBOIS
a
First Draft: February 2002
a Université de Neuchâtel, Pierre-à-Mazel 7, 2000 Neuchâtel, Switzerland
Tel: +41 32 718 13 60 Fax: +41 32 718 13 61
E-mail: [email protected] and [email protected]
Financial support by the Swiss National Science Foundation (grant n°1214-056849.99) and by the
National Centre of Competence in Research “Financial Valuation and Risk Management” is gratefully
acknowledged. The National Centre of Competence in Research are research programmes supported by
the Swiss National Science Foundation.
2
LONG-RUN ABNORMAL STOCK PERFORMANCE: SOME ADDITIONAL EVIDENCE
Abstract: In this research we study the specification and the power of classic test
statistics used in long-term event studies analysis. Using simulations in random samples,
we show that test statistics based on an arbitrary benchmark are well specified and as
powerful as the ones based on the size and book-to-market benchmark. However, when
conditioning the samples on past stock returns performance, we show that a good
matching procedure is required in order to obtain well specified and powerful tests.
Finally, we examine the specification and the power of calendar-time portfolios. The
cross-sectional standardized t-stat is well specified in random samples in which the
frequency of the events is random or depends on the past market returns performance.
However, when the frequency of events is conditioned on past market returns
performance and the stocks are selected among the most extreme returns misspecified
test statistics are obtained.
1
Since Ibbotson (1975), the analysis of the long-run abnormal stock returns has
attracted a lot of interest in corporate finance. This topic is important for at least two
reasons: first, it is a mean to explore whose actions undertaken by the management
create value and, second it help determining the sources of stocks’ misspricing.
Routinely, empirical studies have reported negative long-run abnormal returns for
some events and positive for others1 casting some doubt on market efficiency.
Recently, several models based on non-rational agent’s behavior have tackled the
problem. However, as underlined by Fama (1998) these models have trouble in
explaining empirical facts for which they are not designed. From an epistemological
standpoint they have not gained yet the status of a full theory. Moreover, empirical
findings show that, depending on the information, investors underreact or overreact
half of the time. Therefore, it could be that long term abnormal returns are not
correctly estimated and/or the statistical tests are biased.
Barber and Lyons (1997) and Kothari and Warner (1997) examined extensively
this methodological issue. They identify three potential sources of misspecification:
the survivor bias, the rebalancing bias and the skewness bias. Lyon, Barber and Tsai
(1999) (LBT hereafter) show that cross-sectional dependence and a bad asset-pricing
model are two additional sources of misspecification. In fact, based on simulations a
la Brown and Warner (1980), LBT show that a “Traditional event study framework
and buy-and-hold abnormal returns calculated using carefully constructed reference
portfolios” yiel well-specified test statistics in random samples. However, as
1 Three to five years negative abnormal returns are found after IPOs, SEOs, mergers, dividends
omissions and listing on the NYSE while the converse is obtained for stock repurchases, splits, spin-
offs and earnings announcement; see Ritter (1991), Ritter and Lougrhan (1995), Ikenberry et al.
(1995), Ikenberry et al. (1996) among others.
2
underlined by Fama (1998, p. 290, Table 1) most of the events seem selective so that
the experimentaél design suggested by LBT is misleading.
In this research, we use a benchmark randomly selected because Ferson,
Sarkissian and Simin (1999) show that a portfolio constructed with stocks sorted
alphabetically help explain the cross-sectional dispersion of stock returns even if this
method lack of any financial content. In fact, when applied to the measurement of the
long-run performance of stock returns, the same test statistics calculated from a
“non-financial” benchmark provide identical or superior results in terms of
specification and power compared to the size and book-to-market benchmarking.
Unfortunately from a statistical perspective, financial events are rarely random.
When examining stock returns before the event, previous studies have found that
abnormal performance is likely to occur for a wide variety of events. LBT (p. 185)
suggest that matching firms to firms with similar pre-event returns performance
would also control well for the misspecification of the size and book-to-market
matching-firm method. For that purpose, we split NYSE and AMEX stocks in
quintiles according to their stock returns performance over the past twelve months.
Then, we determine two samples by randomly selecting firms within each quintile.
For each firm in both samples, we select randomly a matching-firm in three different
ways. The matching firm is drawn among a) all NYSE-AMEX stocks, b) the first
quintile (highest returns) and c) the fifth quintile (lowest returns). The simulations
show that a good matching procedure (a firm with prior abnormal returns is matched
with a firm with similar returns) leads to well-specified tests.
In addition, “real event studies” frequently exhibit strong clustering during
specific periods of time. For example, several empirical studies document that SEOs
are by far more frequent when markets are bullish. Under these conditions, cross-
3
sectional dependence of stock returns makes the grouping of event-firms into
portfolios preferable.
Finally, we examine the specification and the power of calendar-time portfolios
with two benchmarks, namely the Fama and French (1993) and Carhart (1997)
models. Abnormal returns are estimated as “in-sample error” (see Mitchell and
Stafford (2000)) and as “forecasted errors” (see Kothari and Warner (1997)) using a
variety of t-statistics (standard t-stat, cross-sectional t-stat, standardized t-stat and
cross-sectional and standardized t-stat) inspired from the short-term event study
setting. When the event-sample is selected randomly, our main results can be
summarized as follows: a) the standardized t-stat has the best specification and is the
most powerful test statistics, b) the period of estimation is not of major concern and,
c) tests constructed with the Carharts’ model (1997) are more conservative and less
powerful than those based the Fama and French (1993) model. When the frequency
of events conditioned on past market returns performance is high, the standardized t-
statistics is still well specified. However, the test statistics are misspecified whenever
the frequency of events conditioned on past stock returns performance is high.
The remainder of the paper is organized as follows. We present the methods used
to calculate abnormal returns and the test statistics in Section I. In Section II, we
examine the specification and the power of various test statistics based on a “non-
financial” matching procedure. In Section III, we study the specification and the
power of these test statistics when the matching is based on past returns. Additional
results using calendar-time portfolios and test statistics adjusted for cross-sectional
heteroscedasticity are provided in Section IV. Section V concludes.
4
I. Abnormal Returns and Statistical Tests In this section we briefly summarize the various calculations of abnormal returns
and of the statistical tests used in the litterature.
A. Cumulated Abnormal Returns over a long-horizon
The model for measuring the normal returns is the following:
( )it t ctE R I R=
where ( it tE R I ) is the monthly expected return for security i during month t, given
the set of information tI , ctR is the monthly return of the matching-firm or the
control portfolio over the same period. The abnormal return over the month t is
calculated as:
itAR
it it ct itAR R R ε= − +
where is an error term independent of i and t, with zero-mean and
constant variance.
( 2~ 0,it Nε )σ
As in others studies, the temporal aggregation of the abnormal returns is done via a
rebalancing strategy (CARs hereafter) and a buy and hold strategy (BHAR
hereafter).
A.1. Cumulated Abnormal Returns
2
1
, ,
T
i h i tt T
CAR AR=
=∑
where is an estimate of the cumulated abnormal returns for stock i, over
the period
,i hCAR
h T= ,1 ,2,...,i iT
The null hypothesis of no average cumulated abnormal returns is stated as
follows:
5
1 , 1,1 1
1 1: 0 :N N
i h A i hi i
H CAR vs H CARN N= =
= ≠∑ ∑ , 0
As it is well known, the average cumulated abnormal returns can be obtained by
rebalancing the portfolio (1 USD long in stock i, 1 USD short in the control c) at the
end of each period (month). Because of transaction costs, average cumulated
abnormal returns are no longer attainable. However, Fama (1998) recommends using
this method because the bad-model problem is less acute compared to the buy-and-
hold abnormal returns.
A.2. Buy and Hold Abnormal Returns
Instead of rebalancing the portfolio at the end of each period, a more realistic
strategy consists in buying a portfolio, which is 1 USD long in stock i and 1 USD
short in the control c. This portfolio is hold until the end of the period
. The abnormal performance of stock i is computed as: ,1 ,2,...,i ih T T=
( ) ( )( ),2 ,2
,1 ,1
, = 1 1i i
i i
T T
i h it it tt T t T
BHAR R E R I= =
+ − +∏ ∏
where ,i hBHAR is the buy and hold abnormal return and h is the holding period.
The null hypothesis of no average buy and hold abnormal returns at the horizon h
is stated as follows:
2 , 2,1 1
1 1: 0 :N N
h i h Ai i
H BHAR BHAR vs H BHARN N= =
= =∑ ∑ , 0i h ≠
2H
B. Common Statistical Tests
The most commonly used statistical test in order to test the null hypothesis of no
abnormal return ( ) is the standard t-test: 1 andH
( )h
h
t statN
ωσ ω
− =
6
where is the sample mean (of the CARs or BHARs) and is the cross-
sectional sample standard deviation for the sample of N firms.
hω ( hσ ω )
However, the distribution of the CARs is often asymmetric and the t-stat must be
adjusted in order to get the proper critical values. This problem is even more acute
for the BHAR. Johnson (1979) proposed the following correction:
21 1ˆ ˆˆ ˆ3 6sat N S skwS skw
N = + +
where ( )
ˆˆ i
S ϖσ ϖ
=
( )( )
3
13
ˆˆ
ˆ
N
ii
i
skwN
ϖ ϖ
σ ϖ=
−=∑
is an estimate of the skewness of the CAR (BHAR)
Sutton (1993) recommend bootstrapping sat in order to obtain a well specified test
statistics. Hence, we proceed as in Lyon, Barber and Tsai (1999): 1000 bootstrapped
samples of size 4bN N=
b
are drawn from the original sample. For each
bootstrapped sample, the sat is calculated as before:
21 1ˆ ˆˆ ˆ3 6
bsa b b b b
b
t N S skw S skwN
= + +
b
( )ˆ
ˆ
b
b bi
S ϖ ϖσ ϖ
−= and ( )
( )
3
13
ˆˆ
ˆ
bNb bi
ib b
b i
skwN
ϖ ϖ
σ ϖ=
−=∑
The critical values *lx (lower bound) and *
ux (upper bound) are obtained from the
empirical distribution of bsat for a given confidence interval : α
* *Pr Pr2
b bsa l sa ut x t x α ≤ = ≥ =
7
C. The Data
In this analysis we use all the NYSE/AMEX firms with available data on the
Daily CRSP files. The period covered goes from July 1962 through December 1996.
In general, the research on long-term stocks’ performance focuses mainly on
ordinary common shares so that CRSP share codes 10 and 11 are eliminated from
our analysis. We use the Daily Files to compute arithmetic monthly returns. This
allows us to swap the matching-firms in “real time” whenever they are delisted.
Nasdaq stocks are excluded to mitigate the new listing bias. However, there is no
specific reason to eliminate those firms having experienced a specific event like new
listing and not those involved in seasoned equity offerings or split which are also
known to produce abnormal returns.
II. Does Book-to-Market and Size Matching Matter? Barber and Lyon (1997) and Lyon, Barber and Tsai (1999) claim that the
matching criterion is crucial. For random samples, they show that size and book-to-
market is required in order to obtain well-specified tests either for matching-firms
and control portfolios. However, Fama and French (1993) show that the market itself
is an important factor, which cast some doubt on a matching procedure relying on
two criteria only. The matching procedure requires well specified and powerful tests
indeed. However, non-relevant matching criteria must lead to opposite results too.
For that purpose, we choose a criterion without any financial content based on the
alphabetical ranking of the stock; see Ferson, Sarkissian and Simin (1999). In order
to compare our results with Lyon, Barber and Tsai (1999), we use two benchmarks: a
matching-firm and a control portfolio.
A. Data and the Sampling Design
A.1. Matching-Firm
8
The firms and the event-dates are drawn randomly from the subset of
NYSE/AMEX firms previously defined and from July 1962 trough December 1991.
Whenever the five-years stock returns series is missing or incomplete for a given
pair, a new pair is drawn. We generate 1000 samples of 200 firms each.
The selection of the matching-firm is based on two criteria. First, the matching-
firm is drawn randomly from the initial population of firms available at the event
date. Second, we select the firm at the event date whose CRSP share code is the next
available in the CRSP File. If the matching-firm disappears during the five-years
period, it is replaced by another firm selected randomly in case 1 and by the next
firm in case 2. The swap is made at the delisting time. Obviously, both criteria lack
of any financial content. The question we address is whether this matching procedure
leads to well specified and powerful test statistics too.
A.2. Control Portfolio
During the period covered by our analysis, 2000 stocks are available on the CRSP
Files so that 50 equally-weighted reference portfolios consisting of 40 securities each
are constructed. In each portfolio, stocks are selected randomly with replacement.
When a firm is delisted before the end of the five-years period, it is not replaced in
the portfolio. Brav, Geczy and Gompers (2000) a similar sampling technique.
B. Results
B.1. Specification of Test Statistics
We study the specification of the four test statistics presented in Section I.B. for
the one year, three years and five years horizons. The results are presented in Table I
for a theoretical rejection rate of 5%.
9
Insert Table I
First, all the test statistics considered here are well specified at the one-year and
the three-years horizons. There is no difference between the random matching and
the criteria based on the following CRSP code firm. Interestingly, the two “non-
financial” matching procedures produce test statistics, which are as well specified in
random samples as the ones based on book-to-market and size criteria. In other
words, there is no gain in using these criteria. From a practical point of view, the
matching of the Compustat Files and the CRSP Files is not necessary. This has two
advantages: the matching is simpler and no bias due to presence in both databases is
introduced.
Second, concerning the five-years horizon, the test statistics based on the control
portfolio remain well specified. Conversely, the test statistics based on the matching-
firm are significantly different at the 1% level from the theoretical rejection rate of
5%. Thus, the control portfolio is the best benchmark for short, medium and long
horizons up to five years.
Third, the correction introduced in the test statistics to account for skewness and
the bootstrapping of the statistics do not out-perform the classic t-stat. In some cases
(BHAR and Matching-firm), the classic t-stat is the unique statistics, which is well-
specified at the five-years horizon.
B.2. Power of Test Statistics
We study the power of the test statistics by adding a constant abnormal return to
each stock. Four different values are examined depending on the horizon. We
consider –20 percent, -10 percent, 10 percent and 20 percent for the one-year horizon
and –50 percent, -20 percent, 20 percent and 50 percent for both the three-years and
10
the five-years horizons. The results of the simulations are presented in Table II and
summarized in Figure 1.
Insert Table II
As far as the power of the test is concerned, our matching criteria (Random
Matching or Next CRSP Code Matching) lead to similar results. In fact, this is not
surprising because these criteria are “independent” from any financial theory. The
Control Portfolio is a better benchmark than the matching-firm. The power of the test
statistics based on the CARs and a control portfolio is always higher than 90% even
with a small additional increment (10 percent for a one year-horizon and 20 percent
for three to five years horizons).
Strikingly, our method produces more powerful tests than a size and book-to-
market based matching. Let us consider two examples. When 10 percent (-10
percent) are added over a one-year period, the standard Student-t applied to the
BHARs has a power of 43% (39%) in LBT compared to 63.4% (58.6%); see Table
II-A. The difference is even more important with the bootstrapped t-test corrected for
the skewness: we find 95.4%(63.3%) against 70% (55%) in LBT.
In general, bootstrapping the statistics increases the specification of the tests
statistics which is even better after correcting for skewness. However, this result
does not hold any longer for the power of the tests. Contrarily to LBT, the
bootstapped statistics are less powerful than their standard counterparts and
sometimes there is a huge difference. In particular, the power of the bootstrapped t-
stat adjusted for skewness (calculate with the BHAR and the control portfolio) is
63.3 percent as opposed to 82.5 percent with the standard t-stat. This is really
embarrassing because this techique was supposed to perform well in that setting.
11
Insert Figure 1
We see that the test statistics constructed from a benchmark based on size and
book-to-market are less powerful than the ones constructed from an arbitrary
benchmark. However, these discrepancies may be explained by the sampling designs
of both studies. Nevertheless, our main conclusion remains: simulations based on
event-firms selected randomly do not help validate criteria in forming matching-firm
or control portfolio benchmarks.
II. Matching and Past Performance A. Conditional Samples Based on Past Returns
The characteristics of our sample remain unchanged. Our analysis is based on
NYSE/AMEX firms from July 1962 through December 1996 (CRSP share codes 10
and 11 excluded). Each month, the securities are sorted according to their twelve-
months prior returns and affected to the corresponding quintile. The quintile Q1 (Q5)
contains the stocks with previous high returns (low returns). Two types of event-firm
samples are determined depending on the previous performance. We construct these
samples by randomly drawing 1000 samples of 200 firms from Q1 and Q5
separately.
In order to measure the abnormal returns, a matching-firm is chosen in three
different ways: a) a random selection over the whole population, b) a random
selection among Q1 firms, and c) a random selection among Q5 firms.
For each event-firm, we draw randomly 40 stocks from Q1 (Q5) and calculate the
buy and hold abnormal returns for the corresponding equally-weighted portfolio at
the one-year, three-years and five-years horizons. When a firm is delisted during the
performance measurement period, it is not replaced beyond that date.
12
B. Results
To assess the specification of the test conditioned on past returns, we use the
bootstrapped t-stat adjusted for skewness. In Table III, we report the critical values at
2.5% and 97.5% in order to highlight the asymmetry of the biases depending on a)
the benchmark (matching-firm or control portfolio), and b) the adequacy of the
matching procedure based on past returns.
Insert Table III
First, the bootstrapped t-stat adjusted for skewness is ill-specified when the
matching-firm or the control portfolio is selected randomly. Not surprisingly, the
results are even worse when firms with high past performance (low) are matched
against a control with low past performance (high). In fact, we reproduce a specific
momentum-type strategy, which yield a positive performance2.
Second, with matching-firms or control portfolios selected to match past stock
returns of event-firms, the test statistics is well specified and nearly symmetric (the
empirical rejection rate is the same for the upper and the lower bound).
The characteristics of the event-firms are generally ignored in empirical studies.
Despite the fact that events are rarely random, the matching-firm or a control
portfolio procedure based on size and book-to-market is chosen routinely. However,
this is not the best procedure whenever the event-sample exhibit previous specific
patterns in stock returns.
2 Cooper (1999) provides evidence that wide varieties of momentum strategies (portfolio weights)
produce significant abnormal returns.
13
III. Abnormal Returns in Calendar Time
A. Data and the Sampling Design
The characteristics of our sample are slightly modified. Our analysis is based on
NYSE/AMEX firms from July 1968 through December 1988 (CRSP share codes 10
and 11 excluded) because of the availability of the size and book-to market factors
which were downloaded from Ken French’s website3.
First, the firms and the event-dates are drawn randomly from the subset of
NYSE/AMEX firms previously defined and from July 1968 through December 1991.
Whenever the three-years stock returns series is missing or incomplete for a given
pair, a new pair is drawn. We construct 1000 samples of 200 firms each, whose
returns are aggregated in order to form 1000 equally-weighted portfolios. The
number of firms may not be constant up to five years within each portfolio because
of delisting. Nevertheless, there is no general agreement on how to circumvent this
problem.
Second, we assume that events are no longer uniformly distributed over time.
Depending on the previous twelve-months market returns ( ), the number of
events within that month is defined as follows:
12mtR
• : no event, 10% 12mtR− ≤
• : one event (normal frequency), 10% 12 30%mtR− ≤ ≤
• : six events (high frequency). 30% 12mtR≤
Firms are drawn randomly from the population in both normal and high frequency
event-periods. This sampling allows us to examine the case of events occurring
mostly during bullish market periods.
3 Update Fama and French Factors are available at
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/
14
Third, in high-frequency periods firms are drawn randomly from the sub-sample
of stocks having experienced high twelve-months returns (above 20%), which
corresponds to firms engaged in the event because they have high past returns.
Fourth, we examine the converse setting in which the frequency of events is
increased when the market was bearish ( ). In that case, stocks are
drawn randomly from the population (bearish random) and from the previous lowest
twelve-months returns (bearish, loser).
12 30%mtR ≤ −
To study the power of the tests, each month we add a specific increment to the
monthly stock returns such that the abnormal return is equal to a given increment at
the end of the five-years holding period4. The incremental abnormal returns take the
following value: -20 percent, -10 percent, 10 percent and 20 percent.
B. Abnormal Returns and Statistical Tests
The general model used to calculate abnormal returns is the following:
( )2
1
i
i
T
it it t ik ikt itk T
R E R I γ δ ε=
= + +∑
where ikγ is the abnormal return of stock over month , i k
iktδ is a dummy variable equal to 1 whenever k and 0 otherwise, t=
1iT is the beginning of the window for the stock i ,
2iT is the end of the window for the stock , i
itε is an error term with zero mean and constant variance . ( )20, iN σ
First, expected returns are described by the Fama and French (1993) model:
( ) ( )it t ft i mt ft i t i tE R I R R R s SMB h HMLβ= + − + +
4 The monthly increment corresponding to a total increment of 30 percent is equal to 60 1 0.30 1 0.004382+ − = .
15
where is the return on the three-month Treasury bills, is the return of the
market portfolio (CRSP value-weighted index), is the return of the size
portfolio, is the return of the book-to-market portfolio,
ftR
HML
mtR
t
tSMB
t I is the information
set provided by , and . mtR tSMB tHML
Brav, Geczy and Gompers (2000) and Jegadeesh (2000) use an extension of the
Fama and French model, namely the four factor model proposed by Carhart (1997).
The fourth factor is the portfolio 12M . It consists in investing an equal amount in
the 30% of stocks which have experienced the highest returns during the last twelve
months (from t through ) and being short in the 30% with the lowest
returns. The equation is:
12− 1t −
( ) ( ) 12it t ft i mt ft i t i t i tE R I R R R s SMB h HML m PRβ= + − + + +
where are the monthly returns of portfolio 12tPR 12M .
Four sets of parameters are estimated for both models and used to forecast both
the conditional mean and the conditional variance5. First, the parameters are
estimated over the event-period ,1 ,2,...,i iT T for each stock i as in Mitchell and
Stafford (2000), abnormal returns being in sample error-estimates. Second, the five-
years period before the event is used, and abnormal returns are forecasted errors.
Third, the model is estimated with a five-years moving window ending at time t ,
and the one-period ahead forecast estimates the abnormal return. Fourth, the window
used for parameter estimation expends until t , and the abnormal return is
calculated as before.
1−
1−
The null hypothesis of no abnormal returns can be written as follows:
5 See Tashman (2000) concerning the forecasting with linear regression models.
16
3 3,1 1 1 1
1 1: 0 :T N T N
it itA
t i t it t
H vs HT n T n
γ γ= = = =
= ≠∑∑ ∑∑ 0
where is the number stocks in month t within the event-portfolio, T is the
total number of periods for which the portfolio is defined. We omit the months for
which the portfolio contains no stocks
tn
6.
Whenever, the series is independent and has finite variance, the hypothesis can be
tested with a standard t-test (see eq. xx). However, the portfolios’ weights are time
varying, which is a potential source of heteroscedasticity because the variance is a
decreasing function of the number of stocks within the portfolio. Therefore, the t-stat
is calculated with the series of abnormal returns standardized by the series of their
own residual variances 21
ˆtnit
i t itn sγ
=∑ . In so doing, low-frequency event-periods are not
over-weighted.
Another potential source of heteroscedasticity comes from both the variances of
the stocks, which is specific and the forecasting horizon, which depends on the
forecasting model7. First, the series of abnormal returns is standardized by the
forecasted variance and aggregated at each time t given portfolios’ weights
producing a new series on which the standard t-test and the cross-sectional t-test are
calculated.
C. Results
C.1. Alternative Test Statistics in Random Samples
6 This feature occurs very seldom. In fact, the probability of having no stock is neglectable. 7 The calculation of these test statistics follow Patell (1976) and Boehmer, Poulsen and Musumeci
(1991) in which they are found to be well-specified and powerful for the short term (typically a ten
days window).
17
The analysis of the empirical specification in random samples is presented in
Table IV. The results concerning the specification correspond to no abnormal returns
(0 percent increment). Anything else being equal, the choice of the benchmark is not
of major concern as the test statistics are very similar. The rejection rate of the null
(no abnormal return) tends to be slightly higher than the theoretical rate of 5 percent
with Carharts’ (1997) model, while the converse is true with Fama and French
(1993) model. Thus, the former is slightly more conservative.
Insert Table IV
The estimation period is not very important either. Nevertheless, the estimation
over the sample-period provides well specified test statistics (with the exception of
the standardized t-test and the Carhart model) at the 1 percent level8. This is
interesting from a practical perspective because no returns are required prior to the
event.
As far as the specification is concerned, and independently of both the model and
the estimation period, the cross-sectional variance adjustment, which accounts for
the time-varying variance of the portfolio (t-cross and t-standard cross) matters. The
standardization (adjustment for the specific residual variance of the stock) has a
minor impact. Even more, as it was the case for the short-term analysis (see
Bohemer, Poulsen and Musumeci (1991)), standardization of the abnormal returns
alone deteriorates the specification of the test. When both corrections are applied
(standardization and cross-sectional adjustment), the empirical specification is close
to its theoretical counterpart whatever the model and the estimation period.
8 Our results apparently contradit Kothari and Warner (1997). However, sampling methods (grouping
in portfolio instead of single stocks) and standardization do not allowed a direct comparison.
18
The results presented above concerning the benchmark and the estimation period
extend to the power of the tests. However, test statistics adjusted for the cross-
sectional variance are less powerful than the classic t-stat and the standardized t-stat.
To conclude, when the abnormal performance is of an unknown form, the t-statistics
based on the standardized abnormal returns calculated with the Fama and French
(1993) model offers a reasonable solution.
C.2. Alternative Test Statistics in Non Random Samples
Following Loughran and Ritter (2000), we allow the frequency of events to
depend strongly on the market past performance. The purpose is to construct a more
realistic sampling design, as many events are driven by past performance. The results
concerning the specification of the test statistics are presented in Table V.
Insert Table V
Strikingly, whenever the event is the consequence of the extreme stock past
returns performance, the results concerning the specification are disastrous. We find
almost surely an abnormal performance. In fact, the simulation portfolios are
momentum type portfolios long (short) in past winners (losers) and short (long) in
the market for the Bullish-Winner (Bearish-Loser) sampling, which are known to
produce abnormal performance. In this setting, the matching-firm procedure is by far
the best solution in order to obtain well-specified test statistics.
When the event-frequency is random, the results (see Table VI) are similar to
what was found in random samples (see Section III C.1.). Once again, the t-test
based on the cross-sectional standardized abnormal returns is well specified. The
power of the test statistics presents also similar patterns.
19
Insert Table VI
In general, previous findings are confirmed. The power of the test statistics in
detecting an abnormal return is higher in bearish market periods and for positive
abnormal returns as well. Test statistics calculated with the Fama and French model
estimated during the event-period are the most powerful. Finally, the cross-sectional
variance adjustment does not produce powerful test statistics.
IV. Conclusion The intent of this research was to study the specification and the power of classic
test statistics used in long-term event study analysis. Using random samples, we
showed that an arbitrary benchmark without any financial content leads to test
statistics that are as well specified as the ones based on the size and book-to-market
benchmark.
However, pre-event abnormal performance has been found which cast some doubt
on the reliability of simulations based on pure random samples. When conditioning
the samples on the past stock returns performance, our simulations showed that a
good matching procedure (a firm with similar returns) leads to well specified tests.
Finally, we examined the specification and the power of calendar-time portfolios.
When the event-sample is selected randomly, our main results can be summarized as
follows. The cross-sectional standardized t-stat has the best specification among the
four tests statistics we examined. The period of estimation is not of a major concern.
As far as the benchmark is concerned, Carhart model (1997) is slightly more
conservative than Fama and French (1993) model. When the frequency of events is
conditioned on past market returns performance, the cross-sectional standardized t-
statistics remains well specified whenever the stocks are selected randomly.
20
However, when the frequency of events is conditioned on past market returns
performance and stocks are selected among the most extreme performers, all the test
statistics examined are misspecified and powerless.
As underlined by LBT, the analysis of long-run abnormal returns is treacherous as
there is no general method, which performs well in the situations frequently
encountered in empirical studies. The pattern of abnormal returns during the one-
year period preceding the event has a strong impact on both the specification and the
power of test statistics. Thus, it is worth paying attention to the specificity of the
sample.
21
References
Barber, B. and J. Lyon, 1997. Detecting long-run abnormal stock returns: the empirical power and specification of test-statistics. Journal of Financial Economics 43, 341-372.
Boehmer, E., J. Musumeci and A. Poulsen, 1991. Event study methodology under conditions of event-induced variance. Journal of Financial Economics 30, 253-272.
Brav, A., C. Geczy and P. Gompers, 2000. Is the abnormal return following equity issuances anomalous?. Journal of Financial Economics 56, 209-250.
Brock, W., J. Lakonishok and B. LeBaron, 1992. Simple technical trading rules and the stochastic properties of stock returns. Journal of Finance 47, 1731-1764.
Brown, S. and J. Warner,1980. Measuring security price performance. Journal of Financial Economics 8, 205-258.
Brown, S. and J. Warner, 1985. Using daily stock returns: the case of event studies. Journal of Financial Economics 14, 3-31.
Carhart, M., 1997. On persistence in mutual fund performance. Journal of Finance 52, 57-82. Cooper, M.,1999. Filter rules based on price and volume in individual security overreaction. Review of Financial Studies 12, 901-935. DeBondt, W. and R. Thaler, 1985. Does the stock market overreact?. Journal of Finance 40, 793-805. Eckbo, E., R. Masulis and O. Norli, 2000. Seasoned public offerings : resolution of the ‘new issues
puzzle’. Journal of Financial Economics 56, 251-292. Fama, E., 1998. Market efficiency, long-term returns, and behavioral finance. Journal of Financial
Economics 49, 283-306. Fama, E. and K. French, 1992. The cross-section of expected returns. Journal of Finance 47, 427-465. Fama, E. and K. French, 1993. Common risk factors in the returns on stocks and bonds. Journal of
Financial Economics 33, 3-56. Ferson, W.E., E. Sarkissian and T. Simin, 1999. The alpha factor asset pricing model: a parable.
Journal of Financial Markets 2, 49-68. Ferson, W.E. and R.W. Schadt, 1996. Measuring fund strategy and performance in changing
economic conditions. Journal of Finance, 425-461. Ibbotson, R., 1975. Price performance of common stock new issues. Journal of Financial Economics
2, 235-272. Ikenberry, D., J. Lakonishok and T. Vermaelen, 1995. Market underreaction to open market share
repurchases. Journal of Financial Economics 39, 181-208. Jaffe, J.F., 1974. Special information and insider trading. Journal of Business 47, 410-428. Jegadeesh, N., 2000. Long-term performance of seasoned equity offerings: benchmark errors and
biases in expectations. Financial Management 29, 5-30. Kothari, S. and J. Warner, 1997. Measuring long-horizon security price performance. Journal of
Financial Economics 43, 301-339. Kothari, S. and J. Warner, 2001. Evaluating mutual fund performance, Journal of Finance,
Forthcoming. Loughran, T. and J. Ritter, 1995. The new issues puzzle. Journal of Finance 50, 23-51. Loughran, T. and J. Ritter, 2000. Uniformly least powerful tests of market efficiency. Journal of
Financial Economics 55, 361-390. Lyon, J., B. Barber and C. Tsai, 1999. Improved methods for tests of long-run abnormal stock returns.
Journal of Finance 54, 165-202. Mandelker, G., 1974. Risk and return: the case of merging firms. Journal of Financial Economics 1,
303-335. Mitchell, M. and E. Stafford, 2000. Managerial decisions and long-term stock price performance.
Journal of Business 73, 287-320. Patell, J., 1976. Corporate forecasts of earnings per share and stock price behaviour: empirical tests.
Journal of Accounting Research 14, 246-276. Ritter, J., 1991. The long-term performance of initial public offerings. Journal of Finance 46, 3-27. Tashman, L., 2000. Out-of-sample tests of forecasting accuracy: an analysis and review. International
Journal of Forecasting 16, 437-450.
22
Tabl
e I:
The
Spec
ifica
tion
of A
ltern
ativ
e Te
st S
tatis
tics B
ased
on
a R
ando
m M
atch
ing
in R
ando
m S
ampl
es
In th
is ta
ble
the
perc
enta
ge o
f 100
0 sa
mpl
es o
f 200
firm
s tha
t rej
ect t
he n
ull h
ypot
hesi
s of n
o an
nual
, thr
ee-y
ears
and
five
-yea
rs a
bnor
mal
retu
rn a
t the
theo
retic
al
leve
l of 5
per
cent
are
pre
sent
ed. T
he sa
mpl
e se
lect
ion
is b
ased
on
a no
n fin
anci
al c
riter
ion
(ran
dom
mat
chin
g or
nex
t CR
SP c
ode)
. Bot
h th
e cu
mul
ativ
e ab
norm
al
retu
rns (
CA
R) a
nd b
uy a
nd h
old
abno
rmal
retu
rn (B
HA
R) a
re u
sed.
Bol
d ita
lic c
hara
cter
s ind
icat
e th
at th
e em
piric
al re
ject
ion
rate
is d
iffer
ent a
t the
1 p
erce
nt
leve
l fro
m th
e th
eore
tical
reje
ctio
n ra
te.
R
ando
m M
atch
ing
Nex
t CR
SP C
ode
Mat
chin
g H
oriz
on
1 ye
ar
3 ye
ars
5 ye
ars
1 ye
ar
3 ye
ars
5 ye
ars
C
AR
Mat
chin
g Fi
rm
t-sta
t
5.
46.
47.
8 5.
36.
67.
7 t-s
tat b
oots
trap
5.3
6.5
7.9
5.4
6.
47.
7 t-s
kew
5.
56.
68.
1 5.
46.
88.
0 t-s
kew
boo
tstra
p 5.
0 6.
4 7.
8 5.
2
6.1
7.4
C
AR
Con
trol P
ortfo
lio
t-sta
t
5.4
5.9
6.1
5.5
5.3
5.5
t-sta
tboo
tstra
p
5.
85.
55.
95.
65.
05.
5t-s
kew
5.
65.
76.
35.
85.
65.
6t-s
kew
boot
stra
p
5.
75.
45.
85.
24.
85.
4
BH
AR
Mat
chin
g Fi
rm
t-sta
t
5.0
4.4
4.8
6.2
4.6
4.6
t-sta
t boo
tstra
p 6.
1 6.
5 7.
8
7.6
6.2
7.0
t-ske
w
6.
17.
2 8.
57.
16.
3 7.
6 t-s
kew
boo
tstra
p 5.
8 5.
9 7.
7 6.
9
6.
16.
8
BH
AR
Con
trol P
ortfo
lio
t-sta
t
5.9
5.2
5.3
6.1
4.5
4.3
t-sta
tboo
tstra
p
6.
06.
26.
86.
35.
95.
4t-s
kew
5.6
6.2
7.3
5.9
6.3
6.0
t-ske
wbo
otst
rap
5.6
4.9
5.2
5.8
5.3
5.3
23
Tabl
e II
-A: T
he P
ower
of T
est S
tatis
tics B
ased
on
a R
ando
m M
atch
ing
in R
ando
m S
ampl
es, o
ne-y
ear h
oriz
on
In th
is ta
ble,
we
pres
ent t
he p
erce
ntag
e of
100
0 sa
mpl
es o
f 200
firm
s dra
wn
rand
omly
that
reje
ct th
e nu
ll hy
poth
esis
of n
o an
nual
(Pan
el A
), th
ree-
year
s (Pa
nel B
) an
d fiv
e-ye
ars (
Pane
l C) a
bnor
mal
retu
rn fo
r var
ious
leve
ls o
f abn
orm
al re
turn
s and
hor
izon
s. Th
e sa
mpl
e se
lect
ion
is b
ased
on
a no
n fin
anci
al c
riter
ion
(ran
dom
m
atch
ing
or n
ext C
RSP
cod
e). B
oth
the
cum
ulat
ive
abno
rmal
retu
rns (
CA
R) a
nd b
uy a
nd h
old
abno
rmal
retu
rn (B
HA
R) a
re u
sed.
R
ando
m M
atch
ing
Nex
t CR
SP C
ode
Mat
chin
g In
crem
ent
-20
%
-10
%
10 %
20
%
-20
%
-10
%
10 %
20
%
C
AR
Mat
chin
g Fi
rm
t-sta
t
100.
075
.081
.410
0.0
100.
077
.984
.610
0.0
t-sta
t boo
tstra
p
99
.977
.378
.299
.810
0.0
75.3
80.6
100.
0t-s
kew
10
0.0
73
.781
.599
.910
0.0
77.6
84.1
100.
0t-s
kew
boo
tstra
p
99
.574
.875
.399
.899
.873
.078
.599
.7
CA
R C
ontro
l Por
tfolio
t-s
tat
10
0.0
96.3
97.2
100.
0 10
0.0
97.2
97.4
100.
0t-s
tat b
oots
trap
100.
093
.497
.110
0.0
100.
094
.396
.410
0.0
t-ske
w
99.9
95.2
97.0
100.
099
.896
.397
.510
0.0
t-ske
w b
oots
trap
99.3
89.7
96.4
100.
099
.391
.294
.810
0.0
B
HA
R M
atch
ing
Firm
t-s
tat
98
.058
.663
.498
.2
98.9
59.3
65.3
98.9
t-sta
t boo
tstra
p
98
.360
.559
.097
.198
.160
.562
.397
.4t-s
kew
96
.6
58
.563
.396
.797
.158
.065
.197
.5t-s
kew
boo
tstra
p
95
.256
.953
.694
.194
.656
.057
.394
.7
BH
AR
Con
trol P
ortfo
lio
t-sta
t
99.2
82.5
91.6
100.
0 99
.483
.193
.310
0.0
t-sta
t boo
tstra
p
98
.171
.296
.010
0.0
98.2
73.9
94.8
100.
0t-s
kew
94
.6
73
.794
.410
0.0
94.1
74.0
95.4
100.
0t-s
kew
boo
tstra
p
88
.863
.395
.410
0.0
88.8
66.2
94.2
100.
0
24
Tabl
e II
-B (c
ontin
ue):
The
Pow
er o
f Tes
t Sta
tistic
s Bas
ed o
n a
Ran
dom
Mat
chin
g in
Ran
dom
Sam
ples
, thr
ee-y
ears
hor
izon
R
ando
m M
atch
ing
Nex
t CR
SP C
ode
Mat
chin
g In
crem
ent
-50
%
-20
%
20 %
50
%
-50
%
-20
%
20 %
50
%
C
AR
Mat
chin
g Fi
rm
t-sta
t
100.
079
.694
.710
0.0
100.
081
.695
.310
0.0
t-sta
t boo
tstra
p
10
0.0
84.4
91.3
100.
010
0.0
86.7
91.8
100.
0t-s
kew
10
0.0
79
.594
.710
0.0
100.
081
.495
.210
0.0
t-ske
w b
oots
trap
100.
083
.288
.910
0.0
100.
084
.689
.610
0.0
C
AR
Con
trol P
ortfo
lio
t-sta
t
100.
099
.299
.710
0.0
100.
099
.299
.410
0.0
t-sta
t boo
tstra
p
10
0.0
98.6
99.6
100.
010
0.0
99.0
99.4
100.
0t-s
kew
99
.9
99
.099
.710
0.0
99.9
98.8
99.3
100.
0t-s
kew
boo
tstra
p
99
.997
.699
.410
0.0
99.9
97.4
98.7
100.
0
BH
AR
Mat
chin
g Fi
rm
t-sta
t
96.5
43.2
50.0
97.8
97
.742
.051
.698
.6t-s
tat b
oots
trap
97.5
45.8
46.0
97.1
97.5
45.3
48.4
97.5
t-ske
w
93.3
44.4
50.2
94.7
94.9
42.3
53.0
95.4
t-ske
w b
oots
trap
92.4
42.3
42.7
91.6
92.9
40.9
44.5
92.2
B
HA
R C
ontro
l Por
tfolio
t-s
tat
98
.058
.192
.310
0.0
98.2
62.5
92.7
100.
0t-s
tat b
oots
trap
96.9
47.6
94.9
100.
097
.550
.594
.010
0.0
t-ske
w
84.4
45.8
95.7
100.
084
.949
.995
.110
0.0
t-ske
w b
oots
trap
78.1
42.1
94.7
100.
078
.944
.594
.810
0.0
25
Tabl
eau
II-C
(con
tinue
) : T
he P
ower
of T
est S
tatis
tics B
ased
on
a R
ando
m M
atch
ing
in R
ando
m S
ampl
es, f
ive-
year
s hor
izon
R
ando
m M
atch
ing
Nex
t CR
SP C
ode
Mat
chin
g In
crem
ent
-50
%
-20
%
20 %
50
%
-50
%
-20
%
20 %
50
%
C
AR
Mat
chin
g Fi
rm
t-sta
t
100.
056
.489
.410
0.0
100.
058
.391
.310
0.0
t-sta
t boo
tstra
p
10
0.0
65.4
81.0
100.
010
0.0
64.0
82.9
100.
0t-s
kew
10
0.0
56
.089
.310
0.0
100.
057
.791
.110
0.0
t-ske
w b
oots
trap
100.
064
.578
.610
0.0
100.
063
.380
.210
0.0
C
AR
Con
trol P
ortfo
lio
t-sta
t
100.
096
.595
.510
0.0
100.
096
.296
.110
0.0
t-sta
t boo
tstra
p
10
0.0
92.2
95.5
100.
010
0.0
94.6
95.7
100.
0t-s
kew
10
0.0
95
.395
.410
0.0
100.
095
.296
.210
0.0
t-ske
w b
oots
trap
100.
090
.094
.510
0.0
99.8
92.1
94.1
100.
0
BH
AR
Mat
chin
g Fi
rm
t-sta
t
72.5
19.7
25.3
74.4
75
.919
.122
.878
.4t-s
tat b
oots
trap
77.2
22.2
21.8
74.8
76.0
23.5
22.1
75.4
t-ske
w
71.2
24.5
28.8
69.9
72.4
23.3
26.2
74.8
t-ske
w b
oots
trap
69.7
21.7
20.8
65.3
67.9
21.7
21.4
67.5
B
HA
R C
ontro
l Por
tfolio
t-s
tat
76
.316
.974
.210
0.0
77.9
18.3
75.9
99.9
t-sta
t boo
tstra
p
72
.014
.773
.999
.975
.516
.976
.010
0.0
t-ske
w
59.0
11.8
85.3
100.
062
.913
.785
.010
0.0
t-ske
w b
oots
trap
62.1
13.5
75.9
100.
062
.414
.677
.910
0.0
26
Figu
re 1
: Pow
er o
f the
Boo
tstra
pped
Stu
dent
t-te
st fo
r the
BH
AR
Stra
tegy
with
a R
ando
m M
atch
ing-
Firm
s
This
Fig
ure
pres
ents
the
empi
rical
per
cent
age
that
reje
ct th
e nu
ll hy
poth
esis
of n
o an
nual
, thr
ee-y
ears
and
five
-yea
rs b
uy a
nd h
old
abno
rmal
retu
rns (
BH
AR
) for
va
rious
leve
ls o
f inc
rem
enta
l abn
orm
al re
turn
s (x-
axis
) and
hor
izon
s. Th
e sa
mpl
e se
lect
ion
cons
ists
of 1
000
sam
ples
of 2
00 ra
ndom
firm
s who
se m
atch
ing
is
base
d on
a n
on fi
nanc
ial c
riter
ion
(the
mat
chin
g-fir
m is
dra
wn
rand
omly
). Th
e re
sults
are
for t
he b
oots
trapp
ed t-
test
adj
uste
d fo
r ske
wne
ss
0102030405060708090100
-50
-40
-30
-20
-10
010
2030
4050
Abn
orm
al R
etur
ns
pow
er
1 ye
ar3
year
s5
year
s
27
Tabl
e II
I: Sp
ecifi
catio
n of
Boo
tstra
pped
-adj
uste
d Te
st S
tats
itics
in M
omen
tum
Bas
ed S
ampl
es
In th
is ta
ble
the
perc
enta
ge o
f 100
0 sa
mpl
es o
f 200
firm
s tha
t rej
ect t
he n
ull h
ypot
hesi
s of n
o an
nual
, thr
ee-y
ears
and
five
-yea
rs a
bnor
mal
retu
rn a
t the
theo
retic
al
leve
l of 2
.5 p
erce
nt a
nd 9
7.5
perc
ent a
re p
rese
nted
. The
sam
ple
sele
ctio
n is
bas
ed th
e tw
elve
-mon
ths p
ast p
erfo
rman
ce o
f frim
s whi
ch a
re c
lass
ified
into
qui
ntile
. Th
e fir
ms a
re ra
ndom
ly d
raw
n fr
om th
e po
pula
tion
(ran
dom
), th
e fir
st q
uint
ile (h
igh
perf
orm
ance
) and
the
fifth
qui
ntile
(low
per
form
ance
). Th
e bu
y an
d ho
ld
abno
rmal
retu
rn (B
HA
R) i
s use
d in
ord
er to
est
imat
e th
e ab
norm
al p
erfo
rman
ce. T
est s
tatis
tics a
re b
oots
trapp
ed st
atis
tics (
mat
chin
g-fir
m) a
djus
ted
for s
kew
ness
(c
ontro
l por
tfolio
). B
old
italic
cha
ract
ers i
ndic
ate
that
the
empi
rical
reje
ctio
n ra
te is
diff
eren
t at t
he 1
per
cent
leve
l fro
m th
e th
eore
tical
reje
ctio
n ra
te.
Hor
izon
1 ye
ar
3 ye
ars
5 ye
ars
C
ontro
l
2.5%
97
.5%
2.5%
97
.5%
2.5%
97
.5%
Sam
ple
with
Pas
t Hig
h Pe
rfor
man
ce
Mat
chin
g-fir
m
Ran
dom
0.3
8.1
2.4
3.3
6.5
1.3
Con
trol P
ortfo
lio
Ran
dom
0.
2
8.0
5.8
0.8
23.1
0.2
Mat
chin
g-fir
m
Hig
h pe
rfor
man
ce
2.3
2.7
2.6
2.5
2.8
2.4
Con
trol P
ortfo
lio
Hig
h pe
rfor
man
ce
2.1
2.8
2.4
2.7
2.8
2.7
Mat
chin
g-fir
m
Low
per
form
ance
0.0
35.3
1.2
8.3
9.8
2.0
Con
trol P
ortfo
lio
Low
per
form
ance
0.
0
50
.64.
21.
9 23
.0
0.0
Sam
ple
with
Pas
t Low
Per
form
ance
M
atch
ing-
firm
R
ando
m12
.90.
52.
1 2.
40.
45.
8C
ontro
l Por
tfolio
R
ando
m
13.6
0.1
0.8
11.2
0.0
45.1
Mat
chin
g-fir
m
H
igh
perf
orm
ance
28.3
0.0
3.4
1.8
0.5
9.6
Con
trol P
ortfo
lio
Hig
h pe
rfor
man
ce
30.8
0.
02.
1 6.
80.
030
.2M
atch
ing-
firm
Lo
w p
erfo
rman
ce
2.0
2.8
2.2
2.7
2.3
2.6
Con
trol P
ortfo
lio
Low
per
form
ance
2.
52.
12.
72.
82.
82.
5
28
Tabl
e IV
: Spe
cific
atio
n an
d Po
wer
of T
est S
tatis
tics w
ith C
alen
dar P
ortfo
lios i
n R
ando
m S
ampl
es
In th
is ta
ble,
we
pres
ent t
he e
mpi
rical
reje
ctio
n ra
te o
f the
nul
l hyp
othe
sis (
no a
bnor
mal
retu
rns)
with
incr
emen
ts ra
ngin
g fr
om –
30 p
erce
nt to
30
perc
ent.
It is
ca
lcul
ated
ove
r 100
0 sa
mpl
es o
f 200
firm
s dra
wn
rand
omly
. The
firm
s are
hol
d a
five-
year
s per
iod
and
aggr
egat
e in
to 1
000
equa
lly-w
eigh
ted
portf
olio
s. Th
e fo
llow
ing
regr
essi
ons a
re e
stim
ated
(
)(
)an
d12
RR
hH
ML
RR
hH
ML
mPR
αit
fti
im
tft
it
it
itit
fti
im
tft
it
it
ii
itR
RsS
MB
RR
sSM
Bβ
εα
βε
−=
++
+−
=+
++
+
−+
−+
whe
re
is th
e m
onth
ly re
turn
on
the
cale
ndar
-tim
e po
rtfol
io,
is th
e re
turn
on
the
thre
e-m
onth
Tre
asur
y bi
lls,
is th
e re
turn
of t
he m
arke
t por
tfolio
(CR
SP
valu
e-w
eigh
ted
inde
x),
is th
e re
turn
of t
he si
ze p
ortfo
lio,
itRft
Rm
tR
tSM
Bt
HM
L is
the
retu
rn o
f the
boo
k-to
-mar
ket p
ortfo
lio a
nd w
here
a
re th
e m
onth
ly re
turn
s of
portf
olio
12
12t
PRM
. The
est
imat
atio
n pe
riod
is th
e fiv
e-ye
ars e
vent
per
iod,
the
five-
year
s per
iod
befo
re th
e ev
ent,
and
an e
xpen
ding
per
iod
begi
nnin
g fiv
e ye
ars
befo
re th
e ve
nt. T
he se
ries
itα
ε+
and
its c
ondi
tiona
l var
ianc
e ar
e us
ed to
cal
cula
te th
e st
atis
tics.
Bol
d ita
lic c
hara
cter
s ind
icat
e th
at th
e em
piric
al re
ject
ion
rate
is
diff
eren
t at t
he 1
per
cent
leve
l fro
m th
e th
eore
tical
reje
ctio
n ra
te.
i
In
crem
ent
-3
0%-2
0%-1
0%0%
10%
20%
30%
Pane
l A: F
ama
and
Fren
ch M
odel
Es
timat
ion
over
the
even
t-per
iod
t-sta
t
99
.085
.833
.2
3.7
36.8
89.1
99.7
t-cro
ss
88.5
74.8
32.6
5.3
17.1
57.5
81.8
t-sta
ndar
d10
0.0
93.3
33.2
4.4
60.7
98.5
99.9
t-sta
ndar
dcr
oss
88.4
72.3
22.2
4.2
36.6
75.4
86.9
Estim
atio
n be
fore
the
even
t-per
iod
t-sta
t
99.1
83.1
30.3
2.
7 26
.684
.099
.1t-c
ross
90.2
73.6
34.1
5.9
10.1
46.4
74.8
t-sta
ndar
d99
.788
.730
.02.
7 36
.792
.799
.7t-s
tand
ard
cros
s88
.574
.431
.84.
417
.560
.981
.0Es
timat
ion
with
an
expe
ndin
g sa
mpl
e pe
riod
t-sta
t
99
.385
.5
31
.33.
0 27
.483
.799
.1t-c
ross
88.9
74.7
35.0
6.5
10.3
47.2
76.2
t-sta
ndar
d99
.588
.528
.62.
3 44
.395
.899
.4t-s
tand
ard
cros
s88
.872
.528
.63.
623
.467
.484
.6
29
Tabl
e IV
(con
tinue
): Sp
ecifi
catio
n an
d Po
wer
of T
est S
tatis
tics w
ith C
alen
dar P
ortfo
lios i
n R
ando
m S
ampl
es
In
crem
ent
-3
0%-2
0%-1
0%0%
10%
20%
30%
Pane
l B: C
arha
rt M
odel
Es
timat
ion
over
the
even
t-per
iod
t-sta
t
98
.681
.721
.0
6.3
51.4
94.3
100.
0t-c
ross
86.7
66.2
24.7
4.5
26.1
66.1
82.9
t-sta
ndar
d99
.991
.525
.77.
8 68
.299
.310
0.0
t-sta
ndar
dcr
oss
87.8
66.7
17.3
6.5
44.9
78.7
88.7
Estim
atio
n be
fore
the
Even
t-per
iod
t-sta
t
95
.864
.612
.3
6.6
47.6
92.8
99.8
t-cro
ss
81.8
58.4
19.7
4.8
22.7
59.8
81.2
t-sta
ndar
d98
.573
.514
.08.
4 59
.398
.099
.9t-s
tand
ard
cros
s83
.761
.316
.15.
233
.170
.985
.3Es
timat
ion
with
an
Expe
ndin
g Sa
mpl
e-pe
riod
t-sta
t
97
.469
.4
11.4
5.8
48.2
93.5
100.
0t-c
ross
83.5
59.6
20.2
3.7
22.7
60.5
81.2
t-sta
ndar
d98
.775
.310
.89.
7 65
.798
.810
0.0
t-sta
ndar
dcr
oss
83.2
58.2
12.9
5.4
38.4
76.6
87.8
30
Ta
ble
V: S
peci
ficat
ion
of A
ltern
ativ
e Te
st S
tatis
tics D
epen
ding
on
the
Freq
uenc
y of
the
Even
t with
Cal
enda
r Por
tfolio
s
In th
is ta
ble,
we
pres
ent t
he e
mpi
rical
reje
ctio
n ra
te o
f the
nul
l hyp
othe
sis (
no a
bnor
mal
retu
rns)
. It i
s cal
cula
ted
over
100
0 sa
mpl
es o
f firm
s whi
ch n
umbe
r is
dete
rmin
ed a
ccor
ding
to th
e tw
elve
-mon
ths p
ast m
arke
t ret
urns
. The
freq
uenc
y is
hig
h w
hen
mar
ket r
etur
ns a
re e
xtre
me.
The
firm
s are
dra
wn
rand
omly
ove
r the
in
itial
pop
ulat
ion
(ran
dom
Bea
rish
and
Ran
dom
Bul
lish)
and
ove
r the
stoc
ks w
hich
exp
erie
nced
an
extre
me
past
per
form
ance
(Bul
lish
Win
ner a
nd B
earis
h Lo
ser)
. The
firm
s are
hol
d a
five-
year
s per
iod
and
aggr
egat
e in
to 1
000
equa
lly-w
eigh
ted
portf
olio
s. Th
e fo
llow
ing
regr
essi
ons a
re e
stim
ated
(
)(
)an
d12
RR
hH
ML
RR
hH
ML
mPR
αit
fti
im
tft
it
it
itit
fti
im
tft
it
it
ii
itR
RsS
MB
RR
sSM
Bβ
εα
βε
−=
++
+−
=+
++
+
−+
−+
whe
re
is th
e m
onth
ly re
turn
on
the
cale
ndar
-tim
e po
rtfol
io,
is th
e re
turn
on
the
thre
e-m
onth
Tre
asur
y bi
lls,
is th
e re
turn
of t
he m
arke
t por
tfolio
(CR
SP
valu
e-w
eigh
ted
inde
x),
is th
e re
turn
of t
he si
ze p
ortfo
lio,
itRft
Rm
tR
tSM
Bt
HM
L is
the
retu
rn o
f the
boo
k-to
-mar
ket p
ortfo
lio a
nd w
here
a
re th
e m
onth
ly re
turn
s of
portf
olio
12
tPR
12M
. The
est
imat
atio
n pe
riod
is th
e fiv
e-ye
ars e
vent
per
iod,
the
five-
year
s per
iod
befo
re th
e ev
ent,
and
an e
xpen
ding
per
iod
begi
nnin
g fiv
e ye
ars
befo
re th
e ve
nt. T
he se
ries
itα
ε+
and
its c
ondi
tiona
l var
ianc
e ar
e us
ed to
cal
cula
te th
e st
atis
tics.
Bol
d ita
lic c
hara
cter
s ind
icat
e th
at th
e em
piric
al re
ject
ion
rate
is
diff
eren
t at t
he 1
per
cent
leve
l fro
m th
e th
eore
tical
reje
ctio
n ra
te.
i
Mod
el
Fam
a an
d Fr
ench
(199
3)
Car
hart
(199
7)
Mar
ket
Bul
lish
Bea
rish
Bul
lish
Bea
rish
Bul
lish
Bea
rish
Bul
lish
Bea
rish
R
ando
m
R
ando
mW
inne
rLo
ser
Ran
dom
Ran
dom
Win
ner
Lose
r
Estim
atio
n ov
er th
e Ev
ent-P
erio
d t-s
tat
5.8
3.3
100.
010
0.0
4.7
8.1
99.9
100.
0t-c
ross
6.0
1.4
85.3
93.3
2.1
3.8
85.3
91.1
t-sta
ndar
d
6.8
6.2
100.
010
0.0
6.7
12.7
100.
010
0.0
t-sta
ndar
d cr
oss
4.7
4.7
87.1
93.4
4.0
6.8
85.8
91.6
Es
timat
ion
befo
re th
e Ev
ent-p
erio
d t-s
tat
4.0
1.3
99.4
100.
0 4.
0 8.
099
.910
0.0
t-cro
ss
7.4
0.9
74.4
91.9
3.0
3.6
79.8
92.1
t-sta
ndar
d
3.6
1.2
99.7
100.
06.
0 9.
210
0.0
100.
0t-s
tand
ard
cros
s 4.
6 0.
9
76
.292
.64.
35.
480
.691
.4
Estim
atio
n w
ith a
n Ex
pend
ing
Sam
ple-
perio
d t-s
tat
4.
21.
999
.9
10
0.0
4.1
7.7
99.9
100.
0t-c
ross
6.6
1.5
81.0
93.6
1.8
3.6
80.3
92.8
t-sta
ndar
d
3.
62.
310
0.0
100.
07.
314
.110
0.0
100.
0t-s
tand
ard
cros
s 3.
9 1.
2
84
.595
.04.
46.
084
.093
.3
31
Tabl
e V
I: Po
wer
of A
ltern
ativ
e Te
st S
tatis
tics D
epen
ding
on
the
Freq
uenc
y of
the
Even
t with
Cal
enda
r Por
tfolio
s
In th
is ta
ble,
we
pres
ent t
he e
mpi
rical
reje
ctio
n ra
te o
f the
nul
l hyp
othe
sis (
no a
bnor
mal
retu
rns)
whe
n an
arti
ficia
l inc
rem
ent i
s add
ed to
the
retu
rns.
It is
ca
lcul
ated
ove
r 100
0 sa
mpl
es o
f firm
s whi
ch n
umbe
r is d
eter
min
ed a
ccor
ding
to th
e tw
elve
-mon
ths p
ast m
arke
t ret
urns
. The
freq
uenc
y is
hig
h w
hen
mar
ket
retu
rns a
re e
xtre
me.
The
firm
s are
dra
wn
rand
omly
ove
r the
initi
al p
opul
atio
n (r
ando
m B
earis
h an
d R
ando
m B
ullis
h) a
nd o
ver t
he st
ocks
whi
ch e
xper
ienc
ed a
n ex
trem
e pa
st p
erfo
rman
ce (B
ullis
h W
inne
r and
Bea
rish
Lose
r). T
he fi
rms a
re h
old
a fiv
e-ye
ars p
erio
d an
d ag
greg
ate
into
100
0 eq
ually
-wei
ghte
d po
rtfol
ios.
The
follo
win
g re
gres
sion
s are
est
imat
ed
()
()
and
12R
Rh
HM
LR
Rh
HM
Lm
PRα
itft
ii
mt
fti
ti
tit
itft
ii
mt
fti
ti
ti
iit
RR
sSM
BR
RsS
MB
βε
αβ
ε−
=+
++
−=
++
++
−
+−
+
whe
re
is th
e m
onth
ly re
turn
on
the
cale
ndar
-tim
e po
rtfol
io,
is th
e re
turn
on
the
thre
e-m
onth
Tre
asur
y bi
lls,
is th
e re
turn
of t
he m
arke
t por
tfolio
(CR
SP
valu
e-w
eigh
ted
inde
x),
is th
e re
turn
of t
he si
ze p
ortfo
lio,
itRft
Rm
tR
tSM
Bt
HM
L is
the
retu
rn o
f the
boo
k-to
-mar
ket p
ortfo
lio a
nd w
here
a
re th
e m
onth
ly re
turn
s of
portf
olio
12
tPR
12M
. The
est
imat
atio
n pe
riod
is th
e fiv
e-ye
ars e
vent
per
iod,
the
five-
year
s per
iod
befo
re th
e ev
ent,
and
an e
xpen
ding
per
iod
begi
nnin
g fiv
e ye
ars
befo
re th
e ve
nt. T
he se
ries
itα
ε+
and
its c
ondi
tiona
l var
ianc
e ar
e us
ed to
cal
cula
te th
e st
atis
tics.
i
Mar
ket
Bul
lish
Bai
ssie
rIn
crem
ent
-20%
-10%
10
%20
%-2
0%-1
0%
10%
20%
Pane
l A: F
ama
and
Fren
ch M
odel
Estim
atio
n ov
er th
e Ev
ent-P
erio
d t-s
tat
85
.533
.035
.588
.8
97.7
54.4
60.4
99.2
t-cro
ss
70.6
32.2
11.9
50.7
82.2
36.0
32.6
76.3
t-sta
ndar
d
92
.933
.161
.097
.999
.250
.485
.199
.9t-s
tand
ard
cros
s
68
.725
.029
.371
.381
.925
.457
.786
.1
Estim
atio
n be
fore
the
Even
t-per
iod
t-sta
t
81.8
31.4
25.2
81.3
97
.047
.148
.298
.3t-c
ross
71
.834
.79.
038
.281
.837
.519
.271
.6t-s
tand
ard
86.3
33.2
35.7
91.0
98.6
52.8
62.0
99.5
t-sta
ndar
dcr
oss
71.5
32.7
13.8
52.3
84.4
38.1
30.5
76.1
Es
timat
ion
with
an
Expe
ndin
g Sa
mpl
e-pe
riod
t-sta
t
83.7
32.2
25.7
82
.697
.752
.145
.398
.3t-c
ross
71
.935
.47.
738
.782
.939
.519
.370
.7t-s
tand
ard
86.6
30.0
45.5
93.7
98.3
46.0
71.7
99.9
t-sta
ndar
dcr
oss
71.6
28.1
17.7
57.7
82.5
32.1
40.7
81.1
32
Tabl
e V
I (co
ntin
ue):
Pow
er o
f Alte
rnat
ive
Test
Sta
tistic
s Dep
endi
ng o
n th
e Fr
eque
ncy
of th
e Ev
ent w
ith C
alen
dar P
ortfo
lios
Mar
ket
Bul
lish
Bai
ssie
rIn
crem
ent
-20%
-10%
10
%20
%-2
0%-1
0%
10%
20%
Pane
l B: C
arha
rt M
odel
Estim
atio
n ov
er th
e Ev
ent-P
erio
d t-s
tat
78
.923
.249
.094
.3
92.5
34.7
82.1
99.9
t-cro
ss
68.3
22.4
20.0
60.1
76.3
21.2
48.7
84.0
t-sta
ndar
d
90
.727
.168
.699
.398
.738
.193
.910
0.0
t-sta
ndar
dcr
oss
67.7
16.8
38.4
74.3
79.0
15.6
66.5
87.6
Es
timat
ion
befo
re th
e Ev
ent-p
erio
d t-s
tat
66
.315
.346
.492
.2
88.7
18.1
77.2
99.5
t-cro
ss
61.9
20.9
18.6
54.4
72.6
13.6
41.2
81.2
t-sta
ndar
d
72
.515
.559
.896
.592
.524
.685
.799
.9t-s
tand
ard
cros
s
61
.318
.226
.465
.374
.915
.351
.383
.3
Estim
atio
n w
ith a
n Ex
pend
ing
Sam
ple-
perio
d t-s
tat
69
.014
.146
.9
91.5
89.3
19.3
77.5
99.8
t-cro
ss
61.0
19.3
19.0
56.0
73.2
13.5
42.0
83.6
t-sta
ndar
d
73
.411
.866
.398
.192
.017
.791
.899
.9t-s
tand
ard
cros
s
58
.313
.131
.869
.274
.29.
959
.285
.0