Pseudo Market Timing and the Stationarity of the Event-Generating Process
Paul Schultz*
April, 2004.
*
University of Notre Dame
A number of researchers have shown that stocks seem to perform poorly in the three to
five years following their IPO. In a recent paper (Schultz (2003)), I suggest that a subtle bias in
the calculation of long-run abnormal returns may be at least partly responsible for the poor longrun performance of IPOs. This bias, which I refer to as pseudo market timing, is a problem
specifically when abnormal returns are calculated in event time.
The idea behind pseudo market timing is that companies are more likely to sell equity
when they can get a high price for it. This emphatically does not mean that companies are able to
determine when their stock is overpriced and issue equity at those times. Instead, it means that
IPOs become more numerous following high market returns and high returns for comparable
firms. In this case, when we as researchers look back on a long time series of IPOs and IPO
returns, we will see that IPOs cluster near market peaks. Because they tend to cluster at what
were, ex-post, market peaks, IPOs tend to perform poorly in the aftermarket. Simulations in
Schultz (2003) suggest that even if managers have no ability to predict future returns, there is a
50 percent chance that cumulative abnormal returns of -18% over five years after the IPO will be
observed.
Viswanathan and Wei (2003) conduct a rigorous analytical examination of pseudo market
timing on measured long-run performance. They show that for fixed sample size, expectations of
cumulative abnormal returns and buy-and-hold abnormal returns will be less than zero if the
number of events increases with past returns. They show further that abnormal performance will
be more negative for longer event periods.
The simulations in Schultz (2003) assume that the number of IPOs is proportional to the
level of stock prices - which makes them a random walk. Viswanathan and Wei (2003) and
Dahlquist and de Jong (2003) correctly point out that if the number of IPOs is stationary, pseudo
market timing is a small sample problem. That is, when the number of months in the sample
period approaches infinity, the expected event period abnormal returns become arbitrarily close
to zero. This raises two questions. First, does the number of IPOs follow a stationary process?
Second, if the number of IPOs is indeed stationary, is the small sample underperformance likely
to be important in practice?
My answer to the first question is that I cannot reject a null hypothesis that the number of
1
IPOs follows a non-stationary process. Like Vishwanathan and Wei (2003), I employ augmented
Dickey-Fuller tests for a unit root. Like them, I reject a unit root when only one lagged value of
the number of IPOs is used. I present evidence though, that the number of IPOs is better
described by a higher order autoregressive process than by an AR(1). When several lags are
included, I cannot reject a unit root at even a ten percent confidence level.
My answer to the second question is yes, even if pseudo market timing is a small sample
problem, it is likely to be important in practice. Using a specification for the number of IPOs that
is similar in form to that of Viswanathan and Wei (2003), I find mean buy-and-hold abnormal
underperformance of 23 percent for the five years following IPOs. The mean underperformance
measured with cumulative abnormal returns is 15.8 percent. The number of IPOs is assumed to
follow a non-stationary process in these simulations. My results are much stronger than
Viswanathan and Wei’s because I use several lags of returns and numbers of IPOs to estimate the
total impact of returns on the number of offerings.
The rest of the paper is organized as follows. In the first section I discuss whether or not
the number of IPOs follows a stationary process and present new evidence that is consistent with
non-stationarity. In Section II I report simulations that reveal the importance of pseudo market
timing in practice. Section III offers conclusions.
I. Does the Number of Initial Public Offerings Follow a Stationary Process
In Schultz (2003), I assume that the number of initial public offerings is proportional to
the level of the market. If the market level is a random walk, this means the number of IPOs is
non-stationary. Is this a reasonable assumption? Consider the following thought experiment.
Right now, the Nasdaq Composite index is approximately 2,000. In 20 years, in 2024, if things
go well this index could be around 20,000. If things go very poorly, the index could be around
500. The first case signifies a vibrant, growing economy that has created much wealth for
investors. The second indicates a contracting economy and large losses for investors. The
2
assumption that the number of IPOs is proportional to the level of the market is equivalent to
saying that the number of IPOs is greater in the vibrant and healthy economy, regardless of stock
returns at the time. If we instead assume that the number of IPOs follows a stationary AR(1)
process, the expected number of IPOs would be about the same in either case as long as returns
were similar in 2022 and 2023. It seems implausible that the number of offerings in the long-run
is independent of the condition of the economy in the long-run.
A. Existing Evidence
Viswanathan and Wei (2003) estimate a first order vector autoregression with current
values of the logarithm of the number of IPOs, the market return, and the return of the portfolio
of IPOs as dependent variables. The explanatory variables are the previous months’ realizations
of the variables. Because there are no IPOs in some months, Viswanathan and Wei add .5 to the
number of IPOs before taking the natural log.
In the regression with the log of the number of IPOs as the dependent variable,
Viswanathan and Wei (2003) find that the coefficient on month t-1’s return on IPOs is 1.7364,
which is significant at the at the five percent level. Higher returns on IPOs in month t-1 leads to
more IPOs in month t. The coefficient on the log of the number of IPOs in the previous month is
0.89, with a t-statistic of over 37.
The coefficient on month t-1's number of IPOs is large and indicates a great deal of
persistence in the series, but doesn’t indicate non-stationarity. Viswanathan and Wei (2003)
conduct Augmented Dickey-Fuller and Phillips-Perron tests for unit roots in the log number of
IPOs. Both of these tests, strongly reject a null hypothesis of a unit root. The same conclusions
are reached regardless of whether or not a trend in the number of IPOs is assumed.
The problem with their tests is that they implicitly assume that the log number of IPOs
follows an AR(1) process. There are reasons however, to believe that higher order processes
should be considered. Suppose that the expected number of IPOs in a month followed an AR(1)
process. Assume further that the actual number of offerings is the expected number plus a
3
random component. If we regress the actual number of IPOs in one month on the actual number
the previous month, we have an errors in variables problem and the coefficient on the lagged
variable will be biased towards zero. If the actual number of IPOs in previous months contains
information on the expected number in month t beyond what is contained in the actual number of
IPOs at t-1, the number of IPOs in previous months should also be included in the regression.
The problem is compounded in their simulations of the effects of pseudo market timing
on returns. Here both the lagged number of IPOs and lagged returns are important. It seems likely
that returns should affect the number of IPOs for several months into the future. Suppose that an
IPO candidate decides to go public after observing high returns for comparable public firms.
Before the offering takes place it needs to choose an underwriter, draft the necessary statements,
file with the SEC, and await SEC approval. The lag between observing that conditions are right
for an offering and actually going public may be several months and will differ from firm to firm.
Hence we would expect the number of IPOs in a month to be influenced by returns in several
previous months.
B. Regressions of the Number of IPOs on Previous Months’ IPOs and Returns
It is easy to demonstrate that the number of IPOs in a month depends on the number of
offerings and their returns in several previous months, not just the most recent. IPOs for January
1993 through December 2002 are obtained from SDC. To be consistent with Viswanathan and
Wei (2003), Schultz (2003) and others, IPOs are only included if: (1) the offer price is $1.00 or
more, (2) Gross proceeds are $1,000,000 or more, (3) the offering included stock only (no units),
(4) the stock appeared on the CRSP tapes within three months of the offering, (5) the IPO was
not a closed-end fund, REIT, foreign company, utility, bank or savings and loan. There are 7,088
remaining IPOs.
In Table I, I report regressions of the natural logarithm of the number of IPOs on several
lagged values of itself, lagged return on the CRSP value-weighted market portfolio, and lagged
excess return from IPOs. Following Viswanathan and Wei (2003), I add ½ to the number of IPOs
each month before taking the natural log. All regressions and all simulations in this paper use the
4
natural logarithm of the number of IPOs plus ½ , although, for brevity, I may refer to the variable
as the number of IPOs. For each month, the returns on IPOs are calculated as an equal-weighted
average of the returns of all IPOs from the previous 36 months. Excess returns on IPOs for each
month are calculated by subtracting the CRSP value-weighted index from the IPO index return.
The first regression in the table replicates the portion of Viswanathan and Wei’s (2003)
vector autoregression with the log of the number of IPOs as the dependent variable. The
coefficient on the lagged logarithm of the number of IPOs is 0.8686, quite close to the 0.89
estimated by Vishwanathan and Wei. The coefficient on the CRSP value-weighted return from
the previous month is 0.9059 and the coefficient on the previous months’ IPO return is 1.6652.
The coefficient on the IPO return is similar to Vishwanathan and Wei’s estimate of 1.79.
The next column presents estimates of a regression using the excess return on IPOs rather
than the raw return. This is just a linear transformation of the previous regression, but I run
remaining regressions in this form because it is easier to compare the impact of market and
excess returns.
The regression reported in the third column includes three lags of each variable, while the
fourth column reports regressions with six lags. When three lags are included, the sum of the
coefficients on lagged values of the log number of IPOs goes from 0.8686 to 0.9248. When six
lags are included, the sum of the coefficients reaches 0.9527, much closer to 1.0 and nonstationarity than Viswanathan and Wei (2003) find. In addition, coefficients on higher order lags
of market returns are positive in five cases and positive and significant at the five percent level in
four cases. Market returns have a long-lasting impact on the number of IPOs. Similarly,
coefficients on higher order lags of returns of past IPOs are positive for five of the six lags, and
positive and significant at the five percent level for three of the six.
The last column of the table reports coefficients from a regression of the log of the
number of IPOs on 12 lags of the log of the number of IPOs, the return on the market, and the
excess returns of past IPOs. To save space, I report the coefficients for the first six lags of each
variable and the sum of the coefficients for lags seven through twelve. F-tests show that the sum
of the coefficients for lags seven through twelve are significantly different from zero at the one
percent level for both the number of IPOs and the return on the market. The sum of the
5
coefficients for lags seven through twelve on the IPO excess returns is positive, but insignificant.
With the inclusion of the additional lags however, the coefficients on the first six lags of past IPO
excess returns increase.
It is clear that examining just one lag for returns and two lags for the number of offerings,
as Viswanathan and Wei (2003) do, fails to capture the full effect of stock returns on the number
of IPOs. When only one lag is included in Table I, the coefficient on lagged IPO excess returns is
1.67. With 12 lags, the sum of the coefficients is 11.19. Thus when only one lag is used, less than
1/6th of the effect of IPO excess returns on the number of offerings is captured. In addition, as
additional lags of the number of IPOs is included, the series looks less and less likely to be
stationary. When one lag is included in the regressions, the coefficient on the lagged number of
IPOs is 0.8686. The number of IPOs is persistent, but stationary. When six lags are included the
coefficient sum reaches 0.9527. When 12 lags are included, the sum of the coefficients reaches
0.9767, a level that suggests non-stationarity is very possible.
C. Formal Tests for Unit Roots
Our evidence suggests that the natural log of the number of IPOs follows a higher-order
autoregressive process. That is,
To test for a unit root, I employ the augmented Dickey-Fuller test. This test involves testing the
significance of the coefficient D on ln(Nt-1) in the regression
6
The test for a unit root involves a one-sided test to see if D is significantly less than zero.
The t-statistic for D has a nonstandard asymptotic distribution because, under the null hypothesis
of a unit root, the number of IPOs is non-stationary and the usual limit theory does not apply. The
critical values for the t-statistic for D are instead based on approximations from MacKinnon
(1994). Note that when the log of the number of IPOs follows a higher order autoregressive
process, multiple lagged values of the change in the log number of offerings should be included
in the test.
Regression estimates, along with p-values from the augmented Dickey-Fuller tests are
shown in Table II. The first regression includes the log of the number of IPOs (plus 1/2) in the
previous month as an explanatory variable, but does not include any changes in the number of
IPOs in previous months. The coefficient on the on the log of the previous month’s IPOs is 0.1360, and the MacKinnon p-value allows us to reject a null hypothesis of a unit root at the .01
percent level. This is consistent with the results of Viswanathan and Wei (2003). But, in a higher
order autoregressive process we would expect lagged changes in the number of IPOs to be
significant. Regressions 2 through 7 in Table II include from one to six lagged changes in the log
of the number of IPOs. When four or more lags are included, we can no longer reject a null
hypothesis of a unit root at the five percent level. When five lags are included, the p-value for the
null hypothesis of a unit root reaches 9.75% When the sixth lag is added, the null hypothesis
cannot be rejected at even the ten percent level. The results in Table I suggest that the log number
of IPOs in months t-7 through t-12 are a significant predictor of the log number of number of
IPOs for month t. When eleven lags are included, as an AR(12) process would suggest, the pvalue of the unit root test is .2565. When a trend is included in the regressions (not shown) pvalues are much higher, indicating even less ability to reject a null hypothesis of a unit root.
II. The Economic Significance of Pseudo Market Timing
Can pseudo market timing result in economically significant underperformance following
7
IPOs? In Viswanathan and Wei (2003), the effect is small. But, by using only one lag of past
returns and two lags of the log number of IPOs in their regressions, they significantly understate
the magnitude and persistence of the effects of returns on the number of IPOs.
I use a series of simulations to examine the economic significance of pseudo market
timing. In these simulations, like those of Vishwanathan and Wei (2003), the log number of
offerings for a month is determined by the lagged number of offerings, lagged market returns,
and lagged IPO returns. The principal difference between these simulations and those of
Vishwanathan and Wei is that I use several lags of returns and number of offerings to generate
the number of IPOs. In each simulation, I draw 300 simulated monthly market returns from a
normal distribution with mean 0.94293% and variance 0.2254%. These are the same as the mean
and variance of monthly returns on the CRSP value-weighted index over 1973-2002. To get a
series of IPO returns, I first regress the returns of the portfolio of companies that had an IPO in
the previous 36 months on the CRSP value weighted index. The slope coefficient of 1.4634
from this regression along with an intercept coefficient of -0.43695% are used to simulate IPO
returns from the simulated market return. This intercept is chosen so that the expected returns on
IPOs are equal to the expected return on the market. In other words, the ex-ante expected excess
return for an IPO is set to zero by construction. I then add an error to the simulated IPO return for
each month. The error is drawn from the normal distribution with a mean of zero and variance of
0.2631%. This is the variance of the residuals from the regression of actual IPO returns on
CRSP value-weighted market returns.
The simulated number of IPOs in a month is generated from the simulated number of
IPOs in previous months, the simulated return on the value-weighted market in previous months,
and past values of the excess return on the IPO portfolio. The coefficients from the regressions
with six or 12 lags, as reported in Table II are used to generate the simulated number of offerings.
Note that in this case, I am not assuming the number of IPOs is non-stationary. The
coefficients from the regressions suggest that the effects of returns on the number of IPOs is
long-lasting, but not permanent.
Results are reported in Table III. For each specification, I run 1,000 simulations of a 25
year sample period. Table III provides a description of the distribution of abnormal returns across
8
the 1,000 simulations. The first four columns of the table report excess returns when the
simulated number of IPOs each month is determined by six lags of simulated market returns, IPO
excess returns, and number of IPOs. The mean cumulative abnormal return (CAR) for a 36
month holding period is -6.84%. The mean CAR is negative for 68.7%of the simulations. When
we examine returns for the full five years following IPOs, we find the mean CAR is -10.07%.
The next two columns describe the buy-and-hold abnormal returns (BHAR) for three and
five years following IPOs. As before, six lags are used to determine the number of IPOs in each
simulated month. As a result of compounding, buy-and-hold abnormal returns suggest a greater
level of underperformance than cumulative abnormal returns. The mean BHAR is -9.08% for a
three year aftermarket period and -16.37% for a five year aftermarket period. Median BHARs are
even lower. For half of the simulations, buy-and-hold abnormal returns are less than -11.11% for
three years or less than -21.91% for five years.
As shown by the regression in Table I, the number of IPOs and returns seven to twelve
months prior to month t are statistically significant predictors of the number of offerings in
month t. Hence in the next four columns of Table III we use 12 lags to determine the number of
offerings each month. Using the additional lags has a big affect. Now, the mean CAR for a five
year period is -15.81% while the mean BHAR is -22.95%. The median BHAR is -28.05%. A
closer look at the percentiles of the distribution reveals that 25% of the simulations had five year
CARs less than -31.12% and BHARs less than -43.74%.
Comparing the results when six and twelve lags are used to generate the number of IPOs
demonstrates two things. First, the effects of pseudo market timing on measurement of long-run
abnormal returns can only be appreciated when the persistence of the effects of returns on the
number of events is accurately estimated. Second, when reasonable estimates of the permanent
effect returns on the number of IPOs is used in simulations, it appears that pseudo market timing
can explain much of the poor long-run performance of IPOs.
III. Summary and Conclusions
Dahlquist and de Jong (2003) and Viswanathan and Wei (2003) point out, correctly, that
9
simulations in Schultz (2003) assume that the number of IPOs follows a random walk. They also
point out that if the number of IPOs follows a stationary process, pseudo market timing is a small
sample problem. This is also correct. The problem with their argument is that the data indicate
that the number of IPOs is non-stationary or at least very close to it.
The sum of the coefficients on the number of IPOs in previous months from a regression
of the number of current IPOs on previous returns and number of IPOs is over 0.97 - quite close
to 1.0. Dickey-Fuller tests fail to reject a unit root in the number of IPOs when a proper number
of lags is used. These results are different from Viswanathan and Wei (2003) because they use
only one lag in testing for unit roots. My tests show quite clearly that the number of IPOs is
related to the previous number of IPOs and previous returns at several lags. The assumption in
Schultz (2003) that the number of IPOs follows a random walk appears reasonable.
Of course, even if the number of IPOs is stationary and pseudo market timing is a small
sample bias, a small sample may still be very large relative to the time series of IPOs available to
researchers. I use simulations with a stationary process for IPOs estimated from the data to
examine the importance of pseudo market timing for 25 year sample periods. I find median buyand-hold underperformance of more than 28% for five years after IPOs. The lesson for
researchers investigating long-run performance remains the same - use calendar time returns.
10
Table I. Regressions of the natural logarithm of the number of IPOs (plus 1/2) on lagged values
of the log of the number of IPOs, lagged values of the return on the CRSP value-weighted market
index, and lagged returns of previous months’ IPOs.
IPO returns are calculated for each month as an equal-weighted average of the returns of all IPOs
from the previous 36 months. Excess returns for IPOs are calculated by subtracting the CRSP
value-weighted market return for the month from the IPO return. The sample period is February
1973 through December 2002. To be included in the count of IPOs, the offering must have a
price of $1.00 or more, gross proceeds must be $1 million or more, must appear on the CRSP
files within three months of the effective date, and the offering must include common stock only
(no units). Offerings by funds, REITS, banks, and utilities are also excluded.
11
1
3
0.8686***
4
5
0.5559***
0.4418***
0.3944***
ln(N t-2)
0.0727
-0.0084
-0.0046
ln(N t-3)
0.2961***
0.1046*
0.1118*
ln(N t-4)
0.2174***
0.1597**
ln(N t-5)
0.0290
-0.0208
ln(N t-6)
0.1683***
0.1080*
ln(N t-1)
0.8686 ***
2
0.2282***
ln(N t-7)+...+ln(Nt-12)
3.6195***
3.4677***
3.2501***
R mkt,t-2
1.8639**
2.7222***
2.6063***
R mkt,t-3
0.5584
1.7090**
1.7679**
R mkt,t-4
2.2193***
1.8644**
R mkt,t-5
1.0288
1.2409
R mkt,t-6
-0.7408
-0.0959
R mkt,t-1
0.9059
2.5711***
6.7641***
R mkt,t-7+...+Rmkt,t-12
R IPO,t-1
1.6652 **
1.6652**
1.7336**
1.6297**
2.0221***
XS IPO,t-2
1.0672
1.4452**
1.7987***
XS IPO,t-3
-0.5149
-0.1599
0.2668
XS IPO,t-4
1.0332
1.6464**
XS IPO,t-5
1.1998*
1.3811**
XS IPO,t-6
1.3447**
1.9898***
XS IPO,t-1
XS IPO,t-7+...+XS IPO,t-12
2.0812
Constant
0.2772
0.2772
0.1164
0.0130
-0.1203
Eln(N t-j)
0.8686
0.8686
0.9248
0.9527
0.9767
Adj. R 2
0.7603
0.7603
0.7950
0.8160
0.8301
***
Statistically significant at the 1% level.
**
Statistically significant at the 5% level.
*
Statistically significant at the 10% level.
12
Table II. Augmented Dickey-Fuller tests for a unit root in the number of IPOs per month.
The sample period is the 359 months from February 1973 through December 2002. The number of IPOs is measured as the natural
logarithm of the number of IPOs during the month plus .5. To be included in the count of IPOs, the offering must have a price of $1.00
or more, gross proceeds must be $1 million or more, the stock must be included in the CRSP files within three months of the IPO and
the offering must include common stock only (no units). In addition, offerings by funds, REITS, banks, and utilities are excluded.
1
N t-1
-0.1360
(-5.09)
)N t-1
2
3
4
5
6
7
-0.1109
(-4.06)
-0.0889
(-3.25)
-0.0765
(-2.82)
-0.0757
(-2.74)
-0.0712
(-2.58)
-0.0678
(-2.43)
-0.1847
(-3.53)
-0.2458
(-4.60)
-0.3026
(-5.56)
-0.3036
(-5.37)
-0.3080
(-5.49)
-0.3024
(-5.30)
-0.2024
(-3.89)
-0.2617
(-4.86)
-0.2636
(-4.61)
-0.3040
(-5.21)
-0.3045
(-5.20)
-0.1644
(-3.09)
-0.1669
(-2.91)
-0.2121
(-3.58)
-0.2010
(-3.28)
-0.0078
(-0.14)
-0.0556
(-0.97)
-0.0479
(-0.79)
-0.1079
(-1.99)
-0.1059
(-1.82)
)N t-2
)N t-3
)N t-4
)N t-5
)N t-6
-0.0073
(-0.13)
Constant
0.3132
(4.30)
0.2559
(3.60)
0.2080
(2.85)
0.1837
(2.54)
0.1816
(2.47)
0.1751
(2.38)
0.1645
(2.22)
Test Statistic
-5.087
-4.064
-3.254
-2.818
-2.744
-2.579
-2.434
Mac Kinnon p-value
0.0000
0.0011
0.0170
0.0557
0.0667
0.0975
0.1325
13
Table III. Simulated event period cumulative abnormal returns and buy-and-hold abnormal returns.
In each simulation, 300 months of simulated market returns are drawn randomly from a normal distribution with mean 0.943% and
standard deviation of 0.225%. These are the mean and standard deviations of the monthly CRSP value-weighted market returns over
1973 - 2002. For each month, the return of recent IPOs is -0.43695% plus 1.4634 times the return on the market plus a normally
distributed error. The error has a mean of 0 and a standard deviation of 0.263%. This is the standard deviation of the residuals for the
regression of the mean return of stocks with an IPO in the previous 36 months on the CRSP value-weighted market return. The excess
returns for IPOs for a month are obtained by subtracting the simulated market returns from the simulated IPO returns. For each month
of the simulation, the number of IPOs is calculated based on the returns of the market in the previous six (or 12) months, the excess
returns on IPOs in the previous six (or 12) months, and the number of IPOs in each of the past six (12) months. The relation between
the number of IPOs and past return and IPOs is estimated with OLS regressions of the logarithm of the number of IPOs on the
logarithms of the number of IPOs in previous months and the CRSP value weighted returns and excess returns of IPOs in previous
months. For each specification, 1,000 simulations are run and the mean event period abnormal return is calculated for the simulation.
The table reports the distribution of abnormal returns across the1,000 simulations.
6 lags
6 Lags
12 Lags
12 Lags
3 Year CAR
5 Year CAR
3 Year BHAR
5 Year BHAR
3 Year CAR
5 Year CAR
3 Year BHAR
5 Year BHAR
Mean
-6.84%
-10.07%
-9.08%
-16.37%
-11.13%
-15.81%
-13.63%
-22.95%
F
14.57%
23.06%
18.31%
35.32%
16.22%
25.43%
18.49%
34.50%
10 th Percentile
-24.33%
-37.98%
-28.85%
-51.17%
-29.68%
-46.14%
-32.71%
-55.76%
25 th Percentile
-16.59%
-25.07%
-20.82%
-38.55%
-21.24%
-31.12%
-24.59%
-43.74%
Median
-7.01%
-10.22%
-11.11%
-21.91%
-12.06%
-15.98%
-16.29%
-28.05%
75 th Percentile
2.27%
3.21%
-0.04%
-1.50%
-1.37%
-1.49%
-5.50%
-10.95%
90 th Percentile
10.91%
19.21%
12.60%
23.40%
8.79%
16.78%
7.48%
13.36%
%#0
68.7%
69.3%
75.0%
75.9%
77.8%
77.1%
83.0%
83.7%
14
References
Dahlquist, Magnus, and Frank de Jong, 2003, Pseudo Market Timing: Fact or Fiction?, Working
paper, Stockholm Institute for Financial Research.
MacKinnon, James G., 1994 Approximate asymptotic distribution functions for unit-root and
cointegration tests, Journal of Business and Economic Statistics 12: 167-176.
Schultz, Paul, 2003, Pseudo market timing and the long-run performance of IPOs, Journal of
Finance 58, 483-517.
Viswanathan, S. And Bin Wei, 2003, Endogenous Events and Long Run Returns, Working
paper, Duke University.
15