Inference in long horizon event studies, a bayesian approach

... they actually appear in the data Since factor loadings and covariance matrices used to simulate individual firm returns are unknown, I also incorporate estimation risk (e.g., Klein and Bawa (1976)... vq and small diagonal elements in '&~1 result in low amount of shrinkage and large variation across the factor lo a d in gs- T he parameter values for the diagonal elements in &-1 are set as... S a m ple X ? fr o m p(Ai | Aj , , A* ) p(Aj | A, 1+1, A3 , , A* ) p{Xd | Aj+1, A} t.) T he vectors A , A1 , , A1 , are a realization from a Markov Chain It can be shown (Geman and Gem an

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INFERENCE IN LONG-HORIZON EVENT STUDIES:

CHICAGO, ILLINOIS AUGUST 1998

Copyright 1998 by Brav, Alon

All rights reserved

UMI Microform 9841496 Copyright 1998, by UMI Company All rights reserved

This microform edition is protected against unauthorized copying under Title 17, United States Code

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I thank my committee members George Constantinides, Mark Mitchell, Nick Poison, Richard Thaler and especially Eugene Fam a for their guidance and encouragement In addition, I would like to thank J.B H eaton for his invaluable insights throughout the w riting of this dissertation.

A C K N O W L E D G M E N T S iii

LIST OF F I G U R E S vi

LIST OF T A B L E S vii

A B S T R A C T viii

Chapter 1 IN T R O D U C T IO N 1

1.1 B ack g ro u n d 3

2 M ETH O D O LO G Y 7

2.1 D ata D e sc rip tio n 7

2.2 Basic Setup and Model E s tim a tio n 9

2.3 Predictive D istribution for Long-Horizon R e tu rn s 18

2.4 Statistical In fere n c es 19

2.4.1 Do the Residual Covariations M a tte r ? 21

2.4.2 A check on Simulation E r r o r 21

3 EMPIRICAL R E S U L T S 23

3.1 Initial Public Offerings 23

3.1.1 Sample Description 23

3.1.2 Statistical Inferences 25

3.1.3 Do the Residual Covariations M a tte r ? 26

3.1.4 A check on Simulation E r r o r 26

3.1.5 Comparison with an Alternative M e t h o d 27

3.2 Stock R e p u rc h a s e s 30

3.2.1 Sample Description 30

3.2.2 Statistical In f e re n c e 31

3.2.3 Are “Value” Repurchasing Firms U ndervalued? 32

3.2.4 Explaining the Difference in Computed R e t u r n s 33

3.3 Dividend I n i t i a t i o n s 35

3.3.1 Sample Description 36

3.3.2 Statistical In f e re n c e 37

3.4 Dividend O m is s io n s 39

3.4.1 Sample Description 39

3.4.2 Statistical In f e re n c e 40

A P R O O F S 44

A l Positive-Definiteness of R 44

A.2 Sampling from the conditional distributions for p and crl 45

B F IG U R E S 47

C T A B L E S 52

R E F E R E N C E S 66

Figure Page

1 Shrinkage Estim ation of Factor L o a d i n g 48

2 Shrinkage Estim ation of Residual Standard D e v ia tio n s 49

3 Shrinkage Estim ation of the common correlation c o e f f ic ie n t 50

4 Comparison of EPO and Replacing Firm s Residual Standard Deviations 51

Table Page

1 D ata description for computer and d a ta processing I P O s 53

2 Aftermarket performance of computer and data processing I P O s 53

3 Regression results for computer and d ata processing I P O s 54

4 Predictive densities for the computer and data processing I P O s 54

5 Predictive densities assuming independence for the com puter and d ata process­ ing I P O s 55

6 Simulation sensitivity c h e c k 55

7 IP O d ata d e s c r ip tio n 56

8 EPO aftermarket p e rfo rm a n c e 56

9 Predictive densities for IPO average abnormal r e t u r n 57

10 Predictive densities assuming independence for IP O average abnorm al return 57 11 Simulation sensitivity c h e c k 57

12 IP O sample bo o tstrap d i s t r i b u tio n 58

13 Industry Classifications for EPO A n a ly sis 58

14 IP O Excess R eturn Relative to the NYSE-AMEX value-weight i n d e x 59

15 Stock repurchases d a ta d e s c r ip tio n 60

16 Predictive densities for stock repurchases average abnorm al r e tu r n 60

17 Predictive densities for repurchase sample sorted by bo o k -to -m ark et 61

18 Comparison of abnorm al return calculations for repurchase s a m p l e 61

19 Industry classifications for stock repurchase a n a ly s is 62

20 Repurchasing firms excess return relative to the NYSE-AMEX value-weight index 62 21 Dividend initiations d a ta d e s c r ip tio n 63

22 Regression results for the dividend initiation s a m p le 63

23 Predictive densities for dividend initiations average abnorm al r e t u r n 64

24 Dividend omissions d a ta description 64

25 Regression results for the dividend omission s a m p l e 65

26 Predictive densities for dividend omissions average abnorm al r e t u r n 65

Statistical inference in many long-horizon event studies has been ham pered by the fact th at abnorm al returns are neither normally distributed nor independent This study presents a new approach to inference th at overcomes these difficulties To illustrate the use of the methodology, long-horizon returns subsequent to various corporate events are examined Inference using the new procedure is shown to be sensitive to both non-normality and cross-correlation and to dom inate other popular testing methods.

IN T R O D U C T IO N

Recent em pirical studies in finance document systematic long-run abnormal price reactions subsequent to numerous corporate activities.1 Since these results imply th a t stock prices react w ith a long delay to publicly available information, they appear to be at odds with the Efficient M arkets Hypothesis (EMH)

Long-run event studies, however, are subject to serious statistical difficulties th a t weaken their usefulness as tests of the EMH In particular, most studies maintain the stan ­dard assumptions th a t abnorm al returns are independent and normally distributed although these assumptions fail to hold even approximately at long horizons F irst, samples of long- horizon abnormal returns are not independently distributed because many of the sample firms overlap in calendar tim e.2 Second, abnormal returns are not normally distributed because long-horizon retu rns axe skewed-right by the compounding of single-period returns The standard calculation of abnormal return - sample firm return minus the return on a well diversified portfolio - results in a distribution of abnormal returns th a t is skewed-right

as well (see B arber and Lyon (1997a) and Kothaxi amd W arner (1997)) Both deviationsfrom the standard assum ptions imply th a t parametric inferences th a t rely on independence

‘See H itter (1991) and L oughran an d R itter (1995) for initial public offerings, Ikenberry, Lakonishok and Vermaelen (1995) for stock repurchases, Speiss and Affleck-Graves (1995) for seasoned equity offerings, Michaely, T haler and W om ack (1995) for dividend initiations and omissions and W omack (1996) for stock recom m endations.

2 Overlap in calendar tim e is associated w ith positive cross-sectional dependence because of unpriced industry factors in retu rn s as this paper docum ents later See also Collins and D ent (1984), Sefcik an d Thom pson (1986) and B ernard (1987).

and normality are incorrect.3

In this paper I propose a methodology th at confronts non-normality and cross- sectional dependence in abnorm al returns The methodology employs a “predictive” ap­proach, advanced in the Bayesian literature (e.g., Box (1980), Rubin (1984), Gelfand, Dey and Chang (1992), Ibrahim and Laud (1994), Laud and Ibrahim (1995), Gelman, Carlin, Stem and R ubin (1995) and G e lm a n , Meng and Stem (1996)) It is a goodness of fit criterion, based on the idea th a t good models among those in consideration should make predictions close to w hat has been observed in the data.4 Therefore, given an asset pricing model and a distribution for firm residuals, the model’s param eters are estim ated for all the sample firms Then, given the estim ated parameters, long-horizon returns for all firms are simulated taking account of the estim ated residual variations and covariations The resulting average abnorm al return is calculated using observed factor realizations These steps are repeated a large num ber of times and the simulated averages are used to construct the null distribution for th e sample mean If the actual abnormal return is extreme relative

to the range of predicted realizations, I reject that it is a realization from the constructed density

In this paper this approach is implemented to assess the validity of a single model relative to observed d a ta noting th a t it can be easily extended to select among competing

3 Asym ptotically, th e norm ality of th e sam ple m ean is guaranteed using a C entral Lim it Theorem argu­

m ent The adequacy of this approxim ation is sam ple specific since it depends on th e ra te of convergence which is negatively related to b o th th e degree of cross-sectional dependency and non-norm ality See also Cowan and Sergeant (1997).

4For sim ilar applications in th e tim e-series literature see Tsay (1992) and P aparoditis (1996) an d in th e health-care research see Stangl an d H u erta (1997).

asset pricing models (using Bayesian posterior odds ratios) Concentrating on a specific asset pricing model is in the spirit of Box (1980) who argued:

“In m a k in g a predictive check it [is] not necessary to be specific about an al­ternative model This issue is of some im portance for it seems a m atter of ordinary h u m a n experience th at an appreciation th a t a situation is unusual does not necessarily depend on the immediate availability of an alternative.”(page 387)

The paper proceeds as follows Section 1.1 discusses recent attem pts to address inference problems in long-horizon event studies C hapter 2 presents the proposed method­ology Results are given in chapter 3 Chapter 4 concludes

1.1 B ackground

Recent papers by K othari and Warner (1997) and B arber and Lyon (1997a) address biases in long-horizon event studies Both document th a t the traditional t-test approach is severely misspecified when applied to samples of randomly chosen firms Specifically, both papers show th a t for randomly chosen firms, tests of abnorm al performance are misspecified and indicate abnormal performance too frequently

B arber and Lyon (1997a) argue that misspecification arises from three possible biases: the “new listing” bias, the “rebalancing” bias and the “skewness” bias The first two explain why the calculated average return on a sample of randomly chosen firms may differ systematically from the average return on their benchmark Specifically, the “new listing” bias arises because sample firms usually have a long pre-event return record while the benchmark portfolio includes firms th at have only recently begun trading and are known

to have abnormally low returns (R itter (1991)) The “rebalancing bias” arises because the

compounded return on the benchm ark portfolio implicitly assumes periodic rebalancing of the portfolio weights while th e sample firm returns are compounded w ithout rebalancing The “skewness” bias refers to the fact th at with a skewed-right d istrib u tio n of abnormal returns, the student-t distribution is asymmetric with a mean smaller th a n the zero null.

Kothari and W arner (1997) discuss additional sources of misspecification First, they argue that param eter shifts in the event-period can severely affect tests of abnormal performance For example, th e increase in variability of abnormal re tu rn s over the event- period needs to be incorporated when conducting inferences Second, th ey stress the issue

of firm survival and its effect b o th on the measured abnormal return a n d its variability

These studies are im portant because they document the possibility of erroneous inferences in long-horizon event studies and thus the need for improved testing procedures

th at can potentially overcome these problems Essentially, two methodologies have been suggested as a remedy

The first, by Ikenbeny, Lakonishok and Vermaelen (1995) (hereafter, ILV), is a non-parametric bootstrap approach W ith this approach, the researcher generates an em­pirical distribution of average long-horizon abnormal return and then infers if the observed performance is consistent w ith this distribution To generate an empirical distribution when all th at we observe is one realization of average abnormal return, ELV argue for the follow­ing procedure First, replace each firm from the original sample w ith another firm that (supposedly) has the same expected return and calculate the latter sam ple’s abnormal per­formance Second, repeat this replacement a large number of times and then compare the observed abnormal retu rn to those generated by the new samples The researcher cam reject

the null of no abnormal performance if it is unlikely th a t the realized re tu rn came from the simulated distribution The appeal of this approach is th a t it is easy to implement Once the dimensions th a t determine expected returns have been specified, th e replacement of the original sample is straightforward and the desired empirical density is easy to generate.

T h is m ethod has two flaws, however The replacement of original sample firms

implies an assumption th a t the two samples are “similar” in every dim ension including but

not limited to expected returns This is unlikely for two reasons F irst, if the two samples have systematically different residual variation then the resulting em pirical distribution will

be biased.5 Second, if the original sample abnormal returns are cross-sectionally correlated then the replacement w ith random samples, which are by construction uncorrelated, will lead to false inferences

The second approach, by Lyon, Barber and Tsai (1998) advocates the use of care­fully constructed benchmark portfolios th a t are free of the “new listing” and “rebalancing” biases mentioned above Moreover, to account for the “skewness” bias, they propose the use

of a skewness adjusted t-statistic.6 Lyon et al show th at for random ly selected samples, their methods yield well-specified test statistics

The fact th a t most of their analysis is conducted on randomly selected samples

implies th a t their proposed m ethods are applicable to studies whose “events” occurred at random While it may be the case th a t certain corporate events are uncorrelated acrossfirms, observation suggests this is not tru e for initial public offerings, seasoned equity offer­5In section 3.1.5 I show th a t th e residual variation subsequent to an initial p u b lic offering is much higher

th a n th e variation of a random ly selected firm Metrick (1997) provides further s u p p o rt for th is criticism 6Lyon e t al also su p p o rt th e use of th e bo o tstrap approach due to ILV.

ings, stock repurchases and mergers, events that are frequently th e subject of long-horizon event studies Indeed, Lyon et al recognize this and attem p t to correct their methods for cross-sectional correlation Regretfully, as they point out, the m ethod proposed by ELV cannot be adjusted while their skewness adjusted t-statistic does not elim inate the misspec- ification in samples w ith overlapping returns.

Next, I present the approach in chapter 2 by applying it directly to a small data set of initial public offerings

M E T H O D O L O G Y

In this section I present a long-horizon event study methodology th a t addresses the statistical issues raised in section 1.1 To ease exposition, I build the methodological approach in an application to a sm all data set of initial public offering (IPO ) firms (a one industry subset, chosen arbitrarily, from the full sam ple used in section 3.1)

The goal is to simulate a density for long-horizon abnormal retu rn against which

we can compare the realized abnormal performance This density is constructed under the null hypothesis of no abnormal performance M ost im portant, the sim ulated density attem pts to capture both the firm-specific residual standard variations th a t induce non­normality and the cross-sectional correlations, reflecting these as they actually appear in the data Since factor loadings and covariance m atrices used to simulate individual firm returns are unknown, I also incorporate estimation risk (e.g., Klein and Bawa (1976) and Jorion (1991)) in constructing the simulated density

2.1 D a ta D escrip tion

The sample comprises 113 initial public offerings conducted over th e period 1975-

1984 from the com puter and data processing services industry The source of the data is

R itter (1991).1

^ h e industry’s definition is based on R itter (1991) (SIC code 737) I use three digit SIC codes provided from R itter to assign IP O s to this industry.

Table 1 Panel (a) provides the mean and median market capitalization (size) and book-to-market ratios for these firms Size is calculated using the first closing price on the CRSP daily tape while pre-issue book values are from R itter (1991) Book values were unavailable for 11 firms Panel (b) gives the allocation of these firms into size and book- to-market quintiles th a t were formed using size and book-to-market cutoffs of NYSE firms Panel (c) provides the a n n u a l volume of issuance.

Panel (a) shows th a t the typical firm in this sample is small w ith median market capitalization equal to 29.7 million dollars and median book-to-market ratio equal to 0.06

In fact, as shown in panel (b), most of the sample firms belong to the bottom quintile of size and book-to-market using NYSE firm breakpoints Prom panel (c) it is evident th at,

a t least in this industry, equity issues were clustered in calendar tim e, mostly in 1981 and 1983

Table 2 provides descriptive statistics regarding these firms’ five-year aftermarket return performance relative to the NYSE-AMEX value-weight index.2 The appropriateness

of this and other benchmarks will be discussed later in this section Also given is the cross- sectional standard deviation of abnormal returns and the skewness of the abnormal return distribution

The five year returns on these firms are striking Investors holding shares of IPOs

in this industry earned 24.5%, on average, over a five year period while the market as a whole nearly doubled in value Note also that the median firm lost 47.2% of its valueover this period Furthermore, the skewness and the standard deviation of the abnormal2If a firm delists prem aturely I calculate th e buy and hold return until th e delisting m onth both for the sample firm and th e benchm ark.

return distribution axe extremely large These estimates reflect the success of a few firms amongst the abysmal performance of most of the sample For example, Legent Corp (IPO

in January 1984) earned 564% in excess of the market, M anufacturing D a ta Sys Inc (IPO

in February 1976) earned 679% excess return and Cullinet Software Inc (IPO in August 1978) earned 866% excess return On the other hand, B.P.I Systems Inc (IPO in June 1982), Interm etrics Inc (IPO in June 1982) and Computone Sys Inc (IPO in November 1981) underperform ed th e market by 305%, 249% and 235% respectively Finally, normality

of the abnormal re tu rn distribution is rejected for any traditional level of significance.3

Even w ith this m agnitude of underperformance, can we reject the hypothesis th at this result is due to sam ple variability? To answer this question it is necessary to construct

a simulated density o f average abnormal return under the null hypothesis of no abnormal performance and th en ask w hether measured abnormal return is extrem e compared to the constructed density

2.2 B asic S etu p and M od el E stim ation

The goal in this p art of the analysis is to estim ate firm factor loadings, on a givenset of benchmark portfolios, as well as residual variations and covariations The estimatedmodel is used later to generate samples of long-horizon returns obeying th e same covariation

and variability as observed in the data I assume th a t asset returns are generated by a

k-3The chi-square te st t h a t I use is described in Davidson and M acKinnon (1993) pages 567-570.

factor model, entertaining the Fam a and French (1993) three-factor model in particular.4,5

Given a sample of N firms w ith T* i = 1 , , N monthly observations, define r, as the (Ti x 1) c o lu m n vector of firm t ’s returns in excess of the risk-free rate, fi the (Ti x k)

m atrix of factor m im ic k in g portfolios’ returns and Pi the (k x 1) vector of factor loadings.

The firm’s excess returns are modeled as follows:

The system of all N assets is w ritten using a Seemingly Unrelated Regressions (SUR) setup

(see Zellner (1962) and Gibbons (1982)):

P i , , Px- E is a ($2i^i Ti x 1) stacked vector of firm residuals.

I assume throughout th e analysis a multivariate normal distribution for E w ith

* The approach can be applied for any m odel For example, a characteristic based asset pricing model in

Daniel, G rinblatt, T itm an and W enners (1997) is easily im plem entable with m inor modifications.

5In an earlier version of th e paper I also estim ated the CAPM of Sharpe (1964) a n d Lintner (1965) T h e model was soundly rejected using th e proposed methodology.

mean zero and a (J^jlLi T x YliLi Ti) variance-covariance m atrix E The residuals are

assumed to be temporally independent and to share a common contemporaneous correlation

denoted p.

Under the null o f no abnormal performance and conditional on having the “tru e”

factor model, regression (2.2) is estim ated without an intercept term The model is estimated

using a shrinkage procedure to exploit s im ila r exposure to the three factors (see for example Lindley and Smith (1972), B lattberg and George (1991), Gelfiand, Hills, Racine-Poon and Smith (1990) and Stevens (1996)) Specifically, I seek to take advantage of the assumption

th at firms within a given industry tend to have similar factor loadings Furthermore, the shrinkage approach is also used to estimate firm specific residual variations incorporating prior beliefs th at firms’ residual variation should be “similar” resulting in extreme estimates being “pooled” towards th e sample average This added information will result in more accurate estimation

Obviously, different prior beliefs may yield different inferences Hence, throughout the analysis below, I report results consistent with varying degrees of strength regarding these prior beliefs.6

Posterior distributions for the model param eters are derived as follows First, I

specify the residuals’ likelihood function and the prior beliefs on B and E.

6See K othari and Shanken (1997) page 184 and Stam baugh (1997) page 323 for discussions regarding the use of informative priors.

Likelihood Function:

The likelihood function 1{E\B,H) is m ultivariate normal,

1{E\B, S) a |£ |- 5 e x p { - i( i2 - FB)'Y,~l (R - F B ) } (2.3)

Prior for B:

The prior for B is formed using a hierarchical m ultivariate norm al setup Each Bi

is assumed to be an independent and identical draw from the following multivariate normal distribution:

where P is a (k x 1) mean vector and A^ is a {k x k) diagonal m atrix with elements (<$i, , 5k)- I add an additional layer of uncertainty by modeling the precision of the prior beliefs regarding P (i.e A ^ 1) as a random draw from th e following W ishart prior,7

As Lindley and Smith (1972) note, the specification of and 'I'-1 determines theamount of shrinkage used Specifically, _1 determines the location of the prior distribution

while i/p, the degrees of freedom, determine its dispersion Furtherm ore, one can interpret

the above prior as a posterior distribution obtained after observing an imaginary sample of

size i / 0 and mean centered at 'k- 1

Since A ^ 1 determines the extent of the shrinkage, setting i / 0 large along w ith alarge location value in 'If_1 results in high degree of shrinkage Conversely, small values for7See Zellner (1971) page 389 for the properties of this distribution.

vq and small diagonal elements in '&~1 result in low am ount of shrinkage and large variation

across the factor lo a d in gs- T h e parameter values for the diagonal elements in \&-1 are set as

follows I estim ate N individual factor loadings from separate OLS regressions and calculate the sum of squares for each of the k sets of loadings about th e ir grand average Shrinkage is

induced by employing either a h a lf or a quarter of these k sum s of squares and then using

the reciprocals as the diagonal entries in \P- 1 These levels o f shrinkage will be referred to

later as “m ild” and “strong” respectively Finally, the degrees of freedom uq is set equal to

N

0 is modeled as a draw from the following m ultivariate normal distribution,

The vector 0 specifies my beliefs about the centred tendency of the factor loadings and

specifies the strength of this prior information Since I have no prior information regarding

0 I let the d a ta determine th e central tendencies Hence, th e elements in are set to zero

Prior for E:

The prior for E is specified following Barnard, M cCulloch and Meng (1997) E

is written in term s of two matrices: E = S R S , where S is diagonal m atrix with standard deviations on its diagonal and R a correlation m atrix, both of dim ension ( ^ ^ T x Tt)

Prior beliefs are specified separately for 5 and R.

1 specify a lognormal prior for the N distinct elem ents of 5 ,

The parameters for this prior are specified as follows First, I estim ate the N residual stan­

dard deviations from separate OLS regressions and calculate their grand mean and variance

Second, I specify s and 5a such th a t my prior beliefs regarding the stan d ard deviations are

centered at the observed mean of the residual standard deviations Shrinkage is induced by using either a half or one-sixteenth of the observed variance of the stan d ard deviations Inthe analysis below results will be reported for b oth levels of shrinkage (denoted by “mild”and "strong” correspondingly).8

Prior specification for R requires specifying a prior for the common correlation coefficient p T he only information th a t I am willing to impound in my inferences is th at

the residual covariance matrix is positive definite In appendix A l I show that the following uniform prior satisfies these beliefs

where p* is given in appendix A l

Using Bayes Theorem, I combine the prior beliefs and likelihood function to obtain the joint posterior distribution for the parameters and hyper-param eters of the model,

tional distributions of the parameters and hyper-parameters:

1 M ultivariate normal distribution for B,

where

5* = ( F ' E ^ F + I S ® A j ' r H C F ' E ^ F ) ^ , , + (7 * ® A ^ 1) ^ ® 0 ) ) (2.6b)

where t/v is a (N x 1) vector of ones, 1^ is an (N x N ) identity m atrix and bgis is a vector

of GLS regression coefficients, namely bgis = (F 'E -1 F ) ~ l F 'E -1 R.

2 M ultivariate normal distribution for the hyper-param eter vector 0,

where 7* is a {k x k) identity matrix.

3 W ishart distribution for the precision m atrix A ^ 1,

T he vectors A°, A1, , A1, are a realization from a Markov Chain It can be shown (G em an and Gem an

(1984)) th a t the joint distribution of (A\ , , A*) converges to p ( A i , , Ad | D A T A ) i.e th e joint posterior

distribution, as t —► 00 u n d er mild regularity conditions For application of th e Gibbs sampler to other problems, see for exam ple Gelfand and Sm ith (1990), G elf and, Hills, R acine-Poon and Sm ith (1990).

where D is a (k x N ) m atrix whose N columns, each of length k, are taken sequentially from the vector B

4 The conditional distribution for p is proportional to,

p\B, S oc |£ ( p ) r * exp{ - \ { R - F B )' {S R { p ) S )~ 1{R - F B )} (2.6d)

Where R(p) is the correlation m atrix and the parentheses emphasize th a t it is a function

of p I draw from this conditional distribution using the griddy Gibbs approach (see R itter and Tanner (1992), Tanner (1996)) The details are given in appendix A 2.

5 The conditional distributions for each <7, Vi = 1 , , N are proportional to,

a - (1+Ti)x exP{ - l [ (Zog(g; ) ~- f l! + (R - F B ) '( 5 _ ifi(p)5_t ) - 1(f? - FB)]}

0(j-Where 5_t denotes the standard deviation m atrix conditional on the other IV — 1 standard

deviation draws As with the conditional distribution for p, I draw from this density using

the griddy Gibbs approach detailed in appendix A.2

I obtain the Gibbs sam pler’s initial values by first running OLS univariate regres­

sions and then setting the initial values for 0 equal to th e average of the OLS param eter estimates and the elements in S equal to the sample stan d ard deviations The initial value for p is set to zero The sampler is iterated 600 times and th e first 100 draws axe discarded.10

Before discussing the regression results, it is worthwhile to explore the effect ofdifferent amounts of s h r in k ag e on param eter estimation Figure 1 shows the distribution of10Convergence of th e Gibbs sam pler in particular, and Markov chain Monte Carlo m ethods in general, has received considerable atten tio n recently (see, for example, Gilks, Richardson and Spiegelhalter (1996)) Convergence was m onitored by com paring th e results of multiple chains started a t different (random ) initial values as well as th e use of tim e series p lo ts of th e param eter draws.

the factor loadings’ posterior means as a function of the degree of shrinkage Starting from the left, where no shrinkage is used (denoted “OLS” ), shrinkage is increased as we move

to the right resulting in the param eter’s posterior means having smaller dispersion W ith strong shrinkage information is shared across different firms and the posterior means are tightly clustered relative to the least squares estimates Consequently, if the basic premise

th a t firms w ithin industries have sim ilar factor loadings is correct, then incorporating this information will benefit the precision of estimation

Figure 2 displays the dispersion of the residual standard deviation posterior means

as a function of increasing shrinkage I start from the left, where no shrinkage is used (de­noted “OLS” ), shrinkage is increased as we move to the right resulting in stronger s h r in k a g e

toward the prior m ean (16% in this case)

Figure 3 presents both th e prior and posterior distributions for p The left panel plots the uniform prior described in equation 2.4e which puts mass only on those values

of p th at guarantee th a t the variance-covariance m atrix is positive definite The right plot gives the marginal posterior d istribution given the data

Table 3 gives the regression results In each row I report the average of the pos­terior means for the factor loadings and residual standard deviations I also report the correlation coefficient’s posterior m ean (standard deviations are provided in parentheses) The two panels below correspond to different s h r in k a g e scenarios of the factor loadings The rows within each panel correspond to different s h r in k a g e scenarios of the residual standard deviations

Consider first the estim ated factor loadings In both panels m arket betas are close

to 0.90 w ith stronger shrinkage of the residual standard deviation causing the posterior means of the market betas to decrease slightly The factor loadings on HML and SMB confirm the descriptive statistics given in Table 1 Firms in th is industry covary with the returns of small and low book-to-market firms Slopes on HML are nearly all equal to —0.75 while loading on SMB are as high as 1.12 Note th a t the effect of the shrinkage on the factor loadings manifests itself in lower standard deviation of the posterior means For example, in

panel i the standard deviation o f the posterior means of the m arket betas is 0.32 (first row) while in pamel ii the standard deviation falls to 0.25 given the stronger shrinkage Last,

the sample average return on SMB is virtually zero over the period of estimation while it is close to 50 basis points for HML In addition, the market betas are all smaller than unity implying th a t the three factor model predicts lower returns, on average, for these EPO firms relative to the NYSE-AMEX value-weight index used in table 2

Next, consider the estim ated monthly residual stan d ard deviations The reported average of 15.5% is large and nearly 50% larger than the residual standard deviation of an average firm traded on the NYSE or AMEX as reported in K othari and Warner (1997).11 The dispersion of the residual standard deviation posterior means is as high as 4.34% (panel

ii first row) and it declines to 2.05% as the shrinkage is increased (panel ii third row).

Finally, consider the estim ated average correlation T he reported range of 2.56% —2.61% is not large compared to prior studies of intra-industry correlation.12 In section 2.4.1

11 See th eir table 5 They rep o rt an average of 10.4% standard deviation of m onthly abnormal returns over th eir te st period.

12 B ernard (1987) reports an average intra-industry correlation equal to 18% for m arket model residuals.

I show that these small correlations still affect the distribution of the sample abnormal mean return.

2.3 P red ictiv e D istrib u tion for Long-H orizon R eturns

In this section I describe how to simulate buy-and-hold returns which axe used later to construct the density of the sample mean abnormal return

Let M denote the number of draws on B and £ retained from the Gibbs sampler

o u tp u t and let B j and Hj be the jth such draws Then, conditional on the factor realizations,

I draw K vectors of firm returns Rj each of length (52iLi Tix 1) from the likelihood function

in equation 2.3 By repeating this procedure M times I obtain a set {/Ex, , R k m } ofdraws from an “averaged” likelihood function th at incorporates the additional parameter uncertainty This density is called the “predictive” distribution in the Bayesian literature since, given the modeling assumptions, it generates all possible realizations of the vector

i ? 13

I calculate long-horizon firm returns by compounding the single-period returns for each firm (113 buy and hold returns) Since my factor regressions were done in excess return form, the realized t-bill rate is added back to each single-period draws Similarly, long-

horizon benchmark returns for each firm are calculated using the factors’ realized returns

and the posterior distributions of the firm factor loadings

The above K x M draws yield a set of K x M abnormal means returns that are

used to construct the predictive distribution of the sample mean In th e analysis below K

I3See Tanner (1996) pages 53 and 101 an d Box (1980) page 385.

is set equal to four (rather than one) to reduce variation due to sim ulation error Since M

is equal to 500, the predictive density is based on 2000 draws

The null of no abnormal performance will be “called into question” at the a percent

level if the average abnormal return obtained from the original sample is greater [smaller] than the (1 — a) [a] percentile abnormal return observed in the constructed d istribution.14

2.4 S ta tistic a l Inferences

It is tim e to ask whether the observed returns are consistent with the three factor model of Fama and French Table 4 reports descriptive statistics for the predictive densities under different shrinkage scenarios I report the 1st, 5th, 50th an d 95th percentiles, as well

as the m ean.15 Panels (i) and («) give the properties of these densities for two different

levels of B shrinkage W ithin each panel I report the effect of residual variance shrinkage

The rightmost column gives the sample abnormal performance calculated using the firms’ factor loadings Since different s h r in k a g e scenarios are employed, the resulting average ab­normal returns differ slightly

Five interesting results emerge The first concerns th e large range of these dis­tributions For example, consider the first row in panel (z) A t the 5% level, we cannot reject the three factor model with a realized abnorm al performance as low as —39.5% or

as high as 51.3% Second, comparison of panels (z) and (ii) reveals th a t the shrinkage of

the factor loadings reduces the measured abnorm al performance This change is of a small

u For an extensive discussion regarding this Bayesian approach to model checking using a significance test see Box (1980).

15In principle, th e m ean of the distribution should be exactly zero Because of sim ulation error, it can be seen th a t th e m ean departs slightly from its theoretical value.

magnitude though Third, the median firm, under all shrinkage scenarios earns a negative abnormal return Recall th at these densities were generated under the null of no abnorm al performance which means th a t for skewed-right distributions we should expect to observe

th a t the median firm underperforms Fourth, it can be seen that the resulting shape of the predictive density is insensitive to the differing shrinkage scenarios Fifth, and more impor­tan t, under all shrinkage scenarios and at the 5% significance level, I cannot reject th a t the observed abnorm al performance is consistent w ith th e Fam a and French three factor model

One of th e motivations for this research was th e possible existence of cross-sectional correlation in firm abnormal returns Hence, in this section I examine whether th e covari­ations that were estimated in table 3 have any effect on inference

Table 5 gives the predictive distributions generated by constraining the residual covariance m atrix estimated earlier to be diagonal, th a t is, imposing independence Com­parison of the results in tables 4 and 5 reveals th a t firm cross-sectional correlations (reported

in table 3) have a large effect on inferences Im posing diagonal covariance matrices resulted

in a reduction in uncertainty regarding the sam ple m ean abnormal return Specifically, the 1th percentile has shrunk by 10% — 15% across th e different shrinkage scenarios while the 5th percentile has shrunk by 8% — 10% across different shrinkage scenarios Similarly, the 95th percentile has decreased by as much as 12%

2-4-2 A check on Sim ulation Error

Finally, how sensitive are these results to simulation error? The extent of simula­tion error is examined as follows For each shrinkage scenario I use the simulated means and resample 100 times, with replacement, samples o f abnorm al mean returns each containing

2000 observations For each such sample I calculate the 1st, 5th, 50th and 95th percentiles Summary statistics regarding the variation of these statistics across the different simulations are presented in table 6

The sensitivity results reveal that all percentiles are measured accurately The accuracy increases for quantiles that are closer to the mode of the distribution The 1st and 5th percentiles as well as th e median are m easured more accurately th an the 95th per­centile which is more sensitive to extreme observations As we shall see in the next section, accuracy will further improve as we increase the sample size

EM PIR IC A L RESULTS

3.1 In itial P u b lic Offerings

In this section I conduct inferences regarding the long-term returns to a sample of

1521 IPOs issued over the period 1975-1984 The sample is the one used in R itter (1991) who finds —27.4% size and industry adjusted three-year abnorm al returns for these firms.1 Below this interesting result is re-visited using the three-factor model of Fama and French Given the evidence in Loughran and R itter (1995) th at abnorm al returns persist for five years after the event, I examine a five-year horizon as well

Table 7 gives the distribution of the sample firms by size and book-to-market For each IPO I use the pre-issue book value reported by R itter (1991) while size is determined using the first closing price available from CRSP The 5x5 size and book-to-market cut­offs were determined using NYSE firm breakpoints Each IPO was first allocated to a size quintile and then allocated to a book-to-market quintile Panel (a) reports the number of observations in each cell T he last row in this panel gives the number of missing book-to- market observations.2 Panel (6) reports the mean market capitalization within each size'T h e d a ta is available a t h ttp ://w w w cb a.u fl.ed u /fire/facu lty /ritter.h tm Three IPO s were deleted since

I could not find m onthly d a ta for those firms on CRSP Two additional IP O s were deleted since I had less than four m onthly observations to use in the regressions below.

2I was unable to calculate m ark et capitalization for 3 IPOs on th eir first trading day on C R SP since either their price or shares o u tstan d in g were missing.

quintile Panel (c) reports the mean book-to-market for each cell.

The evidence in table 7 indicates th a t the m ajority of the sample firms are con­centrated in the smallest size and book-to-market quintiles w ith approximately 80% of the sample belonging to the smallest size quintile Moreover, the extremely low book-to-market ratio indicates either tremendous growth opportunities th a t were available to these firms, investor mispricing, or both The near uniform ity of these attributes across th e sample firms is im portant since the chosen asset pricing model is required to explain realized aver­

age returns to this “type” of firm.

To set the stage for the inferences in the next section, five year returns to these IPOs are calculated versus the five year return on the NYSE-AMEX value-weight index The purpose of this comparison is to provide further information regarding the average long-horizon return of this sample The benchmark return th a t accounts for firm factor loadings is calculated later in this section Table 8 reports the sample average and median return as well as the average market return The last three columns give the average excess return as well as its cross-sectional standard deviation and skewness (disaggregated infor­mation regarding industry performance is given in table 14)

From table 8 we see th a t the five year underperformance relative to this market index is —65.7% F urther inspection of the underperformance by industry (see tab le 14) reveals th a t 15 out of 17 industries underperformed In fact, in only three industries, finan­cial institutions, insurance and drug and genetic engineering, did the median firm earn a positive raw return over this five year period From the last two c o lu m n s we see th a t the standard deviation and skewness of the abnormal returns distribution are extrem ely large

The latter statistic confirms the argument th at long-horizon excess returns are not normally distributed.

3.1.2 Statistical Inferences

The first step is to decompose the sample into industries, conduct inferences within industries and then aggregate the results I form 17 industry classifications based on R itter (1991) and Spiess and Affleck Graves (1995) SIC codes for the IPO sample are from Com-

p u stat and R itter (1991) T he list of the original R itte r 3-digit industry classifications and the additions made are given in table 13

For each industry the methodology outlined in chapter 2 is used to estim ate firm factor loadings, residual variations and cross-correlation Then, using the param eters’ pos­terior distributions and th e procedure outlined in section 2.3, I simulate 2000 long-horizon average returns for each industry.3

Table 9 presents the s u m m a ry statistics regarding the distribution of the sample

m ean aggregated across all 17 industries The table reports the 1st, 5th, 50th and 95th percentiles of the distribution as well as the mean The realized abnormal return corrected for each firm’s factor loadings is given in the last column I present results corresponding

to four different shrinkage scenarios

It can be seen th a t under all shrinkage scenarios the observed IPO returns areinconsistent with the three factor model

3N ote th a t the last in d u stry definition ” O ther” contains IP O s th a t were not associated w ith any of the previous 16 industries Since it was assumed th a t th e source of residual cross-correlation was due to industry factors, the residual correlation for this industry is set equal to zero Similarly, I do not shrink any of th e factor loadings since these IP O s are not associated w ith any distinct industry.

3.1.3 Do the Residual C ovariations M a tte r?

As in section 2.4.1 I now examine whether the covariations th a t were estimated within each industry earlier have any effect on my inferences Table 10 below gives the the predictive distributions generated by constraining all residual covariance matrices to be diagonal, th a t is, imposing independence

Comparison of the results in tables 9 and 10 reveals th a t firm cross-sectional cor­relations have an affect on inferences Imposing diagonal covariance matrices results in a reduction in uncertainty r e g a r d in g th e sample mean abnormal return Specifically, the 1st percentile has s h r a n k by 3.6% — 4.9% across the different shrinkage scenarios while the 5th percentile has s h r a n k by 3.2% — 3.8% across different shrinkage scenarios Similarly, the 95th percentile has decreased by as much as 6.4% Taken together, the range from the 5th

to the 95th percentile is now approximately 20% shorter relative to the ranges reported in table 9 This result underscores the im portance of residual cross-correlation as a source of uncertainty regarding the sample abnorm al performance

I now examine the sensitivity of these results to simulation error For each of the simulated distributions reported in table 9, I use the simulated means and resample 100 times, w ith replacement, samples of long-horizon mean returns each containing 2000 obser­vations For each bootstrapped sample I calculate the 1st, 5th, 50th and 95th percentiles

I now present summary statistics regarding the variation of these statistics across the dif­ferent simulations in table 11

The bootstrap results indicate th at the fractiles of the distribution are accurately measured For example, the statistics in the first row indicate th a t in 100 bo o tstrap simu­lations the average 1st percentile was —25.2% w ith a standard deviation of 0.2%.

In this section I compare the proposed methodology to an alternative, non para­metric, approach advanced by Ikenberry, Lakonishok and Vermaelen (ILV) (1995) This approach was outlined in section 1.1 and for completeness I now explain its im plem entation

in detail.4

The first step is to construct benchmark portfolios based on size and book-to- market ratios Beginning in July 1974,1 form size and book-to-market quintile breakpoints based on NYSE firms information I allocate all NYSE, AMEX and NASDAQ firms into these 5x5 = 25 portfolios based on their known book values and market capitalizations (for additional details, see Fama and French (1993)) As in Mitchell and Stafford (1996) and Lyon, B arber and Tsai (1998), I include in the analysis only firms th a t have ordinary common share codes (CRSP share codes 10 and 11) Moreover, to make sure th a t IPOs are not compared to themselves, I exclude all IP O s from the benchmark construction.0 I repeat this procedure for every July throughout 1989, recording the resulting breakpoints and firm allocations

Next, each sample firm is matched to one of the 25 portfolios based on its size

*1 thank M ark Mitchell and Erik Stafford for the d a ta used in this section.

’ Specifically, I delete th e first five years of retu rn history since going public for all EPOs over th e period 1975-1989 This d a ta comes from two sources For the period 1975-1984 I use R itte r’s (1991) d atab ase and for 1985-1989 I use 1966 common equity IPOs identified from the SDC database.

and book-to-market a ttrib u tes th a t were known at the m onth of the IP O Then, I calcu­late benchmark buy and hold returns by equal weighting the buy and hold returns of all the firms in th a t relevant portfolio I make sure th at the length of th e benchmark return horizon is either 60 m onths or shorter if the IPO delisted prem aturely Hence, for each

of my 1521 sample firms, I obtain a benchmark retu rn th a t is m atched based on size and book-to-market and I calculate the EPO sample abnorm al return

Given th a t each IP O has been assigned to a size and book-to-m arket portfolio

allocation, I randomly select from th a t allocation a replacement firm w ith th e same return

horizon as the original firm.6 This replacement is repeated for all IP O s in the sample, resulting in a new “pseudo” sample I proceed to calculate the la tte r sample abnormal performance relative to th e original size and book-to-market portfolios Repeating this re­placement 2000 times results in 2000 average abnormal returns which are used to construct the bootstrapped density for the sample mean The null of no abnorm al performance is rejected at the a percent level if the average abnormal retu rn obtained from the original

sample is greater [smaller] th a n the (1 — a) [a] percentile abnorm al re tu rn observed in the

bootstrapped distribution

Table 12 presents the bootstrapped distribution The table rep o rts the 1st, 5th, 50th and 95th percentiles as well as its standard deviation, mean and skewness coefficient The abnormal return adjusted by size and book-to-market is given in th e last column

Two im portant results emerge from table 12 First, th e —6.17% average abnormalreturn is much smaller th a n th e abnorm al return reported in table 9 using th e three factor6If a replacem ent firm delists prem aturely, I invest the proceeds from th e delisting firm in another ran­ domly selected firm from the sam e portfolio, for th e remaining period.

model This difference confirms the result in Brav and Gompers (1997) who find th a t five year returns to recent IPOs do not differ from the average retu rn on benchmarks constructed based on size and book-to-market ratios Second, the range and shape of the bootstrapped density are m arkedly different than those reported in section 3.1, table 9 This density is less skewed and th e range between the 5th and 95th percentiles is much shorter.7

The differences between the bootstrapped distribution and the density generated

in the previous section are not surprising As argued in section 1.1, the replacement of IPO firms with non-issuing firms neglects possible differences in residual standard deviations and covariations between the original sample of EPOs and the replacing firms, yielding an empirical distribution th a t does not reflect th e large uncertainty of IPO firms

To verify the conjecture th at the residual standard deviations of IPOs and their replacement firms differ, the following analysis was conducted For all IPOs and replac­ing firms, I calculated the residual standard deviations relative to their respective size and book-to-market b e n c h m a rk s Then, for each IPO , I determ ined the percentile of its stan­dard deviation relative to the standard deviations of its potential replacing firms yielding

1521 percentiles If IPO standard deviations do not differ systematically from replacing firms’ standard deviations, we should expect these percentiles to be equally distributed be­tween 1% — 100% Accordingly, in Figure 4 I plot the histogram of these percentiles and the expected count of these percentiles in each bin (the dashed horizontal bar)

The histogram of percentiles indicates th a t there are disproportionately more IPO swith high residual standard deviations th a n predicted by the matching firm approach The7The com parison o f th e two densities is com plicated by th e fact th a t these densities were generated given different asset pricing formulations (factor model versus a ttrib u te-b ased model).