Cross-Sectional Dependence andthe Use of Generalized Least Squares*

liAVI BHUSHAN Massachusetts Institute of Technology

Abstract. Tbis paper provides additional evidence on tbe relation between the actual andasymptotic standard errors under generalized least squares (GLS). Simulations are con-ducted in the event-study framework using both daily and weekly returns. The case ofsimultaneous calendar time and industry clustering is also considered. The major findingis that the asymptotic standard errors under GLS can significantly underestimate the truestandard errors when the number of parameters estimated is not small compared to thenumber of observations available to estimate them. To minimize the possibility of incor-rect inferences, use of GLS should in general be combined with standard errors obtainedfrom other techniques such as bootstrap.

R4sumd. L'auteur apporte de nouvelles preuves de la relation entre les erreurs-typesr&lles et asymptotiques dans le cadre de I'utilisation des moindres carr^s generalises. IIproc^e a des simulations dans le contexte d'une etude d'^v^nements, en utilisant les ren-dements a la fois quotidiens et hebdomadaires. Le cas de simultanfiite de la date et dugroupage par secteur d'activite sont egalement pris en consideration. Principale conclu-sion de l'auteur : les erreurs-types asymptotiques dans le cadre de I'utilisation des moin-dres carr^s generalises peuvent entratner une sous-estimation importante des erreurs-types vedtables lorsque le nombre de paramStres estimes n'est pas restreint par rapportau nombre d'observations disponibles pour formuler ces estimations. Si l'on veut reduireau minimum le risque d'inf^ences inexactes, I'utilisation des moindres carres general-ises devrait, de fagon gdnerale, Stre combinee aux erreurs-types obtenues a partir d'autrestechniques comme celle de l'amorce.

IntroductionThis paper examines the finite-sample bias in standard errors of generalizedleast squares (GLS) estimation in event studies. GLS is potentially the most effi-cient method when the data are cross-sectionally dependent.' Since the finitesample properties of GLS are not known, the asymptotic standard errors that areused can considerably underestimate the tme standard errors by ignoring estima-tion error.^ Currently, there is limited evidence on the divergence between theasymptotic and the true standard errors under GLS.^ This paper investigates

• The author thanks Andrew Alford, Gary Biddle, Prem Jain, Sok-Hyon Kang, RichardLeftwich, Bill Scott, two anonymous referees, and the workshop participants at UCLA formany helpful comments and suggestions.

Contemporary Accounting Research Vol. 9 No. 2 (Spring 1993) pp 450-462 ®CAAA

('ross-Sectional Dependence and Generalized Least Squares 451

tlirough simulations the impact of sample size on the relation between the actualand the asymptotic GLS standard errors.

Chandra and Balachandran (1990), henceforth ( 3 , argue against the use ofGLS in event studies and recommend using nongeneralized tests (portfoliomethods that do not use information from the residual covariance matrix toplace differential weights on firms). They ciie the lower gains in power fromGLS and the high sensitivity of GLS-based tests to model misspecifications asthe reasons for their recommendations.

In CB, estimation efficiency is not an issue since return correlations consid-ered do not exceed 0.1. These correlations may be typical at the individual secu-rity level, but the correlations are much higher if observations correspond toportlblios of securities (e.g.. Gibbons 1982; Stambaugh 1982) or if return inter-val.s are longer (e.g.. Beaver and Landsman 1983; Bernard and Ruland 1987; seeTable 1 in Collins and Dent 1984; Table 1 in Bernard 1987). In such cases, GLSmay result in considerably higher estimation efficiency.

CB argue that typically a correct modei cannot be specified in an event study.I iowever, researchers in many instances may have enough confidence in a spec-ification that is consistent with the employment of GLS or seemingly unrelatedregression (SUR) estimation. For example, if a regulation affects similar assetsin the same way, then a reasonable model can be specified a priori (e.g., Collins,Rozeff, and Dhaliwal 1981; Schipper, Ihompson, and Weil 1987). Anotherexample is the Smirlock and Kaufold (1987) .study of the reaction of bank stockprices to the Mexican debt crisis when the authors propose that abnormal returnsshould be proportional to exposure {ratio of Mexican loan to bank equity). Someother recent applications of SUR with cross-sectional constraints include Moyer(1990) and Scholes, Wilson, and Wolfson (1990).

The basic point is that researchers arc .still likely to use GLS in instanceswhen it has the potential to offer gains in power, or when a well-formulated the-ory can lead to a model that is amenable lo tJLS or SUR estimation. Situationslike these motivate my examination of the Unite-.sample bias problem in theGLS standard errors.

The major finding of this study is that die asymptotic standard errors underGLS, by ignoring estimation error, can significantly underestimate the true stan-diird errors. Iu the particular event study context considered here, if there aretewer than 10 time-series observations available for each parameter to be esti-mated, then GLS can result in the probability of finding abnormal performance,when there is none, to be substantially higher than expected. If, for example,there arc only four to five time.s as many time-series observations as the numberof parameters, then an application of GLS can lead to actual type I errcff of threeto four limes the expected rate.

An implication of these results is that if GLS is employed to improve effi-ciency, then correct standard errors (e.g., obtained using bootstrap bias-adjustedasymptotic standard errors, as suggested by Marais 1986b, or other computer-intensive resampling techniques), should be used lo minimize the possibility ofincorrect inferences.

452 R. Bhushan

The simulation evidence also indicates (bat improvement in the performanceof GLS can be achieved by reducing the number of parameters to be estimated.A reduction in the number of estimated parameters lowers the estimation errorand alleviates the downward bias in the asymptotic standard errors used in GLS.A way to reduce the number of estimated parameters is to impose reasonablerestrictions on them. Constraining the correlation coefficient to be the same foreach firm pair within an industry would be an example of such a restriction.Such restricted fonns of GLS may thus be less exposed to the finite-sample biasproblem and rejoesent a nice compromise between the desire to increase effi-ciency and to avoid the problems of standard GLS. If estimation efficiency isnot an issue, however, then, as suggested by CB, portfolio methods (e.g., Jaffe1974; Mandelker 1974; Sefcik and Thompson 1986), owing to their unbiasedstandard errors, are preferable to GLS-based techniques.

To the extent that estimation error is an issue in other applications of GLS(SUR and the multivariate regression model), the simulation evidence presentedhere can provide some guidance in the design of these other studies.'* However,this guidance can only be rough since there can be certain key differencesbetween the case considered here and other ^plications of GLS. Examples ofsuch differences are that, in the general SUR estimation, regressors need not beidentical across equations, and there may be no cross-sectional restrictions onthe coefficients.

The rest of the paper is organized as follows. The next section discusses theproblem and the methods employed. The following section presents theapproach adopted for simulations and the results. The final section of the paperoffers some concluding remarks.

The problemOne approach to dealing with cross-sectional dependence is GLS estimation,which, if feasible, is likely to be the most efficient approach. As pointed out byBemard (1987, section 2.1), a potentially serious problem with GLS is that thetrue residual covariance matrix is not known in almost all empirical studies. Theasymptotic standard errws that are used in hypothesis testing ignore the estima-tion error in the estimated residual covariance matrix and may lead to substan-tial understatement of the true standard errors. Estimation error can be consider-able if the residual covariance matrix is estimated using very few observationsper parameter.^

This paper examines the relation between the asymptotic and actual GLSstandard errors in the particular context of event studies. The problem consid-ered can be fonnulated as a special case of the multivariate regression model(MVRM) of Schipper and Thompson (1983) in which all sample fums share acommon reaction (abnormal performance) JJL to a single event.^

Let^,,. denote the retum on security i in period T (the period T can be one dayor one week):

rOSS-Sectional Dependence and Generalized Least Squares 453

where R^^ is the (value-weighted or equally weighted) return on the market inperiod T, e^^ is the idiosyncratic component of the return ff^ , 5^ equals 1 in theevent period and 0 otherwise, and / ranges from 1 to A'. The variance of e-^ isa / , and a, and ;3, are assumed to be firm-specific parameters. The quantity to beestimated is the abnormal performance \L. and the null hypothesis is that i, = 0,that is, there is no abnormal performance.

Generalized least squares (GLS)Let X be the /V X A' variance-covariance matrix of the e^^. X can be estimatedusing the data from the market model regressions. Specifically, let €,^ denote theresidual from the market model for firm t for period T. Then the elements of Xcan be estimated as

co, = l/(T-2)i^,,^^, (2)

where T is the number of periods over which both 4, and ^^ can be computed.1 .ct S represent an estimate of X. S will be nonsingular if T, the number of time-series ob*i£rvations per fum, is greater than N, ihe number of fums in the sam-ple. Nonsingularity of S is a requirement of Gl -S. The GLS estimator of ji. is

fL=l 'S -"e , / ( l 'S - i l ) (3)

and ils estimated asymptotic variance is

var(A) = ( l ' S ~ ' l ) " ' (4)

where e, is the A' X 1 vector of the residuals e-, in tlie event period and 1 is an A'X 1 vector of ones.

To understand better the effect of estimation error on the performance of theGLS, the GLS estimate of the standard error of fi is examined. It is the asymp-totic standard error (l'S~'l)""'''2; cf. equation (4). Assuming that the distur-bances ^ ^ in (1) are normally distributed, it can be shown that in finite samples,this estimator is downward biased. In fact, theorem (2) in Freedman and Peters(1984) implies (hat

) > Var(PLGUS exact) > E[( l 'S- l l ) - l l . (5)S, e.,.ma.ed) > Var(PLGUS. exact

In (5), fXcL. ,,ji^t,^ and t^c,\ji, exact ^^^er to the GLS estimators of ii usingthe estimated and the true residtial covariance matrices S and X, respectively.The asymptotic variance [(l'S ~ ' l ) " ' j is used as an estimate of the true vari-ance of the GLS estimator. Equation (5) shows that the expected value of thisasymptotic variance is smaller than the variance of the exact GLS estimator,which, in tum, is smaller than the true variance of the GLS estimator of ^,.

454 R. Bhushan

Intuitively, the true variance of the GLS estimator is larger than that of theexact GLS estimator since the estimation error in S also contributes to the vari-ance of the GLS estimator of \i, but tbe asymptotic variance assumes that 2 isknown and thus ignores this source of variation. Furthermcffe, the discrepancybetween VarOi^LS exact) ^^ E[(1'S "^l)"^] also increases with the estimationerror in S since ( l 'S~' l)~^ is a ctmcave function of S.

GLS involves estimation of all elements of the residual covariance matrix sothat the ratio of the number of parameters to be estimated to the number ofavailable observations is given by N(N + \)li2NT), which is roughly propor-tional to NIT. An increase in the number of parameters to be estimated com-pared to the number of observations (i.e., in NIT) is likely to increase the estima-tion error in S and cause a more downward bias in the asymptotic standard errorused in GLS. Since estimation error in S is likely to be high when the number ofparameters to be estimated under GLS is quite high compared to the number ofobservations available, imposing plausible restrictions to reduce the number ofestimated parameters has the potential of reducing estimation error. To this end,I consider the following restricted form of GLS.

Generalized least squares—industry basis (GLS-IND)This method is the same as GLS with the only difference being that the contem-poraneous variance-covariance matrix 2 is assumed to be block diagonalinstead of full as in GLS. Firms are assumed to be stacked according to indus-tries, with each block representing one industry. The cross-industry correlationsare assumed to be zero and the within-industry (block) elements of 2 are esti-mated as under GLS. In addition to reducing estimation error, and thus alleviat-ing the downward bias in the asymptotic standard error used, GLS-IND mayoffer other potential advantages over GLS. If most cross-industry correlationsare zero, then taking that fact into account will improve the efficiency of theestimators. Moreover, when X is assumed to be block diagonal, one has to esti-mate fewer parameters than under GLS. Many times GLS is not feasiblebecause of insufficient data while GLS-IND may be feasible.

SimulationsExperimental designThe experiments are conducted using both daily and weekly data. In addition tousing daily data in the simulations, weekly data are also employed since theycan provide a reasonable balance between reducing measurement error inretums (a potential problem in using daily data) and maintaining enough power.Four different sample sizes (10, 20, 50, and 100 firms) are considered for dailydata and three different sample sizes (10,20, and 50 firms) for weekly data. Theexperimental design for the initial experiments is as follows.

Daily dataFor a sample size of N firms, a day is selected at random from the time period

('ross-Seciional l>ependence and (Jeneralizetl Least Squares 455

.Tuly 1, 1963, to December 24, 1986, and denoted day 0. Day 0 and the next fourdays constitute the five separate event days, and the days -250 to +4 composethe estimation period T. All the firms that have CRSP daily return data over theperiod -250 (o +4 constitute the population of potential firms that can form thesample.^ Firms are selected at random without replacement from this populationuntil there are A' firms with no missing returns for each of the event days (days 0through +4) and at least 150 nonmissing returns (wer the estimation interval.For each of these firms, market model parameters as well as the variance-covari-ance matrix of the residuals are estimated over the estimation interval. Firms inthe same two-digit SIC codes are assumed to belong to the same industry whileapplying the GLS-IND method.^ Fven though no abnormal performance isintroduced for any of the event days, each method is used to detect abnormalperformance lor each of those five t;vent ctiys. The whole procedure is repeatedfor 100 indept^ndent trials.

Weekly dataThe procedure followed is similar to that for daily data with the followingchanges. The lime period from which event week 0 is dniwn is from the begin-ning of July 1964 to the end of November 1986. For this study, weekly returnsEire defined as continuously compounded returns over five (trading)-day periods.Week 0 and the following four weeks compose the five separate event weekswhile the estimation interval spans weeks -100 to +4. To be included in thesample, a fum has to have no missing weekly returns for weeks 0 through +4and at least 50 nonmi.ssing weekly returns during the estimation interval.

Results for a random sample of firmsSimulations are conducted using both equally weighted and value-weightedindices. The average beta fw each of the samples considered for both daily andweekly data is close to 1. The cross-firm (irrespective of the firms' industries)correlations in the equally weighted market model residuals are negligible(approximately 0.1 percent for daily data and 0.2 percent for weekly data), whilefor the value-weighted index the corresponding numbers are 1 percent and 3 per-cent, respectively, for daily and weekly data. The magnitudes for within-industrycorrelations are about 1 percent (2 percent) in daily data and about 3 percent (6percent) in weekly data for equally weighted (value-weighted) index.^ Almost allof the within-industry correlations are significant at the 1 percent level, indicatingthat tor a random sample of firms, some within-industry cross-correlation inresidtials remains even when the equally weighted market index is used.

I'or both GLS and GLS-IND methods. Table 1 presents the b(X)tstrap stan-dard deviation of (11, the average asymptotic standard error of |!1L, and the ratio ofthe two. Both the booLstr^ standard deviation and the average are computedacross the 500 trials —five event days (weeks) X 100 trials."^ The numbers arereported for both daily and weekly data for the various sample sizes for theequally weighted index. The results for the value-weighted index are very simi-lar and are, therefore, not reported.

456 R. Bhushan

TABLE 1Comparison of the asymptotic standard error of |iwith the bootstrap standarddeviation of jl for the GLS and GLS-IND methods, with calendar timeclustering of events

Sample size

Daily data102050100

Weekly data102050

Method

GLS

se-est

0.6160.4080.2450.151

1.2550.8190.424

se-bootstrap

0.7280.4680.3290.260

1.6231.0910.916

Ratio

0.8460.8720.7450.581

0.7730.7500.463

GLS-IND

se-est

0.6260.4220.2650.190

1.2990.9050.584

se-bootstrap

0.7230.4500.2970.211

1.5960.9800.702

Ratio

0.8660.9380.8920.900

0.8140.9230.832

Event dates are the same in calendar time for all securities in the sample. For each of the 100replications, a day (week) is selected at random from the 1963-1986 period and designatedevent day (week) 0, and securities composing a sample are also sdected at random (withoutreplacement). se-e.st is the average of the asymptotic standard error of (1, se-bootstrap is thebootstrap standard deviation of (1, and both are reported in percentages. Ratio is the ratio se-est/se-bootstrap. Both se-est and se-bootstrap are computed over 500 trials^—5 event days(we^s)X 100 trials.

To examine whether the standard errors of fl generated by GLS are biased,the average asymptotic standard error of ji, denoted se-est, is compared with thebootstrap standard deviation of pi, denoted se-bootstrap.^' If a method producesan unbiased estimate of the standard error of ji, then the ratios in the table corre-sponding to that method should be close to LO. As an example, a sample size of100 and with daily data, for the GLS method, se-bootstrap is 0.26 percent so thatan estimate of the true standard error of (1 is 0.26 percent. This estimate is muchhigher than the asymptotic standard error used in GLS, which is only 0.151 per-cent on average (se-est). Thus, in this example, the standard error used by GLSis only 58.1 percent of the true standard error's point estimate.

Table 1 suggests that although there appears to be some downward bias inGLS-IND standard errors also, the bias in GLS-based standard errors can besubstantial. 2 The rejection frequencies (not reported) computed using one-tailed tests at the 0.05 level of significance are also consistent with these results.The average rejection frequencies over the five days (weeks) suggest that fra- theGLS method, except for a sample size of 10 using daily data, the mean fre-quency of type I error exceeds the upper limit of the 95 percent interval aroundthe expected rate of 5 percent.'^ For example, for daily data, a sample size of100 finns with 250 time-series observations leads to an empirical rejection Ire-

Ooss-Sectional Dependence and Generalized Least Squares 457

quency of 17.6 percent, more than three times the expected rate of 5 percent. ForGLS-IND method, however, almost all of the rejection frequencies are withinthe 95 percent confidence interval around the expected rate of 5 percent.

The reduction in downward bias through the use of GLS-IND suggestsGLS-IND alleviates some of the problems of tjLS. Thus, imposing reasonablerestrictions on the parameters to reduce the number of estimated parameters canbe useful in eliminating some of the downward bias in asymptotic standard errorthat arises by ignoring estimation error.

Tlie degree of downward bias in GLS appears to increa.se with NIT—which isabout twice the ratio of the number of parameters estimated in GLS to tlie num-ber of available time-series observations. A decrease in the number of time-series observations available per estimated parameter introduces more estima-tion error in the estimated covariance matrix S and increases the downward biasin tbe GLS-asymptotic standard error. To illustrate these points more suc-cinctly. Table 2 presents another set of simulation results on GLS.

Tliese simulations are for weekly dala ;uid for sample sizes of 20 and 50 withUie number of observations in lhe estimation period allowed to vary. Results aretabulated for se-est, se-bootstrap, and the ratios of the two. For a given samplesize, as T increases, the bootstrap standard deviation of the GLS estimator declines.This is consistent with lhe true variance of the CiLS estimator approaching thevariance of the exact GLS estimator as the number of observations increases. Forlarge NITicg > 0.5), estimation errors can increase the UT.ie standard deviation of

TABLE 2Comparison of the asymptotic standard error of p. v 'ith the bootstrap standarddeviation of ft as the number of time-series observations available to estimate

the parameters in GLS is varied

Estimation weeks

Asymptotic standarderror (%). se-estBootstrap standarddeviation (%),se-bootstrapRatio (= se-est/se-bootstrap)

Sample size

20

30

0.50*)

3.080

0.165

50

0730

1.231

0.593

100

1.091

t>-751

50

100

0,424

0.916

0.463

150

0.492

0.833

0.591

200

0.534

0.771

0.693

Event dates are the same in calendar time for all securities in the sample. For each of the100 replications, a week is selected at random and designated event week 0, and securi-ties composing the sample are also selected at random (without replacement). The aver-age of the asymptotic standard error of fi (se-est) and tbe bootstrap standard deviation of0- are computed over 500 trials—5 weeks X 100 trials

458 R. Bhushan

the GLS estimator by a ^ t o r of two or more. For example, when TV = 20, T = 30results in an estimate of tbe bootstr^ standard deviation of 3.08 percent ccxnparcdto 1.09 percent {(xT= 100. An examin^(»i of the row ccsrespcHiding to the aver-age asymptotic standard error (se-est) sbows that the avwage value of the asymp-totic standard errcff decreases as M r increases, thus illustrating that the downwardbias in the asymptotic standard enor increases with N/T.

When the number of available time-series observations per estimated parametersis about 10 (i.e., N/T=: 0.2), the GLS-asymptotic standard tttor is about 75 percentof the true standard ^ror. If thrae are (Hily four times as many time-series observa-tions available as the estimated parameters (i.e., N/T= 0.5), the GLS-asymptoticstandard error is likely to be less than 50 percent of the true standard error.

The rejection frequencies (not tabulated) also corroborate these findings.They increase with N/T, and even when N/T is about 0.2, which corresponds tohaving about 10 time-series observations per estimated parameter, the rejectionfrequencies are twice the nominal rate. The rejection frequencies are muchhigher for higher values of N/T . For example, for a sample size of 20 and 30weeks in the estimaticm period, the average rejection rate is about 32 percent.These results suggest tiiat the lower the number of available time-series observa-tions per estimated parameter, the higher the discrepancies between Var((i,(3Ls

estimated) ^"d Var(tiGLS. exact) ^ ^ WBrii^^is^ exact) ^nd ERl'S -^ly^] will be!Both these factors contribute to the higher rejection frequencies for GLS.

Results for industry clusteringTo examine the case of simultaneous industry and calendar time clustering,another set of simulations is conducted. For these simulations, firms are assumedto belong to the same industry if they have the same three-digit (rath» than two-digit) SIC codes, since within-industry cross-cctfrelation in residuals is likely to behigher for firms in the same three-digit SIC codes than two-digit SIC codes.

To construct a sample, first an event day (week) is selected at random, then afirm is selected at random Crcxn tbe potential population, and the sample is con-structed to include all firms in this firm's industry if the industry has more than10 firms. If an industry has more than 40 firms, then 40 firms are selected atrandom to compose the sample. The procedure is repeated 100 times. The 100different samples do not all have the same number of firms. Sample size is con-strained to lie between 10 and 40 so that meaningful comparisons can be madeacross observations. Tests are conducted using both equally weighted and value-weighted indices. For the simulations with the equally weighted index, themedian sample sizes are 27 for daily data and 28 for weekly data, and with thevalue-weighted index, these figures are 24 and 27, respectively.

Most of the market model statistics are similar for these simulations as for theearlier ones and hence are not reported. The average values of cross-correlationin residuals using tbe equally weighted market index are 2.8 percent fOT dailydata and 5.8 percent for weekly data. Correlations using the value-weightedindex are 2.6 percent for daily data and 5.7 percent for weeWy data. All thesecorrelations are significant at tbe 1 percent level. As expected, these averages

(Voss-Sectional Dependence and Generalized Least Squares 459

are higher than the within-industry cross-correlation averages for the previoussimulations since those averages are based on two-digit SIC codes.

Since all firms in a sample come from the same industry, the GLS-INDmethod reduces to GLS. However, I include another method—restricted GLS(CiLS-rest), in which the cross-correlations for all firm pairs are constrained tobe equal. Based on the evidence in the previous section in which some of theproblems of GLS were eliminated by reducing the number of parameters to beestimated through the use of GLS-IND, a determination of whether otherrestrictions can also be deemed reasonable is useful.

The results using the equally weighted index are reported in Table 3. The cor-respcmding results for the value-weighted index are similar and are, therefore.not reported. The organization of this table is similar to that of the previoustables: it presents a comparison of the average asymptotic standard error of fXwith the bootstrap standard deviation of p..

The results in Table 3 are similar io those in tables 1 and 2. This table reiteratesthat the asymptotic standard error under GLS (and to a lesser extent underGLS-rest) underestimates the true standard error of (1. The evidence on the rejec-tion frequencies is consistent with these results. For daily data, the average rejec-tion irequencies are 5.6 percent and 5 percent for GLS and GLS-^"est, respectively.For weekly data, these frequencies are 11 percent and 7 percent, respectively.

In summary, (he results for the case of simultaneous calendar time and industryclustering confirm the results of the previous section that asymptotic standardenors under GLS are downward biased. Also confmned is the finding that this biascan be reduced by imposing reasonable restrictions on data to reduce the number ofestimated parameters.

TABLE 3Comparison of GI5 and restricted GLS for detecting abnormal performance with indus-try clustering using an equally weighted index as the market index

Daily data Weekly data

GLS CIS-rest

Asymptotic standarderror (%), se-estBootstrap standarddeviation. (%),se-bootstrap

0.483

0.558

Ratio (=se-est/se-bootstrap) 0.866

0.5 ] 6

0.561

0.920

1.005

1.584

0.634

GLS-rest

1 448

1.619

0.894

Event dates are the same in calendar time for all securities in the sample. For each of the 1(K)replications, a day (week) is selected at random from the 1963-986 period and designatedevent day (week) 0, and the sample consists of all securities belonging to a randomlyselected 3-digit SIC code. The average of the asymptotic standard error of fl (se-est) and thebootstrap standard deviation of jl are computed over 500 trials—5 days (weeks) X 1(X) trials,

460 R. Bhushan

Conclusions

The major conclusions of this research can be summarized as follows.Estimation error plays an important role when GLS is used. In tbe contextexamined here, if the number of time-series obso^ations available is less than10 times the number of parameters to be estimated, then GLS is likely to lead torejection frequencies that are considerably higher than the nominal rate. Theperformance of GLS gets worse as NIT ino'eases. The downward bias in theasymptcHic standard errors can be alleviated by using restricted fonns of GLSwhen reasonable restrictions are imposed on the parameters so that the numberof estimated parameters can be considerably reduced. If estimation efficiency isnot an issue, then portfolio methods, owing to their unbiased standard errors,should in general be preferred to GLS-based techniques.

Endnotes1 Cross-sectional dependence can arise in many accounting and finance studies, espe-

cially in those in which the dependent variables are stock returns, some of which comefrom common time periods. For example, studies assessing the impact of regul Uoryevents and SEC or FASB pronouncements (e.g.. Beaver, Christie, and Griffin 1980;Collins, Rozeff, and Dhaliwal 1981; Leftwich 1981, Schipper and Thompson 1983;Hughes and Rides 1984), as well as those examining stock price reactions to account-ing data (e.g.. Ball and Brown 1968; Biddle and Lindahl 1982; Ricks 1982; Beaverand Landsman 1983; Bernard and Ruland 1987) are likely to display cross-sectionaldependence.

2 Phillips (1985) derives the exact finite sample distribution of the GLS estimator. Hisanalysis, however, is not easily applicable in the empirical context (see footnote 4 inBernard 1987).

3 For example, Marais (1986a) finds, for the specific case of the data of Schipper andThompson (1983), that the asymptotic standard errors for GLS-based techniquescould be as little as 0.10 of the bootstr^ standard errors.

4 There is evidence of the seriousness of the finite sample bias problem in other contextsas well (e.g., Bernard and Ruland 1987).

5 Some examples in which there were less than 10 observations available per estimatedparameter are Gibbons (1982), Schipper and Thompson (1983), and Hughes and Ricks(1984).

6 The MVRM method in general allows for firm reactions to vary cross-sectionally dur-ing the event period. If in the true model firm reactions do vary, then one can examinejoint hypotheses about these reactions by using tests developed in Binder (1985) andSchipper and Thompson (1985). In such a situation, GLS would yield biased andinconsistent estimates. These tests, however, generally apply only when the cross-sec-tional constraints are not built into the estimation procedure. If the identical firm reac-tion assumption is a reasonable one, then an explicit recognition of it as a constraintand GLS estimation can improve estimation efficiency.

7 Brown and Warner (1980,1985), Dyckman, Philbrick, and Stephan (1984), and Jain(1986) also use actual returns data in their simulations, but none of these papers exam-ines the performance of GLS since the focus in these papers is not on calendar timeclustering.

8 Use of a two-digit instead of three-digit (Bernard 1987; Collins and Dent 1984) SICcode to define an industry implies broader industry categories. If a narrower (three-digit) definition of an industry is used, then a random sample of firms is unlikely to

Cross-Sectional Dependence and Generalized Least Squares 461

have too many firms coming from common industries, unless the sample size is verylarge, and the GLS-IND method would essentially be reduced to a weighted leastsquares method. Three-digit SIC codes are used later in the paper when the focus is onsimultaneous calendar time and industry clustering.

9 Bernard (1987. Table 1) reports values of 4 percent and 9 percent for daily and weeklyintraindustry correlations, which are larger than the numbers reported here. One rea-son for this difference is my use of two-digit SIC codes as opposed to three-digit SICcodes in Bernard.

lOTlie separate estimates based on the 100 trials for each of event day (week) are similaracross event days (weeks) and are close to those computed for all the 500 trials.

11 'Iliis procedure assumes that for any method, the estimated bootstrap standard devia-tion of p. is a reasonably precise estimate of the true standard error of fl. This assump-tion appears justified since, as noted earlier, the separate estimates of the bootstrapstandard deviation of L for the various event days (weeks) are quite similar across theevent days (weeks) and also are close lo those computed over all the 500 trials.

12'1'hese results corroborate the evidence in Freedman and Peters (1984) and Marais(1986a,1986b) on the downward bias in the .standard errors generated by GLS.

13 Using a normal approximation to the binomial distribution, the general formula for the95 percent confidence interval concerning expected rejection rate can be written as r±] .96'^[pil -p)/N] where ris the expected rejection rate,/> is the probability of rejec-tion, and N is the number of trials. Assuming independence across the five days(weeks). A' equals 500 in this calculation, and the 95 percent confidence interval forthe mean rejection rate over the five days (weekxl with no abnormal performanceintroduced is 5 percent ±2 percent.

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