zOSIPP Discussion Paper : DP-2001-E-005

Small Sample Evidence on the Impact of Generated Variables
in Event Studies
30 November 2001

C.R. McKenzie
Professor, Osaka School of International Public Policy, Osaka University【Keywords】Bias; Event studies; Generated variables; Hypothesis testing; Power; Size.
【Summary】This paper provides some small sample evidence on the appropriateness of ordinary least squares(OLS) and instrumental variable (IV) estimators of the structural equation, and on the appropriate method of
performing hypothesis testing in event studies when generated variables are present. In event studies, the numberof observations used to estimate the auxiliary equations (and compute the generated variables) and the structural
equation can differ quite substantially. In certain circumstances, this means the appropriate estimator of thestructural equation is the IV estimator rather than OLS estimator. Some Monte Carlo suggests that an IV
estimator of the parameters of interest can lead to considerably smaller biases than the biases of the OLS
estimator. Sizes and powers of tests associated with the coefficient of the generated variable do not seem to beaffected by the presence of the generated variable. In contrast, the sizes of tests associated with the constant areconsiderably distorted when the generated variable should be included in the structural equation.

1

Small Sample Evidence on the Impact of Generated
Variables in Event Studies
C.R. McKenzie
Osaka School of International Public Policy, Osaka University, 1-31 Toyonaka, Osaka 560-0043, JAPAN(mckenzie@osipp.osaka-u.ac.jp)

Abstract: This paper provides some small sample evidence on the appropriateness of ordinary least squares(OLS) and instrumental variable (IV) estimators of the structural equation, and on the appropriate method of
performing hypothesis testing in event studies when generated variables are present. In event studies, the numberof observations used to estimate the auxiliary equations (and compute the generated variables) and the structural
equation can differ quite substantially. In certain circumstances, this means the appropriate estimator of thestructural equation is the IV estimator rather than OLS estimator. Some Monte Carlo suggests that an IV
estimator of the parameters of interest can lead to considerably smaller biases than the biases of the OLS
estimator. Sizes and powers of tests associated with the coefficient of the generated variable do not seem to beaffected by the presence of the generated variable. In contrast, the sizes of tests associated with the constant areconsiderably distorted when the generated variable should be included in the structural equation.
Keywords: Bias; Event studies; Generated variables; Hypothesis testing; Power; Size
1.

this means the appropriate estimator of thestructural equation is the instrumental variable (IV)estimator rather than the ordinary least squares(OLS) estimator [see McKenzie and McAleer,
1998]. The purpose of this paper is to provide somesmall sample evidence in an event study context on:(a) the appropriateness of OLS and IV estimators of
the structural equation; and (b) the appropriate
method to take account of heteroscedasticity when
performing hypothesis testing. It is found that thebiases of OLS are quite considerable compared toan IV estimator. The results for hypothesis testingsuggest that the impact of generated variables
differs depending on the parameter involved in the
hypothesis test.

INTRODUCTION

In the typical examples of generated variables in
economics, the presence of generated variables
leads to errors that are serially correlated andheteroscedastic causing estimators ignoring thegenerated variables problem to be inefficient andhave problems with hypothesis testing [Pagan,1986]. To rectify the problems with hypothesis
testing, Smith and McAleer’s [1994] Monte Carloevidence indicates that it is preferable to use test
statistics computed using the known form of thecovariance matrix of the estimators rather thanusing Newey-West’s estimate of the covariancematrix. In event studies, the presence of generated
variables usually only causes heteroscedasticity.
However, the number of observations used toestimate the auxiliary equations (and compute thegenerated variables) and the structural equation candiffer quite substantially. In certain circumstances,

2.

EVENT STUDIESThe equations estimated in a typical event study arebased on the market model. In the estimation2

correlated with the error term given in (5) since

window, this model can be stylised as follows:yitα= β
i + i x it + eit , i = 1,.., N, t = 1,.., T,= ei2 ( x iT +1δ
+E(λBi iT +1)σ(1)

where y it is the rate of return on the ith firm’s_

As T → ∞ ,an error term which is assumed to be independently

In addition,λ2 Zi + eiT +1 ,null hypotheses of interest areδ 1 = 0 and δ2 = 0 ,↓ 0.

iT +1)

iT +1 is heteroscedastic with variance2 ↓σ
As T → ∞ , λE ( iT+1)that is, the announcement at time T+1 has no impact
on the firm’s rate of return.

estimates are then used to rewrite (2) as(3)

2
ei . That is, the form of the( yiT +1 - Ai - Bi x iT +1) / w i =(8)δ 1 / w iδ
+ 2 Bi / w iλ
+ iT +1 / w i .
σThree choices of w i are considered: (A)

ei ;

_

σ ei [1 + (1 / T) + ( x) 2 / V xi ]1 / 2 ; and(B)
iT +1 - x i
σ ei [1 + (1 / T) + ( x δ(C)iT +1 +_
2 - xi )2/ V xi ]1 / 2 . Choice(A) ignores both the heteroscedasticity arising fromthe presence of generated variables in both thedependent and explanatory variables. Choice (B)ignores the heteroscedasticity arising from thepresence of generated variables in the explanatory
variables. OLS applied to (8) with one of these three
w i is referred to as GLS1，GLS2 and GLS3,
respectively. The required estimate of σ ei is

(4)
2).

/ V xi] .(7)Given the heteroscedasticity of the error in (4)，it isnatural to consider a GLS transformation of (4):Equation (3) (or (2)) is referred to as the structural
equation and (1) is referred to as the auxiliary
equation (or first stage model). When Zi isobserved, this is a standard event study model.
However, as McKenzie and McAleer [1998]
observe, quite often the explanatory variables usedin event studies are generated in some way. In thispaper, analysis is focused on the case where= i [see McKenzie and McAleer, 1998, Table 1
Ziβ
for some examples]. Since the explanatory variable
Zi is unobservable, it needs to be estimated say as
Bi . In this case, (3) can be rewritten as
y iT +1 − A i − Bi x iT +1 = δ1 + δ 2 Bi + λ iT +1 ,2

correlation between the regressors and the error in(4), and the degree of the heteroscedasticity can be
expected to increase.

obtain estimates of αi and βi , Ai and Bi . TheseyiT +1 - Ai - Bi x iT +1δ
= 1δ
+ 2 Zi + eiT +1αβ+ ( i - Ai) + ( i - Bi) x iT +1 , i = 1,.., N.

_
2 - xi )heteroscedasticity simplifies greatly as the numberof observations used at the first stage goes toinfinity. Asδ 2 increases in size, both the

Typically, (1) is estimated by OLS for each i toδ

so that E(λ
Bi

2
λ
σ 2
δ
E( iT
+1) = ei [1 + (1 / T ) + ( x iT +1 +where Z i is a characteristic of firm i. In (2), theiT +1 +

V xi → ∞(2)

i = 1,.., N,

λ iT +1 = eiT +1 α
+ ( i - Ai) β+ ( i - Bi) ( x_That is, the correlation disappears as the number ofobservations used at the first stage goes to infinity.

2
ei . In theevent window, it is assumed thatδ δ
yiT +1α= βi + i x iT +1 + 1 +(6)where x i∑
= Tt =1 x it / T, and V xi∑= Tt =1 ( x it - x i )2 .

equity at time t, x it is the rate of return on themarket portfolio at time t for the ith firm, and e it isdistributed with zero mean and varianceσ_
2 - x i ) / V xi ,(5)

obtained from the OLS estimates of （1）, and theIt should be noted that the regressor in (4) will be

3

OLS estimate ofδ2from (4) is used to compute
The market returns, x it , are generated as a first-

GLS3.

order autoregression= x it -1 + vit ,x itρThe correlation between the regressors and the error
term in (4) prompted McKenzie and McAleer[1998] to suggest that it may be more appropriate toestimate (4) using an IV estimator rather than OLS.
The difficulty with IV estimation is finding anappropriate instrument for Bi . In their empiricalwith vit ~ niid (σ0,set atρ = 0.2 andσgenerated from a uniform distribution over therange (0.5, 1.0). In any one experiment, the valuesofβ i σ, ei2 and x it are fixed. Observations on
yit are generated for i=1,..,N according to (1) fort=-(T-1),..,T, and according to (2) with Ziβ= i fort=T+1 assuming the eit are normally distributed.
Observations t=-(T-1),..,0 are used to obtain theestimates ofβ i used as instruments in the_σ 2 [1 + (1 / T) + ( x)2 / V xi]; and(B)ei
iT +1 - x i_

2/ V xi].

2
v = 1.0 to loosely replicate the

distribution over the range (0,1), andσ ei2 are

Variances of the OLS estimator are also computed
assuming homoscedasticity, and heteroscedasticity
with the assumed variance of the errors in (4) being:

2 - xi )2v are

α i =∀
In (1)，i, i are generated from a uniform
0 β

two sets of instruments to give estimators denoted
as IV1-G1, IV1-G2，IV1-G3, IV2-G1, IV2-G2, andIV2-G3, respectively.

σ 2 [1 + (1 / T) + ( x δ(C)ei
iT +1 +

andσdaily returns on the Japanese Nikkei index in 1996.

example, McKenzie and McAleer [1998] use therank of Bi as an instrument. This estimator isdenoted as IV1. Here, an estimate ofβ i based on asample prior to the estimation window is also used,
and this estimator is denoted by IV2. IV is also
applied to the three choices of w i in (8) for the

σ 2;
(A)
ei

2
ρ
v ). The values of

estimator IV2. The observations t=1,..,T are used as
the estimation window. In (2),δ 1 = 0.0 andδ 2

Tests

takes the values 0.0, 0.1, 0.5, 1.0 and 5.0. The
number of observations was varied as N=30，60,
100, and T=30 ， 60. For each experiment, thenumber of replications is 5000. Therefore, themaximum standard errors of the type 1 errors andrejection
frequenciesare[0.5(1-

using these variance estimators are denoted HOM,
HET1,
HET2
and
HET3,
respectively.
Heteroscedastic-consistentestimatesofthevariances of the OLS estimator are also computed
using White’s estimator and tests using thisestimator are denoted WHITE. For IV1 and IV2,
corresponding estimates of the variances are also
used computed to compute test statistics.

0.5)/5000 ]0.5 =0.007. The nominal sizes of all tests
are set equal to 5%.
4. RESULTS

3. MONTE CARLO EXPERIMENT
Table 1 presents estimates of the biases of various
estimators ofδ 2 for various values ofδ 2 , N and

In examining the finite sample performance of
estimators and test statistics used in event studies, it
is quite common to use actual returns data for boththe firm’s return and the market return [see Binder,
1998]. In contrast, in this paper the data are
generated artificially.

T. Results for GLS2, IV1-G2 and IV2-G2 are notpresented because they are very similar to theresults for GLS1, IV1-G1 and IV2-G1, respectively.
The important finding from Table 1 is that thebiases of IV2 and IV2-G1 are considerably smaller

4and 1.0 as T increases, the biases fall; and (c) the
impact of increasing N is mixed. Although notreported in detail to save space, a similar pattern ofbiases is observed for the corresponding estimatorsδ
of 1 .

than the biases for the other estimators whenδ ≠2 0 . Surprisingly, the IV estimator using therank of Bi as an instrument for Bi does not
perform any better than the OLS estimator. For the
OLS and IV1 related estimators, (a) forδ 2 > 0 as
δ 2 increases, the biases increase; (b) forδ 2 =0.5
Table 1: Bias of Estimators ofδ2__________________________________________________________________________________
T=30
OLS
GLS1
GLS3
IV1
IV1-G1 IV1-G3 IV2
IV2-G1 IV2-G3__________________________________________________________________________________
N δ

2

30

0.0
0.1
0.5
1.0

-0.0005-0.0205-0.0912-0.2291-0.0045-0.0238-0.0903-0.2136-0.0049-0.0212-0.0861-0.2028-0.0069-0.0189-0.0896-0.2239-0.0112-0.0220-0.0856-0.1966-0.0110-0.0214-0.0793-0.1887-0.0033
0.0001
0.02160.02050.0010-0.00400.01090.0198-0.0112-0.0203-0.0822-0.1872

60

0.0
0.1
0.5
1.0

0.0598-0.0536-0.1034-0.1798

0.0601-0.0456-0.1117-0.1833

0.0558-0.0445-0.1082-0.1746

0.0618-0.0412-0.0901-0.16440.0631-0.0383-0.0941-0.16900.0617-0.0378-0.0931-0.1639-0.0076
0.00680.00470.0086-0.0069
0.00630.00600.00530.0605-0.0378-0.0937-0.1669

100 0.0 0.0534
0.04880.04680.0432 0.0376 0.0368 0.0008 0.0012
0.0362
0.1 -0.0398 -0.0331 -0.0332 -0.0440 -0.0406 -0.0407 0.0004-0.0012 -0.0403
0.5 -0.0969 -0.1017 -0.0972 -0.0819 -0.0859 -0.0851 0.0037
0.0044-0.0849
1.0 -0.2287 -0.2211 -0.2120 -0.2171 0.2131-0.2061 -0.0021 -0.0002 -0.2079
T=60
N δ 2
60 0.0 -0.0099 -0.0105 -0.0103 -0.0129 -0.0151 -0.0148 -0.0087 -0.0100 -0.0144
0.1 -0.0333 -0.0272 -0.0266 -0.0302 -0.0260 -0.0255 0.0014
0.0007-0.0257
0.5 -0.0641 -0.0630 -0.0610 -0.0599 -0.0618 -0.0603 -0.0010 -0.0005 -0.0603
1.0 -0.1372 -0.1328 -0.1298 -0.1266 -0.1240 -0.1220 -0.0002 -0.0017 -0.1230__________________________________________________________________________________
assuming homoscedasticity (HOM) perform well.
For the OLS estimator, test statistics computed
using information about the known form of theheteroscedasticity (HET2 and HET3) always havetype 1 errors close to their nominal size.

Estimates of the type 1 errors for t-tests of the null
hypothesis ofδ 2 = 0 for the OLS and IV2
estimators using various estimates of the covariancematrix are presented in Table 2. For IV2, in all but
one case the estimated type I errors are notsignificantly different from the nominal size of thetest. Despite the presence of heteroscedastic errors,
t-tests based on an estimate of the covariance matrix

Rejection frequencies of the false null hypothesis ofδ 2 = 0 are presented in Table 3. For most5

combinations of N, T andδ

2,

there is little

difference in the rejection frequencies across theδTable 2: Type 1 Errors for t-tests of the Null Hypothesis

2 = 0 (Nominal size =5%)_________________________________________________________________________________________
OLS
IV2_____________________________________________________________________________T N HOM
White
HET1
HET2
HET3
HOM
White
HET1
HET2
HET3
__________________________________________________________________________________________
30 30 0.0698* 0.0758* 0.0626
0.05340.05080.05820.0726* 0.0606
0.05080.046030 60 0.0576
0.0662* 0.0642* 0.0574
0.05540.04480.06000.06340.0526
0.046830 100 0.0578
0.06200.06340.0556
0.05520.04380.05240.05760.04720.046660 60 0.0522
0.06100.04820.0454 0.0448 0.0518 0.0596 0.0508
0.0476 0.0466__________________________________________________________________________________________
Note: A * indicates the value is significantly different from 0.05.
Table 3: Rejection Frequencies for t-tests of the Null Hypothesisδ 2 = 0(Nominal size =5%)________________________________________________________________________________________
OLS
IV2_____________________________________________________________________________
T=30
HOM
White
HET1
HET2
HET3
HOM
White
HET1
HET2
HET3
N δ 2
30

0.1 0.0654
0.5 0.1328
1.0 0.3768

0.07720.1446
0.40780.06020.12740.4066

0.05360.11180.38900.04980.10480.3746

0.05400.10800.38220.06520.12920.41100.05960.11900.43220.05200.1056
0.41200.0476
0.0916
0.394460

0.1 0.0578
0.5 0.2228
1.0 0.6014

0.06820.23280.6032

0.0616
0.23980.62680.0546
0.21960.6046

0.05320.21280.60180.04880.2092
0.52240.06140.21400.5276

0.05780.22700.54180.05160.20960.5196

0.0466
0.1986
0.5136100 0.1 0.0660
0.5 0.2984
1.0 0.8196
T=60
N δ 2

0.06680.29480.81680.06780.31480.84020.05980.28520.82340.05860.28280.82080.06200.26300.81600.06780.26460.81900.07020.28500.83500.06000.25700.81900.05780.25120.816460

0.1 0.0558
0.0592
0.05640.05180.05120.0606
0.06900.06540.05900.0574
0.5 0.2146
0.21860.21140.2024 0.2000 0.2048 0.2074 0.2040
0.1948 0.1900
1.0 0.6930
0.69600.71000.69680.69080.71980.71540.73260.72040.7156___________________________________________________________________________________________
two estimators and the five estimates of theincreases, the rejection frequencies fall. Again
covariance matrix. This is perhaps a little surprising
increases in N have a mixed impact.
given the large differences in the biases of the OLS
Rejection frequencies for t-tests of the true null
hypothesisδ 1 = 0 when the value ofδ 2 is varied
and IV2 estimators observed in Table 1. As is
expected when δ 2 increases, the rejection
are displayed in Table 4. Since the null hypothesis
frequencies increase. Forδ 2 =0.5 and 1.0 as T
is true, these rejection frequencies should be close6

to the nominal size of the test，0.05. For many of thetest statistics using the OLS estimator，it is foundthat the rejection frequencies are significantly
higher than 0.05. For the IV estimates, the only
estimates of the covariance matrix that consistently
give test statistics with rejection frequencies that are
not significantly different from 0.05 are HOM and
HET3.

5. ACKNOWLEDGEMENTS
The author wishes to thank Michael McAleer for
helpful discussions and acknowledge the financial
support of the Japanese Ministry of Education’s
Science Research Grant No. 12630099.

δ 1 = 0 (Nominal Size =5%)
Table 4: Rejection Frequencies for t-tests of the True Null Hypothesis_________________________________________________________________________________________
OLS
IV2_____________________________________________________________________________
T=30
HOM
White
HET1
HET2
HET3
HOM
White
HET1
HET2
HET3
N δ 2
30 0.01.0
5.0

0.0674* 0.0750*
0.0730* 0.0962*
0.2824* 0.3094*

0.06220.0546
0.05300.06380.0816* 0.0766* 0.0746* 0.0490
0.4120* 0.3916* 0.2948* 0.0594

0.0736* 0.0640
0.05440.05000.0684* 0.0664* 0.0578 0.0492
0.0724* 0.1460* 0.1344* 0.0504

60 0.01.0
5.0

0.06180.0686*0.0728* 0.0842*
0.6646* 0.6998*

0.06140.05180.05020.04580.0816* 0.0732* 0.0710* 0.0516
0.8350* 0.8232* 0.6898* 0.0530

0.06120.0606
0.0582

0.05620.0642*0.1794*0.04760.0046
0.0562 0.0490
0.1676* 0.0466

100 0.0 0.0562
0.06181.0 0.1208* 0.1258*
5.0 0.8422* 0.8658*
T=60
N δ 2

0.0644* 0.0574 0.0568 0.0502
0.1346* 0.1226* 0.1210* 0.0542
0.9410* 0.9324* 0.8684* 0.0430

0.0576
0.0566
0.05600.06000.0660*0.1562*0.0522 0.0506
0.0586
0.05220.1396* 0.0480

60

0.0 0.0572
0.06340.05340.05120.0506
0.0556
0.0622
0.05320.05080.0484
1.0 0.0710* 0.0778* 0.0724* 0.0680* 0.0654* 0.0582 0.0652* 0.0614
0.0570 0.0512
5.0 0.2044* 0.2214* 0.3068* 0.2966* 0.2128* 0.0476 0.0548 0.1000* 0.0958* 0.0430___________________________________________________________________________________________
Note: A * indicates the value is significantly different from 0.05.
their applications, Review of Economic Studies,
53(4), 517-538, 1986.
Smith, J. and M. McAleer, Newey-West covariancematrix estimates for models with generated
regressors, Applied Economics, 26, 635-640,
1994.

6. REFERENCES
Binder, J.J., The event study methodology since1969, Review of Quantitative Finance andAccounting, 11, 111-137, 1998.
McKenzie, C.R. and M. McAleer, An econometricanalysis of event studies in finance, mimeo,
Osaka University, 1998.
Pagan, A., Two stage and related estimators and7