Conditional Metbods in Event Studies regression E(E I E)=Bo+B'x=Bo+B1x++Baxe (12) estimated for a sample of firms announcing event E(firms announcing NE are not considered by the conventional procedure),where we have dropped firm-specific subscript i for notational ease.The linear model,Equation (12),is clearly misspecified,given the conditional model,Equation (7).What sort of inferences might it yield,if used anyway?That is,are regression coefficients B related in some way to the true cross-sectional parameters 0 of Equation (7)? Such a relationship does exist and,under fairly weak conditions, it takes a simple form:each linear regression coefficient Bi is propor- tional to true coefficient 0j.Additionally,every B is biased towards zero,relative to 0j. The underlying intuition is illustrated by the following observation: the true slope sy of Equation (7)is attenuated relative to 0.10 Formally, S= 8Ee|E=-0,π80)， (13) oxi where y=E'x/aand6(y)=λE(y)[入e()+yl.Since(1)|π|<1(it is a correlation),and (2)0<(y)<1,1 it follows immediately from Equation (13)that I s<0 I.One might conjecture on this basis that each regression coefficient B,is biased towards zero,relative to 0, with the opposite sign if >0.12 Further,Equation (13)also suggests that downward bias should be greater when 1.is small.Here,announcement effects are less sensitive to conditioning information.Hence,regression coefficients B,should be smaller. 2.8(y)is small.This happens when y='x/a is large [Goldberger (1983)1,that is,for highly anticipated events.Here,little information is contained in firms'announcements of E or resultant abnormal returns Once again,estimated regression coefficients B,should be smaller. Precisely these results obtain when regressors x are multivariate normally distributed.Proposition 1 contains the formal statement. o In a different setting.Lanen and Thompson (1988)also suggest that slope s may be attenuated due to partial anticipation of the event. Interpreting as a correlation involves the normalization var(e)=a.The bounds on ()follow from two properties of the standard normal variable z-(1)E(z Iz>-y)=()is decreasing in y,that is,(y)=-8(y)<0;and (2)var(z>-y)=1-8(y)>0 [Greene (1993)l. The linear regression itself does not necessarily estimate the slope of the nonlinear function (see. e.g.,Stoker (1986),White (1980)L 11
Conditional Methods in Event Studies regression E(² | E) = β0 + β0 x = β0 + β1x1 +···+ βkxk (12) estimated for a sample of firms announcing event E (firms announcing N E are not considered by the conventional procedure), where we have dropped firm-specific subscript i for notational ease. The linear model, Equation (12), is clearly misspecified, given the conditional model, Equation (7). What sort of inferences might it yield, if used anyway? That is, are regression coefficients β related in some way to the true cross-sectional parameters θ of Equation (7)? Such a relationship does exist and, under fairly weak conditions, it takes a simple form: each linear regression coefficient βj is proportional to true coefficient θj . Additionally, every βj is biased towards zero, relative to θj . The underlying intuition is illustrated by the following observation: the true slope sj of Equation (7) is attenuated relative to θj . 10 Formally, sj = ∂E(² | E) ∂xj = −θjπδ(y), (13) where y = θ0 x/σ and δ(y) = λE (y)[λE (y) + y]. Since (1) | π |< 1 (it is a correlation), and (2) 0 < δ(y) < 1,11 it follows immediately from Equation (13) that | sj |<| θj |. One might conjecture on this basis that each regression coefficient βj is biased towards zero, relative to θj , with the opposite sign if π > 0.12 Further, Equation (13) also suggests that downward bias should be greater when 1. | π | is small. Here, announcement effects are less sensitive to conditioning information. Hence, regression coefficients βj should be smaller. 2. δ(y) is small. This happens when y = θ0 x/σ is large [Goldberger (1983)], that is, for highly anticipated events. Here, little information is contained in firms’ announcements of E or resultant abnormal returns. Once again, estimated regression coefficients βj should be smaller. Precisely these results obtain when regressors x are multivariate normally distributed. Proposition 1 contains the formal statement. 10 In a different setting, Lanen and Thompson (1988) also suggest that slope sj may be attenuated due to partial anticipation of the event. 11 Interpreting π as a correlation involves the normalization var(²) = σ. The bounds on δ(y) follow from two properties of the standard normal variable z — (1) E(z | z > −y) = λE (y) is decreasing in y, that is, λ0 E (y) = - δ(y) < 0; and (2) var(z | z > −y) = 1 − δ(y) > 0 [Greene (1993)]. 12 The linear regression itself does not necessarily estimate the slope of the nonlinear function [see, e.g., Stoker (1986), White (1980)]. 11
The Review of Financial Studies /v 10 n 1 1997 Proposition 1.Suppose (1)event E occurs if and only if 0o+ ∑29-y+业>0，and(2 information and abnormal return e are bivariate normal with correlation nt and marginal distributions N(0,1);and regressors (x....,x)are multivariate normal,indepen- dent of3 Then,coefficients (B1.....B)in the linear model,Equa- tion (12),estimated for a sample of firms announcing E are given by (1-R2)(1-t) 月=-π1+1-R2)1-0 =-8πu, (14) wbere 1.t=var(T E)/var(),=0'x+. 2.R2 coefficient of determination ("explained variance")in the population regression oft on (1,x1,...,x). 3u"2 See the Appendix for the proof. To interpret the proportionality factor u,observe that (1)the term (1-R2)represents the variance of t not explained by public informa- tion x,that is,the unexpected component of information t;and (2) term(1-t)proxies the information revealed by event E.Therefore, the product of the two-and hence the term u-represents tbe unexpected component of information t revealed by event E.Another way of viewing this is to consider the fraction of information t that is lost by restricting oneself to event E.Part of information t is lost to (1)pre-event expectations and (2)the nonevent NE.The constant u represents the fraction of information t that remains in event E. Thus,u is small when the event reveals little information;conversely, u is large for highly surprising events.This intuition is formalized in Lemma 1. Lemma 1.Let u be as defined above.Then (1)0<u <1,and (2)u is small wben event E is,on average,bigbly anticipated. See the Appendix for the proof. n the choice model underlying Proposition 1,firms choose between E and Ng based on latent information t.Condition (2)specifies how the latent information maps into stock-return informa- tion since it is the latter that causes observed announcement effects.Multivariate normality of r)is stronger than what is needed for Proposition 1 to obtain.All we need is that the conditional expectation E(I r)be linear in r.Multivariate normality is sufficient,though not necessary for this condition to hold.Finally,note that while firms have two choices (E or NE)in the event modeled here,Proposition 1 also applies to events in which each announcing firm has more than two choices-such as dividend announcements,wherein firms have three choices (increase, keep unchanged,or decrease dividends). 12
The Review of Financial Studies / v 10 n 1 1997 Proposition 1. Suppose (1) event E occurs if and only if θ0 + Pk j=1 θjxj + ψ > 0; and (2) information ψ and abnormal return ² are bivariate normal with correlation π and marginal distributions N(0,1); and regressors (x1,..., xk ) are multivariate normal, independent of ψ. 13 Then, coefficients (β1,...,βk ) in the linear model, Equation (12), estimated for a sample of firms announcing E are given by βj = −θjπ (1 − R2)(1 − t) t + (1 − R2)(1 − t) = −θjπµ, (14) where 1. t = var(τ | E)/var(τ ), τ = θ0 x + ψ. 2. R2 = coefficient of determination (“explained variance”) in the population regression of τ on (1, x1,..., xk ). 3. µ = (1−R2)(1−t) t+(1−R2)(1−t) See the Appendix for the proof. To interpret the proportionality factor µ, observe that (1) the term (1−R2) represents the variance of τ not explained by public information x, that is, the unexpected component of information τ ; and (2) term (1 − t) proxies the information revealed by event E. Therefore, the product of the two — and hence the term µ — represents the unexpected component of information τ revealed by event E. Another way of viewing this is to consider the fraction of information τ that is lost by restricting oneself to event E. Part of information τ is lost to (1) pre-event expectations and (2) the nonevent NE. The constant µ represents the fraction of information τ that remains in event E. Thus, µ is small when the event reveals little information; conversely, µ is large for highly surprising events. This intuition is formalized in Lemma 1. Lemma 1. Let µ be as defined above. Then (1) 0 <µ< 1, and (2) µ is small when event E is, on average, highly anticipated. See the Appendix for the proof. 13 In the choice model underlying Proposition 1, firms choose between E and N E based on latent information τ . Condition (2) specifies how the latent information maps into stock-return information since it is the latter that causes observed announcement effects. Multivariate normality of (x, τ ) is stronger than what is needed for Proposition 1 to obtain. All we need is that the conditional expectation E(x | τ ) be linear in τ . Multivariate normality is sufficient, though not necessary for this condition to hold. Finally, note that while firms have two choices (E or N E) in the event modeled here, Proposition 1 also applies to events in which each announcing firm has more than two choices — such as dividend announcements, wherein firms have three choices (increase, keep unchanged, or decrease dividends). 12
Conditional Metbods in Event Studies With these results in hand.one can readily establish useful com- parative statics about regression coefficients B,: Downward bias in Bi's.This is an immediate consequence of 0<u<1 (Lemma 1),<1,and Equation (14);together,these imply that IBI<0 l. .Opposite Sign.Each Bi is signed opposite to 0j,provided >0,as seen from Equation (14).To understand this result,note that 0;reflects the marginal impact of an increase in regressor x on the probability of event E,while B;reflects the marginal impact on the announcement effect associated with E.Since an increase in the probability of event E decreases the expected announcement effect upon announcing E ifπ>0，0,and Bi have the opposite sign whenπ>0. More attenuation wben n is small.This follows directly from Equation (14). More attenuation wben events are bigbly anticipated.For highly anticipated events,u is small,from part 2 of Lemma 1.From Propo- sition 1,this implies that B,I is small. Summarizing,Proposition 1 has the interesting implication that the traditional cross-sectional procedure may be used for cross-sectional inferences in event studies.Specifically,a statistical test for significance of regression coefficient B,(=1,...,k),is equivalent to a test for significance of the corresponding cross-sectional parameter e,of the conditional model. However,for practical purposes,two questions remain.One,while Proposition 1 provides an interpretation of the linear regression co- efficients,are the usual OLS standard errors appropriate for use in significance tests?Second,how robust is Proposition 1 to the assump- tion that regressors x are multivariate normal?Simulation evidence needs to be developed on these issues. 3.Issues in Choosing Event-Study Methodology Section 2 suggests that under certain conditions,both conditional and traditional methods are valid means of inference.How might one choose between the two approaches in practice?We address this issue in the context of cross-sectional inferences,as conditional methods are likely to be useful only when cross-sectional hypotheses are being tested.14 One's choice between the two approaches would depend primarily on the performance of each method (i.e.,the likelihood of making 4Simulation evidence on the FFJR procedure (reported in earlier versions of this article)attest to this point.These results are available upon request. 13
Conditional Methods in Event Studies With these results in hand, one can readily establish useful comparative statics about regression coefficients βj : • Downward bias in βj ’s. This is an immediate consequence of 0 <µ< 1 (Lemma 1), | π |< 1, and Equation (14); together, these imply that | βj |≤| θj |. • Opposite Sign. Each βj is signed opposite to θj , provided π > 0, as seen from Equation (14). To understand this result, note that θj reflects the marginal impact of an increase in regressor xj on the probability of event E, while βj reflects the marginal impact on the announcement effect associated with E. Since an increase in the probability of event E decreases the expected announcement effect upon announcing E if π > 0, θj and βj have the opposite sign when π > 0. • More attenuation when | π | is small. This follows directly from Equation (14). • More attenuation when events are highly anticipated. For highly anticipated events, µ is small, from part 2 of Lemma 1. From Proposition 1, this implies that | βj | is small. Summarizing, Proposition 1 has the interesting implication that the traditional cross-sectional procedure may be used for cross-sectional inferences in event studies. Specifically, a statistical test for significance of regression coefficient βj, (j = 1,..., k), is equivalent to a test for significance of the corresponding cross-sectional parameter θj of the conditional model. However, for practical purposes, two questions remain. One, while Proposition 1 provides an interpretation of the linear regression coefficients, are the usual OLS standard errors appropriate for use in significance tests? Second, how robust is Proposition 1 to the assumption that regressors x are multivariate normal? Simulation evidence needs to be developed on these issues. 3. Issues in Choosing Event-Study Methodology Section 2 suggests that under certain conditions, both conditional and traditional methods are valid means of inference. How might one choose between the two approaches in practice? We address this issue in the context of cross-sectional inferences, as conditional methods are likely to be useful only when cross-sectional hypotheses are being tested.14 One’s choice between the two approaches would depend primarily on the performance of each method (i.e., the likelihood of making 14 Simulation evidence on the FFJR procedure (reported in earlier versions of this article) attest to this point. These results are available upon request. 13