Tbe Review of Financial Studies/v 10 n 1 1997 where x,denotes a vector of firm-specific variables in the pre-event market information set,and 6 is a vector of parameters.Given Equa- tion (1),firm i's private information is given by :=T1-E1(t), (2) where E()=0,with no loss of generality. In what follows next,we model an event as an announcement that each firm chooses to make (or not to make),depending on the nature of its information ti.Our goal is to develop econometric models for the resultant announcement effect. To fix matters,consider a situation in which each firm i must choose between two mutually exclusive and collectively exhaustive alterna- tives on the event date:either the firm must announce the event(E) or the nonevent (NE).Suppose that the firm's decision depends on its information t,as follows: E台t1≥0台：十且x;≥0 (3) NE台t1<0台：+旦x;<0 (4④ The choice model,Equations (3)and (4),reflects that the decision to announce an event is an endogenous choice of firms:here,event E is announced if and only if conditioning information t is "large enough." Otherwise,the "nonevent"NE is announced.6 What do markets learn from firm i's announcement?Given Equa- tions (3)and (4),firm i's choice between E and NE partially reveals its private information vi,and thereby leads markets to form revised ex- pectations about the value of i.The revised expectation,E(C), Ce[E,NE),constitutes the unexpected information on the event date. If there is an information effect (i.e.,the information revealed has a stock-price effect),we should find abnormal returns (say e)to be related to unexpected information.This relationship is linear under the following (jointly sufficient)assumptions. Assumption 4.Risk Neutrality:Investors are risk-neutral towards the event risk. Assumption 5.Linearity:Conditioning information is a linearsignal ofexpected stock return.That is,E(ri)=ni,wbere ri stands for stock return,and i for conditioning information. 6 Conditioning on t being"large enough,"-equivalently,a sample selection bias-characterizes all voluntary corporate events.For instance,takeovers plausibly occur if and only if the personal or corporate gains ()from acquiring are positive;dividend increases are announced only when future earnings (T)are "large enough"to sustain higher dividends,and so on.The fact that t is a function of elements x in the pre-event information set captures the effect that some firms are more likely to announce the event than others. 6
The Review of Financial Studies / v 10 n 1 1997 where xi denotes a vector of firm-specific variables in the pre-event market information set, and θ is a vector of parameters. Given Equation (1), firm i’s private information ψi is given by ψi = τi − E−1(τi), (2) where E−1(ψi) = 0, with no loss of generality. In what follows next, we model an event as an announcement that each firm chooses to make (or not to make), depending on the nature of its information τi. Our goal is to develop econometric models for the resultant announcement effect. To fix matters, consider a situation in which each firm i must choose between two mutually exclusive and collectively exhaustive alternatives on the event date: either the firm must announce the event (E) or the nonevent (N E). Suppose that the firm’s decision depends on its information τi, as follows: E ⇔ τi ≥ 0 ⇔ ψi + θ0 xi ≥ 0 (3) N E ⇔ τi < 0 ⇔ ψi + θ0 xi < 0 (4) The choice model, Equations (3) and (4), reflects that the decision to announce an event is an endogenous choice of firms: here, event E is announced if and only if conditioning information τ is “large enough.” Otherwise, the “nonevent” N E is announced.6 What do markets learn from firm i’s announcement? Given Equations (3) and (4), firm i’s choice between E and N E partially reveals its private information ψi, and thereby leads markets to form revised expectations about the value of ψi. The revised expectation, E (ψi | C), C∈{E, N E}, constitutes the unexpected information on the event date. If there is an information effect (i.e., the information revealed has a stock-price effect), we should find abnormal returns (say ²) to be related to unexpected information. This relationship is linear under the following (jointly sufficient) assumptions. Assumption 4. Risk Neutrality: Investors are risk-neutral towards the event risk. Assumption 5. Linearity: Conditioning information is a linear signal of expected stock return. That is, E(ri | ψi) = πψi, where ri stands for stock return, and ψi for conditioning information. 6 Conditioning on τ being “large enough,” — equivalently, a sample selection bias — characterizes all voluntary corporate events. For instance, takeovers plausibly occur if and only if the personal or corporate gains (τ ) from acquiring are positive; dividend increases are announced only when future earnings (τ ) are “large enough” to sustain higher dividends, and so on. The fact that τ is a function of elements x in the pre-event information set captures the effect that some firms are more likely to announce the event than others. 6
Conditional Metbods in Event Studies Thus,if the event has an information effect,x should be significant in the nonlinear cross-sectional specifications: E(∈：|E)=πE(:IE)=πE(:|巴'x;+1≥0)， (5) and E(e|NE)=πE(1|NE)=πE(:I旦's+:<O), (6) where ei is the event-date abnormal return for firm i.Intuitively,when firm i makes an announcement C,it signals that the expected return, given its information,is E(n C)=E(i C)(via Assumption 5). Under risk neutrality,E(rI C)also equals the expected event-date abnormal return E(C).? If private information v is distributed normally,N(0,o2),the above models may be rewritten as n(e'x:)/a E(∈i|E)=πo N(但'x/o) =πo入E(gx/o), (7) and -n(e'xi/a) E(e:lNE)=πo1-Nex/o) =πo入NE(g'x/o), (8) where n()and N()denote the normal density and distribution,re- spectively,and Ac()denotes the updated expectation of private in- formation v,given the firm's choice Ce[E,NE). Equation(7),our first"conditional"specification for announcement effects,was introduced by Acharya (1988).The model admits to two sets of hypothesis tests: 1.Test for existence of information effect:A test for significance of indicates whether announcement effects (e)are related to the information revealed in the event [Ac()],that is,whether there exists an information effect. 2.Factors explaining announcement effects:A test for significance of coefficients (j=1,2,...,k)identifies from the setx(= 1,2,...,k)those factors that explain the cross-section of announce- ment effects. With risk aversion,we have two cases of potential interest.For firm-specific events of the sort analyzed here,announcement effects will be shifted upwards since (priced)uncertainty is resolved on the event date.In other words,E()>0.An interesting second case relates to events aggregate in character (such as federal interventions in fixed-income markets)and in which event risk is priced.Here Ac(),the "private information"is aggregate,and may be interpreted as a zero mean innovation in a priced APT factor.If the event risk is priced under a linear pricing operator,the risk-premium for the event could be estimated using cross-sectional and time-series data,much as in standard empirical APT studies le.g.,McElroy and Burmeister (1988)l. 7
Conditional Methods in Event Studies Thus, if the event has an information effect, π should be significant in the nonlinear cross-sectional specifications: E(²i | E) = πE(ψi | E) = πE(ψi | θ0 xi + ψi ≥ 0), (5) and E(² | N E) = πE(ψi | N E) = πE(ψi | θ0 xi + ψi < 0), (6) where ²i is the event-date abnormal return for firm i. Intuitively, when firm i makes an announcement C, it signals that the expected return, given its information, is E(ri | C) = πE(ψi | C) (via Assumption 5). Under risk neutrality, E(ri | C) also equals the expected event-date abnormal return E(²i | C). 7 If private information ψi is distributed normally, N (0, σ2), the above models may be rewritten as E(²i | E) = πσ n(θ0 xi)/σ N (θ0 xi/σ ) = πσλE (θ0 xi/σ ), (7) and E(²i | N E) = πσ −n(θ0 xi/σ ) 1 − N (θ0 xi/σ ) = πσλN E (θ0 xi/σ ), (8) where n(·) and N (·) denote the normal density and distribution, respectively, and λC (·) denotes the updated expectation of private information ψ, given the firm’s choice C²{E, N E}. Equation (7), our first “conditional” specification for announcement effects, was introduced by Acharya (1988). The model admits to two sets of hypothesis tests: 1. Test for existence of information effect: A test for significance of π indicates whether announcement effects (²) are related to the information revealed in the event [λC (·)], that is, whether there exists an information effect. 2. Factors explaining announcement effects: A test for significance of coefficients θj (j = 1, 2,..., k) identifies from the set xj (j = 1, 2,..., k) those factors that explain the cross-section of announcement effects. 7 With risk aversion, we have two cases of potential interest. For firm-specific events of the sort analyzed here, announcement effects will be shifted upwards since (priced) uncertainty is resolved on the event date. In other words, E−1(²i) > 0. An interesting second case relates to events aggregate in character (such as federal interventions in fixed-income markets) and in which event risk is priced. Here λC (·), the “private information” is aggregate, and may be interpreted as a zero mean innovation in a priced APT factor. If the event risk is priced under a linear pricing operator, the risk-premium for the event could be estimated using cross-sectional and time-series data, much as in standard empirical APT studies [e.g., McElroy and Burmeister (1988)]. 7
Tbe Review of Financial Studies /v 10 n 1 1997 That the model is consistent with equilibrium follows from (1)risk neutrality towards event risk,and (2)the fact that the ex ante expected abnormal return is zero: Ek=E.NEE(Eil k)*Prob(k) =πo{2E()*N(E'x,/o)+入NE()*[1-N(E'x/o} =0. (9) With this discussion in hand,it is fairly straightforward to develop binary event models under the alternate information structures,As- sumptions 2 and 3.We present these models next and close Section 1 with a discussion on how one might choose between the three spec- ifications in practice. 1.4 Model II:information arrival not known prior to event Equation (7)was based on Assumption 1.under which markets knew ex ante about the arrival of information t.Suppose instead that the framework is Assumption 2:markets do not know ex ante about in- formation arrival.8 We now consider the conditional model for this situation. Given Assumption 2,pre-event expectations about ti were not formed.Hence,ti itself (as opposed to i in the previous section) is firm i's private information.As before,the conditional expectation of private information (here t,given event E,constitutes the infor- mation revealed by E.This variable must be related to announcement effects,linearly so under Assumptions 1 and 2,if the event has an information effect.That is,should be significant in the model E(e:|E)=πE(t:|E)=πE(t;It1≥0) =π[E'x;+入(@'x/o)】, (10) where the last equality is obtained by using ti=x+i.Equa- tion (10)-hereafter,the EMW model-was,in essence,introduced by Eckbo,Maksimovic,and Williams (1990). For some intuition,compare the EMW model,Equation (10),with the Acharya model,Equation (7).The EMW model has the extra term @'x-the unconditional expectation of ti.In the Acharya model,pre- event expectations of t led to its unconditional expectation,() being incorporated into the stock price prior to the event.Here,pre- event expectations were not formed (under Assumption 2);hence, s This is the case,for instance,in takeover announcements involving bidders with no history of acquisitions or targets not previously in play.Here,markets plausibly do not know,prior to the actual takeover announcement,that the acquirer had identified the relevant target and that some announcement related to the acquisition was forthcoming. 8
The Review of Financial Studies / v 10 n 1 1997 That the model is consistent with equilibrium follows from (1) risk neutrality towards event risk, and (2) the fact that the ex ante expected abnormal return is zero: 6k=E,N E E(²i | k) ∗ Prob(k) = πσ{λE (·) ∗ N (θ0 xi/σ ) + λN E (·) ∗ [1 − N (θ0 xi/σ )]} = 0. (9) With this discussion in hand, it is fairly straightforward to develop binary event models under the alternate information structures, Assumptions 2 and 3. We present these models next and close Section 1 with a discussion on how one might choose between the three specifications in practice. 1.4 Model II: information arrival not known prior to event Equation (7) was based on Assumption 1, under which markets knew ex ante about the arrival of information τ . Suppose instead that the framework is Assumption 2: markets do not know ex ante about information arrival.8 We now consider the conditional model for this situation. Given Assumption 2, pre-event expectations about τi were not formed. Hence, τi itself (as opposed to ψi in the previous section) is firm i’s private information. As before, the conditional expectation of private information (here τi), given event E, constitutes the information revealed by E. This variable must be related to announcement effects, linearly so under Assumptions 1 and 2, if the event has an information effect. That is, π should be significant in the model E(²i | E) = πE(τi | E) = πE(τi | τi ≥ 0) = π £ θ0 xi + λE (θ0 xi/σ )¤ , (10) where the last equality is obtained by using τi = θ0 xi + ψi. Equation (10) — hereafter, the EMW model — was, in essence, introduced by Eckbo, Maksimovic, and Williams (1990). For some intuition, compare the EMW model, Equation (10), with the Acharya model, Equation (7). The EMW model has the extra term θ0 xi — the unconditional expectation of τi. In the Acharya model, preevent expectations of τ led to its unconditional expectation, (θ0 xi), being incorporated into the stock price prior to the event. Here, preevent expectations were not formed (under Assumption 2); hence, 8 This is the case, for instance, in takeover announcements involving bidders with no history of acquisitions or targets not previously in play. Here, markets plausibly do not know, prior to the actual takeover announcement, that the acquirer had identified the relevant target and that some announcement related to the acquisition was forthcoming. 8
Conditional Metbods in Event Studies the term 'x,appears in the expression for the abnormal return on the event date.Thus,contrary to a claim in Acharya (1993),we note that the EMW model is not nested within the Acharya model.The two models differ in their assumptions about the underlying information structure. Both models are,in fact,limiting cases of a binary event model based upon Assumption 3.We derive this encompassing specification next and clarify the sense in which it nests the Acharya and EMW models 1.5 Model III:information arrival partially known Suppose now that the information structure is described by Assump- tion 3:markets assess a probability p that information ti has arrived at firm i.Given Equation (1),the stock-price reaction in light of the assessed probability p is given by E(e)=pre'x. Now,if event E does occur,it conveys two pieces of information. First,it confirms that information t has arrived at firm i,that is,the probability of information arrival is raised from p to 1.Second,it conveys via choice model Equations (3)and (4)that 'x+>0. Together,the two pieces of information lead to an announcement effect given by E(e:|E)=π (1-p)'x+a (11) It is easily seen that Equation (11)nests the EMW and Acharya models as the special cases p=0 and p=1,respectively.The traditional event-study methods never obtain as the appropriate specifications, for any value of p. How does one choose between these conditional specifications in practice?The preceding discussion demonstrates that this choice is es- sentially a matter of picking the informational assumption appropriate to one's context.Specifically,the EMW model is probably a good ap- proximation of Equation (11)for nonrepetitive announcements whose timing is not well-identified ex ante.On the other hand,when mar- kets are reasonably certain that some event-related announcement is forthcoming,the Acharya model is appropriate.For intermediate sit- uations,Equation (11)is appropriate.Its practical value is not known and awaits empirical applications,as all received work is based on the EMW and Acharya models. For the discussion that follows,we focus on the Acharya model, Equation (7),(i.e.,the case when p1)since the EMW model,Equa- 9
Conditional Methods in Event Studies the term θ0 xi appears in the expression for the abnormal return on the event date. Thus, contrary to a claim in Acharya (1993), we note that the EMW model is not nested within the Acharya model. The two models differ in their assumptions about the underlying information structure. Both models are, in fact, limiting cases of a binary event model based upon Assumption 3. We derive this encompassing specification next and clarify the sense in which it nests the Acharya and EMW models. 1.5 Model III: information arrival partially known Suppose now that the information structure is described by Assumption 3: markets assess a probability p that information τi has arrived at firm i. Given Equation (1), the stock-price reaction in light of the assessed probability p is given by E−1(²i) = pπθ0 xi. Now, if event E does occur, it conveys two pieces of information. First, it confirms that information τ has arrived at firm i, that is, the probability of information arrival is raised from p to 1. Second, it conveys via choice model Equations (3) and (4) that θ0 xi + ψ > 0. Together, the two pieces of information lead to an announcement effect given by E(²i | E) = π · (1 − p)θ0 xi + σ λE µθ0 xi σ ¶¸ . (11) It is easily seen that Equation (11) nests the EMW and Acharya models as the special cases p = 0 and p = 1, respectively. The traditional event-study methods never obtain as the appropriate specifications, for any value of p. How does one choose between these conditional specifications in practice? The preceding discussion demonstrates that this choice is essentially a matter of picking the informational assumption appropriate to one’s context. Specifically, the EMW model is probably a good approximation of Equation (11) for nonrepetitive announcements whose timing is not well-identified ex ante. On the other hand, when markets are reasonably certain that some event-related announcement is forthcoming, the Acharya model is appropriate. For intermediate situations, Equation (11) is appropriate. Its practical value is not known and awaits empirical applications, as all received work is based on the EMW and Acharya models. For the discussion that follows, we focus on the Acharya model, Equation (7), (i.e., the case when p ≈ 1) since the EMW model, Equa- 9
The Review of Financial Studies /v 10n 1 1997 tion (10),a"truncated regression"specification [see Greene (1993)or Maddala (1983)]has been treated fairly extensively in the economet- ric literature.By contrast,the properties of Equation (7)are not as well-understood:they are related to,but differ in interesting ways from,those of standard "selectivity"models.Hence,we focus on the Acharya model,Equation (7),next and through the rest of this article. 2.On Inferences Via Traditional Methods The conditional specifications developed in Section 1 are quite differ- ent from traditional event-study procedures.How might one interpret inferences via traditional methods in light of this difference? Working in the context of the Acharya model,Equation (7),we make two points.Specifically,we argue that even when event-study data are generated exactly as per Equation(7),(1)the FFJR procedure is well-specified as a test for existence of information effects (i.e.,the hypothesis =0),whether or not any of the factors x explain an- nouncement effects;and (2)the traditional cross-sectional procedure yields regression coefficients proportional to the true cross-sectional parameters a,under conditions to be described shortly.Thus,while traditional techniques are indeed misspecified in the sense discussed before,the implications of such misspecification are probably not as serious as the previous literature [Acharya (1988,1993),Eckbo,Mak- simovic,and Williams(1990)]suggests.Conventional methods do al- low one to conduct significance tests for both a and cross-sectional parameters 6,despite these parameters being potentially estimated inconsistently. It is relatively straightforward to establish that the FFIR procedure may be viewed as a test of the hypothesis m=0.9 The cross-sectional results need some argument though,and we present these in what follows next. 2.1 The conventional cross-sectional procedure The conventional cross-sectional procedure may be written as a test of significance of regression coefficients (B1,...,B)in the linear 9 Take expectations over conditioning factors x in Equation (7).The unconditional (over x)an- nouncement effect,given event E,is given by Er(E)=xaE()l,which is nonzero if and only if is nonzero [since ()Ol.Hence,detecting a nonzero unconditional announcement effect,as in the FFJR procedure,is equivalent to an observation that is nonzero.Variants of the FFJR procedure,such as those introduced in Schipper and Thompson (1983),possess a similar interpretation. 10
The Review of Financial Studies / v 10 n 1 1997 tion (10), a “truncated regression” specification [see Greene (1993) or Maddala (1983)] has been treated fairly extensively in the econometric literature. By contrast, the properties of Equation (7) are not as well-understood: they are related to, but differ in interesting ways from, those of standard “selectivity” models. Hence, we focus on the Acharya model, Equation (7), next and through the rest of this article. 2. On Inferences Via Traditional Methods The conditional specifications developed in Section 1 are quite different from traditional event-study procedures. How might one interpret inferences via traditional methods in light of this difference? Working in the context of the Acharya model, Equation (7), we make two points. Specifically, we argue that even when event-study data are generated exactly as per Equation (7), (1) the FFJR procedure is well-specified as a test for existence of information effects (i.e., the hypothesis π = 0), whether or not any of the factors x explain announcement effects; and (2) the traditional cross-sectional procedure yields regression coefficients proportional to the true cross-sectional parameters θ, under conditions to be described shortly. Thus, while traditional techniques are indeed misspecified in the sense discussed before, the implications of such misspecification are probably not as serious as the previous literature [Acharya (1988, 1993), Eckbo, Maksimovic, and Williams (1990)] suggests. Conventional methods do allow one to conduct significance tests for both π and cross-sectional parameters θ, despite these parameters being potentially estimated inconsistently. It is relatively straightforward to establish that the FFJR procedure may be viewed as a test of the hypothesis π = 0.9 The cross-sectional results need some argument though, and we present these in what follows next. 2.1 The conventional cross-sectional procedure The conventional cross-sectional procedure may be written as a test of significance of regression coefficients (β1,...,βk ) in the linear 9 Take expectations over conditioning factors x in Equation (7). The unconditional (over x) announcement effect, given event E, is given by Ex (²i | E) = πσEx [λE (·)], which is nonzero if and only if π is nonzero [since λE (·) > 0]. Hence, detecting a nonzero unconditional announcement effect, as in the FFJR procedure, is equivalent to an observation that π is nonzero. Variants of the FFJR procedure, such as those introduced in Schipper and Thompson (1983), possess a similar interpretation. 10