Conditional Methods in Event Studies and an Equilibrium Justification for Standard Event-Study Procedures N.R.Prabhala Yale University The literature on conditional event-study metb- ods criticizes standard event-study procedures as being misspecified ifevents are voluntary and investors are rational.We argue,bowever,tbat standard procedures (1)lead to statistically valid inferences,under conditions described in this article;and (2)are often a superior means ofinference,even wben event-study data are gen- erated exactly as per a class of rational expec- tations specifications introduced by the condi- tional metbods literature.Our results provide an equilibrium justification for traditional event- study metbods,and we suggest bow tbese simple procedures may be combined witb conditional metbods to improve statistical power in event studies. I am grateful to Stephen Brown,my dissertation chairman,for stimulating my interest in the topic,his guidance,and numerous valuable suggestions. Thanks are also due to Franklin Allen (the executive editor),Yakov Ami- hud,Mitchell Berlin,Silverio Foresi,Robert Hansen,Kose John,L.Misra Robert Whitelaw,and especially Bent Christensen,William Greene,Chester Spatt (the editor),and an anonymous referee whose extensive feedback has greatly improved the article.I have also benefitted from discussions with my colleagues at NYU,and from comments of seminar participants at var- ious universities:British Columbia,Columbia,Cornell,Georgia Institute of Technology,Michigan,New York University,Purdue,Rutgers,Strathclyde, Universiry of California,Los Angeles,Virginia Polytechnic Institute,and Yale.Any errors that remain are solely mine.Address correspondence to N.R.Prabhala,School of Management,Yale University,135 Prospect Street, New Haven,CT 06520. The Review of Financial Studies Spring 1997 Vol.10.No.1,pp.1-38 C 1997 The Review of Financial Studies 0893-9454/97/$1.50
Conditional Methods in Event Studies and an Equilibrium Justification for Standard Event-Study Procedures N. R. Prabhala Yale University The literature on conditional event-study methods criticizes standard event-study procedures as being misspecified if events are voluntary and investors are rational. We argue, however, that standard procedures (1) lead to statistically valid inferences, under conditions described in this article; and (2) are often a superior means of inference, even when event-study data are generated exactly as per a class of rational expectations specifications introduced by the conditional methods literature. Our results provide an equilibrium justification for traditional eventstudy methods, and we suggest how these simple procedures may be combined with conditional methods to improve statistical power in event studies. I am grateful to Stephen Brown, my dissertation chairman, for stimulating my interest in the topic, his guidance, and numerous valuable suggestions. Thanks are also due to Franklin Allen (the executive editor), Yakov Amihud, Mitchell Berlin, Silverio Foresi, Robert Hansen, Kose John, L. Misra, Robert Whitelaw, and especially Bent Christensen, William Greene, Chester Spatt (the editor), and an anonymous referee whose extensive feedback has greatly improved the article. I have also benefitted from discussions with my colleagues at NYU, and from comments of seminar participants at various universities: British Columbia, Columbia, Cornell, Georgia Institute of Technology, Michigan, New York University, Purdue, Rutgers, Strathclyde, University of California, Los Angeles, Virginia Polytechnic Institute, and Yale. Any errors that remain are solely mine. Address correspondence to N. R. Prabhala, School of Management, Yale University, 135 Prospect Street, New Haven, CT 06520. The Review of Financial Studies Spring 1997 Vol. 10, No. 1, pp. 1–38 °c 1997 The Review of Financial Studies 0893-9454/97/$1.50
The Review of Financial Studies /v 10 n 1 1997 Event studies are widely used to study the information content of corporate events.Such studies typically have two purposes:(1)to test for the existence of an "information effect"(i.e.,the impact of an event on the announcing firm's value)and to estimate its magnitude,and (2)to identify factors that explain changes in firm value on the event date. To test for the existence of an information effect,empirical finance has primarily employed the technique developed in Fama,Fisher, Jensen,and Roll (1969)(referred to as FFJR hereafter).FFJR suggest that if an event has an information effect,there should be a nonzero stock-price reaction on the event date.Thus,inference is based on the statistical significance of the average announcement effect for a sample of firms announcing the event in question.The FFJR test is usually followed by a linear regression of announcement effects on a set of firm-specific factors to identify those factors that explain the cross-section of announcement effects.Most event studies in the applied literature have been based on the above methods.2 However,recent literature on conditional event-study methods [Acharya (1988,1993),Eckbo,Maksimovic,and Williams (1990)]ar- gues that the traditional methods are misspecified in a rational ex- pectations context.Briefly,the argument is that corporate events are voluntary choices of firms and are typically initiated when firms come to possess information not fully known to markets.The unexpected portion of such information should determine the stock-price reaction to the event. When events are modeled in this manner within simple equilib- rium settings,the resulting specifications are typically nonlinear cross- sectional regressions3 that bear little resemblance to the simple mod- els conventionally used in event studies.Hence,it has been suggested that the conventional methods are misspecified and lead to unreliable inferences,implying that such methods should not be used in prac- tice.More generally,this debate does raise the important issue that though the standard event-study procedures have been widely used in empirical work,little is understood about their consistency and power Announcement effect (or abnormal return)denotes the excess of the actual event-date stock return over the unconditional expected return for the stock.The latter is usually estimated via the market model,calibrated on pre-event data [see Brown and Warner(1985)for a more complete discussionl. 2A partial list of applications includes studies of (1)equity and debt issues [Asquith and Mullins (1986),Eckbo (1986)1,(2)timing of equiry issues [Korajczyk,Lucas,and McDonald (1991)1,(3) takeovers [Asquith,Bruner,and Mullins (1983),(4)dividends [Bajaj and Vijh (1990),and (5)stock repurchases [Vermaelen (1984). 3The nonlinearity stems from the endogeneity of events.Endogeneiry truncates the statistical dis- tribution of announcement effects. 2
The Review of Financial Studies / v 10 n 1 1997 Event studies are widely used to study the information content of corporate events. Such studies typically have two purposes: (1) to test for the existence of an “information effect” (i.e., the impact of an event on the announcing firm’s value) and to estimate its magnitude, and (2) to identify factors that explain changes in firm value on the event date. To test for the existence of an information effect, empirical finance has primarily employed the technique developed in Fama, Fisher, Jensen, and Roll (1969) (referred to as FFJR hereafter). FFJR suggest that if an event has an information effect, there should be a nonzero stock-price reaction on the event date. Thus, inference is based on the statistical significance of the average announcement effect1 for a sample of firms announcing the event in question. The FFJR test is usually followed by a linear regression of announcement effects on a set of firm-specific factors to identify those factors that explain the cross-section of announcement effects. Most event studies in the applied literature have been based on the above methods.2 However, recent literature on conditional event-study methods [Acharya (1988, 1993), Eckbo, Maksimovic, and Williams (1990)] argues that the traditional methods are misspecified in a rational expectations context. Briefly, the argument is that corporate events are voluntary choices of firms and are typically initiated when firms come to possess information not fully known to markets. The unexpected portion of such information should determine the stock-price reaction to the event. When events are modeled in this manner within simple equilibrium settings, the resulting specifications are typically nonlinear crosssectional regressions3 that bear little resemblance to the simple models conventionally used in event studies. Hence, it has been suggested that the conventional methods are misspecified and lead to unreliable inferences, implying that such methods should not be used in practice. More generally, this debate does raise the important issue that though the standard event-study procedures have been widely used in empirical work, little is understood about their consistency and power 1 Announcement effect (or abnormal return) denotes the excess of the actual event-date stock return over the unconditional expected return for the stock. The latter is usually estimated via the market model, calibrated on pre-event data [see Brown and Warner (1985) for a more complete discussion]. 2 A partial list of applications includes studies of (1) equity and debt issues [Asquith and Mullins (1986), Eckbo (1986)], (2) timing of equity issues [Korajczyk, Lucas, and McDonald (1991)], (3) takeovers [Asquith, Bruner, and Mullins (1983)], (4) dividends [Bajaj and Vijh (1990)], and (5) stock repurchases [Vermaelen (1984)]. 3 The nonlinearity stems from the endogeneity of events. Endogeneity truncates the statistical distribution of announcement effects. 2
Conditional Metbods in Event Studies in rational expectations settings,such as those underlying conditional methods. This article has three purposes.First,we present a simple exposition of conditional methods that focuses on their economic content.We show that all conditional models have essentially the same economic intuition,and derive all received models within a common framework that reflects this perspective.This synthesis reconciles different speci- fications proposed in the literature,clarifies their shared intuition,and suggests how one might choose between or extend such models in practice. Our second point is that while traditional event-study techniques are indeed misspecified in the conditional methods context,they still lead to valid inferences,under certain statistical conditions described in this article.Specifically,even when event-study data are generated exactly as per conditional models of the sort introduced by Acharya (1988),(1)the FFJR procedure remains a well-specified test for de- tecting the existence of information effects;and (2)the conventional cross-sectional procedure yields parameter estimates proportional to the true conditional model parameters,under the conditions men- tioned before.The proportionality factor has a simple interpretation in terms of the informational parameters of the event.These results provide,for the first time,an equilibrium justification for the proce- dures conventionally used to conduct event studies. Finally,if both traditional and conditional methods lead to equiv- alent inferences,how does one choose between the two in practice? Working in the context of the conditional model proposed by Acharya (1988),we develop simulation evidence on this issue.Our evidence suggests that one's choice would depend mainly on whether one has, besides the usual event-study data,an additional sample of"nonevent" firms,that is,firms that were partially anticipated to announce but did not announce the event in question.If such nonevent data are avail- able,conditional methods are powerful means of conducting event studies and may be implemented effectively using a simple"two-step" estimator.Absent nonevent data,conditional methods appear to offer little value relative to traditional procedures. This article is organized along the above lines.Section 1 presents conditional methods for event studies.Section 2 presents and dis- cusses the main analytic results,regarding the equivalence of infer- ences via conditional and traditional event-study methods.Section 3 motivates the question of choosing between the two approaches,and Section 4 outlines the structure of the simulations conducted to ad- dress this question.Simulation results are presented in Section 5,and Section 6 offers concluding comments. 3
Conditional Methods in Event Studies in rational expectations settings, such as those underlying conditional methods. This article has three purposes. First, we present a simple exposition of conditional methods that focuses on their economic content. We show that all conditional models have essentially the same economic intuition, and derive all received models within a common framework that reflects this perspective. This synthesis reconciles different speci- fications proposed in the literature, clarifies their shared intuition, and suggests how one might choose between or extend such models in practice. Our second point is that while traditional event-study techniques are indeed misspecified in the conditional methods context, they still lead to valid inferences, under certain statistical conditions described in this article. Specifically, even when event-study data are generated exactly as per conditional models of the sort introduced by Acharya (1988), (1) the FFJR procedure remains a well-specified test for detecting the existence of information effects; and (2) the conventional cross-sectional procedure yields parameter estimates proportional to the true conditional model parameters, under the conditions mentioned before. The proportionality factor has a simple interpretation in terms of the informational parameters of the event. These results provide, for the first time, an equilibrium justification for the procedures conventionally used to conduct event studies. Finally, if both traditional and conditional methods lead to equivalent inferences, how does one choose between the two in practice? Working in the context of the conditional model proposed by Acharya (1988), we develop simulation evidence on this issue. Our evidence suggests that one’s choice would depend mainly on whether one has, besides the usual event-study data, an additional sample of “nonevent” firms, that is, firms that were partially anticipated to announce but did not announce the event in question. If such nonevent data are available, conditional methods are powerful means of conducting event studies and may be implemented effectively using a simple “two-step” estimator. Absent nonevent data, conditional methods appear to offer little value relative to traditional procedures. This article is organized along the above lines. Section 1 presents conditional methods for event studies. Section 2 presents and discusses the main analytic results, regarding the equivalence of inferences via conditional and traditional event-study methods. Section 3 motivates the question of choosing between the two approaches, and Section 4 outlines the structure of the simulations conducted to address this question. Simulation results are presented in Section 5, and Section 6 offers concluding comments. 3
Tbe Review of Financial Studies/v 10 n 1 1997 1.On Conditional Methods Section 1 develops conditional models for event studies.The main point made here is that all conditional models have essentially the same economic intuition:they relate announcement effects to the un- expected information revealed in events.While the notion of relating announcement effects to unexpected information is not new,we show here that it is the common theme that underlies all conditional models We demonstrate that all received models may be derived in terms of this framework,and that the models differ only because they make dif- ferent assumptions about the information structure underlying events. The exposition proceeds as follows.Section 1.1 opens with a dis- cussion of the intuition underlying conditional methods.Section 1.2 discusses alternative ways of modeling the information structure in events,and Sections 1.3 through 1.5 develop econometric models for announcement effects for each of these information structures. 1.1 Intuition underlying conditional methods To begin,note the potential dichotomy between the fact of an event and the information it reveals.For instance.the event "takeover"is plausibly less surprising for a bidder with announced acquisition pro- grams than for a bidder with no history of acquisitions.Similarly,the event"dividend increase"is less surprising for a firm with an unusually good spell of earnings than for a firm with flat or declining earnings. Thus,a given event may convey less information for some firms and more for others.Further,it should be the unexpected information revealed in events that causes the stock-price changes around event dates.4 This discussion suggests the following empirical procedure for car- rying out event studies:(1)estimate for each firm the unexpected information that the event reveals:(2)compute the cross-sectional correlation between information and abnormal return and test for its significance.A nonzero correlation would indicate that abnormal return is systematically related to information revealed in the event (i.e.,there exists an information effect).Conversely,zero correlation implies lack of an information effect.This intuition underlies every conditional specification analyzed in this article. Central to the conditional paradigm is the notion of"information revealed in events.Next,we discuss how this might be modeled in the event-study context. Malatesta and Thompson (1985),Thompson (1985),and Chaplinsky and Hansen (1993)also recognize the role of partial anticipation of events and examine its implications for event studies based on FFJR-style procedures
The Review of Financial Studies / v 10 n 1 1997 1. On Conditional Methods Section 1 develops conditional models for event studies. The main point made here is that all conditional models have essentially the same economic intuition: they relate announcement effects to the unexpected information revealed in events. While the notion of relating announcement effects to unexpected information is not new, we show here that it is the common theme that underlies all conditional models. We demonstrate that all received models may be derived in terms of this framework, and that the models differ only because they make different assumptions about the information structure underlying events. The exposition proceeds as follows. Section 1.1 opens with a discussion of the intuition underlying conditional methods. Section 1.2 discusses alternative ways of modeling the information structure in events, and Sections 1.3 through 1.5 develop econometric models for announcement effects for each of these information structures. 1.1 Intuition underlying conditional methods To begin, note the potential dichotomy between the fact of an event and the information it reveals. For instance, the event “takeover” is plausibly less surprising for a bidder with announced acquisition programs than for a bidder with no history of acquisitions. Similarly, the event “dividend increase” is less surprising for a firm with an unusually good spell of earnings than for a firm with flat or declining earnings. Thus, a given event may convey less information for some firms and more for others. Further, it should be the unexpected information revealed in events that causes the stock-price changes around event dates.4 This discussion suggests the following empirical procedure for carrying out event studies: (1) estimate for each firm the unexpected information that the event reveals; (2) compute the cross-sectional correlation between information and abnormal return and test for its significance. A nonzero correlation would indicate that abnormal return is systematically related to information revealed in the event (i.e., there exists an information effect). Conversely, zero correlation implies lack of an information effect. This intuition underlies every conditional specification analyzed in this article. Central to the conditional paradigm is the notion of “information revealed in events.” Next, we discuss how this might be modeled in the event-study context. 4 Malatesta and Thompson (1985), Thompson (1985), and Chaplinsky and Hansen (1993) also recognize the role of partial anticipation of events and examine its implications for event studies based on FFJR-style procedures. 4
Conditional Metbods in Event Studies 1.2 Specifying the information structure As argued before,events reveal the information that their announce- ment is conditioned on.Suppose that this information consists of a variable ti,which arrives at firm i on an information arrival date. Information ti is subsequently revealed to markets,via the event,on an event date. What do markets learn from the revelation of t?Clearly,this de- pends on what markets knew,prior to the actual event date,about the arrival of information t;at firm i.Here,we allow for three possibilities: Assumption 1.Markets know,prior to the event,that the event-related information t;bas arrived at firm i (but not its exact content). Assumption 2.Markets do not know,prior to the event,that infor- mation ti bas arrived at firm i. Assumption 3.Markets assess a probability p (0,1)that informa- tion ti bas arrived at firm i. Under Assumption 1,information arrival is common knowledge prior to the event;under Assumption 2,markets do not know about information arrival prior to the event-date.Finally,Assumption 3 is the encompassing case that permits markets to make probabilistic assessments about information arrival.5 Each assumption leads to a different econometric specification for announcement effects,as we show below. For expositional ease and because previous work has been based on Assumptions 1 and 2,we first develop the methodology under Assumptions 1 and 2,in Sections 1.3 and 1.4.We then present an encompassing specification,based on Assumption 3,in Section 1.5. 1.3 Model I:information arrival known prior to event We begin by making Assumption 1:that markets know,prior to the event,that the event-related information t;has arrived at firm i.In general,this leads markets to form expectations about ti.Suppose these expectations are given by E1(t)=Ex1= (1) =1 5 The following example illustrates the distinction between the three assumptions.Consider the event "takeover"and suppose takeovers occur if and only if the acquirer-bidder synergy (T)is positive.Assumption 1 implies that markets always know,prior to each takeover announcement, that the bidder had identified the target in question.The only uncertainty is whether t is positive or not.In contrast,Assumption 2 implies that markets do not know,prior to each announcement, that the target had been identified.Under Assumption 3,markets assign probabiliry ps(0,1)that the target had been identified. 5
Conditional Methods in Event Studies 1.2 Specifying the information structure As argued before, events reveal the information that their announcement is conditioned on. Suppose that this information consists of a variable τi, which arrives at firm i on an information arrival date. Information τi is subsequently revealed to markets, via the event, on an event date. What do markets learn from the revelation of τi? Clearly, this depends on what markets knew, prior to the actual event date, about the arrival of information τi at firm i. Here, we allow for three possibilities: Assumption 1. Markets know, prior to the event, that the event-related information τi has arrived at firm i (but not its exact content). Assumption 2. Markets do not know, prior to the event, that information τi has arrived at firm i. Assumption 3. Markets assess a probability p ∈ (0, 1) that information τi has arrived at firm i. Under Assumption 1, information arrival is common knowledge prior to the event; under Assumption 2, markets do not know about information arrival prior to the event-date. Finally, Assumption 3 is the encompassing case that permits markets to make probabilistic assessments about information arrival.5 Each assumption leads to a different econometric specification for announcement effects, as we show below. For expositional ease and because previous work has been based on Assumptions 1 and 2, we first develop the methodology under Assumptions 1 and 2, in Sections 1.3 and 1.4. We then present an encompassing specification, based on Assumption 3, in Section 1.5. 1.3 Model I: information arrival known prior to event We begin by making Assumption 1: that markets know, prior to the event, that the event-related information τi has arrived at firm i. In general, this leads markets to form expectations about τi. Suppose these expectations are given by E−1(τi) = θ0 xi = Xn j=1 θjxij (1) 5 The following example illustrates the distinction between the three assumptions. Consider the event “takeover” and suppose takeovers occur if and only if the acquirer-bidder synergy (τ ) is positive. Assumption 1 implies that markets always know, prior to each takeover announcement, that the bidder had identified the target in question. The only uncertainty is whether τ is positive or not. In contrast, Assumption 2 implies that markets do not know, prior to each announcement, that the target had been identified. Under Assumption 3, markets assign probability p ε (0, 1) that the target had been identified. 5
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+: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. 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
