Selection bias and econometric remedies in accounting and finance research

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1 Selecton bas and econometrc remedes n accountng and fnance research Jennfer Wu Tucker Fsher School of Accountng Warrngton College of Busness Admnstraton Unversty of Florda (352) (Offce) (352) (Fax) E-mal: January 2011 Forthcomng n Journal of Accountng Lterature I thank Fard AtSahla, Joel Demsk, Mke Donohoe, Mark Flannery, Bll Greene, Marcus Krk, Tom Omer, Doug Schroeder, and Senyo Tse for dscusson and comments. I thank the Lucano Prda, Sr. Term Professorshp Foundaton for fnancal support. I thank Wayne Losano for edtoral assstance. All errors are mne.

2 Selecton bas and econometrc remedes n accountng and fnance research ABSTRACT Whle managers accountng and fnancal decsons are, for many, fascnatng topcs, selecton bas poses a serous challenge to researchers estmatng the decsons effects usng non-expermental data. Selecton bas potentally occurs because managers decsons are non-random and the outcomes of choces not made are never observable. Selecton bas due to observables arses from sample dfferences that researchers can observe but fal to control. Selecton bas due to unobservables arses from the unobservable and thus uncontrolled sample dfferences that affect managers decsons and ther consequences. In ths artcle I revew two econometrc tools developed to mtgate these bases the propensty score matchng (PSM) method to mtgate selecton bas due to observables and the Heckman nverse-mlls-rato (IMR) method to address selecton bas due to unobservables and dscuss ther applcatons n accountng and fnance research. The artcle has four takeaways. Frst, researchers should select the correct method to allevate potental selecton bas: the PSM method mtgates selecton bas due to observables, but does not allevate selecton bas due to unobservables. Second, n applyng PSM researchers are advsed to restrct ther nferences to frms whose characterstcs can be found n both the sample and control groups. Thrd, the IMR method, though popular, s lmted to stuatons n whch the choces are bnary, the outcomes of choces are modeled n a lnear regresson, and the unobservables n the choce and outcome models follow multvarate normal dstrbuton. Researchers can overcome these constrants by usng full nformaton maxmum lkelhood estmaton. Last, when the IMR method s used, specal attenton should be pad to the formulas n calculatng IMRs. The artcle also calls for researchers attenton to other approaches to evaluatng the effects of managers decsons. Keywords: selecton bas, propensty score matchng, nverse mlls rato, Heckman model

3 1. Introducton Many key decsons made at a frm can be categorzed as choces. The decsons range from operatng, nvestng, and fnancng to fnancal reportng, voluntary dsclosure, and executve compensaton. For example, a retaler may adopt a just-n-tme nventory system or a tradtonal one. Frms ncrease or decrease research-and-development expendtures, open new stores or close exstng stores, and ncrease hrng or lay off employees. Managers may ssue debts or equty to rase captal and further decde what partcular types of securty to ssue. Frms may dstrbute dvdends or repurchase stocks to return nvestments to nvestors. Managers may manpulate reported earnngs. Antcpatng an earnngs shortfall, some managers warn nvestors but others do not. Frms may hre compensaton consultants for CEO pay. These decsons fascnate accountng and fnance researchers, who are nterested n evaluatng ther consequences. A serous challenge for researchers, however, s that, for a gven frm, researchers observe only the outcome of the choce made but not the outcomes of choces not made. Therefore, researchers are unable to compare the outcome dfference of choces for a gven frm to evaluate the effects of ts decson. To overcome ths problem, researchers often attempt to select a control frm that s dentcal, except for the decson choce, to the frm that has made the choce of nterest. Ths task s readly accomplshed n controlled experments, where subjects can be randomly assgned to treatment (.e., the choce of nterest) vs. non-treatment (.e., the alternatve choce) so that researchers can make nferences about the average effect of treatment. The task s problematc outsde the realm of controlled experments, where frms are heterogeneous and corporate decsons are observed ex post. What makes evaluatons more challengng s that researchers cannot 1

4 observe all the nformaton that managers and nvestors use n decson makng. In other words, researchers use smaller sets of nformaton to evaluate managers decsons than the nformaton sets used by managers and nvestors. It s crucal for researchers to account for the observable and unobservable dfferences between a selected control frm and the deal control frm (that s, the sample frm tself) n evaluatng treatment effects. Absent controls for these dfferences, selecton bas, whch s one form of endogenety problem, can lead to napproprate nferences about treatment effects. Examples of observable dfferences are frm sze and growth. Selecton bas due to observables results from a falure to control for dfferences researchers can observe. Unobservable dfferences arse because researchers are restrcted to smaller nformaton sets than managers and market partcpants. Examples of unobservables are nformaton revealed durng a fnancal audt that are known to some market partcpants or other nformaton that s publcally dsclosed by the company but s too costly for researchers to collect. 1 Selecton bas due to unobservables results from a falure to control for the effect of dfferences researchers cannot observe. Both types of bas have been serous concerns for labor economsts, who evaluate a varety of tranng and welfare programs. Statstcal and econometrc tools have been developed to address them. Matchng a partcpant wth a non-partcpant wth smlar (of course, observable) characterstcs was a common tool used n the 1980s to mtgate selecton bas due to observables [Heckman, Ichmura, and Todd 1998]. Matchng on covarates (e.g., frm characterstcs) s deal when the number of characterstcs over whch partcpants and non-partcpants dffer s lmted and the varables are categorcal. 1 Throughout the artcle, the term unobservables means the factors affectng both treatment selecton and treatment outcome. If an unobservable factor affects ether process but not both, t does not cause an estmaton bas. 2

5 Matchng, however, s dffcult or nfeasble when the number of characterstcs to match s large and the sample sze s lmted. Rosenbaum and Rubn [1983] propose matchng by a functon of covarates rather than by each covarate. The functon they choose s the probablty of an ndvdual beng selected nto the program (and thus to be treated). Ths matchng method s referred to as propensty score matchng (PSM). PSM has been wdely used and dscussed n numerous dscplnes, ncludng statstcs, economcs, and medcne, n the past three decades and has been ncreasngly appled n accountng and fnance research n the past few years. For example, Journal of Accountng Economcs had no publcatons usng PSM before 2010, but two publcatons n 2010; Journal of Fnancal Economcs publshed ts frst two artcles that use PSM n 2004 and three artcles durng , but publshed or accepted for publcaton 10 artcles alone n The concern of selecton bas due to unobservables was frst thoroughly addressed by Lee [1978] and Heckman [1979]. Heckman proposes a two-stage approach to evaluatng programs for whch the treatment choces are bnary and the program outcomes depend on a lnear combnaton of observable and unobservable factors. Hs approach s to estmate the choce model n the frst stage and add a bas correcton term n the second-stage regresson. After further restrctng unobservables to multvarate normal dstrbutons, he derves the bas correcton varable n the form of nverse Mlls rato (IMR), that s, a rato of standard normal p.d.f. over standard normal c.d.f. (or 1 mnus the c.d.f.) The IMR method s easy to mplement and requres lttle computng power, and thus has become mmensely popular. The method has been ncreasngly used n accountng and fnance research n recent years. For example, Journal of Accountng Research publshed only one study that uses the IMR method by 1997, but nne studes snce Journal of 3

6 Accountng and Economcs publshed only one study wth the IMR method by 1999, but 15 studes snce The use of IMR n Journal of Fnancal Economcs also surged n 2010, followng a steady pace of applcatons snce The growng nterest n PSM and IMR n accountng and fnance research warrants a survey dscussng the condtons for each method and the confuson and perhaps mstakes n ther applcatons. In ths artcle I share my observatons regardng these ssues. The PSM method requres condtonal ndependence, whch means that the selecton or selfselecton of partcpant (treated frm) vs. non-partcpant (untreated frm) can all be explaned by observable factors. The estmated treatment effect usng PSM can only be generalzed to common support, meanng the porton of the populaton that can meanngfully decde whether to partcpate (unless all observatons are used wth weghts that ncrease wth the closeness of ther match wth the treated frm a approach known as kernel weghtng ). The IMR method, on the other hand, deals wth selecton on unobservable factors. Because IMRs are derved from truncated bnormal dstrbutons, t s only approprate f the frst-stage choce decson s modeled n probt, and the secondstage outcome s modeled n a lnear regresson, and f the unobservables n the two stages are bnormally dstrbuted. 2 When these condtons are not met, addng IMR to the second stage does not correct the selecton bas that researchers ntend to correct. 3 Despte the popularty of PSM and IMR, there s confuson n accountng and fnance research about the approprate use of each. One msconcepton s that PSM can address 2 Wooldrdge [2002, p ] states that the bnormal dstrbuton assumpton can be relaxed for the second-stage error term f ts mean condtonal on the frst-stage error term (whch has to be normal) s lnear. However, except for bnormal dstrbutons, few bvarate dstrbutons satsfy ths condton. 3 In ths artcle I assume that all relevant observable varables are dentfed and relably measured and that the form of ther relatons s correctly specfed. The success of mtgatng selecton bas depends on model specfcaton and varable measurement even f the estmaton method s approprate. See Tucker [2007, p.1079] for an example of result senstvty to model specfcatons. 4

7 selecton bas due to observables as well as unobservables. It s mportant for researchers to understand the generatng process of the non-expermental data and confrm that unobservables are not the prmary concern before choosng PSM. 4 Even when the prmary concern s selecton bas due to observables, researchers usng PSM stll need to dentfy the common support and check whether the treated and control frms matched by propensty scores are n fact close regardng the covarates. In addton, studes are advsed to check the senstvty of fndngs to the effects of unobservables; so far, very few do so. The problems n IMR applcatons are twofold. Frst, some studes use the IMR method even when t s not applcable. The IMR method has been popular n accountng and fnance research because of ts tradton and smplcty. The method, however, s only applcable n a lmted number of stuatons stated prevously. When the IMR method s not applcable, as long as the model s parametrc researchers can estmate treatment effects by full nformaton maxmum lkelhood estmaton (FIML). FIML s maxmum lkelhood estmaton appled to a system of equatons. It s more effcent than the IMR method even when the latter s applcable because FIML uses all nformaton at once rather than n two steps as under the IMR approach. Second, some studes mght have used wrong formulas n calculatng IMR for treated and untreated (control) frms. Ths artcle contrbutes to the accountng and fnance lteratures n several ways. Frst, the artcle dscusses two popular econometrc tools PSM and IMR n one unfed framework. Ths structure allows researchers to compare and contrast two types of selecton bas and the econometrc tools that mtgate them. Second, the artcle provdes a 4 Even though PSM s wdely used n labor economcs n examnng mandatory (.e., selecton by a program manager) and voluntary selectons, the technque mght not be as useful n accountng and fnance where managers have more dscreton, the decsons nvolve more partes, and the decson-makng process s more opaque to researchers. 5

8 revew of the applcatons of PSM at a tme when the method has pqued researchers nterest. Although Roberts and Whted [2011] nclude a secton on matchng methods n ther revew of endogenety n corporate fnance research, ther coverage s techncal and does not survey PSM applcatons. Thrd, ths artcle dscusses several ssues n applyng the IMR method to correct selecton bas due to unobservables. In contrast, the revew of Francs and Lennox [2008] focuses solely on the ssue of havng the same covarates n both the frst and second stages. Fnally, compared wth extant econometrc readngs (e.g., Heckman et al. [1998], Heckman [2001], Greene [2003], Schroeder [2010]), ths artcle emphaszes ntuton and applcatons and avods unnecessary techncal detals, thus appealng to a broader audence n accountng and fnance. Overall, the artcle focuses on drawng researchers attenton to the dscussed ssues rather than servng as a how-to manual. For the same reason, the survey of applcatons n the artcle s representatve, but not exhaustve. The rest of ths artcle s organzed as follows. Secton 2 presents the necessary econometrc framework for dscussng the two types of selecton bas. Secton 3 ntroduces PSM and dscusses the confuson and nadequacy n applyng the method. Secton 4 ntroduces the IMR method and dscusses the confuson and mstakes n usng the method. Secton 5 concludes. 2. Selecton Bas Ths secton ntroduces the econometrc framework wth bnary treatment choces and defnes selecton bas. I use managers decson of whether to ssue a warnng about a forthcomng earnngs dsappontment as an example. In econometrc termnology, the decson to ssue a warnng s the treatment and warnng frms are the treated group; 6

9 the decson not to ssue a warnng s the non-treatment and non-warnng frms are the untreated group. The outcome of nterest s nvestors prce reacton to the decson to warn. Dependng on whether researchers are nterested n makng nferences about the whole populaton, the subpopulaton of the treated, or the subpopulaton of the untreated, they may examne the average treatment effect (ATE), average treatment effect on the treated (ATT), or average treatment effect on the untreated (ATUT). All the three terms answer hypothetcal questons. In the warnng example, ATE estmates on average how stock returns dffer f frms warn versus f they do not warn, regardless of whether n realty a frm warns or not. ATT estmates on average how the observed stock returns of the frms that have warned dffer from the hypothetcal returns had these frms not warned. ATUT estmates on average how the stock returns of the frms that dd not warn would have been dfferent had they warned than ther observed returns wthout warnng. The dffculty n answerng these questons s that the outcome of warnng from a warnng frm s of course observed, but the outcome of non-warnng for the same frm s never observed and s referred to as a counterfactual outcome. Smlarly, the outcome of a non-warnng frm had t warned s also counterfactual. In non-expermental settngs, researches attempt to use an observed outcome to proxy for a counterfactual outcome n estmatng treatment effects. Selecton bas arses when the proxy s not close to the counterfactual that s proxed for. I use the followng framework to demonstrate bas n estmatng ATT, snce among the three treatment effects ATT draws researchers nterest most often. Equaton (1) models the outcome (Y) of treatment (wth subscrpt 1) and Equaton (2) models the outcome of non-treatment (wth subscrpt 0), where X s the factors beyond the 7

10 treatment decson that affect the outcome and are observable to researchers and v s a collecton of relevant factors unobservable to researchers. 5 Equaton (3) models the treatment decson, where researchers merely observe whether a frm s treated ( T 1 when T * 0 ) or untreated ( T 0 when T * 0 ), not the cost-beneft analyss by the decson maker (.e., T* s latent). In ths decson, some factors (Z) are observable to researchers and others (ε) are not. Y X (Data are observed only when T * 0, that s, T =1) (1) Y0 0 X 0 (Data are observed only when T * 0, that s, T =0) (2) T* Z (3) By defnton, ATT = E [ATT(x)], where ATT( x) EY ( ) EY ( ) (4) 1 T 1 0 T 1 observed counterfactual Researchers need a proxy for the counterfactual. A ready canddate for the proxy s to use an outcome observed from the untreated group (or a subset of the group). If researchers smply compare the average dfference n the outcome of the treated vs. the proxed counterfactual, the estmated ATT s ATT E[ ATT ( x)], where ATT ( x) EY ( ) EY ( ) (5) 1 T 1 0 T 0 proxy The dfference between true ATT and estmated ATT s the estmaton bas, due to some frms beng selected (or self-selected) nto one group to be treated and others nto the 5 I assume that the loadngs on the observables n the two outcome equatons are the same, consstent wth the popularly used treatment effect model n Greene [2003]. Ths assumpton lkely holds n general settngs. For example, n the warnng example, there s no reason to beleve that the sze and growth mmckng factors n asset prcng play dfferent roles for warnng frms than for non-warnng frms. Later, I dscuss studes that allow some of the loadngs to vary across the outcome equatons. 8

11 untreated group such that proxes have to be used for counterfactual outcomes. Ths bas s referred to as the selecton bas n the econometrc lterature. Selecton bas ATT ATT E[ E( Y ) E( Y )] (6) 0 T 1 0 T 0 counterfactual proxy It s up to a researcher to decde whether observables or unobservables are the prmary cause of the selecton bas, because whether a varable s observable depends on the researcher s nformaton set. If observables, denoted by the vector of X, are the prmary cause and the bas caused by unobservables can be gnored, the researcher s goal s to best control for the effects of X on the outcome ether (1) by controllng and removng the effects usng a regresson approach or (2) by choosng control frms from the untreated group closest to sample frms regardng these observables usng a matchng approach. PSM s a partcular matchng method. On the other hand, researchers may decde that the effects of unobservables on the estmaton bas cannot be gnored. Ths means that the unobservables n the outcome equatons, v 1 and v 0, are correlated wth the unobservables n the decson (choce) model, ε (See Secton 4). In other words, after researchers consder all factors that they can observe, they stll beleve that some unobservable factors are contrbutng both to the choce and the outcome of the treatment. For example, n the earnngs warnng example, perhaps managers wth worse news that s unobservable to researchers but observable to some nvestors are more lkely to warn to avod lawsuts for wthholdng bad news. Although stock prce plummets after the warnng, leavng the mpresson that the act of warnng has caused the prce to drop, n fact the prce would declne as soon as nvestors observe the bad news regardless of the warnng. The magntude of the prce declne regardless of 9

12 warnng s the selecton bas due to unobservables. The IMR method s an ntutve, smple, and hghly parameterzed method addressng ths type of selecton bas. The method also convenently controls for selecton bas due to observables n the second-stage regresson. 3. Propensty Score Matchng and ts Applcatons The regresson approach to mtgatng selecton bas due to observables mposes a lnear relaton between the observables ( covarates ) and the outcome of nterest. In addton, hghly correlated covarates may nduce multcollnearty n regresson estmaton. Matchng by covarates could avod both problems. Matched-sample desgns have a long hstory n accountng and fnance research (Cram, Karan, and Stuart [2009]; Loughran and Rtter [1997]). For example, researchers dentfy a control frm from the untreated group that s n the same ndustry as, and has the closest frm sze to, the treated frm, assumng that ndustry and frm sze are key determnants of the outcome of nterest. Researchers then compare the outcomes of treated frms wth those of control frms. Sometmes researchers further control for factors that are not bases of matchng n a multvarate regresson after ntal matchng. Covarate matchng s deal when treated and untreated frms only dffer on a few dmensons and the dfferences are represented by categorcal varables. For each treated frm, researchers can fnd a control frm wth the exact covarates. When covarate matchng s feasble, t produces the best estmate of ATT (Rosenbaum and Rubn [1983]; Zhao [2004]). As the number of dmensons grows, however, covarate matchng wll become dffcult or nfeasble. That s, researchers may not fnd any untreated frm that shares the same characterstcs as a gven treated frm. PSM s one way to overcome ths dmensonalty problem by aggregatng all covarates nto one score usng a lkelhood 10

13 functon. The use of the lkelhood functon may appear more sophstcated than tradtonal covarate matchng to an audence not famlar wth PSM because PSM nvolves estmatng a parametrc choce mode and calculatng propensty scores. Matchng by an aggregate score of treatment propensty, however, s drven by statstcal concerns (e.g., exact covarate matchng s nfeasble) rather than economc concerns. Ths s why PSM was frst proposed by statstcans and not by econometrcans. The beneft of reducng dmensons from a large number of covarates to one aggregate score comes wth a cost: PSM provdes a coarse match (Rosenbaum and Rubn [1983]). Stll, the ultmate goal of matchng s to fnd a control group that resembles (n terms of dstrbutonal smlarty) the treated group on the observable characterstcs when exact matchng by covarates s nfeasble. Thus, t s mportant for researchers to check how well the treated and control frms are matched on the covarates after matchng them by propensty scores. If the two groups are poorly matched on covarates, researchers may need to reconsder the specfcaton of the lkelhood functon that s used n PSM n the frst place. In the framework presented n Secton 2, nstead of estmatng the outcome regressons, researchers usng PSM compare the outcomes of treated frms wth those of control frms, where the control frm (or group) s dentfed as a frm (or subgroup) n the untreated group wth a propensty score close to that of the treated frm. Here, the propensty score s the predcted lkelhood of a frm beng selected for treatment often based on the observables, X, assumng covarates X=Z, or on the observables that affect both the treatment and outcome (Calendo and Kopeng [2008], p.38). Researchers use several crtera to dentfy the control frm (or group) wth a propensty score close to the treated frm s: (1) the smallest dstance, (2) a group n the nearest neghborhood, or (3) kernel 11

14 weghtng (where every observaton n the untreated group s used wth hgher weghts for closer observatons and lower weghts for more dstant observatons) (Daz and Handa 2006). The PSM estmator for ATT s often defned as the mean outcome dfference of treated and control frms matched by PSM. In other words, the counterfactual outcome n Equaton (4) s proxed by the average outcome of control frms selected by PSM. The estmator s unbased under three condtons. The frst condton requres that after matchng by propensty scores, the selecton of treatment and non-treatment can be consdered random. Intutvely, t means that the selecton bas s caused by observables, not unobservables (that affect both treatment selecton and treatment outcome). Ths condton has been referred to as the fundamental dentfcaton condton, condtonal ndependence (or a weaker condton of mean ndependence ), unconfoundedness, gnorablty condton, and selecton on observables (Zhao [2004]; Heckman et al. [1998]). The second condton requres that at the propensty scores used n matchng, both treatment and non-treatment selectons are possble. The condton fals at a gven score f only treated frms are observable at that score. Ths condton s referred to as the common support condton. Intutvely, outsde the common support, one cannot reasonably fnd a match for the treated frm. The thrd condton s balancng, that s, the dstrbutons of covarates are approxmately smlar for the treated and control groups after PSM. 6 These condtons have mplcatons for applyng PSM and drawng nferences. Frst, PSM s applcable to settngs n whch selecton bas due to unobservables s not a major 6 The lterature of matchng estmators n general lsts the frst two condtons. Zhao [2004, p.92] ponts out that these two general condtons hold under PSM only f the balancng property s satsfed. Therefore, PSM requres three condtons. 12

15 concern. There appears to be confuson among researchers about when to use PSM. The confuson s lkely due to dfferent nterpretatons and usage of selecton bas and endogenety. Selecton bas s techncally meanngful both for selecton on observables and selecton on unobservables, but t s orgnally and more frequently used for problems of selecton on unobservables (Heckman [1979]; Vella [1998]). Endogenety also has dfferent nterpretatons among researchers. In econometrcs, endogenety merely means that the covarates are correlated wth the error term (Wooldrdge [2002], p.50) and thus endogenety exsts n cases of selecton on observables and selecton on unobservables as well as n other omtted-correlated-varable stuatons. In other branches of economcs, endogenety means that a varable s determned wthn the context of a model t s a choce (Wooldrdge [2002], p.50). When researchers state that they use PSM to mtgate selecton bas or to address endogenety wthout qualfcatons, the statement s techncally correct, but t mght be nterpreted by some readers as meanng that PSM can address selecton bas due to unobservables. Here are a few ncdences of the confuson. Frst, the term selecton models n econometrcs refers to the econometrc models dealng wth selecton on unobservables, not the technques dealng wth selecton on observables, even though the word selecton appears n both problems. In another example, Hamlton and Nckerson [2003] revew the applcatons of selecton models n management and suggest that selecton bas s the result of treatment s beng a choce. In fact, f the choce can be fully explaned by varables observable to researchers, selecton models are not needed even though selecton bas and endogenety techncally exst. In some nstances, researchers use endogenety as a reason to employ PSM wthout argung that the endogenety problem s due to observables, not 13

16 unobservables (Hale and Santos [2009]; Lee and Wahal [2004]). 7 What adds to the confuson s a statement n a wdely crculated lterature survey by Armstrong, Guay, and Weber [2010, p.207]: Although a propensty score research desgn s one technque to address dentfcaton dffcultes posed by endogenous matchng on unobservable varables, there are a number of other recent advances n the statstcs and econometrcs lteratures that seem well suted to addressng ths and other smlar research questons. In fact, PSM and the IMR method are vewed as substtutve alternatves n some studes. For example, Doyle, Ge, and McVay [2007] report results usng PSM and the IMR method sde by sde as alternatves to address frms self-selecton, even though ther dscusson apparently alludes to selecton on unobservables. Lee and Wahal [2004] use an endogenous swtchng model, whch s a varant of models addressng selecton on unobservables, as an alternatve to ther PSM-based prmary analyss. As Peel and Makepeace [2009] note, the ncreased popularty of PSM n recent years s perhaps due to researchers belefs that PSM s an alternatve method to tradtonal Heckman procedures (for whch the IMR method s the most popular mplementaton) for estmatng treatment effects, after the latter was crtczed for lackng robustness to model specfcatons 7 Hale and Santos [2009, p.187] state, In realty, ths decson s lkely to be endogenous. We use a propensty score matchng to control for potental endogenety of the set of frms that ssues publc bonds and the tmng of ther bond IPO. Lee and Wahal [2004, p.377] wrte, Although controls are undenably mportant, the crux of our problem s that venture backng s not randomly dstrbuted, but represents an endogenous choce. Ths ntroduces a selectvty bas, one that can easly reverse nferences. To account for ths bas, we use matchng methods that endogenze the recept of venture fnancng and do not mpose lnearty of functon from restrctons. 14

17 (Stolzenberg and Relles [1997]). 8 However, PSM does not cure the dsease that tradtonal Heckman procedures are supposed to cure, even though the latter s mperfect. 9 To reduce the confuson n the lterature, t s mportant for researchers who use PSM to be explct that they use the technque to addresses selecton on observables, not on unobservables. It s also helpful to readers f researchers could argue why selecton on unobservables s not a serous concern n ther settng. Appendx A lsts the studes publshed n Journal of Accountng and Economcs and Journal of Fnancal Economcs that use PSM. It s surprsng that half of the studes do not even menton that the PSM technque s for addressng selecton on observables. Furthermore, when PSM s used, researchers could provde a valdty check f they test the senstvty of the fndngs to smulated unobservables followng the procedures n Rosenbaum [2010] and Peel and Makepeace [2009]. Unfortunately, wth the excepton of Armstong, Jagolnzer, and Larcker [2010], most accountng and fnance PSM studes do not do so. Second, nferences from PSM are vald only for the range of propensty scores of common support. Researchers are advsed to dentfy ths range and generalze ther fndngs to only ths proporton of the populaton rather than to the whole populaton. Among accountng and fnance studes that use PSM n man analyses, few check for common support and qualfy the fndngs (Appendx A). Thrd, t s mportant for researchers to check the balancng property, at least the means of the covarate dstrbutons, after matchng by propensty scores. The dmenson 8 Peel and Makepeace [2009] state n ther openng paragraph, Whle Heckman two-step model procedures are employed n ths context n an endeavor to control for unobserved selecton bas, researchers are becomng aware of ther potental senstvty and are ncreasngly turnng to propensty score (PS) matchng to nvestgate treatment effects. In fact, PSM s also senstve to specfcatons of the choce model. 9 PSM can be effectvely used together wth the dfference-n-dfference (DID) approach, where DID removes selecton bas due to tme-nvarant unobservables so that the frst condton for PSM s satsfed (Calendo and Kopeng [2008, p.55]; Krk [2010]; Mclnns and Collns [2010]; Chava and Purnanandam [2011]). 15

18 reducton by PSM s worthwhle only f the covarates of treated and control frms have smlar dstrbutons after beng matched by propensty scores. If not, researchers perhaps need to modfy the specfcaton of the choce model (Calendo and Kopeng [2008], p.43). Among the studes surveyed n Appendx A, only 58% check the balancng property. Other ssues n applyng PSM relate to the specfcatons of the outcome and choce models. Although a bnary choce s used n the framework n Secton 2, there can be more than two choces as long as they are categorcal. Accordngly, propensty scores can be calculated from other dscrete choce models than probt (see Armstrong et al. [2010]). Regardng the outcome equatons, most studes estmate ATT merely from the average outcome dfference between treated and control frms after matchng; that s, the analyss s based on a unvarate comparson after matchng. In most accountng and fnance settngs, the treatment outcome may be determned by factors that do not affect treatment selecton. For example, n Lee and Wahal [2004], the authors examne the dfferences n frst-day IPO returns between venture-captal-based frms and non-vc-based frms by comparng the returns of the former wth those of control frms selected from the latter group based on propensty scores of venture-captal-backng choce. General nvestors (other than the venture captalsts) n the IPO market probably consder other factors, such as a frm s age, busness complexty, recent product development, and sales growth beyond the consderatons of venture captalsts, who have provded fundng at an earler stage of the frm. Although unvarate outcome comparsons after PSM produce consstent estmators of treatment effects, controllng for factors that affect treatment outcome even f they do not affect treatment selecton would yeld more effcent estmators Some studes nclude n the choce model the varables that affect the treatment outcome but not selecton (e.g., underwrter ranks n Lee and Wahal [2004]). Includng rrelevant varables n the choce model would 16

19 4. The Inverse Mlls Rato Method and ts Applcatons Whle PSM addresses selecton bas due to observables by fndng a control frm (T =0) from the untreated group that s closest to the treated frm (T =1) to mnmze the dfference between the rght-hand terms of Equaton (6), the IMR method addresses selecton bas due to unobservables by estmatng a bas correcton term n the frst stage through the choce model and addng t n the second-stage outcome regresson. As the label suggests, selecton bas due to unobservables has much to do wth the unobservables n the outcome model and the unobservables n the choce model. In Secton 2 the true average treatment effect (ATE) n the parametrc model s 1 0. The IMR method n fact estmates ATE, not ATT, based on observed data. Next, I wll dscuss how the IMR method estmates ATE and then show how researchers can nfer ATT from an estmate of ATE. Stll, a crude ATE estmator could be calculated by comparng the average outcome dfferences of treated and untreated frms,, because ths s all researchers can observe. Here, EY ( ) X E ( ) X E ( Z ) 1 T T E ( Y ) X E ( ) X E ( Z ) 0 T T (7) (7) (8) Assume bnormal dstrbutons of (v 1, ε) and (v 0, ε) wth 0 means and covarances and 0 v 1 v and normalze at 1 as n bnary probt models (herenafter probt ). Followng the propertes of truncated bnormal dstrbutons (Greene 2003, pp.759 and 788), we have: ( Z ) ( Z ) E( 1 Z ) v 1 v1 1 ( Z ) ( Z ) lead to poor performance of the choce model and ncrease the varance of the PSM estmator (Calendo and Kopeng [2008], p.38). 17

20 ( Z ) ( Z ) E( 0 Z ) v 0 v0 ( Z ) 1 ( Z ) Thus, Equatons (7) and (8) become (9) and (10). Dfferencng them yelds (11): EY ( ) X E( ) X 1 T T v 1 ( Z ) ( Z ) (9) EY ( ) X E( ) X 0 T T v 0 ( Z ) 1 ( Z ) (10) ( Z ) ( Z ) EY ( 1 T 1) ( 0 T 0) ( 1 0) [ ] EY v 1 v0 ( Z ) 1 ( Z ) (11) True ATE Selecton bas due to unobservables Equaton (11) ndcates that our crude estmator on the left-hand sde estmates ATE wth bas due to unobservables. Note that the dfferences n observables have already been controlled for and removed by X. To correct for the bas due to unobservables, researchers usng the IMR method would estmate γ of the choce model (Equaton (3)) n the frst stage and add ( Z ˆ ) to Equaton (1) and ( Z ˆ ) ( Z ˆ ) 1 ( Z ˆ ) to Equaton (2), where ˆ s the estmated γ, n the second-stage least-squares regresson estmatons. In ths way, even though treatment outcomes are observed only for a partal sample for Equaton (1) and non-treatment outcomes are observed only for a partal sample for Equaton (2), 1 and 0 can each be consstently estmated from the observed data as ndcated n Equatons (9) and (10). The two ratos added n the second stage are referred to as the nverse Mlls rato (IMR) for treated frms and untreated frms, respectvely. The approach s referred to as the two-stage least squares estmaton usng IMR, shortened as the IMR method n ths artcle. 18

21 To mplement ths approach, researchers could ether estmate the augmented Equatons (1) and (2) separately or stack them nto one equaton, Equaton (12), usng an ndcator varable T to dstngush treatment from non-treatment ( treatment takes the value of 1) and settng the dependent varable to be Y = Y 1 * T + Y 0 * (1-T ). The coeffcent on T s then the estmated ATE and the coeffcents on the IMR varables are the estmated covarance between the unobservables n the treatment decson and those n the treatment outcome regresson. Equaton (12) s presented n Tucker [2007] and s modfed from the standard treatment-effect model of Greene [2003, p.788], whch constrans the coeffcents on the IMR for treated and untreated frms (.e., v1 and v 0 ) to be the same. Wooldrdge [2002, p.631] nstead presents a more flexble model than Equaton (12) by allowng the coeffcents on the exogenous observables to dffer across the treatment outcome and non-treatment outcome equatons. Y ( )* T X IMR * T IMR *(1 T) w (12) v1 v0 ATE Wth the estmate of ATE, ATT can be estmated accordng to the relaton: 11 ( Z ) ATT ATE E[ E( Y1 T 1) ( 0 T 1)] ( )* [ ] E Y ATE v 1 v E 0 ( Z ) It mght appear odd that bases caused by unobservables can ever be estmated and controlled. The ntuton behnd t s as follows. In estmatng average treatment effects, 11 ATT( x) EY ( ) EY ( ) [ X E( )] [ X E( )] 1 T 1 0 T T T 1 ( Z ) ( Z ) ( ) E( ) E( ) ( ) T 1 0 T v v 1 0 ( Z ) ( Z ) ( Z ) ATE ( ) v v 1 0 ( Z ) See Wooldrdge [2002, pp ] and Schroeder [2010, p.215]. 19

22 even though researchers do not observe ε, v 1, and v 0, all researchers need to know s the mean effect of the unobservable factors n the treatment decson on the treatment outcome gven observed data. Ths effect can be calculated from truncated bvarate dstrbutons of the unobservables as long as the dstrbutons are specfed. The latter condton s automatcally satsfed n parametrc analyss. For example, n ordnary least squares, even though the error term s not observable, we make specfc assumptons about ts dstrbuton. Thus, ths mean effect s estmated from the frst stage and added to the second stage for error correcton. Ths s the ntuton of all two-stage estmatons of selecton models, where the IMR method s a specal, albet restrctve, case. As Equaton (11) shows, the selecton bas to be corrected by the IMR method has two components: one related to the treated group and the other related to the untreated group. The magntude of each component ncreases wth the covarance between the unobservables n the choce model and the unobservables affectng treatment outcome. Of course, when these unobservables are not correlated, there s no selecton bas from unobservables. These covarances are estmated from the second-stage regresson. It mght be surprsng that the IMRs contrbute to the bas even though they are made of observables n the choce model. The contrbuton s because the IMRs reveal nformaton about the unobservables of the treatment decson. In partcular, gven the observed choce, one can nfer about the unobservable based on the observables because the two together n Equaton (3) determne whether the net benefts cross a threshold for managers to select the treatment The IMRs are monotonc transformatons of Z wth reversng orderng (See Tucker [2007]). 20

23 Treatment-effect models are by far the most common selecton models n accountng and fnance research. Researchers occasonally use other selecton models, for example, the tradtonal Heckman model (Heckman [1979]) when they are nterested n the determnants of the outcome after treatment or non-treatment but not the dfference n outcome between treatment and non-treatment. When the treatment decson depends on the perceved outcome of treatment vs. non-treatment, researchers use the endogenous swtchng model. The IMR method could be used for all the three models. The IMR method s smple but hghly parameterzed. Its smplcty perhaps explans why t s wdely used n accountng and fnance research. 13 However, the method requres strong assumptons for both the outcome regresson and the choce model. The error correcton varable s n the form of IMR only when (1) the outcome regresson s lnear, (2) the choce model s probt, and (3) the unobservables n the frst and second stages follow bvarate normal dstrbutons. When these requrements are not satsfed, the error correcton varable wll be n other forms and addng IMR to the outcome regresson wll not correct the selecton bas that researchers ntend to correct. In Fgure 1 I summarze varous ways n whch researchers use selecton models. The IMR method s applcable to only the frst three settngs: treatment-effect model, tradtonal Heckman, and endogenous swtchng. In these settngs the IMR estmators are consstent, but ther standard errors must be adjusted for samplng errors that occur n the frst stage (Greene [1981]; Maddala [1983]). The adjustments are nontrval and are best done usng statstcal software. Moreover, the IMR method s less effcent than the full nformaton maxmum lkelhood estmaton (FIML), where the latter uses maxmum 13 A keyword search of nverse mlls rato yelds 24 artcles n The Accountng Revew, 12 n Journal of Accountng Research, 16 n Journal of Accountng and Economcs, 9 n Journal of Fnance, 36 n Journal of Fnancal Economcs, and 9 artcles n Revew of Fnancal Studes. 21

24 lkelhood estmaton on a system of equatons. 14 In settngs for whch the IMR method s not approprate, FIML s applcable as long as the models are parametrc. 15 I dscuss examples of accountng applcatons below to hghlght these ponts. Some studes use the IMR method even though the second-stage outcome model s dscrete and therefore nonlnear. For example, Weber and Wllenborg [2003] examne n probt n the second stage whether a frm s pre-ipo audtor s opnon s more predctve of ts post-ipo stock survval when the audtor s a Bg-6 frm than when t s not and model the Bg-6 decson n the frst stage. Wu and Zang [2009] model fnancal analysts departures vs. stays after brokerage mergers n the frst stage and examne n two separate logt models n the second stage the nternal promoton of analysts to research executve postons after they stay and the external promoton of analysts after they leave the orgnal brokerage frm. Cohen and Zarown [2010] examne n probt n the second stage the lkelhood of real earnngs management at frms that have already been dentfed n the frst stage as havng managed earnngs through ether real or accrual management. All the three studes estmate a probt model n the frst stage and add IMR to the second stage, ntendng to correct for selecton bas due to unobservables However, the IMR term(s) does not correct for the bas because the second-stage model s nonlnear. A bvarate probt model wth sample selecton would be approprate (Greene [2002], p.e17-19). The model can be estmated by FIML Among the accountng studes surveyed, Omer, Bedard, and Falsetta [2006] and L and Ramesh [2009] use FIML. The latter acknowledges the effcency of FIML over the IMR method. 15 FIML does not need addtonal assumptons beyond those for two-stage estmatons and s not necessarly less robust to volatons of assumptons than are two-stage estmatons. Two-stage estmatons, on the other hand, requre less computng power than FIML. Perhaps for ths reason, two-stage estmatons were more popularly used than FIML n the early years. 16 Terza [2009, p.563] demonstrates that two-stage estmatons are feasble even when the second-stage model s nonlnear. The error correcton varables to be added n the second-stage are n complcated forms, not n the form of IMR. Nonlnear least squares estmaton s used for the augmented second-stage equaton. 22

25 Some studes use the IMR method even though the choce model s logt, not probt. For example, Feng and Koch [2010], Engle, Hayes, and Wang [2007], and Khurana and Raman [2004] use logt n the frst stage and add IMR to the second-stage lnear regresson. Ths s napproprate because IMR requres the arguments n the p.d.f. n the numerator and c.d.f. n the denomnator of the rato to be normally dstrbuted. When logt s used n the frst stage, one cannot smply use the IMR formulas, but needs to transform Z by an nverse standard normal c.d.f. functon to ensure that the arguments n the numerator and denomnator functons are normally dstrbuted, not logstcally dstrbuted (Greene [2002], p.e23-71). In some studes there are n fact more than two choces n the frst stage; therefore, a probt model s nsuffcent. For example, Rogers [2008] examnes whether dsclosure qualty dffers f managers subsequently trade stocks than f managers tradng ncentves are absent and, more mportantly, whether dsclosure qualty s hgher before nsder sellng than before nsder buyng. Insders have three choces: sell, hold, and buy the frm s stocks. Perhaps for econometrc convenence, n the frst stage the author assumes that managers consder ether sell vs. hold or buy vs. hold and models sell vs. hold and buy vs. hold separately n probt. Ths research desgn essentally changes managers decson of three choces to two sequental decsons, the frst of whch (.e., decdng to go down the path of sell vs. hold or the other path of buy vs. hold ) s skpped and never modeled n Rogers [2008]. More than two choces of treatment are common n accountng and fnance research. In these stuatons, researchers often use ordered probt for ordered choces and multnomal logt for unordered choces n the frst stage and model treatment outcome n a lnear 23

26 regresson n the second stage. When the second-stage model s lnear, researchers can estmate the choce model n the frst stage, calculate the expected outcome resdual gven each choce, and add the varables to the second stage as long as the outcome of the treatment choce s observed. These new varables are ratos but are not IMRs (see Vella [1998], pp ; Greene [2002], p.e23-79). In addton, an nverse standard normal c.d.f. transformaton s requred when multnomal logt s used n the frst stage (see Greene [2002], pp. E23-72 to 73). There are four other ssues n applyng the IMR method. Frst, the formula to calculate IMR for treated frms s dfferent from that for untreated frms, even though one varable label s used for both groups n standard treatment-effect models. The IMRs for treated and untreated frms have opposte sgns. Ths s why the IMR varable s hghly correlated wth the treatment ndcator varable. Ths hgh correlaton s not a weakness of the selecton model. Second, there have been two sets of formulas for IMR n the econometrc lterature and the dfference between them s the sgn. The IMR formulas descrbed above can be found n Heckman [1979], Vella [1998], and Greene [2003]. But n some other studes the formulas are ( Z ) ( Z ) for treated frms and ( Z ) for untreated frms (Lee [1978]; 1 ( Z ) Maddala [1983], p ). The dfferences are due to the sgn of the error term n the choce model: t s postve n the former lterature but negatve n the latter lterature (.e., T* Z ). Because popular software (SAS, Stata, and Lmdep) uses a postve sgn for the error term, the formulas n ths artcle are consstent wth model estmatons from modern software. If researchers are merely nterested n the average treatment effect, the sgn dfferences n IMR calculatons wll not affect the estmate of the effect. But f researchers also make nferences about the correlaton of the unobservables n the 24

27 treatment decson and treatment outcome, a wrong sgn for IMR wll result n a wrong sgn for the estmated correlaton. Ths detal of IMR formulas s sometmes mssed by researchers. For example, Hamlton and Nckerson [2003] survey and propose sophstcated selecton model technques for the management lterature, but the formulas they provde n ther text and Appendx 1 are ncorrect. They specfy a postve sgn for the error term n the frst stage but follow the formulas n the latter econometrc lterature that uses the error term wth a negatve sgn. Early accountng studes that use the IMR method provde correct formulas (Core and Guay [1999]; Verreccha and Leuz [2000]). Most subsequent studes, wth the exceptons of Chaney, Jeter, and Shvakumar [2004] and Tucker [2007], are not explct about the formulas for IMR calculatons and thus there s no tellng whether the formulas used for treated and untreated frms are dfferent and correct. From the glmpse of dscusson provded, confuson about the IMR formulas seems to contnue. For example, Chen, Matsumoto, and Rajgopal [2010] note that ther IMR for treated frms s a negatve functon of ftted probabltes from the choce model, mplyng that ther formula s ncorrect because they use a postve sgn for the error term n the frst stage (they do make nference about the correlaton of unobservables). Gven nconsstent notatons n the econometrc lterature and confuson n applcatons, future research could be explct and cautous about IMR calculatons Haw, Hu, Hwang, and Wu [2004, p.439] note, Includng the expected probablty of beng n the fnal sample (nverse Mlls rato) as another explanatory varable n the regressons does not alter the fndngs. Ths statement mples msunderstandng of IMR. Hogan [1997] uses an endogenous swtchng model to examne a frm s tradeoff between the benefts of less IPO underprcng and the cost of hgher audt fees from hrng Bg-6 audtors. Her hypotheses rely upon the sgns of the estmated correlatons of the unobservables, but the sgns of her IMRs are ncorrect, because the formula for IMR n her Equaton (1) s based on choce model T* Z, but her emprcal choce model, her Equaton (4), s T* Z. 25

28 The thrd ssue s other varatons of the treatment-effect model. One varaton s to allow the coeffcents on the exogenous covarates, X, to dffer for treated frms and untreated frms. Ths can be acheved by addng nteracton terms of the treatment ndcator and the exogenous covarates. Examples of such models are Gvoly, Hayn, and Katz [2010], Lous [2005], Leone, Rock, and Wllenborg [2007], and Chung and Wynn [2008]. Another varaton s to examne two treatments (not two choces of one treatment). When the treatments are ndependent, the second-stage lnear regresson can nclude both ndcator varables for the treatments and both IMRs, each calculated separately from a probt choce model for the treatment. When the treatments are dependent, the frst stage requres a bprobt model, estmated smultaneously. Two ratos wll then be calculated nvolvng margnal bnormal p.d.f. n the numerator and bnormal c.d.f. n the denomnator (Greene [2002], pp.e23-83, E17-15), not n the form of IMRs. The ratos are then added to the second stage, correspondng to the two treatment ndcator varables capturng average treatment effects, for the two-stage least squares estmaton. Of course, the model can be estmated by FIML. Examples of two-treatment-effect studes are Muller and Redl [2002] and Asquth, Beatty, and Weber [2005]. Muller and Redl [2002] examne the choce of Bg 6 vs. non- Bg 6 and the choce of external property appraser vs. nternal appraser together n the frst stage. They then add two IMR ratos to the second stage. Asquth, Beatty, and Weber [2005] examne the choces of nterest-decreasng performance prcng and nterestncreasng performance prcng bank debt contracts n the frst stage and nterest rate spreads n the second stage. They estmate the two stages smultaneously n the prmary 26

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