Fixed-effects binary choice models with three or more periods
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Davezies, Laurent; D'Haultfœuille, Xavier; Mugnier, Martin Article Fixed-effects binary choice models with three or more periods Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Davezies, Laurent; D'Haultfœuille, Xavier; Mugnier, Martin (2023) : Fixed-effects binary choice models with three or more periods, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 14, Iss. 3, pp. 1105-1132, https://doi.org/10.3982/QE1991 This Version is available at: https://hdl.handle.net/10419/296331 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Quantitative Economics 14 (2023), 1105–1132 1759-7331/20231105 Fixed-effects binary choice models with three or more periods Laurent Dav e z i e s CREST, ENSAE—Institut Polytechnique de Paris Xav i e r D’Haultfœuille CREST, ENSAE—Institut Polytechnique de Paris Martin Mugnier CREST, ENSAE—Institut Polytechnique de Paris We consider fixed-effects binary choice models with a fixed number of periods Tand regressors without a large support. If the time-varying unobserved terms are i.i.d. with known distribution F,Chamberlain (2010) shows that the common slope parameter is point identified if and only if Fis logistic. However, he only considers in his proof T=2. We show that the result does not generalize to T≥3: the common slope parameter can be identified when Fbelongs to a family including the logit distribution. Identification is based on a conditional moment restriction. Under restrictions on the covariates, these moment conditions lead to point identification of relative effects. If T=3 and mild conditions hold, GMM estimators based on these conditional moment restrictions reach the semiparametric efficiency bound. Finally, we illustrate our method by revisiting Brender and Drazen (2008). Keywords. Binary choice models, panel data, point identification, conditional moment restrictions. JEL classification. C14, C23, C25. 1. Introduction In this paper, we revisit the classical binary choice model with fixed effects. Specifically, let Tdenote the number of periods and let us suppose to observe, for individual i, (Yit,Xit )t=1,,Twith Yit =1X itβ0+γi−εit ≥0, (1.1) Laurent Davezies: [email protected] Xavier D’Haultfœuille: [email protected] Martin Mugnier: [email protected] We would like to thank Pascal Lavergne and three anonymous referees for their helpful comments. Xavier D’Haultfœuille thanks the hospitality of the Paris School of Economics where part of this research was conducted. He also gratefully acknowledges financial support from the research grants Otelo (ANR-17-CE26- 0015-041). This research is supported by a grant of the French National Research Agency (ANR), “Investissements d’Avenir” (LabEx Ecodec/ANR-11-LABX-0047). ©2023 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1991
1106 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) where β0∈RKis unknown and εit ∈Ris an idiosyncratic shock. The nonlinear nature of the model and the absence of restriction on the distribution of γiconditional on Xi:=(X i1,,X iT )renders the identification of β0difficult. Rasch (1960)showsthat if the (εit )t=1,,Tare i.i.d. with a logistic distribution, a conditional maximum likelihood can be used to identify and estimate β0.Chamberlain (2010) establishes a striking converse of Rasch’s result: if the (εit )t=1,,Tare i.i.d. with distribution Fand the support of Xiis bounded, β0is point identified only if Fis logistic. Other papers have circumvented such an impossibility result by either considering large support regressors (see in particular Manski (1987), Honore and Lewbel (2002)) or allowing for dependence between the shocks (see Magnac (2004)). It turns out, however, that Chamberlain (2010) only proves his result for T=2. And in fact, we show that his result does not generalize to T≥3. Specifically, we consider distributions Fsatisfying F(u) 1−F(u)= τ k=1 wkexp(λku)or 1−F(u) F(u)= τ k=1 wkexp(−λku), (1.2) with T≥τ+1, (w1,,wτ)∈(0, ∞)τand 1 =λ1<···<λ τ. We study the identification of β0, assuming that λ:=(λ1,,λτ)is known. The weights w1,,wτremain unknown, thus allowing for much more flexibility on the distribution of εit than in the logit case. In particular, it may either be left- or right-skewed, platykurtic or leptokurtic. Our main insight is that for any Fsatisfying (1.2), a conditional moment restriction holds. We also obtain some results on the corresponding identified set B. For instance, if, roughly speaking, Xiis continuous, we show that Bincludes at most T!−1 points (2 if T=3) and relative marginal effects are point identified. Note that Johnson (2004) considers the same family with τ=2andT=3. However, he does not study the general case and does not show any formal identification result based on the corresponding moment conditions. Obviously, the conditional moment condition can be used to construct GMM estimators. This means, in particular, that √n-consistent estimation is possible beyond the logit case when T>2, overturning again the impossibility results of Chamberlain (2010) and Magnac (2004). Further, we show that if T=3 and mild additional restrictions hold, the optimal GMM estimator based on our conditional moment conditions reaches the semiparametric efficiency bound of the model. Hence, at least when T=3, these moment conditions contain all the information of the model. Finally, we showcase the empirical relevance of our approach by studying whether budget deficits and economic growth affect reelection, revisiting Brender and Drazen (2008). The authors investigate this issue using simple and fixed-effects logit models. However, the assumption of logistic errors is not warranted, so we consider whether the results are robust to this assumption on the unobserved terms. Our results suggest that the relative effects of budget deficits and economic growth or other variables are fairly robust to the logistic assumption. Our paper is related to the seminal work of Bonhomme (2012), who develops a unified approach for models where the conditional distribution of (Y1,,YT)given
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1107 (Xi,γi)is parametrized by β0, but no restriction on the distribution of γi|Xiis imposed. In such set-ups, he shows that the identification and estimation of β0depends on the existence of functions m=0 satisfying Em(Y,X,β0)|X,γ=0. This approach has been fruitfully applied to the dynamic logit model by Kitazawa (2022) and Honoré and Weidner (2020). Our paper may be seen as yet another application of this approach, focusing on static models but dropping the logistic assumption. The remainder of the paper is organized as follows. Section 2describes the moment condition we use for identification of β0and establishes some properties of the identified set based on these moments. Section 3discusses GMM estimation of β0, links it with the semiparametric efficiency bound of the model, and discusses the case of unbalanced panel data. Section 4is devoted to the application. Section 5concludes. All the proofs are collected in the Appendix. The Online Supplementary Material (Davezies, D’Haultfoeuille, and Mugnier (2023)) contains data used in our application and codes for replication. 2. Identification 2.1 The model and moment conditions We drop the subscript iin the absence of ambiguity and let Y=(Y 1,,Y T),X= (X 1,,X T),Xt=(X1,t,,XK,t),Xk,·=(Xk,1,,Xk,T),X−k=(Xk,t)k=k,t=1,,T, Xk,−t=(Xk,s)s=t,andX−k,t=(Xk,t)k=k.Supp(X)⊂RKT denotes the support of the random variable X. For any set A⊂Rp(for any p≥1), we let A∗:=A\{0}and denote by |A|the cardinal of A. Hereafter, we maintain the following conditions. Assumption 1 (Binary choice panel model). Equation (1.1)holds and: 1. (X,γ)and (εt)1≤t≤Tare independent and the (εt)1≤t≤Tare i.i.d.with a known cumulative distribution function (cdf)F. 2. For all (k,t),E[X2 k,t]<∞. 3. β0∈RK∗. The first condition is also considered in Chamberlain (2010). The second condition is a standard moment restriction on the covariates. Finally, we exclude in the third condition the case β0=0 here. This case can be treated separately, as the following proposition shows. Proposition 2.1. Suppose that Assumption 1holds,Fis strictly increasing on Rand there exist (t,t)∈{1, ,T}2such that E[(Xt−Xt)(Xt−Xt)]is nonsingular.Then β0= 0if and only if P(Yt=1, Yt=0|Yt+Yt=1, Xt,Xt)=1 2a.s. (2.1)
1108 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) Condition (2.1) can be tested by a specification test on the nonparametric regression of D=Yt(1−Yt)on (Xt,Xt), conditional on the event Yt+Yt=1; see, for example, Bierens (1990)orHong and White (1995). Turning to identification on RK∗, we first recall the impossibility result of Chamberlain (2010). We say below that Fis logistic if G(u):=F(u)/(1−F(u)) =wexp(λu)for some (w,λ)∈R+∗2. Theorem 2.2. Suppose that T=2, Xtincludes 1{t=2},Assumption 1.1 holds,Fis strictly increasing on Rwith bounded,continuous derivative and Supp(X)is compact. If Fis not logistic,there exists β0∈RK∗,a distribution of γ|Xand an open ball B⊂RK such that β0is not identified compared to β∈B. This result implies in particular that when T=2andFis not logistic, relative effects β0j/β0k,forksuch that β0k=0, may not be identified. Such relative effects are important as they are equal to relative marginal effects if both Xj,tand Xk,tare continuous. If only Xk,tis continuous (say), −β0j/β0kstill corresponds to a compensating variation.1 The key step in Chamberlain’s proof is that if β0is identified for all data generating process satisfying the restrictions of the theorem, the conditional probabilities (conditional on Xand γ) of the four possible trajectories for (Y1,Y2)are necessarily affinely dependent. Moreover, by letting |γ|tend to infinity, the stable trajectories (0, 0)and (1, 1)disappear from this relationship. This leads to the following functional equation for G: ψ1(α)G(u)+ψ2(α)G(u+α)=0, (2.2) for all u∈R,αin an open subset of Rand some functions ψ1(·),ψ2(·)such that for all α, (ψ1(α),ψ2(α)) =(0, 0). The result follows by noting that the solutions necessarily have the form u→wexp(λu). Equation (2.2) relies on the time dummy variable 1{t=2}. However, the proof of Theorem 2 of Chamberlain (2010) shows that even without such a dummy variable, (2.2) is necessary for the semiparametric efficiency bound not to be zero, or equivalently, for the existence of regular, root-n consistent estimators of β0.Inthiscase,αcorresponds to (x2−x1)β0,for(x1,x2)in a set of positive measure. In any case, the same reasoning with T=3 leads to the following equation for G: ψ1(α)G(u)+ψ2(α)G(u+α1)+ψ3(α)G(u+α2)+ψ4(α)G(u)G(u+α1) +ψ5(α)G(u)G(u+α2)+ψ6(α)G(u+α1)G(u+α2)=0, (2.3) for all u∈R,α:=(α1,α2)in an open subset of R2and some functions ψk(·),k=1, ,6, such that for for all α,(ψ1(α),,ψ6(α)) = (0, ,0 ).Wenowhave6=23−2terms 1To see the first point, note that under Assumptions 1–2, μk,t(x):=∂P(Yt=1|Xk,t=xk,t,Xk,−t=xk,−t,X−k=x−k) ∂xk,t=β0kEFx tβ0+γ|X=x, and thus μj,t(x)/μk,t(x)=β0j/β0k.Also,−β0j/β0kcorresponds to the change in Xk,tnecessary to keep P(Yt=1|Xt,α)constant when Xj,tincreases by one unit.
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1109 instead of just 2 =22−2, and thus we can expect to have other solutions than just u→ wexp(λu). And indeed, one can check that if Ghas the form u→ w1exp(λ1u)+ w2exp(λ2u),wecanconstruct(ψ1(α),ψ2(α),ψ3(α)) = (0, 0, 0)such that (2.3)holds, with ψ4(α)=ψ5(α)=ψ6(α)=0. Similarly, if 1/G has the form u→ w1exp(λ1u)+ w2exp(λ2u),wecanconstruct(ψ4(α),ψ5(α),ψ6(α)) = (0, 0, 0)such that (2.3)holds, with ψ1(α)=ψ2(α)=ψ3(α)=0. Note that there may still be other solutions to (2.3) that are increasing and have a limit of ∞(resp., 0) at ∞(resp., at −∞). The question of identifying all such solutions is left for future research. Generalizing this reasoning to any T>2, we see that combinations of at most T−1 exponential functions satisfy the functional restrictions tantamount to (2.3) and which render identification of β0possible. This suggests that identification may be achieved for the corresponding family of distribution, which we now formally introduce. Hereafter, τdenotes a subset of {(λ1,,λτ)∈Rτ:1=λ1<···<λ τ}. Assumption 2 (“Generalized” logistic distributions). 2There exist known τ∈{1, ,T− 1}and λ:=(λ1,,λτ)∈τand unknown w:=(w1,,wτ)∈(0, ∞)τsuch that: Either F(u)/1−F(u)= τ j=1 wjexp(λju)(First type), or 1−F(u)/F(u)= τ j=1 wjexp(−λju)(Second type). Fixing min{λ1,,λτ}to 1 is without loss of generality, as we can always multiply β0, γiand εit by this factor. If Fis of the second type, then one can show that the cdf of −εit is of the first type. Thus, up to changing (Yt,Xt)into (1−Yt,−Xt), we can assume without loss of generality, as we do afterwards, that Fis of the first type. We shall see that τ+1 periods are sufficient to achieve identification. Hence, we assume, again without loss of generality, that T=τ+1: if T>τ+1, we can always focus on τ+1periods. Before describing our identification strategy of β0when Fis a generalized logistic distribution, two remarks are in order. First, we obtain our results below irrespective of the vector w.3Hence, in contradistinction with the fixed-effect logistic model, we do not fix the distribution of ε, but simply impose that it belongs to a family of distributions indexed by two parameters. Members of this family differ in particular by their skewness and kurtosis. In linear regressions, the residuals are often found to have a skewed distribution with either positive or negative excess kurtosis. Then there is no reason why the latent variables corresponding to Yit would not exhibit a similar pattern. Note however that we do fix λ. Identification of λcould also be of interest but is not addressed in this paper. 2Though we use the same name, our family of distributions should not be confused with those introduced by Balakrishnan and Leung (1988) and Stukel (1988). 3We do impose, however, that all the components of ware nonzero, for normalization purposes. Otherwise, the model with w=(w1,0 )and β0, for instance, would be equivalent to the model with w=(0, w1) and β0/λ2. A similar issue arises with, for example, w=(w1,w2,0 )if λ3/λ2=λ2/λ1.
1110 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) Now, the idea behind the identification of β0is to construct a function m= 0such that E(m(Y,X,β0)|X,γ)=0 almost surely. Thus, as mentioned in the Introduction,we apply Bonhomme (2012)’s general idea of functional differencing. The function mis related to the functions ψkin (2.3)whenT=3, and the generalization of (2.3)whenT>3. For any x=(x 1,,x T)∈RKT ,letx−t s=xsif s<t,x−t s=xs+1else. We let Mt(x;β)=(−1)t+1det⎛ ⎜ ⎝ expλ1x−t 1β expλ1x−t T−1β . . .. . . expλT−1x−t 1β expλT−1x−t T−1β⎞ ⎟ ⎠. Then define, for any (y,x,β)∈{0, 1}T×Supp(X)×RK∗, m(y,x;β):= T t=1 1yt=1, yt=0∀t=tMt(x;β). Our first result shows that m, indeed, satisfies a conditional moment restriction. Theorem 2.3. If Assumptions 1–2hold,we have almost surely Em(Y,X;β0)|X,γ=Em(Y,X;β0)|X=0. (2.4) Theorem 2.3 shows there exists a known moment condition, which potentially identifies β0in a more general model than the logistic one. Also, as the number of periods T increases, the class of distributions Ffor which β0can be point identified increases. This is consistent with the idea that if T=∞,β0is point identified for any F, by using variations in Xtof a single individual. Note, however, that the class of generalized logistic distribution is not dense for the set of all cdf’s: any cdf Fbelonging to the closure of this class should be such that either F/(1−F)or (1−F)/F is convex. Theorem 2.3 also complements the results of Chernozhukov, Fernández-Val, Hahn, and Newey (2013) showing that bounds on β0for general Fshrink quickly as Tincreases. Theorem 2.3 holds with T=τ+1=2. In such a case, the conditional moment condition can be written E1{Y1>Y 2}expX 2β0−1{Y2>Y 1}expX 1β0|X=0. This conditional moment generates the first-order conditions of the maximization of the theoretical conditional likelihood, since these the first-order conditions are equivalent to E(X1−X2) expX 1β0+expX 2β01{Y1>Y 2}expX 2β0−1{Y2>Y 1}expX 1β0=0. 2.2 Necessary and sufficient conditions for identification The discussion above implies that with T=τ+1=2, β0is identified by (2.4)assoonas E[(X1−X2)(X1−X2)]is nonsingular. We now turn to the more difficult case where T−
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1111 1=τ>1. Let Bdenote the identified set of β0obtained with our conditional moment conditions, namely B:=b∈RK∗:Em(Y,X;b)|X=0a.s. . We also denote by Bk:={bk:∃b=(b1,,bk,,bK)∈B}(k=1, ,K) the identified set of β0k. Our first result shows that Bis included in a set depending on the distribution of Xonly. To define this set, let us introduce Dj(x;b):=det⎛ ⎜ ⎜ ⎜ ⎜ ⎝ expλjx 1β0 expλjx Tβ0 expλ1x 1b expλ1x Tb . . . expλT−1x 1b expλT−1x Tb ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ and, for all b∈RK∗,let D(b)=x∈Supp(X):max j=1,,T−1Dj(x;b)>min j=1,,T−1Dj(x;b)≥0 or min j=1,,T−1Dj(x;b)<max j=1,,T−1Dj(x;b)≤0. Because Dj(x;β0)=0forallx∈Supp(X),wehaveP(X∈D(β0)) =0. The following lemma shows that Bis actually included in the set of b’s satisfying this property. Lemma 2.4. Suppose that Assumptions 1–2hold.Then B⊂ B:=b∈RK∗:PX∈D(b)=0. This result follows because the moment condition can be written as a weighted sum of the Dj(x;b)’s, with positive weights. It shows that β0is identified if for all nonzero b=β0, we can find some x∈Supp(X)such that all nonzero Dj(x;b)havethesamesign, and the set of such nonzero determinants is not empty. The set Bis convenient in that it does not depend on the unknown distribution of γ|Xbut it is hard to characterize in general. Nevertheless, we are able to obtain results under either of the conditions below. Assumption 3. For all k∈{1, ,K},P(|{Xk,1,,Xk,T}|=T,X−k=0)>0.4 Assumption 4. There exists (s,t,x)∈{1, ,T}2×RK,s<tand a neighborhood Vof x such that Supp(X)∩[R(s−1)K×V×R(t−s−1)K×V×R(T−t)K]has a nonempty interior. The first assumption corresponds to a case where all components of Xare discrete. It imposes that for all kand t, the support of Xk,tincludes 0 and at least T−1 additional elements. Because we can always replace Xk,·by Xk,·−ckfor any ck∈RT, the condition 0∈Supp(Xk,t)for all k,tholds as long as ∩T t=1Supp(Xk,t)is not empty (for all k). The 4When K=1, the condition X−k=0 should simply be omitted.
1112 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) second condition imposes that all components of Xtare continuous. It also imposes that for at least two periods sand t,Supp(Xs)∩Supp(Xt)is not empty. This last condition holds for instance if (Xt)t≥1is strictly stationary. Theorem 2.5. Suppose that Assumptions 1–2hold.Then: 1. If Assumption 3also holds,|B|<∞and Bk⊂{cβ0k:c∈{0}∪(1/λT−1,λT−1)}. 2. If Assumption 4also holds, B⊂ B⊂R:=cβ0:c∈(1/λT−1,λT−1). (2.5) Moreover,|B|≤T!−1and |B|≤2when T=3. All relative effects β0j/β0k,for ksuch that β0k=0, are point identified.5 Whether Assumptions 3or 4hold, Theorem 2.5 shows that underidentification is at most finite, namely |B|<∞. This implies that β0is locally identified in the sense that there exists a neighborhood of β0in which the unique solution to the equation E[m(Y,X;b)|X]=0isb=β0.Further,thefirstresultofTheorem2.5 shows that with discrete regressors satisfying Assumption 3, the “length” of the identified set on β0k, defined as max (b1k,b2k)∈B2 k|b1k−b2k|, cannot exceed β0k(λT−1−1/λT−1)if 0 /∈Bk. Note that under Assumption 3, we can actually identify whether or not β0k=0 without relying on our conditional moments, since the sign of β0kis equal to that of E[Yt−Ys|X−k,s=X−k,t,Xk,t>X k,s]. The second result on continuous regressors is stronger. It shows that if Assumption 4holds, β0is identified up to a scale c,withcbelonging at most to (1/λT−1,λT−1). This directly implies point identification of relative marginal effects. The second result also states that Bincludes at most T!−1 points, and even only 2 points when T=3. Importantly, all of these results hold for any possible distribution of γ|X. Thus, point identification may actually hold for many distributions of γ|X, a point we shall come back to below. The proof of Theorem 2.5 relies on the following ideas. In the first case, when bk/∈ {cβ0k:c∈{0}∪(1/λT−1,λT−1)},weconstructasubsetofSupp(X)of positive probability such that all nonzero Dj(x;b)have the same sign. The result then follows by Lemma 2.4. We use a similar reasoning to prove (2.5). To establish the upper bounds on |B|,weexploit the fact that the family of exponential functions (v→ exp(ζkv))k=1,,Kwith distinct coefficients ζkforms a Chebyshev system (see, e.g., Krein and Abramovich Nudelman (1977), Chapter II for the formal definition of such systems). This property implies that some key determinants do not vanish, and any nonzero “exponential polynomial” v→K k=1αkexp(ζkv)does not have more than K−1zeros. We now turn to necessary conditions for identification. The following result is a partial converse of Lemma 2.4 and Theorem 2.5 above. 5The set of indices ksuch that β0k= 0 is identified since by (2.5), Bk={0}when β0k=0, and 0 /∈Bk otherwise.
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1119 A.3 Lemma 2.4 Let b∈ Bcand let us prove that b/∈B.Fixx∈D(b)and let Jx:={j∈{1, ,T−1}: Dj(x;b)=0}and aj(x):=wjE⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ exp(λjγ) T t=11+ T−1 k=1 wkexpλkx tβ0+γX=x⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (A.1) Then Jx=∅and Em(Y,X;b)|X=x= j∈Jx aj(x)Dj(x;b). (A.2) Moreover, aj(x)>0andalltheDj(x;b)for j∈Jxhave the same sign. Thus, E[m(Y,X; b)|X=x]= 0. Because b∈ Bc, we have by definition of B,P(X∈D(b)) >0. Thus, E[m(Y,X;b)|X=x]=0 with positive probability, implying b/∈B. A.4 Theorem 2.5 Part 1 a. Bk⊂Rk:={cβ0k:c∈{0}∪(1/λT−1,λT−1)}. Let us fix k∈{1, ,K},b=(b1,,bK)and define X0k:=x∈Supp(X):xj,1 =···=xj,T=0∀j=k,{xk,1,,xk,T}=T. First, suppose that β0k=0andbk=0. Then Dj(x;b)does not depend on j.Moreover, because x 1b,,x Tb={xk,1bk,,xk,Tbk}=T, we have Dj(x;b)=0 by properties of Chebyshev systems. Thus, x∈D(b), implying that X0k⊂D(b). By Assumption 3,P(X∈X0k)>0. Hence, P(X∈D(b)) >0. By Lemma 2.4, bk/∈Bkand Bk⊂{0}=Rk. Now, suppose β0k=0. Then any bk∈Rcanbewrittenascβ0k.Weprovethatifc/∈ {0}∪(1/λT−1,λT−1),thenX0k⊂D(b). By Lemma 2.4 again, this shows that Bk⊂Rk. Let us first suppose that c/∈{1/λT−1,λT−1}and fix x∈X0k. Let us show that for each (j,j)∈{1, ,T−1}2, signDj(x;b)=signDj(x;b)=0. (A.3) If c∈(−∞,0 ),wehave cλT−1<···<cλ 2<c<0<1<λ 2<···<λ T−1. (A.4) If c∈(0, 1/λT−1),wehave 0<c<cλ 2<···<cλ T−1<1<λ 2<···<λ T−1. (A.5)
1120 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) Else, c∈(λT−1,+∞)and we have 1<λ 1<···<λ T−1<c<cλ 2<···<cλ T−1. (A.6) Let pjdenote the number of transpositions (i.e., permutations exchanging two elements, leaving the others fixed) needed to sort ˜ λj:=(λj,c,cλ2,,cλT−1)in ascending order. It is clear from equations (A.4)–(A.6)thatpj=pj=pfor all (j,j)∈{1, ,T−1}2. Let ˜ λsj denote the sorted version of ˜ λjand define Dj(x;b,λ)=:det⎛ ⎜ ⎜ ⎜ ⎜ ⎝ expλjx 1β0 expλjx Tβ0 expλ1x 1b expλ1x Tb . . . expλT−1x 1b expλT−1x Tb ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, so that Dj(x;b)=Dj(x;b,λ). Because x∈X0k,wehave Dj(x;b,λ)=det⎛ ⎜ ⎜ ⎜ ⎜ ⎝ exp(λjxk,1β0k) exp(λjxk,Tβ0k) exp(cλ1xk,1β0k) exp(cλ1xk,Tβ0k) . . . exp(cλT−1xk,1β0k) exp(cλT−1xk,Tβ0k) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠=Djx;β0,˜ λj. Hence, for all j∈{1, ,T−1}, sgnDj(x;b)=sgnDjx;β0,˜ λj=(−1)psgnDjx;β0,˜ λsj . Now, let pbe the number of pairwise coordinates permutations needed to sort the vector (x 1β0,,x Tβ0)in ascending order, and let xsdenote a rearrangement of xsuch that xs 1β0<···<x s Tβ0. It follows that, for all j∈{1, ,T−1}, sgnDj(x;b)=(−1)psgnDjx;β0,˜ λsj =(−1)p+psgnDjxs;β0,˜ λsj =(−1)p+p, where the last equality follows by properties of Chebyshev systems. The last equality implies that (A.3)holds.Hence,x∈D(b), implying P(X∈D(b)) >0. Finally, consider the case where b=cβ0with c∈{1/λT−1,λT−1}. By continuity of the determinant and (A.3), we either have 0 ≤minj=1,,T−1Dj(x;b)≤maxj=1,,T−1Dj(x;b) or 0 ≥maxj=1,,T−1Dj(x;b)≥minj=1,,T−1Dj(x;b).Moreover,DT−1(x;β0/λT−1)= 0 and D1(x;λT−1β0)= 0. Therefore, whatever the value of c(1/λT−1or λT−1), we have x∈D(b). Then, again, P(X∈D(b)) >0. The result follows. b. |B|<∞. Because |B|≤K k=1|Bk|, it suffices to prove that for each k,|Bk|<∞.Fixk.If β0k=0, then Bk={0}and we have nothing to prove. Otherwise, let b=(b1,,bK)∈B and fix x=(x1,,xT)∈X0k.Letc∈{0}∪(1/λT−1,λT−1)be such that bk=cβ0k.By
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1121 equation (A.2), we have T−1 j=1aj(x)Dj(x;b)=0, where aj(x)is defined by (A.1). Moreover, by definition of X0k,wehaveDj(x;b)=Dj(x;cβ0).Thencsatisfies T−1 j=1 aj(x)Dj(x;cβ0)=0, (A.7) Developing Dj(x;cβ0)with respect to the first line, and using the definition of the determinant, we obtain T t=1 (−1)t+1 T−1 j=1 aj(x)expλjx tβ0 σ∈St ε(σ)exp! s=t λσ(s)x sβ0"c=0, (A.8) where Stis the set of bijections from {1, ,T}\{t}to {1, ,T−1}and ε(σ)denotes the parity of σ(we can assimilate σto a permutation by assimilating {1, ,T}\{t}with {1, ,T−1}, keeping the natural ordering of both sets). The left-hand side of (A.8)is afunctionofcof the form K k=1dkexp(bkc),withK≤T!(the inequality arises because some coefficients in the exponential monomials may be equal). Let us show that dk=0 foratleastonek.First,remarkthatx tβ0=xk,tβ0,k.Then,because|{xk,1,,xk,T}|=T and β0k=0, we can assume without loss of generality, up to a rearrangement of periods that x 1β0<···<x Tβ0.LetItbe the element of Stsuch that It(s)=s−1{s≥t+1}.Bythe rearrangement inequality, for all σ∈St\{It}, s=t λσ(s)x sβ0< s=t λIt(s)x sβ0. Moreover, for all t∈{1, ,T},let g(t):= s=t λIt(s)x sβ0. Because It(s)=It−1(s)for all t>1ands≤t−2ors≥t+1, we have g(t)−g(t−1)=λt−1x t−1β0−λt−1x tβ0<0. Hence, the exponential monomial with highest coefficient in (A.8)is exp! s=1 λs−1x sβ0"c and we can obtain it only by letting t=1andσ=I1. Because aj(x)>0forallj,the coefficient of this monomial is T−1 j=1aj(x)exp(λjx 1β0)>0. Therefore, at least one dk in the exponential polynomial K k=1dkexp(bkc)satisfies dk=0. Then, by Lemma A.1, the equation K k=1dkexp(bkc)=0 has at most T!−1 solutions. Thus, |Bk|≤T!−1. The result follows.
1122 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) Part 2 The point identification of relative marginal effects is obvious given the other results, which we prove in turn. a. Equation ( 2.5)holds. Let us define, for all b∈RK∗, X1(b)=x=(x1,,xT)∈Supp(X):∃(s,t)∈{1, ,T}2:x sb=x tb,x sβ0=x tβ0, and ∀s,t∈{1, ,T}2,s=t,s,t={s,t}:x sb=x tb. The proof of is divided into three steps. First, we prove that X1(b)⊂D(b),forallb∈ RK∗\lin(β0). In a second step, we prove that B⊂lin(β0). Finally, the third step shows that B⊂R. First step: X1(b)⊂D(b)for all b∈RK∗\lin(β0). Let x∈X1(b)and (s,t)be as in the definition of X1(b). Developing Dj(x;b)according to the first row, we obtain, for all j∈{1, ,T−1}, Dj(x;b)= T =1 (−1)+1expλjx β0D−{1,} j(x;b), where D−{1,} j(x;b)denotes the determinant of the matrix in Dj(x;b)once its first row and th column have been removed. Remark that, for all j∈{1, ,T−1},forall∈ {1, ,T}\{s,t},D−{1,} j(x;b)=0, and D−{1,t} j(x;b)=(−1)|s−t|−1D−{1,s} j(x;b). As a result, Dj(x;b)=(−1)s+1expλjx sβ0D−{1,s} j(x;b)+(−1)t+1expλjx tβ0D−{1,t} j(x;b) =D−{1,s} j(x;b)(−1)s+1expλjx sβ0+(−1)t+1expλjx tβ0(−1)|s−t|−1 =D−{1,s} j(x;b)(−1)s+1expλjx sβ0−expλjx tβ0, wherewehaveused(−1)|s−t|+t=(−1)s.Now,D−{1,s} j(x;b)does not depend on jand by definition of Chebyshev systems, D−{1,s} j(x;b)= 0. Also, the sign of the term inside brackets is equal to the sign of (xs−xt)β0, and thus does not depend on j.Hence,for all (j,j)∈{1, ,T−1}2, sgnDj(x;b)=sgnDj(x;b)=0, which shows that x∈D(b). Second step: B⊂lin(β0). Fix b/∈lin(β0),b=0andletusprovethatP(X∈D(b)) >0. The result will then follow by Lemma 2.4. Suppose without loss of generality that (s,t)in Assumption 4is equal to (1, 2).By that assumption, there exists x:=(x,x,x 3..., x T)∈Supp(X)and a neighborhood Vof xincluded in Supp(X). Since band β0are not collinear, there exists (u 1,u 2)∈R2Ksuch
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1123 that (u1−u2)b=0and(u1−u2)β0=0. Moreover, up to replacing (u 1,u 2)by c(u 1,u 2) with c=0, (u 1,u 2)canbechosenofarbitrarilysmallnorm. Now, let x1=x2=xand A(u1,u2)=u 3,,u T∈RK(T−2):∀(s,t)∈{1, ,T}2,s=t,{s,t}={1, 2}: (us−ut+xs−xt)b=0, The set A(u1,u2)is dense as the intersection of open, dense subsets of RK(T−2).Hence, there exists (u 3,,u T)∈A(u1,u2)with arbitrarily small norm. Then we can ensure that u:=(u 1,,u T)satisfies x∗:= x+u∈ V. Moreover, by construction, x∗∈X1(b). Then Step 1 implies x∗∈D(b)and Dj(x;b)=0forallj. By continuity of the map x→ Dj(x;b)and Assumption 4, there exists a neighborhood of x∗,V⊂D(b)such that P(X∈ V)>0. Hence, P(X∈D(b)) >0. Third step: B⊂R. We just have to prove that if b=cβ0with c∈(−∞,1/λT−1]∪[λT−1,+∞)and c=0 (since β0=0), then b/∈B. The reasoning is exactly the same as in Part 1.a, with just one change: Instead of considering x∈X0k,weconsiderx∈X0,with X0:=x∈Supp(X):x 1β0,,x Tβ0=T. b. |B|≤T!−1. The reasoning is exactly the same as in Part 1.b, with just two changes. First, we reason directly on B, not on Bk. Second, instead of considering x∈X0k,weconsiderx∈X0. c. |B|≤2when T=3. For any b=cβ0∈B, we have, as in equation (A.7), a1(x)D1(x;cβ0)+a2(x)D2(x;cβ0)=0 (A.9) for almost all x∈Supp(X). Suppose there exist three distinct solutions 1, c1,c2to equation (A.9), with 1/λ2<c 1<c 2<λ 2.Multiplyequation(A.9), evaluated at c=c1,by D2(x;c2β0). Similarly, multiply equation (A.9), evaluated at c=c2,byD2(x;c1β0).Subtracting the two expressions, we obtain, since a1(x)>0, D1(x;c1β0)D2(x;c2β0)−D1(x;c2β0)D2(x;c1β0)=0. (A.10) For any x∈X0,letut:=x tβ0. Fixing u2and u3,(A.10) may be written as P(u1):= 13 k=1 αkexp(ζku1)=0, (A.11) where the αkand ζkare functions of (u2,u3). Suppose first that c2>1. Some tedious algebra shows that the smallest ζkis 1 +c1, and its associated coefficient is equal to αk=expc2(u2+λ2u3)−expc2(u3+λ2u2) ×expλ2(u2+c1u3)−expλ2(u3+c1u2).
1124 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) Because u2= u3(as x∈X0), αk= 0. Hence, Pis nonzero and by Lemma A.1, it has at most 12 zeros. However, under Assumption 4.2 and the second part of Assumption 4.3, SuppX 1β0|X 2β0=u2,X 3β0=u3\{u2,u3}>12. Thus, in view of (A.11), Phas strictly more than 12 zeros, a contradiction. Second, suppose that c2<1. Then the largest ζkis λ2(1+c2), and its associated coefficient is equal to αk=−exp(u2+c2u3))−exp(u3+c2u2) ×expc1(u2+λ2u3)−expc1(u3+λ2u2). Again, αk=0 and we reach a contradiction as before. The result follows. A.5 Theorem 2.6 Part 1 Let us suppose that P(|{X1,,XT}|=T)=0. Let T1and T2>T 1denote the two random dates, functions of Xonly, such that XT1=XT2almost surely. For all t∈ {1, T},letetdenote the vector of T−1 zeros and a 1 at coordinate t.Letf(x;b):= E[m(Y,X;b)|X=x]. By definition, f(X;b)= y∈{0,1}T P(Y=y|X)m(y,X;b). (A.12) Moreover, almost surely, P(Y=eT1|X)=#FX T1β0+γ t=T11−FX tβ0+γdFγ|X(γ) =#FX T2β0+γ t=T21−FX tβ0+γdFγ|X(γ) =P(Y=eT2|X). (A.13) Next, remark that the matrices in MT1(X;b)and MT2(X;b)have the same columns but in different order, with T2−T1−1 transpositions needed to obtain the same ordering. Thus, by definition of the determinant, MT1(X;b)=−MT2(X;b), which implies m(eT1,X;b)=−m(eT2,X;b). (A.14) Moreover, for all s/∈{T1,T2},m(es,X;b)=0 because Msincludes two identical columns (given that XT1=XT2). Finally, if tyt= 1, we also have m(y,X;b)=0. These last points, combined with (A.12)–(A.14), imply f(b)=0. Thus, b∈Band the result follows. Part 2 The proof is in two steps. First, we show that for all b∈R, sgnD1(X;b)=−sgnD2(X;b)a.s. (A.15)
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1125 Second, we show that whenever (A.15) holds, we can construct a distribution of γ|X such that (2.4) holds. The result then follows. First step: ( A.15)holds. First, the result holds for b=β0since then Dj(X;b)=0forj∈{1, 2}.Otherwise, fix b=cβ0∈Rand let ˜ λ:=(1, c,cλ2)and ˇ λ:=(λ2,c,cλ2).Letp(resp., p)denotethe minimal number of pairwise coordinate permutations needed to sort the vector ˜ λ(resp., ˇ λ)andlet˜ λs(resp., ˇ λs) be the corresponding vector, sorted in ascending order. If c∈ (1/λ2,1 ),wehavep=1andp=2, whereas if c∈(1, λ2),p=0, and p=1. Hence, in all cases, p=p+1. Now, for any x∈Supp(X),noticethat D1(x;b,λ)=D1(x;β0,˜ λ)=(−1)pD1x;β0,˜ λs, (A.16) D2(x;b,λ)=D2(x;β0,ˇ λ)=(−1)pD2x;β0,ˇ λs. (A.17) Let p be the minimal number of pairwise coordinates permutations needed to sort the vector (x 1β0,x 2β0,x 3β0)in ascending order, and let xsdenote the corresponding vector, that is, such that x s1β0≤x s2β0≤x s3β0.Then D1x;β0,˜ λs=(−1)pD1xs;β0,˜ λs, (A.18) D2x;β0,ˇ λs=(−1)pD2xs;β0,ˇ λs. (A.19) Now, by properties of Chebyshev systems, D1(xs;β0,˜ λs)and D2(xs;β0,ˇ λs)are both nonnegative. Moreover, both are nonzero if and only if |{x 1β0,x 2β0,x 3β0}|=3. The result follows by (A.16)–(A.19)and(−1)p=−(−1)p. Second step: if ( A.15) holds, there exists a distribution of γ|Xsuch that ( 2.4)holds. Let us define ai(γ,x)=wiexp(λiγ) T t=11+ T−1 j=1 wjexpλjx tβ0+γ. (A.20) Then we have Em(Y,X,b)|X=x=Ea1(γ,x)|X=xD1(x,b) +Ea2(γ,x)|X=xD2(x,b). (A.21) Hence, if D1(x,b)=D2(x,b)=0, any distribution of γ|X=xsatisfies E[m(Y,X,b)|X= x]=0. Now, suppose that sgn(D1(x,b)) =−sgn(D2(x,b)) =0. Then R(x):=−D1(x,b)/ D2(x,b)>0. Let us define γ0:=lnw1R(x)/w2 λ2−1. Consider for γ|X=xthe Dirac distribution at γ0. Then, from (A.21), we obtain that E[m(Y,X,b)|X=x]=0. The result follows.
1126 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) A.6 Theorem 3.1 Let us first summarize the proof. We link the current model with a “complete” model where γis also observed. This model is fully parametric, and thus can be analyzed easily. Specifically, we show in a first step that this complete model is differentiable in quadratic mean (see, e.g., van der Vaart (2000), pp. 64–65 for a definition) and has a nonsingular information matrix. In a second step, we establish an abstract expression for the semiparametric efficiency bound. This expression involves in particular the kernel Kof the conditional expectation operator g→E[g(X,Y)|X,γ]. In a third step, we show that K=(x,y)→q(x)m(x,y;β0),Eq2(X)<∞. (A.22) The fourth step of the proof concludes. First step: The complete model is differentiable in quadratic mean and has a nonsingular information matrix Let p(y|x,g;β):=P(Y=y|X=x,γ=g;β). We check that the conditions of Lemma 7.6 in van der Vaart (2000) hold. Under Assumptions 1–2,wehave p(y|x,g;β)= t:yt=1 Fx tβ+g t:yt=01−Fx tβ+g, where Fis C∞on Rand takes values in (0, 1). This implies that β→ lnp(y|x,g;β)is differentiable. Let Sβ:=∂lnp(Y|X,γ;β)/∂β and let Sβk denote its kth component. We prove that E[S2 βk]<∞. First, remark that Sβk = T t=1 Xk,tFX tβ+γ FX tβ+γ1−FX tβ+γYt−FX tβ+γ. Next, we have |Sβk|≤ T t=1|Xk,t|FX tβ+γ FX tβ+γ1−FX tβ+γ = T t=1|Xk,t| T−1 j=1 wjλjeλj(X tβ+γ) T−1 j=1 wjeλj(X tβ+γ) ≤λT−1 T t=1|Xk,t|, (A.23) where we have used the triangle inequality and |Yt−F(X tβ+γ)|≤1toobtainthefirst inequality. Equation (A.23) and Assumption 1.2 imply that E[S2 βk]<∞.Bythedominated convergence theorem and again (A.23), β→ E[SβS β]is continuous. Therefore,
Quantitative Economics 14 (2023) Fixed-effects binary choice models 1127 the conditions in Lemma 7.6 in van der Vaart (2000) hold, and the complete model is differentiable in quadratic mean. Moreover, ESβS β=EV(Sβ|X,γ)= T t=1 E! FX tβ+γ FX tβ+γ1−FX tβ+γ"2 XtX t. Then, if for some v∈RK,vE[SβS β]v=0, we would have X tv=0 almost surely for all t∈ {1, ,T}. By Assumption 5.1, this implies v=0. Hence, the information matrix E[SβS β] is nonsingular. Second step: Vdepends on the orthogonal projection of E[Sβ0|X,Y]on KLet ψ= ( ψ1,, ψK)denote the efficient influence function, as defined on page 363 of van der Vaart (2000). Then V=E[ ψ ψ]and E[ ψ]=0. Let S:=span(Sβ0),G:={q:E[q2(X,γ)] < ∞,E[q(X,γ)] =0}and for any closed convex set Aand any h=(h1,,hK),letAdenote the orthogonal projection on Aand A(h)=(A(h1),,A(hK)).Byequation (25.29), Lemma 25.34 (since the complete model is differentiable in quadratic mean by the first step) and the same reasoning as in Example 25.36 of van der Vaart (2000), ψis the function of (X,Y)of minimal L2-norm satisfying χ=S+G( ψ), (A.24) where χis the efficient influence function of the large model. Because this large model is parametric, we have χ=ESβ0S β0−1Sβ0. (A.25) Equation (A.24)impliesE[( ψ−χ)χ]=0. Thus, defining β0=E[Sβ0|Y,X],weget E ψ β0=E ψS β0=Id, (A.26) Moreover, because E[Sβ0|X,γ]=0, Sand Gare orthogonal. Thus, (A.24) is equivalent to S(χ)=S( ψ)and G(χ)=G( ψ).Moreover,(A.25) implies that G(χ)=0. Hence, ψ∈KK. Now, because Kis an orthogonal projector, we have E ψK(β0)=EK( ψ) β0=E ψ β0=Id, where the last equality follows by (A.26). Hence, if K(β0)λ=0 a.s., we would have λ=0. In other words, E[K(β0)K(β0)]is nonsingular. Now, consider the set F:=EK(β0)K(β0)−1K(β0)+v:EvK(β0)=0. Fis thus the set of vector-valued functions ψsatisfying the equation E[ψK(β0)] =Id. Hence, ψbeing the element of Fwith minimum L2-norm, we obtain ψ=EK(β0)K(β0)−1K(β0). Finally, because V=E[ ψ ψ], V=EK(β0)K(β0)−1. (A.27)
1128 Davezies, D’Haultfœuille, and Mugnier Quantitative Economics 14 (2023) Third step: (A.22)holds Let r∈Kand let us prove that r(y,x)=q(x)m(y,x;β0)for some q. First, by definition of K, we have for almost all (g,x)∈Supp(γ,X), 0=r(0, 0, 0),x0+r(1, 0, 0),x0Gx 1β0+g+r(0, 1, 0),x0Gx 2β0+g +r(0, 0, 1),x0Gx 3β0+g+r(1, 1, 0),x0Gx 1β0+gGx 2β0+g +r(1, 0, 1),x0Gx 1β0+gGx 3β0+g +r(0, 1, 1),x0Gx 2β0+gGx 3β0+g +r(1, 1, 1),x0Gx 1β0+gGx 2β0+gGx 3β0+g. (A.28) Let at:=x tβ0for t∈{1, 2, 3}, and for the sake of conciseness, let us remove the dependence of ron x. Then, using Assumption 2, we obtain, for almost all (g,x), 0=A1e0×g+A2eg+A3eλ2g+A4e2g+A5e2λ2g+A6e(1+λ2)g+A7e3g+A8e(2+λ2)g +A9e(1+2λ2)g+A10e3λ2g, where A1:=r(0, 0, 0), A2:=w1r(1, 0, 0)ea1+r(0, 1, 0)ea2+r(0, 0, 1)ea3, A3:=w2r(1, 0, 0)eλ2a1+r(0, 1, 0)eλ2a2+r(0, 0, 1)eλ2a3, A4:=w2 1r(1, 1, 0)e(a1+a2)+r(1, 0, 1)e(a1+a3)+r(0, 1, 1)e(a2+a3), A5:=w1w2r(1, 1, 0)ea1+λ2a2+ea2+λ2a1+r(1, 0, 1)ea1+λ2a3+ea3+λ2a1 +r(0, 1, 1)ea2+λ2a3+ea3+λ2a2, A6:=w2 2r(1, 1, 0)eλ2(a1+a2)+r(1, 0, 1)eλ2(a1+a3)+r(0, 1, 1)eλ2(a2+a3), A7:=w3 1r(1, 1, 1)ea1+a2+a3, A8:=w2 1w2r(1, 1, 1)ea1+a2+λ2a3+ea1+λ2a2+a3+eλ2a1+a2+a3, A9:=w1w2 2r(1, 1, 1)ea1+λ2(a2+a3)+ea2+λ2(a1+a3)+ea3+λ2(a1+a2), A10 :=w3 2r(1, 1, 1)eλ2(a1+a2+a3). Since λ2=2 is excluded by assumption, there are three cases left depending on the number of different exponents in equation (A.28). First, we consider λ2/∈{3/2, 3}. By Lemma A.1 and because |Supp(γ|X)|≥10, we obtain Ak=0forallk∈{1, ,10 }.A1=A7=0 imply that r(0, 0, 0)=r(1, 1, 1)=0. Next, A4=A6=0 implies that either r(1, 0, 1)=r(1, 1, 0)=r(0, 1, 1)=0or $r(1, 1, 0)=−r(1, 0, 1)eλ2(a3−a2)−r(0, 1, 1)eλ2(a3−a1), r(1, 1, 0)=−r(1, 0, 1)e(a3−a2)−r(0, 1, 1)e(a3−a1).(A.29)