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Identification in a Binary Choice Panel Data Model with a Predetermined Covariate
{JEL Codes:} C23, C33
{Keywords:} {Feedback, Panel Data, Incidental Parameters, Partial Identification.}
\pagenumbering{arabic} \onehalfspacing
Empirical researchers utilizing panel data generally maintain the assumption that covariates are strictly exogenous: realized values of past, current, and future explanatory variables are independent of the time-varying structural disturbances or “shocks”.\footnote{Dependence between the covariates and the time-invariant heterogeneity -- the so-called “fixed effects” -- is, of course, allowed.} In many settings this assumption is unrealistic. If the covariate is a policy, choice or dynamic state variable, then agents may adjust its level in response to past shocks (as when, for example, a firm adjusts its current capital expenditures in response to past productivity shocks).
When strict exogeneity is untenable, sequential exogeneity -- sometimes called predeterminedness -- may be palatable. A predetermined covariate varies independently of current and future time-varying shocks, but general feedback, or dependence on past shocks, is allowed. Assumptions of this type play an important role in, for example, production function estimation (Olley_Pakes_EM96, Blundell_Bond_ER00).
In two seminal papers, arellano1991some and Arellano_Bover_JE95, Manuel Arellano and his collaborators presented foundational analyses of questions of identification, estimation, efficiency and specification testing in linear panel data models with feedback. Today such models are both well-understood and widely-used (see arellano2003panel for a textbook review).
In contrast, the properties of nonlinear models with feedback are much less well-understood. In this paper we study binary choice. Most existing work in this area focuses on the case where the covariate is either strictly exogenous or a lagged outcome. Under strict exogeneity, rasch1960studies and andersen1970asymptotic show that the coefficient on the covariate is point-identified using two periods of data when shocks are logistic. chamberlain2010binary provides conditions under which the logit case is the only one admitting point-identification with two periods (davezies2020fixed provide extensions of this result to the case of $T>2$). In the dynamic case, where the covariate is a lagged outcome, Cox_JRSS58, chamberlain1985heterogeneity and honore2000panel derive conditions for point-identification of the coefficient on the lagged outcome in the logit case, while honore2006bounds show how to compute bounds on coefficients for probit and other models.
Results for binary choice panel models with predetermined covariates are limited. Chamberlain_JOE2022 studies identification and semiparametric efficiency bounds in a class of non-linear panel data models with feedback; he provides both positive and negative results. In an hitherto unpublished section of an early draft of that paper (Chamberlain_1993), he proves that the coefficient on a lagged outcome is not point-identified in a dynamic logit model when only three periods of outcome data are available. Arellano_Carrasco_JOE2003 and honore2002semiparametric study binary choice models with predetermined covariates. Arellano_Carrasco_JOE2003 assume that the dependence between the time-invariant heterogeneity and the covariates is fully characterized by its conditional mean given current and lagged covariates. honore2002semiparametric assume that one of the covariates is independent of the individual effects conditional on the other covariates. In a recent contribution, pigini2022conditional show that one can accommodate specific forms of feedback while maintaining point-identification in binary choice models with pretermined covariates.\footnote{In this paper we focus on panel data with a fixed number $T$ of time periods. The large-T literature has also considered models with dynamics and feedback, see for example carro2007estimating, hahn2002asymptotically, and fernandez2009fixed.}
In what follows we pose two questions. First, under what conditions is the coefficient on a predetermined covariate in a binary choice panel data model point-identified? Second, when the coefficient is only set-identified, how extreme is the failure of point-identification; i.e., what is the width of the identified set?
Our analyses leave the dependence between the (time-invariant) unit-specific heterogeneity and the covariates unrestricted. We focus on the special case of a single binary predetermined covariate, leaving the feedback process from lagged outcomes, covariates and the unit-specific heterogeneity onto future covariate realizations fully unrestricted. This is a substantial relaxation of the strict exogeneity assumption.
Regarding point-identification, we provide a simple condition on the model which guarantees that point-identification fails when $T$ periods of data are available (and $T$ is fixed). The condition is satisfied in most familiar models of binary choice, including the logit one. This finding contrasts with the prior work on logit models cited above, where point-identification typically holds for a sufficiently long panel. As a notable exception, the exponential binary choice model introduced by Al-Sadoon_et_al_ER2017 does not satisfy our condition. In fact, point-identification holds in that case.
Regarding identified sets, we first show that sharp bounds on the coefficient can be computed using linear programming techniques. Our method builds on honore2006bounds, however, in contrast to their work, we allow for heterogeneous feedback. While the regressor coefficient is our main target parameter, we also derive the identified set for an average partial effect. This set can be computed using linear programming techniques as well.
Second, we numerically compute examples of identified sets. We find that, relative to the strictly exogenous case, allowing for a predetermined covariate tends to increase the width of the identified set. However, our calculations also suggest that the identified set can remain informative under predeterminedness, even in panels with as few as two periods, for both the coefficient and the average partial effect. Finally, as is true under strict exogeneity, the widths of the identified sets decrease quickly as the number of periods increases. These observations are based upon sets computed under a particular data generating processe (DGP). It is possible that identified sets may be larger under certain types of feedback.
The outline of the paper is as follows. In Section (ref) we present the model. In Section (ref) we provide a condition that implies that the common parameter in this model is not point-identified when $T=2$. In Section (ref) we show that our condition implies failure of point-identification for all (finite) $T$. In Section (ref) we show how to compute identified sets on coefficients and average partial effects, and we report the results of a small set of numerical illustrations. In Section (ref) we describe potential restrictions one could impose on the feedback process. These restrictions may restore point-identification or shrink the identified set. We conclude in Section (ref). Proofs are contained in the appendix. Lastly, \href{https://github.com/kevindano/Bonhomme-Dano-Graham-SERIES}{replication codes} are available as supplementary material.
Available to the econometrician is a random sample of $n$ units, each of which is followed for $T\geq2$ time periods. We focus on short panels, and keep $T$ fixed. The sampling process asymptotically reveals the joint distribution of $(X_1,\ldots,X_T,Y_1,\ldots,Y_T)$.
For any sequence of random variables $Z_{t}$ and any non-stochastic sequence $z_t$, we use the shorthand notation $Z^{t:t+s}=(Z_{t}',...,Z_{t+s}')'$ and $z^{t:t+s}=(z_{t}',...,z_{t+s}')'$. In addition, we simply denote $Z^t=Z^{1:t}$ and $z^{t}=z^{1:t}$ when the subsequence starts in the first period.
Let $Y_{it}\in\{0,1\}$ and $X_{it}\in\{0,1\}$ denote a binary outcome and a binary covariate, respectively. We assume that $$\Pr(Y_{it}=1\,|\, Y_{i}^{t-1},X_{i}^{t},\alpha_i;\theta)=F(\theta X_{it}+\alpha_i), \quad t=1,\ldots,T,$$ where $\alpha_i\in{\cal{S}}\subset \mathbb{R}$ is a scalar individual effect, $F(\cdot)$ is a known differentiable cumulative distribution function, and $\theta\in\Theta$ is a scalar parameter.
Let $\pi_{x_1}(\alpha)$ denote the distribution of heterogeneity given the initial condition $X_1=x_1$; i.e., the distribution of $\alpha_i\,|\, X_{i1}$. We leave this distribution unrestricted on ${\cal{S}}$. When ${\cal{S}}$ is a discrete subset of the real line, $\pi_{x_1}(\alpha)$ belongs to the unit simplex on ${\cal{S}}$, however it is otherwise unrestricted. We denote as $\Pi$ the collection of all $\pi_{x_1}(\alpha)$, for all $x_1\in\{0,1\}$ and $\alpha\in{\cal{S}}$.
For each $t\geq 2$, let $$\Pr\left(X_{it}=1\,|\, Y_{i}^{t-1}=y^{t-1},X_{i}^{t-1}=x^{t-1},\alpha_i=\alpha\right)=G^t_{y^{t-1},x^{t-1}}(\alpha), \quad t=2,\ldots,T,$$ denote the feedback process through which lagged outcomes, past covariates and heterogeneity affect the current covariate. We leave this distribution unrestricted as well. We denote as $G\in {\cal{G}}_T$ the collection of all $G^t_{y^{t-1},x^{t-1}}(\alpha)$, for all $t\in\{2,...,T\}$, $y^{t-1}\in\{0,1\}^{t-1}$, $x^{t-1}\in\{0,1\}^{t-1}$, and $\alpha\in{\cal{S}}$.
The (integrated) likelihood function conditional on the first period's covariate is
for some (discrete or continuous) measure $\mu$ on ${\cal{S}}$.
A key feature of a model with predetermined covariates is the dependence of the feedback process on lagged outcomes, as reflected in the dependence of $G^t$ on $y^{t-1}$ in ((ref)). When this dependence is ruled out, the covariate is strictly exogenous, and the likelihood function simplifies.\footnote{Under strict exogeneity, the likelihood function factors as
where $\pi_{x^T}(\alpha)$ denotes the distribution of heterogeneity given all periods' covariates $x_1,...,x_T$. } Dynamic responses of covariates to lagged outcome realizations are central to many economic models, including those where $X_{it}$ is a choice variable, policy, or a dynamic state variable.
For any $(\theta,\pi,G)\in\Theta\times\Pi\times{\cal{G}}_T$, and any $(y^T,x^{2:T})\in \{0,1\}^{2T-1}$, let $Q_{x_1}(y^T,x^{2:T};\theta,\pi,G)$ denote the right-hand side of ((ref)). Moreover, let $Q_{x_1}(\theta,\pi,G)$ denote the $2^{2T-1}\times 1$ vector collecting all those elements, for all $(y^T,x^{2:T})\in \{0,1\}^{2T-1}$. Finally, let $Q(\theta,\pi,G)$ denote the $2^{2T}\times 1$ vector stacking $Q_{1}(\theta,\pi,G)$ and $Q_{0}(\theta,\pi,G)$. For a given (population) $(\theta,\pi,G)\in \Theta\times\Pi\times{\cal{G}}_T$, we define the identified set of $\theta$ as
The set in (ref) includes all $\widetilde\theta\in\Theta$ where, for that $\widetilde\theta$, it is possible to find a heterogeneity distribution $\widetilde\pi\in\Pi$, and a feedback process $\widetilde G\in \cal{G}_T$, such that the resulting conditional likelihood assigns the same probability to each of the $2^{2T-1}$ possible data outcomes as the true one (given both $X_{i1}=0$ and $X_{i1}=1$).
In the first part of the paper, we provide conditions on the model under which $\Theta^I$ is not a singleton. This corresponds to cases where $\theta$ is not point-identified. In the second part of the paper, we report numerical calculations of $\Theta^I$ under particular DGPs.
Our focus on $\theta$ is motivated by the extensive literature on the identification of coefficients in binary choice models. However, in applications, average effects may also be of interest. In the second part of the paper, we will also report numerical calculations of identified sets for an average partial effect associated with a change in the binary predetermined covariate.
We first present an analysis of point-identification in the two-period case, since this leads to simple and transparent calculations. In the next section, we will then generalize this result to accommodate $T\geq 2$ periods.
To keep the formal analysis simple, in this section and the next we assume that $\alpha_i$ takes a finite number of values, with known support points.
Assumption (ref) makes the model fully parametric. However this is not a limitation as our aim in this section and the next is to derive conditions under which point-identification fails. The conditions we provide will require sufficiently many support points.\footnote{The analysis is essentially unchanged if one instead assumes that $\mu=\sum_{k=1}^K\lambda_k\delta_{\underline{\alpha}_k}$, for some $\lambda_k>0$.}
We rely on the parameterization given by the $2(K-1)\times 1$ vector $\pi=(\pi_1',\pi_0')'$, where, for all $x_1\in\{0,1\}$, $\pi_{x_1}=(\pi_{x_1}(\underline{\alpha}_1),\ldots,\pi_{x_1}(\underline{\alpha}_{K-1}))'$ and $\pi_{x_1}(\underline{\alpha}_K)=1-\sum \limits_{k=1}^{K-1}\pi_{x_1}(\underline{\alpha}_k)$. The vector $\pi\in\Pi$ is unrestricted, except for the fact that $\pi_{x_1}(\alpha)$, for $\alpha\in{\cal{S}}$, belongs to the unit simplex. This parameterization handles the fact that probability mass functions sum to one.
We next impose the following assumption on the population parameters.
Assumption (ref) places restrictions on the underlying parametric binary choice model and heterogeneity distribution. It rules out heterogeneity distributions that induce a point mass of “stayers” (i.e., units with such extreme values of $\alpha$ that they either always take the binary action or they never do).\footnote{In some microeconometric datasets a substantial fraction of units never alter their value of $X_t$. For example, in CardEM1996 few workers join or leave a union during the sample period.} Assumption (ref) also rules out the “staggered adoption” design common in difference-in-differences analyses. Exploring the implications of non-interior feedback processes is left for future work.
Finally, we assume that the parameter point is regular in the sense of rothenberg1971identification.
The assumption of regularity is standard in the literature on the identification of parametric models (rothenberg1971identification). If $F(\cdot)$ is analytic, the irregular points of $\nabla Q(\theta,\pi,G)$ (i.e., the points $(\theta,\pi,G)$ such that Assumption (ref) is not satisfied) form a set of measure zero (bekker2001identification). Thus, Assumption (ref) is satisfied almost everywhere in the parameter space in many binary choice models, including the probit and logit ones.
We aim to provide a simple condition under which point-identification of $\theta$ fails when $T=2$. We start by observing that, when $T=2$, the $2^{2T-1}=8$ model outcome probabilities given $X_{i1}=x_1$ are
which, given the structure of the model, coincide with
With this notation in hand we present the following lemma.
The proof of Lemma (ref) exploits the fact that, if $\theta$ is point-identified, then it is also locally point-identified. Together with the assumption that the parameter is regular, this allows us to apply a result of bekker2001identification regarding the identification of subvectors, which guarantees the existence of some $x_1\in\{0,1\}$ such that $\nabla_{\theta'} Q_{x_1}$ does not belong to the range of the matrix $
$. We then show, using (\ref{eq_Q_T2}), that this implies the existence of $\phi_{x_1}\neq 0$ such that (ref) and (ref) hold.
When the population parameter $\theta$ is point-identified, Lemma (ref) suggests a method-of-moments approach to estimation. In such settings, $\phi_{X_{i1}}\left(Y_{i1},Y_{i2},X_{i2}\right)$ will generally be a non-trivial function of $\theta$. Let $\phi_{X_{i1}}\left(Y_{i1},Y_{i2},X_{i2};\theta\right)$ be this function. Next, note that condition $\eqref{eq_1}$ in Lemma (ref) corresponds to the conditional moment restriction
while -- continuing to maintain $\eqref{eq_1}$ -- equation $\eqref{eq_2}$ implies the additional requirement that
Analog estimators in point-identified models with feedback, based on these observations, are explored in our companion paper BDG_WP.
This formulation clarifies that a necessary condition for point-identification of $\theta$ is the existence of a non-zero moment function, $\phi_{X_{i1}}\left(Y_{i1},Y_{i2},X_{i2};\theta\right)$, with a mean that is invariant to $X_{i2}$ given $\alpha_{i}$ and the past (i.e., the first period's covariate and outcome). Such a moment function is “feedback robust”, in the sense that it remains valid across all possible feedback processes. This is the content of condition $\eqref{eq_1}$ in Lemma (ref), while $\eqref{eq_2}$ imposes a similar invariance to the distribution of unobserved heterogeneity.
To show that point-identification fails, our focus here, we need to show that no such non-zero moment function exists. It turns out that there is a very simple condition for this in our model. Specifically, from Lemma (ref) we obtain the following corollary.
Corollary (ref) shows that a necessary condition for identification of $\theta$ is that $1$, $F(\alpha)$, and $F(\theta+\alpha)$, for $\alpha\in {\cal{S}}$, are linearly dependent. This condition arises directly from condition ((ref)), which requires the existence of a moment function that is robust to unknown feedback. Indeed, one can show that $1$, $F(\alpha)$, and $F(\theta+\alpha)$ are linearly dependent if and only if there exists a non-constant function $\phi$ such that
However, the condition that $1$, $F(\alpha)$, and $F(\theta+\alpha)$ be linearly dependent is restrictive, as we show in the next subsection.\footnote{While here we focus on a discrete ${\cal{S}}$ under Assumption (ref), note that, when $\theta\neq 0$ and $F$ is strictly increasing on $\mathbb{R}$, $1$, $F(\alpha)$, and $F(\theta+\alpha)$, for $\alpha\in \mathbb{R}$, cannot be linearly dependent. If that were the case, then for some non-zero triplet $(A,B,C)$ we would have $AF(\theta+\alpha)+BF(\alpha)+C=0$ for all $ \alpha\in {\mathbb{R}}$. This would imply, by taking $\alpha\rightarrow \pm \infty$ that $C=0$ and $A+B=0$, which would then imply $A=B=C=0$ and contradict the assumption that $(A,B,C)$ is non-zero.}
Consider the logit model with a binary predetermined covariate, which corresponds to $F(u)=\frac{e^u}{1+e^u}$. In this case, the linear dependence condition of Corollary (ref) requires that, for some non-zero triplet $(A,B,C)$, $$A\frac{e^{\theta +\alpha}}{1+e^{\theta +\alpha}}+B \frac{e^{\alpha}}{1+e^{\alpha}}+C=0,\quad \text{for all } \alpha\in{\cal{S}}.$$ However, this implies $$Ae^{\theta }e^{\alpha}(1+e^{\alpha})+Be^{\alpha}(1+e^{\theta}e^\alpha)+C(1+e^{\alpha})(1+e^{\theta}e^\alpha)=0,\quad \text{for all } \alpha\in{\cal{S}},$$ which is a quadratic polynomial equation in $e^{\alpha}$. Therefore, provided that there are $K\geq 3$ values in ${\cal{S}}$, this implies $$Ae^{\theta }+Be^{\theta }+Ce^{\theta }=0,\quad Ae^{\theta }+B+(1+e^{\theta})C=0,\quad C=0,$$ which, provided that $\theta\neq 0$, entails $$A=B=C=0,$$ contradicting the assumption that $(A,B,C)$ is non-zero.
We have thus proved the following corollary.
A precedent to Corollary (ref) is given in the unpublished working paper by Chamberlain_1993 mentioned in the introduction. In the model he considers, $X_{it}=Y_{i,t-1}$ is a lagged outcome, and $T=2$ (hence, outcomes are observed for three periods). His model also includes an additional regressor: an indicator for period $t=2$.
Suppose now that, for $u\geq 0$, $F(u)=1-e^{-u}$. This corresponds to the exponential binary choice model of Al-Sadoon_et_al_ER2017. Note that here the support of $F(\cdot)$ is a strict subset of the real line. In this case, letting $$A=e^{\theta}, \,B=-1, \,C=1-e^{\theta},$$ we have $$A[1-e^{-(\theta+\alpha)}]+B[1-e^{-\alpha}]+C=0.$$ Hence the non point-identification condition of Corollary (ref) is not satisfied in the exponential binary choice model.
In fact, in this case ((ref)) and ((ref)) are satisfied for $$\phi_{x_1}(y_1,y_2,x_2;\theta)=(1-y_2)e^{\theta x_2}-(1-y_1)e^{\theta x_1},$$ and $\theta$ satisfies the conditional moment restriction $$\mathbb{E}[\phi_{X_{i1}}(Y_{i1},Y_{i2},X_{i2};\theta)\,|\, X_{i1}]=0,$$ that is,
See wooldridge1997multiplicative for several related results. Furthermore, one can show formally that $\theta$ is point-identified based on ((ref)), see Appendix (ref).
In this section we generalize our analysis to an arbitrary number of periods and state our main result.
The arguments laid out in the previous section extend to an arbitrary number of time periods, $T\geq2$. Indeed, using a similar strategy to the proof of Lemma (ref) and proceeding by induction, we obtain the following lemma.
Similarly to Lemma (ref), Lemma (ref) implies the existence of a moment function, with (generally) non-trivial dependence on $\theta$, which is “feedback robust”, in the sense that, for all $ s\in\{0,...,T-2\}$, $$ \mathbb{E}\left[\phi_{X_{i1}}(Y_i^{T},X_i^{2:T};\theta)\,|\, X_i^{T-s},Y_i^{T-(s+1)},\alpha_i\right]=\mathbb{E}\left[\phi_{X_{i1}}(Y_i^{T},X_i^{2:T};\theta)\,|\, X_i^{T-(s+1)},Y_i^{T-(s+1)},\alpha_i\right],$$ while also requiring that $$ \mathbb{E}\left[\phi_{X_{i1}}(Y_i^{T},X_i^{2:T};\theta)\,|\, X_{i1},\alpha_i\right]=0.$$
From Lemma (ref) we obtain the following corollary, which we also prove by induction. This is our main result.
Using that, when $\theta\neq 0$, $1$, $F(\alpha)$, and $F(\theta+\alpha)$, for $\alpha\in {\cal{S}}$, are linearly independent in the logit model, Corollary (ref) implies that in the logit model with a binary predetermined covariate, $\theta$ is not point-identified irrespective of the number of time periods available.
This non point-identification result contrasts with prior work on logit panel data models. Under strict exogeneity, rasch1960studies and andersen1970asymptotic have established that $\theta$ is point-identified under mild conditions on $X_{it}$ whenever $T\geq 2$. In the dynamic logit model when $X_{it}=Y_{i,t-1}$, Chamberlain_1993 shows that $\theta$ is not point-identified when $T=2$ (a result also obtained as an implication of Corollary (ref)). However, chamberlain1985heterogeneity, and honore2000panel in a model with covariates, show that $\theta$ is point-identified under suitable conditions whenever $T\geq 3$.\footnote{Since in the dynamic logit model $X_{it}=Y_{i,t-1}$ is a lagged outcome, $T\geq 2$ (respectively, $T\geq 3$) requires that individual outcomes be available for at least three (resp., four) periods.} By contrast, Corollary (ref) shows that, when the feedback process through which current covariates are influenced by lagged outcomes is unrestricted, the failure of point-identification is pervasive irrespective of $T$, despite the logit structure.
The previous sections show that point-identification often fails in binary choice models with a predetermined covariate. In this section, we explore the degree of identification failure by presenting numerical calculations of the identified set $\Theta^I$ for specific parameter values. In the last part of the section we present calculations of the identified set for an average partial effect.
We show that the identified set $\Theta^I$, defined by set (ref) above, can be represented as a set of $\theta$ values for which a certain linear program has a solution. This characterization facilitates numerical computation of the identified set.
To present our construction, let us first focus on the $T=2$ case, and suppose that Assumption (ref) holds, so $\alpha_i$ has discrete support. For any hypothetical values $(\widetilde{\theta},\widetilde{\pi},\widetilde{G})\in \Theta\times \Pi\times {\cal{G}}_2$, we define
The right-hand-side of (ref) is determined by the unknown heterogeneity distribution, the parametric likelihood for $Y_1$ (given $X_1$ and $\alpha$), and the unknown feedback process for $X_2$. Finding $\Theta_I$ essentially involves repeatedly asking whether, for a given $\widetilde{\theta}$, there exists a valid feedback process and heterogeneity distributions consistent with the observed data distribution (and the parametric part of the model).
Specifically we first require that $\psi_{x_1}(x_2,y_1,\alpha)$ is a valid probability mass function:
Second, we check that it is consistent with the parametric likelihood model for $Y_1$ given $X_1$ and $\alpha$:
Finally, we conclude that $\widetilde{\theta}\in \Theta^I$ if and only if
for some vectors $\psi_{x_1}$ also satisfying ((ref)) and ((ref)) for $x_1\in\{0,1\}$. Condition (ref) ensures compatibility with the likelihood contribution for the period $2$ outcome, $Y_2$.
Since all of the equalities and inequalities in ((ref)), ((ref)) and ((ref)) are linear in $\psi_{x_1}$, it follows that one can verify whether $\widetilde{\theta}\in \Theta^I$ by checking the existence of a solution to a finite-dimensional linear program.\footnote{Note that, to compute the identified set under the assumption of strict exogeneity, one can simply modify this approach by adding to ((ref)), ((ref)) and ((ref)) the additional restriction
which is also linear in $\psi_{x_1}$. The fact that, under strict exogeneity, $\Theta^I$ can be computed using linear programming was first established by honore2006bounds.} We provide details about computation in Appendix (ref).
The characterization of $\Theta^I$ in ((ref)), ((ref)) and ((ref)) remains valid when Assumption (ref) does not hold, and $\alpha_i$ has continuous support. In that case, one needs to interpret $\psi_{x_1}$ in ((ref)) as the product between the density of $\alpha_i$ conditional on $(X_{i2},Y_{i1})$ and the probability of $(X_{i2},Y_{i1})$, both of them conditional on $X_{i1}$ and for hypothetical parameter values. The resulting linear program is infinite-dimensional in that case.
The linear programming representation of $\Theta^I$ extends to any number $T\geq 2$ of periods. To see this, let, for some $(\widetilde{\theta},\widetilde{\pi},\widetilde{G})\in \Theta\times \Pi\times {\cal{G}}_T$, $$\psi_{x_1}(x^{2:T},y^{T-1},\alpha)=\Pr(X_{i}^{2:T}=x^{2:T},Y_{i}^{T-1}=y^{T-1},\alpha_i=\alpha\,|\, X_{i1}=x_1;\widetilde{\theta},\widetilde{\pi},\widetilde{G}),$$ with a similar definition when the support of $\alpha_i$ is not discrete and Assumption (ref) does not hold. In Appendix (ref) we derive the following characterization of the (sharp) identified set $\Theta^I$.
Proposition (ref) shows that one can verify whether $\widetilde{\theta}\in \Theta^I$ by checking the feasibility of a (finite- or infinite-dimensional) linear program. In a setting with lagged outcomes and strictly exogenous covariates, honore2006bounds provided an analogous linear programming representation of the identified set. By contrast, in Proposition (ref) we characterize the identified set of $\theta$ in the general predetermined case where the Granger condition fails; i.e., when $G_{y^{t-1},x^{t-1}}(\alpha)$ may depend on $y^{t-1}$, a situation that honore2006bounds did not consider but anticipated in their conclusion.
In this section we compute identified sets $\Theta^I$ in logit and probit models for a set of example data generating processes (DGPs). In the DGPs, $X_{it}$ follows a Bernoulli distribution on $\{0,1\}$ with probabilities $(\frac{1}{2},\frac{1}{2})$, independent over time, and $\alpha_i$ takes $K=31$ values with probabilities closely resembling those of a standard normal (a specification we borrow from honore2006bounds), and is drawn independently of $(X_{i1},...,X_{iT})$. In the logit case, $F(u)=\frac{e^u}{1+e^u}$, and in the probit case, $F(u)=\Phi(u)$ for $\Phi$ the standard normal cdf. Lastly, we vary $\theta$ between $-1$ and $1$. Note that $X_{it}$ is strictly exogenous in this data generating process. We characterize identified sets in two scenarios: assuming that $X_{it}$ are strictly exogenous, and only assuming that $X_{it}$ are predetermined.
In Figure (ref) we report our numerical calculations of the identified set $\Theta^I$ for the logit model (in the left column panels) and for the probit model (in the right column panels). The three vertical panels correspond to the $T=2,3,4$ cases, respectively. In each graph, we report two sets of upper and lower bounds: those computed while maintaining the strict exogeneity assumption (in dashed lines) and those computed maintaining just predeterminedness (in solid lines). We report the true parameter $\theta$ on the x-axis. To compute the sets, we assume that $\alpha_i$ has the same $K=31$ points of support as in the DGP. We also experimented with fewer and additional support points, as we report below.
Focusing first on the logit case, shown in the left column of Figure (ref), we see that the identified set $\Theta^I$ under strict exogeneity is a singleton for any value of $\theta$ and irrespective of $T$. This is not surprising since $\theta$ is point-identified in the static logit model. In contrast, the upper and lower bounds of the identified set do not coincide in the predetermined case, consistent with our non point-identification result. At the same time, the identified sets appear rather narrow, even when $T=2$, and the width of the set tends to decrease rapidly when $T$ increases to three and four periods. This is qualitatively similar to the observation of honore2006bounds, who focused on dynamic probit models and found that the width of the identified set tends to decrease rapidly with $T$.
Focusing next on the probit case, shown in the right column of Figure (ref), we see that the identified set $\Theta^I$ under strict exogeneity is not a singleton. Moreover, allowing the covariate to be predetermined increases the width of the identified set. However, as in the logit case, the sets appear rather narrow, even when $T=2$, and their widths decrease quickly as $T$ increases. Of course, these observations are specific to a particular data-generating process and the corresponding bounds may be wide for other DGPs.
The results in Figure (ref) are obtained by assuming that the researcher knows the (finite) support of $\alpha_i$. This approach is similar to the one in honore2006bounds. Alternatively, one may wish to characterize the identified set in a class of models where $\alpha_i$ is continuous, e.g., when ${\cal{S}}=\mathbb{R} $ and $\mu$ is the Lebesgue measure. Doing so, as noted earlier, requires approximating an infinite-dimensional linear program. In Appendix Figure (ref), we go take a heuristic step in this direction by reporting numerical approximations to the identified sets, for $T=2$, obtained by taking $K=5$, $K=50$, and $K=500$ points of support for $\alpha_i$, respectively, where the points of support are equidistant percentiles of a standard normal distribution. We find very minor differences compared to the case $K=31$ that we report in Figure (ref). While we do not provide a formal analysis of numerical approximation properties, this suggests that identified sets under continuous $\alpha_i$ may not be markedly different from the ones in Figure (ref).
Overall, these calculations suggest that, while relaxing strict exogeneity tends to increase the widths of the bounds, the identified sets under predeterminedness can be informative even when the number of periods is very small. To reiterate, these conclusions are based on a particular set of example DGPs.
Although our focus in this paper is on the parameter $\theta$, in applications researchers are often interested in average partial effects such as
where the expectation is taken with respect to the distribution of $\alpha_i$.
The identified set for $\Delta$ can also be characterized as the solution to a linear program. Indeed, it follows from Proposition (ref) that $\widetilde{\Delta}$ is in the identified set of $\Delta$ if and only if there exists $\widetilde{\theta}$, $\psi_{0}$ and $\psi_1$ such that ((ref)), ((ref)), and ((ref)) hold, and
where $q_{x_1}=\Pr(X_{i1}=x_1)$. For any given $\widetilde{\theta} \in\Theta^I$, we can therefore compute the set of $\widetilde{\Delta}$ parameters in the identified set by solving a linear program. We provide details about computation in Appendix (ref).
In Figure (ref) we report our computations of the identified set for the average partial effect $\Delta$, relying on the same parameter values and DGP as before. Focusing first on the logit case, shown in the left column of the figure, we see that the identified set under strict exogeneity is not a singleton, except when the true $\theta$ and $\Delta$ are equal to zero. This is not surprising, since average partial effects generally fail to be point-identified in binary choice models, even when covariates are strictly exogenous. Yet, the sets seem rather narrow, even when $T=2$. Allowing the covariate to be predetermined increases the widths of the sets, however the increase is relatively moderate. Moreover, the sets under predeterminedness are very tight whenever $T\geq 3$.
Focusing next on the probit case, shown in the right column of Figure (ref), we see that although the sets appear wider than in the logit case, relaxing strict exogeneity only moderately increases the widths of the sets, especially when $T\geq 3$.
Lastly, while we compute the sets in Figure (ref) under the assumption that $\alpha_i$ has the same $K=31$ points of support as in the DGP, in Appendix Figure (ref) we report approximations of the sets, for $T=2$, obtained using $K=5$, $K=50$, and $K=500$ points of support for $\alpha_i$. The sets appear very similar to the ones based on $K=31$ points of support shown in Figure (ref). However, in this case as well, we do not formally analyze the numerical approximation of the identified sets under continuous $\alpha_i$.
Our analysis suggests that failures of point-identification are commonplace in binary choice models with a predetermined covariate. In this section we describe possible restrictions on the model that can strengthen its identification content. We focus on restrictions on the feedback process, since restrictions on individual heterogeneity are rarely motivated by the economic context.
In some applications one may want to restrict the feedback process to not depend on time-invariant heterogeneity; that is, to impose that
is independent of $\alpha$. For example, in structural dynamic discrete choice models, researchers may be willing to model the law of motion of state variables such as dynamic production inputs as homogeneous across units. kasahara2009nonparametric show how this assumption can help identification in these models. Here we study how a homogeneity assumption can lead to tighter identified sets in our setting.
To proceed, we focus on the case where $T=2$. Given ((ref)), the likelihood function takes the form
where the likelihood factors due to the fact that the feedback process does not depend on $\alpha$. Hence, under Assumption (ref) (which avoids division by zero) we have
A key observation to make about (ref) is its right-hand-side coincides with the likelihood function of a binary choice model with a strictly exogenous covariate (where in addition $\alpha_i$ is independent of $X_{i2}$ given $X_{i1}$). In turn, the left-hand side is weighted by the inverse of the feedback process. This is similar to the inverse-probability-of-treatment-weighting approach to dynamic treatment effect analysis in Jamie Robins' work (e.g., robins2000marginal), with the difference that here we focus on panel data models with fixed effects.
The similarity between ((ref)) and the strictly exogenous case directly delivers point-identification results and consistent estimators. For example, suppose that $F$ is logistic. Given that the left-hand side of ((ref)) is point-identified, it follows from standard arguments (rasch1960studies, andersen1970asymptotic) that $\theta$ is point-identified. Moreover, a consistent estimator of $\theta$ is obtained by maximizing the weighted conditional logit log-likelihood $$\sum_{i=1}^n\widehat{\omega}_i\boldsymbol{1}\{Y_{i1}+Y_{i2}=1\}\left\{ Y_{i1}\ln\left(\frac{\exp(\widetilde{\theta}X_{i1} )}{\exp(\widetilde{\theta}X_{i1} )+\exp(\widetilde{\theta}X_{i2} )}\right)+Y_{i2}\ln\left(\frac{\exp(\widetilde{\theta}X_{i2} )}{\exp(\widetilde{\theta}X_{i1} )+\exp(\widetilde{\theta}X_{i2} )}\right)\right\},$$with weights $$\widehat{\omega}_i=\left\{[\widehat{G}^2_{Y_{i1},X_{i1}}]^{X_{i2}}[1-\widehat{G}^2_{Y_{i1},X_{i1}}]^{1-X_{i2}}\right\}^{-1},$$for $\widehat{G}^2_{y_{1},x_1}$ a consistent estimate of the homogeneous feedback probabilities.\footnote{The analysis in this subsection is not restricted to the binary covariate case. However, when $X_{it}$ are continuous, demonstrating $\sqrt{n}$ consistency of $\widehat{\theta}$ would generally require imposing rate-of-convergence and other requirements on the first-step estimation of the $\widehat{\omega}_i$ weights.}
Another possible restriction on the feedback process is a Markovian condition, such as
is independent of $(y^{t-2},x^{t-2})$. Such a condition may be natural in models where $X_{it}$ is the state variable in the agent's economic problem (as in rust1987optimal and kasahara2009nonparametric, for example).
In order to characterize the identified set $\Theta^I$ with the Markovian condition ((ref)) added, we augment the restrictions ((ref)), ((ref)) and ((ref)) with the fact that, for all $s\in\{2,...,T\}$,
does not depend on $(y^{s-2},x^{s-2})$.\footnote{When $s=T$, this requires that $\frac{\psi_{x_1}(x^{2:T},y^{T-1},\alpha)}{\sum_{x_T=0}^1\psi_{x_1}(x^{2:T},y^{T-1},\alpha)}$ does not depend on $(y^{T-2},x^{T-2})$.}
A difficulty arises in this case since this additional set of restrictions is not linear in $\psi_{x_1}$. As a result, one would need to use different techniques to characterize the identified set in the spirit of Proposition (ref), and to establish conditions for (the failure of) point-identification in the spirit of Corollary (ref). Given this, we leave the analysis of identification in models with Markovian feedback processes to future work.
In this paper we study a binary choice model with a binary predetermined covariate. We find that failures of point-identification are widespread in this setting. Point-identification fails in many binary choice models, with apparently only a few exceptions (such as the exponential model). At the same time, our numerical calculations of identified sets suggest that the bounds on the parameter can be narrow, even in very short panels. This suggests that, while the strict exogeneity assumption has identifying content, models with predetermined covariates and feedback may still lead to informative empirical conclusions, both for the coefficients of the covariates and for average partial effects.
Our analysis of models with a binary covariates can easily be extended to handle general discrete covariates with finite support. In particular, for $\theta$ to be regularly point-identified there need to exist $x_1\neq x_2$ in the support of $X_{it}$ such that $1$, $F(\theta'x_1+\alpha)$, and $F(\theta'x_2+\alpha)$, for $\alpha\in{\cal{S}}$, are linearly dependent. This condition fails in many popular specifications such as the logit. In turn, when $X_{it}$ has finite, non-binary support, the identified set can still be computed as a solution to a linear program, analogously to Proposition (ref). However, the extension to continuous covariates is not straightforward in our setting, in particular since the notion of regularity maintained by Assumption (ref) no longer applies.
Finally, although we have analyzed a binary choice model, our techniques can be used to study other models with stronger identification content, such as models for count data (e.g., Poisson regression, wooldridge1997multiplicative, blundell2002individual) and models with continuous outcomes (e.g., censored regression, honore2004estimation, and duration models, chamberlain1985heterogeneity). Deriving sequential moment restrictions in such nonlinear models was considered by Chamberlain_JOE2022 and is further explored in our companion paper (BDG_WP).
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