Identification of Nonlinear Dynamic Panels under Partial Stationarity

Identification in Nonlinear Dynamic Panel Models under Partial Stationarity

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Introduction

This paper provides a general and unified identification approach for a wide range of panel data models with limited dependent variables, including various discrete (binary, multinomial, and ordered) choice models and censored outcome models. In particular, our approach accommodates dynamic models with any number of lagged dependent variables and contemporaneously endogenous covariates. Moreover, the identification approach does not impose parametric distributions on unobserved heterogeneity, nor on the exact form of endogeneity, thus allowing for more flexible model specifications.

To fix ideas, we start with the following dynamic binary choice model, which is on its own of considerable theoretical and applied interest. Section (ref) generalizes the approach to other limited dependent variable models. Specifically, consider

where $Y_{it}\in\left\{ 0,1\right\} $ denotes a binary outcome variable for individual $i=1,2,...$ and time $t=1,...,T$, $W_{it}\in\mathcal{R}^{d_{w}}$ denotes a vector of observed covariates, $\alpha_{i}\in\mathcal{R}$ denotes the unobserved fixed effect for individual $i$, and $\epsilon_{it}$ denotes the unobserved time-varying error term for individual $i$ at time $t$. The objective is to identify the parameter $\theta_{0}$\footnote{We discuss in Appendix (ref) how our results can be used to derive bounds on certain counterfactual parameters.} using a panel of observed variables $\left(Y_{i},W_{i}\right){}_{i=1}^{n}$, where $W_{i}:=\left(W_{i1},...,W_{iT}\right)$, and similarly for $Y_{i}$. We focus on short panels, where the number of time periods $T\geq2$ is fixed and finite.

The identification of model (ref) has been explored in the literature under various assumptions. For example, chamberlain1980 examines identification under the logistic distribution of $\epsilon_{it}$ and the independence of $\epsilon_{it}$ with respect to $\left(\alpha_{i},W_{i}\right)$. Subsequently, manski1987 relaxes the distributional assumption and employs the following conditional stationarity of $\epsilon_{it}$ to achieve identification:

This condition is also referred to as “group stationarity” or “group homogeneity” and has been exploited in studies such as chernozhukov2013, \citet*{shi2018} and pakes2019.\footnote{To be precise, condition (ref) is often stated in the following weaker “pairwise” version in the literature, \[ \epsilon_{is}\,\sim\,\epsilon_{it}\mid\alpha_{i},W_{is},W_{it},\ \quad\forall s,t=1,...,T, \] where only covariate realizations from the two periods $\left(s,t\right)$ are conditioned on. However, the difference between condition (ref) and the pairwise version above usually only leads to a minor adaptation of the results in the aforementioned papers (as well as in the current one). See Remark (ref) for a follow-up discussion.} Condition (ref) does not impose parametric restrictions on the distributions of $\epsilon_{it}$ and allows for dependence between the fixed effect $\alpha_{i}$ and the covariates $W_{i}$. However, condition (ref) does impose substantial restriction on the dependence between $W_{i}$ and the time-varying error term $\epsilon_{it}$: it effectively requires that all covariates in $W_{i}$ are exogenous with respect to the time varying error $\epsilon_{it}$.\footnote{For instance, suppose $W_{it}=\left(Z_{it},X_{it}\right)$ and $\mathbb{E}\left[\rest{\epsilon_{it}}W_{i}\right]=X_{it}^{'}\eta$, then the conditional distributions of $\epsilon_{it}$ and $\epsilon_{is}$ cannot be the same as long as $X_{it}^{'}\eta\neq X_{is}^{'}\eta$. Hence condition (ref) fails in general.}

In many economic applications, certain components of the observable covariates $W_{i}$, namely $X_{i}$, may exhibit endogeneity issues in violation of condition (ref). For example, in a dynamic setting where $X_{it}$ includes the lagged outcome variable $Y_{i,t-1}$, then the dependence of $Y_{i,t-1}$ on $\epsilon_{i,t-1}$ arises immediately, violating the condition (ref).\footnote{We note that lagged outcome variable $Y_{it}$ are sometimes referred to as “predetermined” or “weakly or sequentially exogenous” covariates in panel data literature. We clarify that in this paper, we use the words “exogenous” and “endogenous” in the stronger “all-period” sense, or more precisely, in the sense of whether condition (ref) holds when conditioned on the covariate in question.} For another example, if $X_{it}$ includes “price” or other variables that may be endogenously chosen by economic agents after observing $\epsilon_{it}$, then $X_{it}$ would be correlated with the contemporaneous $\epsilon_{it}$, so the exogeneity restriction imposed by condition (ref) will again fail to hold.

In this paper, we instead impose and exploit a weaker version of condition (ref) by excluding all endogenous components of $W_{i}$ from the conditioning set. To be precise, from now on we suppose that we can decompose $W_{it}$ as: \[ W_{it}\equiv\left(Z_{it},X_{it}\right), \] where $Z_{it}$ is of dimension $d_{z}$, and $X_{it}$ is of dimension $d_{x}$ with $d_{w}=d_{z}+d_{x}$. Our “partial stationarity” assumption is then formulated as follows:

Our partial stationarity condition (ref), as its name suggests, only requires that the errors are stationary conditional on the realizations of a subvector of the covariates (i.e., the exogenous covariates denoted by $Z_{i}$) while allowing the remaining covariates (denoted by $X_{i}$) to be endogenous in arbitrary manners.\footnote{Our identification strategy and results can be easily adapted under the alternative “pairwise partial stationarity” condition $\epsilon_{is}\sim\epsilon_{it}\mid\alpha_{i},Z_{is},Z_{it}.$ See Remarks (ref) and (ref) for follow-up discussions.} In short, condition (ref) imposes exogeneity conditions only on a subset of covariates, which we label as “$Z$” and will thereafter be referred to as the “exogenous covariates”.\footnote{Condition (ref) also accommodates the standard stationarity assumption conditional on all covariates.}

We describe how to exploit the partial stationarity condition (ref) to derive the identified set on the model parameters $\theta_{0}$ through a class of conditional moment inequalities, which take the form of lower and upper bounds for the conditional distribution $\epsilon_{it}+\alpha_{i}\mid Z_{i}$, solely as functions of observed variables and the model parameters $\theta_{0}$. We show that these bounds must have nonzero intersections over time under the partial stationarity assumption, thereby forming a class of identifying restrictions for the parameter $\theta_{0}$. Conditional on the exogenous covariates $Z_{i}$, our class of inequalities is indexed by a scalar $c\in\mathcal{R}$, which implicitly traces out all possible values that the parametric index $Z_{it}^{'}\beta_{0}+X_{it}^{'}\gamma_{0}$ can take. That said, we show how the effective number of identifying restrictions can be reduced to be finite when $X_{it}$ has finite support, a condition naturally satisfied in the important special case of “$p$-th order autoregressive” dynamic binary choice models, where $X_{it}$ consists of lagged outcome variables $Y_{i,t-1},Y_{i,t-2},...,Y_{i,t-p}$ that are by construction discrete.

We demonstrate the sharpness of the identified set we derived for binary choice models. More precisely, we show that for any $\theta$ that satisfies all the conditional moment inequalities we derived, we can construct an observationally equivalent joint distribution of the observed and unobserved variables in our model. Our proof of sharpness consists of three main steps: we begin by demonstrating “per-period” sharpness, and then progressively generalize the result from “per-period” to “all-period” sharpness, and from discrete $X_{it}$ to general $X_{it}$. A key innovation in our proof technique is using an explicit, simple, and general construction that shows how marginal/aggregate stationarity restrictions and joint choice probability restrictions can be satisfied simultaneously, which might be of independent and wider use.

Our identification strategy based on partial stationarity applies more broadly beyond the context of dynamic binary choice models. In Section (ref), we demonstrate its applicability in a general nonseparable semiparametric model, and show how it can be applied to a range of alternative limited dependent variable models, such as ordered response models, multinomial choice models, and censored outcome models. The results of our approach accommodate both static and dynamic settings across all these models.

We characterize the identified set using a collection of conditional moment inequalities, based on which estimation and inference can be conducted using established econometric methods in the literature, such as \citet*{chernozhukov2007}, andrews2013, and \citet*{chernozhukov2013}. Through Monte Carlo simulations, we demonstrate that our identification method yields informative and robust finite-sample confidence intervals for coefficients in both static and dynamic models.

Literature Review

Our paper contributes directly to the line of econometric literature on semiparametric panel discrete choice models. Dating back to manski1987, a series of work exploits “full” stationarity conditions for identification, such as abrevaya2000rank, \citet*{chernozhukov2013}, \citet*{shi2018}, pakes2019, \citet*{khan2021inference}, gao2020, wang2022semiparametric, and \citet*{botosaru2023identification}. As discussed above, full stationarity conditions given all observable covariates effectively require that all covariates are exogenous with no dynamic effects (i.e., lagged dependent variables). In contrast, we exploit the “partial” stationarity condition, allowing for lagged dependent variables, as well as other endogenous covariates.

In the literature on dynamic discrete choice models, our paper is most closely related to \citet*[KPT thereafter]{khan2023identification}, who studies the following dynamic panel binary choice model

where the one-period lagged dependent variable $Y_{i,t-1}\in\{0,1\}$ serves as the endogenous covariate, and $Z_{it}$ are exogenous covariates. KPT exactly imposes the “partial stationarity” condition (ref) in the specific context of (ref), and derives the sharp identified set for $\theta_{0}$ by explicitly enumerating the realizations of the one-period lagged outcome variable $Y_{i,t-1}$ (across two periods $t,s$). In contrast, our model (ref), along with the “partial stationarity” condition, is stated with more general specifications of the endogenous covariates $X_{it}$. The covariates $X_{it}$ can include more than one lagged dependent variables (e.g. $Y_{i,t-1},Y_{i,t-2},$ ...) and other endogenous variables (such as “price” if $Y_{it}$ represents the purchase of a particular product), which may be continuously valued. Consequently, our identification strategy is substantially different from that of KPT, and can be applied more broadly to various other dynamic nonlinear panel models. In the specific model (ref), we show that the identifying restrictions we derived are equivalent to those derived in KPT and thus both approaches lead to sharp identification. Relatedly, mbakop2023identification proposes a computational algorithm to derive conditional moment inequalities in a general class of dynamic discrete choice models (potentially with multiple lags). The focus of mbakop2023identification is on scenarios where the lagged discrete outcome variables are the only endogenous covariates in the model, and the proposed algorithm relies on the discreteness of these variables. Relative to these works, our paper introduces an analytic approach that directly applies to a more general class of dynamic binary choice models, as well as other types of models with continuous limited dependent variables and any number of endogenous covariates, regardless of whether they are discrete or continuous and whether they take the form of lagged outcome variables or not.

Our identification strategy relies on a type of stationarity condition, while alternative approaches utilize other notions of exogeneity. For example, honore2000panel provides identification by exploiting events where the exogenous covariates stay the same across two periods: they consider both the logit case and a semiparametric case, but both under the independence between time-changing errors and the lagged dependent variable, as well as the intertemporal independence of errors. Additionally, aristodemou2021semiparametric exploits an alternative type of “full independence” assumption for identification in dynamic binary choice models. The “full independence” assumption essentially requires that the time-varying errors from all time periods and the exogenous variables from all time periods are independent (conditional on initial conditions), but does not make intertemporal restrictions on the errors (such as stationarity). Hence, such `full independence” assumption and the partial stationarity assumption in our paper do not nest each other as special cases. \citet*{chesher2023identification} applies the framework of generalized instrumental variables chesher2017generalized to the context of various dynamic discrete choice models with fixed effects, and utilizes a similar “full independence” assumption aristodemou2021semiparametric for identification.\footnote{Our identification strategy shares some conceptual similarity with the idea of generalized instrumental variable (GIV) in \citet*{chesher2017generalized}, who proposes a general approach for representing the identified set of structural models with endogeneity. \citet*{chesher2017generalized}, \citet*{chesher2020generalized}, and \citet*{chesher2023identification} demonstrate how the GIV framework can be applied to various settings, but focus mostly on the use of exclusion restrictions and/or full independence assumptions. In this paper, we neither impose exclusion restrictions nor independence assumptions but instead explore identification under a partial stationarity condition.} More differently, some other papers work with sequential exogeneity in various dynamic panel models and provide (non-)identification results under different model restrictions. For example, shiu2013identification imposes a high-level invertibility condition along with a restriction that rules out the dependence of covariates on past dependent variables. More recently, \citet*{bonhomme2023identification} investigates panel binary choice models with a single binary predetermined covariate under sequential exogeneity, whose evolution may depend on the past history of outcome and covariate values. The sequential exogeneity condition considered in these papers and the partial stationarity condition in ours again do not nest each other as special cases: in particular, our current paper accommodates contemporaneously endogenous covariates that violate sequential exogeneity. In summary, the key assumptions, identification strategy, and identification results of these studies are substantially different from and thus not directly comparable to those in our current paper.

Our paper is also complementary to the literature that studies dynamic logit models with fixed effects. This literature typically assumes that time-varying errors are conditionally independent across time, independent from all other variables, and follow the logistic distribution. The logit assumption in panel settings has long been studied, such as in \citet*{chamberlain1984panel} and \citet*{chamberlain2010binary}. We do not impose the logit assumption, nor require conditional independence across time, and our identification strategy is very different from those based on the logit assumption.

Our paper also contributes to the general panel data literature on linear and nonlinear models with and without endogeneity and dynamics. Most relatedly, botosaru2017binarization proposes a binarization strategy for general panel data models with fixed effects without requiring time homogeneity, but focuses on static settings. botosaru2022 considers a model where the outcome variable is generated as a strictly monotone (and thus invertible) transformation of a linear model, and they exploit time homogeneity in conditional means (instead of the whole distributions) for identification. Our current paper, with a focus on discrete choice models, does not require strict monotonicity and invertibility.

It should be acknowledged that our partial stationarity condition (ref), as well as the full stationarity condition (ref), cannot handle random coefficients as in berry1995automobile and mcfadden2000mixed. On the other hand, our approach can be naturally extended to handle nonadditive and functional fixed effects as discussed in Remark (ref) and Section (ref). It would be interesting in future research to investigate whether our approach can be further adapted to incorporate heterogeneous coefficients.

The rest of the paper is organized as follows. Section (ref) studies the sharp identification of panel binary choice models with endogenous covariates. Section (ref) demonstrates how our key identification strategy generalizes to a wide range of dynamic nonlinear panel data models, such as ordered response models, multinomial choice models, and censored outcome models. Section (ref) presents simulation results about the finite-sample performances of our approach, and Section (ref) explores the empirical application of income categories using various ordered response models. We conclude with Section (ref).

Dynamic Binary Choice Model

Model

To explain the partial stationarity and our key identification strategy, we start with the canonical binary choice model, which is of wide theoretical interest itself. In Section (ref), we explain how our identification strategy can be applied more generally.

Specifically, consider the same binary model as introduced in (ref): $$ Y_{it}=\mathbf{\mathbbm1}\left\{ W_{it}^{'}\theta_{0}+\alpha_{i}+\epsilon_{it}\geq0\right\}. $$ Recall that we decompose $W_{it}\equiv\left(Z_{it}^{'},X_{it}^{'}\right)^{'}$, and, throughout this paper, we will refer to $Z_{i}$ as “exogenous covariates”, and refer to $X_{i}$ as “endogenous covariates”. The exact difference between $Z_{i}$ and $X_{i}$ is formalized through the “partial stationarity” condition, which we now state as a formal assumption:

Assumption (ref) essentially requires that the (conditional) distribution of $\epsilon_{it}$ stays the same across all time periods $t=1,...,T$ even if $Z_{i}$ realize to different values. To illustrate, suppose that there are only two periods $t=1,2$, and that $Z_{i1},Z_{i2}$ realize to two values $z_{1},z_{2}$, respectively, with $z_{1}<z_{2}$. Then Assumption (ref) requires that $\epsilon_{i1}$ and $\epsilon_{i2}$ still have the same (conditional) distributions: in particular, $\epsilon_{i1}$ cannot be stochastically smaller (or larger) than $\epsilon_{i2}$ because of $z_{1}<z_{2}$. Hence, Assumption (ref) can be thought as a definition of the “exogeneity” of the covariates $Z_{it}$ in our context.

In contrast, Assumption (ref) imposes no such restrictions on the (potentially) endogenous covariates $X_{i}$. In fact, since $X_{i}$ does not appear in Assumption (ref) at all, here we are completely agnostic about the dependence structure between $\epsilon_{i}$ and $X_{i}$: in particular, the conditional distribution of $\epsilon_{it}$ is allowed to vary across $t$ arbitrarily for any particular realization of $X_{i}$. As a result, different forms of endogeneity in $X_{i}$ can be incorporated under our framework in a unified manner, as we illustrate in the examples below.

Key Identification Strategy

We now explain our key identification strategy based on the partial stationarity condition. Write $v_{it}:=-\left(\epsilon_{it}+\alpha_{i}\right)$ so that model (ref) can be rewritten as \[ Y_{it}=\mathbf{\mathbbm1}\left\{ v_{it}\leq W_{it}^{'}\theta_{0}\right\} . \] For any constant $c\in\mathcal{R}$, consider first the event \[ Y_{it}=1\text{ and }W_{it}^{'}\theta_{0}\leq c. \] Whenever the event above happens, we must have $v_{it}\leq W_{it}^{'}\theta_{0}\leq c$, implying that $v_{it}\leq c$. Formally, the above can be summarized by the following inequality:

Symmetrically, we can also consider the “flipped” event \[ Y_{it}=0\text{ and }W_{it}^{'}\theta_{0}\geq c, \] which implies $v_{it}>c$:

Rearranging the above, we have

Next, taking conditional expectations of (ref) and (ref) given $Z_{i}=z$, we have

where “$|z$” is a shorthand for “$Z_{i}=z$” that we will use throughout the paper. Note that the middle equality of (ref) follows from the partial stationarity condition (Assumption (ref)).\footnote{Specifically, observe that Assumption (ref) implies the partial stationarity of $v_{it}$ given $Z_{i}$, since

for any $c$, and hence $v_{it}\mid Z_{i}\stackrel{d}{\sim}v_{is}\mid Z_{i}.$} Essentially, in the above we exploit the joint occurrence of $v_{it}\leq W_{it}^{'}\theta_{0}$ and $W_{it}^{'}\theta_{0}\leq c$ to deduce an implication on the composite error $v_{it}\leq c$ that is free of the endogenous covariates $X_{it}$, and then leverage the partial stationarity of $v_{it}$ given $Z_{i}$ for intertemporal comparisons.

Since the lower and upper bounds in (ref) hold for any $t$ and $s$, we summarize the identifying restrictions (ref) across all time periods in the following proposition.

Proposition (ref) characterizes the identified set $\Theta_{I}$ for $\theta_{0}$ as restrictions on the conditional joint distribution of $Y_{it}$ and $X_{it}$ given $z$. More specifically, the restrictions in (ref) can be regarded as a collection of conditional moment inequalities that relate $\mathbf{\mathbbm1}\left\{ Y_{it}=1,\ W_{it}^{'}\theta\leq c\right\} $ and $\mathbf{\mathbbm1}\left\{ Y_{it}=0,\ W_{it}^{'}\theta\geq c\right\} $ conditional on $z$.

Proposition (ref) holds regardless of whether the endogenous covariates $X_{it}$ are discrete or continuous. When $X_{it}$ are continuous (taking a continuum of values), then Proposition (ref) requires that condition (ref) hold for a continuum of constants $c\in\mathcal{R}$, so that (the information in) the whole joint distribution of the binary variable $Y_{it}$ and the continuous variable $W_{it}^{'}\theta=z_{t}^{'}\beta+X_{it}^{'}\gamma$ can be captured by the collection of joint distributions of $\left(Y_{it},\mathbf{\mathbbm1}\left\{ W_{it}^{'}\theta\leq c\right\} \right)$ across all possible cutoff values $c$.

However, when $X_{it}$ are discrete, such as in the AR$\left(p\right)$ dynamic model where $X_{it}$ consists of $p$ lagged binary outcome variables, there is no need to evaluate (ref) at all possible values of $c\in\mathcal{R}$, since the inequalities in (ref) can only bind at finitely many values of $c$. We formalize this observation via the following Proposition.

Proposition (ref) shows that the discreteness of the endogenous covariates $X_{it}$ help reduce the infinite number of inequality restrictions in Proposition (ref) to finitely many, or more precisely, $KT$ ones (conditional on $z$).

The case of discrete $X_{it}$ is conceptually important, since it nests the dynamic AR$\left(p\right)$ model widely studied in the literature as a special case. Clearly, when $X_{it}$ consists of $p$ (finitely many) lagged binary outcome variables $Y_{i,t-1},...,Y_{i,t-p}$, then $X_{it}$ by construction can only take $K=2^{p}$ discrete values. Specialized further to the AR(1) model in KPT, Proposition (ref) shows that the identified set $\Theta_{I}$ is characterized by $2T$ conditional restrictions, which is drastically smaller than the $9T\left(T-1\right)$ conditional restrictions listed in KPT (even when $T$ is small).

Sharpness

So far we have only shown that $\Theta_{I}$ is a valid identified set for $\theta_{0}$. However, it is not yet clear whether it has incorporated all the available information for $\theta_{0}$ under the current model specification. We now proceed to establish the sharpness of $\Theta_{I}$ under appropriate conditions.

We start with the discrete case where the support of $X_{it}$ is assumed to be finite. Remarkably, the sharpness of our identified set can be established without any additional assumptions in this case.

The formal definition of sharpness, along with the complete proof of Theorem (ref), are available in Appendix (ref). In short, we show (by construction) that, for each $\theta\in\Theta_{I}\backslash\left\{ \theta_{0}\right\} $, there exists a data generating process (DGP) that satisfies Assumption (ref) and produces the same joint distribution of observable data $\left(Y_{i},W_{i}\right)$ under model (ref) with parameter $\theta$. Theorem (ref) demonstrates that our key identification strategy based on the bounding of (endogenous) parametric index by arbitrary constants, as described in Section (ref), is able to extract all the available information for $\theta_{0}$ from the model and the observable data, and thus it is impossible to further differentiate $\theta_{0}$ from alternatives in the identified set $\Theta_{I}$ under model (ref) and our assumption of partial stationarity (without further restrictions).

Theorem (ref) immediately implies that, in the special case of dynamic AR$\left(p\right)$ models where $X_{it}$ consists of discrete lagged outcomes, our characterization of the identified set $\Theta_{I}^{disc}$ in Proposition (ref) is sharp. In particular, our result generalizes the corresponding result in KPT, which focuses on the AR(1) model. Furthermore, KPT characterizes the sharp identified set via $9T\left(T-1\right)$ conditional restrictions, the derivation of which is based on an exhaustive enumeration of lagged outcome realizations $Y_{i,t-1}$. In this paper we adopt an entirely different (and much more general) identification strategy, and arrive at a characterization of the identified set by $2T$ conditional restrictions, which we also show to be sharp by Theorem (ref). Since our model and assumption specialize exactly to that in KPT under the AR(1) specification, it follows that our $2T$ restrictions must be able to reproduce all the $9T\left(T-1\right)$ restrictions in KPT. This demonstrates that our identification strategy not only applies more generally than the one in KPT, but also leads to a more elegant characterization of the sharp identified set with much fewer restrictions. We provide a more detailed explanation about this point in the next subsection.

Another conceptually remarkable, or surprising, feature of Proposition (ref) and Theorem (ref) is that they are established without reference to the exact nature, or interpretation, of the endogenous covariates $X_{it}$. The identified set $\Theta_{I}$ we characterized is valid and sharp regardless of whether $X_{it}$ are specified as lagged outcome variables, contemporaneously endogenous covariates, or a combination of both.

Our proof of sharpness consists of two main steps. First, we show for each $\theta\in\Theta_{I}\backslash\left\{ \theta_{0}\right\} $ how to construct the per-period marginal distributions of errors that match the per-period marginal choice probabilities. Second, we show how to combine the $T$ per-period marginal distributions into an all-period joint distribution that matches the all-period joint choice probabilities, so that observational equivalence holds.

The proof techniques we exploited are also different from, and thus novel relative to, those used in the related work that leverages stationarity-type conditions for partial identification, such as \citet*{pakes2019} for static multinomial choice model and KPT for dynamic AR(1) model. Instead of showing existence only, we provide a more explicit construction of the joint distribution of the latent variables, which is valid regardless of the exact type of endogeneity in $X_{it}$. In particular, a key challenge in proving sharpness based on stationarity-type conditions lies in that stationarity imposes only aggregate restrictions (via integrals/sums) of the joint distribution of errors, which is rather implicit to work with. A key innovation in our proof technique is to show how marginal/aggregate stationarity restrictions and joint choice probability restrictions can be satisfied simultaneously by an explicit, simple and general construction, which might be of independent and wider use.

Next, we seek to establish the sharpness of our identification set in the case where certain or all components of $X_{it}$ may be continuous. Below we present an additional set of regularity conditions for the continuous case and the corresponding sharpness result, followed by a discussion of the conditions and the result.

Theorem (ref) establishes the sharpness of our identification set under the additional regularity conditions imposed in Assumption (ref). The proof, presented in Appendix (ref), follows the general construction strategy used in the discrete-case proof, with some key adaptions to handle several continuity and measure-zero issues arising in the continuous case. Such adaptions utilize the conditions in Assumption (ref), which we now explain in more details.

Assumption (ref)(a) can be effectively regarded as a setup of the continuous-case model. In our current context, given $z$, the induced index $W_{it}^{'}\theta_{0}=z_{t}^{'}\beta_{0}+X_{it}^{'}\gamma_{0}$ is what enter most directly into our model, rather than $X_{it}$ per se. Part of Assumption (ref)(a) states that $W_{it}^{'}\theta_{0}$ is continuously distributed on a bounded interval, which can be satisfied with various lower-level conditions on $X_{it}$. For example, if $X_{it}|z$ is continuously distributed on a bounded and connected support with nonempty interior, and if $\gamma_{0}$ is restricted to lie within a bounded set (which can be imposed as a scale normalization without loss of generality), then $X_{it}^{'}\gamma_{0}|z$ is continuously distributed on a bounded connected interval. If in addition $X_{it}|z$ is assumed to have a density that is strictly positive (almost) everywhere on its support, then the induced density of $X_{it}^{'}\gamma_{0}|z$ will also be (almost) everywhere strictly positive. Note also Assumption (ref)(a) may also be satisfied if some (but not all) components of $X_{it}$ are discrete, as long as some other component(s) of $X_{it}$ is continuously distributed with nonzero coefficient and a sufficiently large support.

Assumption (ref)(b), along with the assumptions of connectedness (interval representation of the support) and strictly positive densities (strictly increasing CDFs) for $X_{it}^{'}\gamma_{0}|z$ in Assumption (ref)(a), are imposed mainly as simplifying restrictions that are not conceptually necessary but allow for a more convenient notation. Essentially, they jointly imply that the per-period CCPs on the left-hand and right-hand sides of (ref) are continuous and strictly increasing in $c$ on connected intervals, leading to simpler notation in the proof via the use of the inverse function and the intermediate value theorem. Without these conditions, we would need to handle “flat regions”, “jump points”, and “continuously increasing regions” separately and then combine them together to produce the final result, which should be achievable using a combination of the proof techniques in the discrete case and the continuous case.

Assumption (ref)(c) is a key condition for the validity of our adapted construction in the continuous case, but it is admittedly the most nonstandard and implicit one, which warrants further explanation. Effectively, Assumption 2(c) rules out certain “knife-edge” degenerate DGPs that result in a “flat region of contact” between $\ol L$ and $\ul U$, though the exact form of such degeneracy can be rather complicated given the nonlinear nature of the binary choice model and the generality of the endogeneity we incorporate. Below we provide some intuition for why Assumption (ref)(c) should be regarded as a relatively mild condition. For more detail, see Appendix (ref) for two illustrative examples where Assumption (ref)(c) (almost) trivially holds, as well as a general sufficient condition for Assumption (ref)(c).

Note that under Assumption (ref)(a)(b), we have $L_{t}\left(c|z,\theta_{0}\right)<U_{t}\left(c|z,\theta_{0}\right)$ with strict inequality, so $\ol L\left(\rest cz,\theta_{0}\right)\leq\ul U\left(\rest cz,\theta_{0}\right)$ can only hold with equality if there exist two different periods $t\neq s$ such that $L_{t}\left(c|z,\theta_{0}\right)=U_{s}\left(c|z,\theta_{0}\right)$. If this holds for all $c$'s in a small open interval, i.e. with positive Lebesgue measure in violation of Assumption 2(c), then we can deduce that their derivatives in $c$ must also match, i.e.,

on an open interval, with $L_{t}^{'}$ and $U_{s}^{'}$ given by

where $\pi_{t}\left(\rest{\cdot}z\right)$ denotes the conditional pdf of $X_{it}^{'}\gamma_{0}$ given $Z_{i}=z$.

Consequently, $L_{t}^{'}=U_{s}^{'}$ on an open interval essentially means that the density-weighted CCPs on the right-hand sides of (ref) must change continuously in $c$ in exactly the same functional form on a continuum, despite all of the following: (i) $L_{t}^{'}$ is defined on the event $Y_{it}=1$ while $U_{s}^{'}$ is defined as on the event $Y_{is}=0$, which are “flipped” events that may generally vary with $c$ in different manners, (ii) the values of $z_{t}^{'}\beta_{0}$ and $z_{s}^{'}\beta_{0}$ can be different, so the conditioning events are generally different for $L_{t}$ and $U_{s}$ as well, (iii) the conditional distribution of $X_{it}$ given $Z_{i}=z$ may be different (nonstationary) across periods $t$, so $\pi_{t}\left(\rest{c-z_{t}^{'}\beta_{0}}z\right)$ and $\pi_{s}\left(\rest{c-z_{s}^{'}\beta_{0}}z\right)$ may be different even if $z_{t}^{'}\beta_{0}=z_{s}^{'}\beta_{0}$, (iv) the dependence structure between $X_{it}$ and $\epsilon_{it}$ may vary across $t$. For all these reasons, it appears unlikely that (ref) can hold for a continuum of $c$, except under carefully designed “knife-edge” DGPs. Hence, we consider Assumption 2(c) as a mild condition. See Appendix (ref) for more examples and details.

Reconciliation with Related Work

Our identifying restrictions in (ref) and (ref) have a somewhat “nonstandard” representation in terms of (conditional) joint probabilities of $Y_{it}$ and $X_{it}$ (given $Z_{i}$), instead of conditional probabilities of $Y_{it}$ given $X_{it}$ (such as lagged outcomes), which are more usually found in the related literature. Hence, we provide a more detailed discussion about the content and interpretation of our identifying restrictions, as well as a more explicit explanation of how they relate to existing results in the related literature.

Reconciliation with manski1987

Consider first the special case where there are no endogenous covariates $X_{it}$, or in other words, $X_{it}$ is degenerate. In this case, our “partial stationarity” condition specializes to the “full stationarity” condition (ref) as in manski1987. However, our identifying restriction (ref) still has a different form than the identifying restriction in manski1987. To illustrate, focus on any two periods $\left(t,s\right)$, and observe that our identifying restriction becomes:

while the “maximum-score-type” identifying restrictions in manski1987 are of the form

The “maximum-score-type” identifying restriction (ref) has a quite intuitive and interpretable representation: across two periods $\left(t,s\right)$ under full stationarity, the conditional choice probability at period $s$ is larger if and only if the index $z_{s}^{'}\beta_{0}$ is larger. In contrast, our restriction (ref) has a somewhat twisted representation even in this simple setting.

However, a closer look reveals that our (ref) reproduces Manski's “maximum-score-type” identifying restrictions in the current context. To see this, notice that, by setting $c=z_{t}^{'}\beta_{0}$ in (ref), we obtain

Hence, if $z_{s}^{'}\beta_{0}\geq z_{t}^{'}\beta_{0}$, i.e., the left-hand side of (ref) holds, then the above implies that \[ \mathbb{P}\left(\rest{Y_{it}=1}z\right)\leq1-\mathbb{P}\left(\rest{Y_{is}=0}z\right)=\mathbb{P}\left(\rest{Y_{is}=1}z\right), \] which becomes exactly the right-hand side of (ref). Switching $t$ with $s$ in the argument above produces the other implication $z_{s}^{'}\beta_{0}\leq z_{t}^{'}\beta_{0}$ $\Rightarrow$ $\mathbb{P}\left(\rest{Y_{is}=1}z\right)\leq\mathbb{P}\left(\rest{Y_{it}=1}z\right)$. Together these exactly constitute the “if-and-only-if” restriction in (ref). Hence, even though our inequality restriction (ref) looks different from the more intuitive “maximum-score-type” restriction, they both incorporate the same information.

Reconciliation with KPT

Now, consider the dynamic AR(1) model as studied in KPT, where the only endogenous covariate is the one-period lagged outcome variable, i.e., $X_{it}:=Y_{i,t-1}$.

To illustrate, first focus on any two periods $\left(t,s\right)$ only, and observe that in this case our identifying restriction becomes

Under the same model and assumption, KPT derives the following 9 inequality implications for $\left(t,s\right)$:\footnote{We adapt the notation in KPT to our current notation, and state these 9 inequalities as strict inequalities, which lead to a simpler and more focused explanation. The equivalence between our restriction and the KPT restrictions still hold if their inequalities are stated in the weak form.}

KPT(i): $\mathbb{P}\left(\rest{Y_{it}=1}z\right)>\mathbb{P}\left(\rest{Y_{is}=1}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}+\left|\gamma_{0}\right|>0.$

KPT(ii): $\mathbb{P}\left(\rest{Y_{it}=1}z\right)>1-\mathbb{P}\left(\rest{Y_{i,s}=0,Y_{i,s-1}=1}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}-\min\left\{ 0,\gamma_{0}\right\} >0.$

KPT(iii): $\mathbb{P}\left(\rest{Y_{it}=1}z\right)>1-\mathbb{P}\left(\rest{Y_{i,s}=0,Y_{i,s-1}=0}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}+\max\left\{ 0,\gamma_{0}\right\} >0.$

KPT(iv): $\mathbb{P}\left(\rest{Y_{it}=1,Y_{it-1}=1}z\right)>\mathbb{P}\left(\rest{Y_{is}=1}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}+\max\left\{ 0,\gamma_{0}\right\} >0.$

KPT(v): $\mathbb{P}\left(\rest{Y_{it}=1,Y_{it-1}=1}z\right)>1-\mathbb{P}\left(\rest{Y_{is}=0,Y_{i,s-1}=1}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}>0.$

KPT(vi): $\mathbb{P}\left(\rest{Y_{it}=1,Y_{it-1}=1}z\right)>1-\mathbb{P}\left(\rest{Y_{is}=0,Y_{i,s-1}=0}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}+\gamma_{0}>0.$

KPT(vii): $\mathbb{P}\left(\rest{Y_{it}=1,Y_{it-1}=0}z\right)>1-\mathbb{P}\left(\rest{Y_{is}=0}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}-\min\left\{ 0,\gamma_{0}\right\} >0.$

KPT(viii): $\mathbb{P}\left(\rest{Y_{it}=1,Y_{it-1}=0}z\right)>1-\mathbb{P}\left(\rest{Y_{is}=0\text{\ensuremath{,}}Y_{i,s-1}=1}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}-\gamma_{0}>0.$

KPT(ix): $\mathbb{P}\left(\rest{Y_{it}=1,Y_{it-1}=0}z\right)>1-\mathbb{P}\left(\rest{Y_{is}=0,Y_{i,s-1}=0}z\right)$ $\Rightarrow$ $\left(z_{t}-z_{s}\right)^{'}\beta_{0}>0.$

In a way, the 9 inequality restrictions in KPT above are similar to the “maximum-score restrictions”, in the sense that all of them take the form of logical implications between intertemporal comparisons of various conditional probabilities and intertemporal differences of covariate indexes.

Using a very different identification strategy than the one in KPT, we arrived at our inequality restriction (ref), which looks very different from the collection of 9 inequality restrictions in KPT. At first sight it is not clear how (ref) relates to and compares with the 9 KPT restrictions. However, a closer look again reveals that our restriction (ref) can reproduce all the 9 restrictions in KPT, and thus incorporate all the information therein in a unified format.

Take KPT(v) as an illustration and suppose that the left-hand side of KPT(v) holds, then it implies

With $X_{it}=Y_{i,t-1}$, our inequality restriction (ref) can be equivalently rewritten as follows,

where the realization of $Y_{i,t-1}$ is explicitly enumerated as in KPT.

Note that we can further relax condition (ref) by dropping the two probabilities $\mathbb{P}\left(\rest{Y_{it}=1,\ Y_{i,t-1}=0}z\right)\mathbf{\mathbbm1}\left\{ z_{t}^{'}\beta_{0}\leq c\right\} $ and $\mathbb{P}\left(\rest{Y_{is}=0,\ Y_{i,s-1}=0}z\right)\mathbf{\mathbbm1}\left\{ z_{s}^{'}\beta_{0}\geq c\right\} $ as it makes the lower bound smaller and the upper bound larger:

Then, the statement that $\mathbf{\mathbbm1}\left\{ z_{t}^{'}\beta_{0}+\gamma_{0}\leq c\right\} $ and $\mathbf{\mathbbm1}\left\{ z_{s}^{'}\beta_{0}+\gamma_{0}\geq c\right\} $ both hold is precisely equivalent to the following statement of \[ z_{t}^{'}\beta_{0}\leq z_{s}^{'}\beta_{0}\ \Rightarrow\ \mathbb{P}\left(\rest{Y_{it}=1,\ Y_{i,t-1}=1}z\right)\leq\ 1-\mathbb{P}\left(\rest{Y_{is}=0,\ Y_{i,s-1}=1}z\right). \] By contraposition, it leads to exactly the same implication of KPT(v): \[ \mathbb{P}\left(\rest{Y_{it}=1,\ Y_{i,t-1}=1}z\right)>\ 1-\mathbb{P}\left(\rest{Y_{is}=0,\ Y_{i,s-1}=1}z\right)\Longrightarrow z_{t}^{'}\beta_{0}>z_{s}^{'}\beta_{0}. \] Hence, we have shown that (ref) implies KPT(v).

Similarly, it is shown in Appendix (ref) that (ref) implies all 9 restrictions in KPT. In fact, the representation (ref) reveals why there are precisely 9 KPT-type restrictions. The two period-$t$ indicators $\mathbf{\mathbbm1}\left\{ z_{t}^{'}\beta_{0}+\gamma_{0}\leq c\right\} $ and $\mathbf{\mathbbm1}\left\{ z_{t}^{'}\beta_{0}\leq c\right\} $ in the upper expression of (ref) may take 3 “useful”\footnote{The 4th combination, $\mathbf{\mathbbm1}\left\{ z_{t}^{'}\beta_{0}+\gamma_{0}\leq c\right\} =\mathbf{\mathbbm1}\left\{ z_{t}^{'}\beta_{0}\leq c\right\} =0$, will make the upper expression of (ref) equal to $0$, so that the inequality (ref) holds trivially. Hence, this $\left(0,0\right)$ combination is not useful.} combinations $\left(1,0\right),\left(0,1\right)$ and $\left(1,1\right)$, while the two period-$s$ indicators $\mathbf{\mathbbm1}\left\{ z_{s}^{'}\beta_{0}+\gamma_{0}\geq c\right\} $ and $\mathbf{\mathbbm1}\left\{ z_{s}^{'}\beta_{0}\geq c\right\} $ in the lower expression of (ref) may also take 3 useful combinations. Consequently, in total there are $3\times3=9$ useful combinations, which exactly correspond to the $9$ left-hand-side suppositions in the 9 KPT restrictions.

Hence, while our restriction (ref) appears very different from the 9 KPT restrictions, it actually automatically incorporates all the KPT restrictions. In particular, by treating the endogenous covariate $X_{it}=Y_{i,t-1}$ as a random variable, our restriction (ref) automatically aggregates the identifying information across all possible realizations of $Y_{i,t-1}$, without the need to explicitly consider each possibility separately.

Now, consider a general setting with $T\geq2$ periods. By our Proposition (ref) and Theorem (ref), the sharp identified set can be characterized by $2T$ restrictions, which are generated by evaluating (ref) at each $c$ of the $2T$ points in $\left\{ z_{t}^{'}\beta,z_{t}^{'}\beta+\gamma:t=1,...,T\right\} $. In contrast, across $T$ periods the KPT approach produces $9T\left(T-1\right)$ restrictions, which are generated by imposing the $9$ KPT restrictions across all ordered time pairs $\left(t,s\right)$. Hence, our approach provides a much simpler characterization of the sharp identified set, using a significantly smaller number of restrictions. For example, with $T=2$ periods, we have $4$ restrictions while KPT has $18$; with $T=3$, we have $6$ restrictions while KPT has 54. Hence, the reduction in the number of restrictions relative to KPT is quite remarkable.

In summary, while the representation of our identifying restrictions in Propositions (ref) and (ref) may appear somewhat unusual in the first place, it actually becomes equivalent to the more familiar representations in the specialized settings of manski1987 and KPT.

Extensions

The key idea underlying our identification strategy extends beyond the binary choice model and can be applied to a wide range of nonlinear panel data models with dynamics and endogeneity. We first present our general identification strategy in a generic semiparametric model (Section 3.1), and then demonstrate how this strategy can be applied and adapted to ordered response (Section 3.2), multinomial choice (Section 3.3) and censored outcome (Appendix B.3) settings.

Identification Strategy in a Generic Model

We start with a generic semiparametric model that helps convey the generality of our key identification strategy

where $Y_{it}\in\mathcal{Y}$ can be either a discrete or continuous variable, $\alpha_{i}$ is the individual fixed effect of arbitrary dimension, $\epsilon_{it}$ is the time-varying error of arbitrary dimension, $W_{it}$ is a vector of observable covariates, $\theta_{0}\in\mathcal{R}^{d_{w}}$ is a conformable vector of parameters, and the function $G$ is allowed to be unknown, nonseparable but assumed to satisfy the following:

Note that, we can obtain the binary choice model in Section (ref) by setting $\alpha_{i},\epsilon_{it}$ to be scalar-valued, and $G\left(W_{it}^{'}\theta_{0},\,\alpha_{i},\,\epsilon_{it}\right)=\mathbf{\mathbbm1}\left\{ W_{it}^{'}\theta_{0}+\alpha_{i}+\epsilon_{it}\geq0\right\} $, where $G$ is by construction weakly increasing in $W_{it}^{'}\theta_{0}$.

As before, we decompose $W_{it}$, and correspondingly $\theta_{0}$, into two components, $W_{it}=\left(Z_{it}^{'},X_{it}^{'}\right)^{'}$ and $\theta_{0}=\left(\beta_{0}^{'},\gamma_{0}^{'}\right)^{'}$, and impose the partial stationarity condition (Assumption (ref)). We now show how partial stationarity can be exploited in conjunction with weak monotonicity (Assumption (ref)) to obtain identifying restrictions in the presence of endogeneity.

Let $\mathcal{Y}$ denote the support of $Y_{it}$. For any $c\in\mathcal{R}$ and $y\in\mathcal{Y}$, observe that \[

\] where the inequality holds by the monotonicity of the function $G$. Symmetrically, we have \[

\] which is equivalent to \[ \mathbf{\mathbbm1}\left\{ G\left(c,\,\alpha_{i},\,\epsilon_{it}\right)\leq y\right\} \leq1-\mathbf{\mathbbm1}\left\{ Y_{it}>y,\ Z_{it}^{'}\beta_{0}+X_{it}^{'}\gamma_{0}\leq c\right\} . \] The partial stationarity assumption $\epsilon_{it}\mid Z_{i},\alpha_{i}\sim\epsilon_{is}\mid Z_{i},\alpha_{i}$ implies the stationarity of the transform function $G$: $G\left(c,\alpha_{i},\epsilon_{it}\right)\mid Z_{i},\alpha_{i}\sim G\left(c,\alpha_{i},\epsilon_{is}\right)\mid Z_{i},\alpha_{i}$. After integrating out $\alpha_{i}$, the stationarity condition persists conditioned on $Z_{i}$ alone: \[ G\left(c,\alpha_{i},\epsilon_{it}\right)\mid Z_{i}\sim G\left(c,\alpha_{i},\epsilon_{is}\right)\mid Z_{i}. \] Combining the above derived bounds on $\mathbf{\mathbbm1}\{G\left(c,\,\alpha_{i},\,\epsilon_{it}\right)\leq y\}$, we have

The key difference of the above and the corresponding identifying restrictions in Section 2 lies in that the “middle term” in (ref) is no longer the conditional CDF of $\alpha_{i}+\epsilon_{it}$, but the conditional probability of $G\left(c,\,\alpha_{i},\,\epsilon_{is}\right)\leq y$, with the latter representation not dependent on scalar-additivity of fixed effect $\alpha_{i}$ and time-varying errors $\epsilon_{it}$.

We summarize the identifying restrictions derived above by the following proposition:

Note that in the binary choice setting of Section (ref), it suffices to set $y=0$ in (ref), which then coincides with the identifying results in Proposition (ref). This also shows that the identified set does not change at all, regardless of whether scalar-additivity of $\alpha_i$ and $\epsilon_{it}$ is imposed or not in the binary choice model.

The results in Proposition (ref) generally hold regardless of whether the dependent variable and the endogenous covariate are discrete or continuous. The next proposition shows that additional discreteness in either the dependent variable or endogenous covariates can further simplify and reduce the number of the identifying conditions in (ref).

Moreover, when $Y_{it}\in\left\{ \ol y_{1},...,\ol y_{K}\right\} $ with $\ol y_{j}\leq\ol y_{j+1}$ for any $t$, then $\Theta_{I,gen}=\Theta_{I,gen}^{disc_{y}},$ where $\Theta_{I,gen}^{disc_{y}}$ consists of all $\theta=\left(\beta^{'},\gamma^{'}\right)^{'}$ that satisfy condition (ref) for any $y\in\left\{ \ol y_{1},...,\ol y_{K-1}\right\} $.

Proposition (ref) shows that for the general model, when both the outcome and the endogenous variable are discrete, it is sufficient to focus on a finite number of identifying restrictions. The number of these restrictions is determined by the support of the outcome variable and the covariate index. The proof of Proposition (ref) follows the same reasoning as Proposition (ref), so it is omitted here. The central idea is that for any point $c$ or $y$ outside the range specified in Proposition (ref), we can find a point within the specified range that provides weakly more informative results. Therefore, the inclusion of these outside points would not provide additional information for the identified set.

Ordered Response Model

Consider that the outcome variable $Y_{it}$ takes $J$ ordered values: $Y_{it}\in\left\{ y_{1},..,y_{J}\right\} $ with $y_{j}<y_{j+1}$. Examples of such ordered outcomes include various income categories, health outcomes, or levels of educational attainment. We study the following panel ordered choice model:

where $Y_{it}^{*}$ denotes the latent dependent variable, and $Y_{it}$ denotes the ordered outcome which takes value $y_{j}$ when $Y_{it}^{*}\in(b_{j},b_{j+1}]$. The threshold parameters satisfy $b_{1}=-\infty,b_{J+1}=+\infty$, and the remaining threshold parameters $b_{j}$ (where $b_{j}\leq b_{j+1}$) can be either known or unknown for $2\leq j\leq J-1$. The binary choice model in (ref) is nested with $J=2$ and $b_{2}=0$.

While the ordered response model (ref) here can be regarded as a special case of the generic model (ref), the special “ordered cutoffs” structure in (ref) contains more information than an unknown generic $G$ function in (ref). As a result, even though the general identification strategy in Section (ref) still applies, we can adapt the identification argument to the special additional structure imposed here, obtaining a sharper result than a direct application of Proposition (ref). In particular, we explain why the line of our identification arguments help us find such an adaption that exploits the special model structure.

We now explain this in more details. Following the arguments in Section (ref), we have \[ \mathbf{\mathbbm1}\left\{ Y_{it}\leq y_{j},\ b_{j+1}-W_{it}^{'}\theta_{0}\leq c\right\} \leq\mathbf{\mathbbm1}\left\{ v_{it}\leq c\right\} \] For a given $c$, the above inequality holds for any response index $j$. This immediately implies that we can take the largest one to get a tighter lower bound: \[ \max_{j}\mathbf{\mathbbm1}\left\{ Y_{it}\leq y_{j},\ b_{j+1}-W_{it}^{'}\theta_{0}\leq c\right\} \leq\mathbf{\mathbbm1}\left\{ v_{it}\leq c\right\} . \] In addition, an inspection of the LHS reveals that the maximum is attained at \[ j=\ol j_{c}\left(W_{it}\right):=\max\left\{ j:b_{j+1}-W_{it}^{'}\theta_{0}\leq c\right\} \] since such a (random) $j$ would maximize $\mathbf{\mathbbm1}\left\{ Y_{it}\leq y_{j}\right\} $ subject to $b_{j+1}-W_{it}^{'}\theta_{0}\leq c$. Consequently, we obtain

where the last equality holds since $\mathbf{\mathbbm1}\left\{ b_{j+1}-W_{it}^{'}\theta_{0}\leq c\right\} =0$ for any choice $j>\ol j_{c}\left(W_{it}\right)$.

The final expression (ref) is particularly nice for three reasons: First, it aggregates the information aggregated from different $y_{j}$ together to produce a tighter lower bound. Second, the expression circumvent the need to compute the maximizer cutoff $\ol j_{c}$. Third, it is represented as a linear sum (instead of a maximum) so that conditional expectation of (ref) remains a linear sum of conditional expectations.

To see the advantage of the third point above, we take conditional expectation of (ref) given $z$ as before, obtaining \[ \mathbb{P}\left(\rest{v_{it}\leq c}z\right)\geq\sum_{j=1}^{J}\mathbb{P}\left(\rest{Y_{it}=y_{j},\ b_{j+1}-W_{it}^{'}\theta_{0}\leq c}z\right). \] where the RHS can be computed as a simple sum of CCPs about each ordered value $y_{j}$.

Similarly, we can derive an upper bound \[ \mathbb{P}\left(v_{is}\leq c\mid z\right)\leq1-\sum_{j=1}^{J}\mathbb{P}\left(\rest{Y_{is}=y_{j},b_{j}-W_{is}^{'}\theta_{0}\geq c}z\right), \] which can be combined with the lower bound to yield the following result.

We emphasize again that Proposition (ref) is not a direct application of Proposition (ref), since Proposition (ref) explicitly utilizes the special model structure of the order response model to aggregate information from all response index $j$ together to form tighter bounds for each $c$. In contrast, a naive application of Proposition (ref) would yield

which remains valid but is a collection of bounds imposed on each $j$ separately, thus is generally not as tight as the bounds in (ref).

Multinomial Choice Model

In this subsection, we apply our key identification strategy to panel multinomial choice model with endogeneity. Specifically, consider a set of unordered choice alternatives $\mathcal{J}=\left\{ 0,1,...,J\right\} $. Let $u_{ijt}$ denote the latent utility for individual $i$ of selecting choice $j$ at time $t$, which depends on the three components: observed covariate $W_{ijt}=(Z_{ijt}',X_{ijt}')'$, unobserved fixed effects $\alpha_{ij}$, and unobserved time-varying preference shock $\epsilon_{ijt}$. Let $Y_{it}\in\mathcal{J}$ denote individual $i$'s choice at time $t$. We study the following panel multinomial choice model: \[

\] and impose the same partial stationarity assumption: \[ \epsilon_{is}\mid Z_{i},\alpha_{i}\stackrel{d}{\sim}\epsilon_{it}\mid Z_{i},\alpha_{i}\quad\text{for any}\ s,t\leq T. \] with $Z_{it}:=\{Z_{ijt}\}_{j\in\mathcal{J}}$,$\alpha_{i}:=\{\alpha_{ij}\}_{j\in\mathcal{J}}$ and $\epsilon_{it}:=\{\epsilon_{ijt}\}_{j\in\mathcal{J}}$ defined to collect terms across all $J$ choice alternatives.

We emphasize that this model is not a special case of the generic model (ref) in Subsection (ref), since in the current model the $J$ outcome values are unordered, and the model involves multiple indexes and multivariate monotonicity. Hence, we cannot directly apply Proposition (ref) to the current setting. That said, we explain how the key idea from Subsection (ref) can again be adapted to obtain identification result in the panel multinomial choice setting.

We start by looking at the indicator variable $Y_{it}^{j}:=\mathbf{\mathbbm1}\{Y_{it}=j\}$ of choosing alternative $j$, which maintains a similar monotone structure with Assumption (ref):

and the new variable $Y_{it}^{j}$ is increasing in $W_{ijt}'\theta_{0}-W_{ikt}'\theta_{0}$ $\forall k\in\mathcal{J}$

More generally, for any subset $K\subset\mathcal{J}$, the indicator variable $Y_{it}^{K}:=\mathbf{\mathbbm1}\{Y_{it}\in K\}$ represents individual $i$'s choice belonging to the subset $K$, given by

and the variable $Y_{it}^{K}$ is increasing in $W_{ijt}'\theta_{0}-W_{ikt}'\theta_{0}$ for any $j\in K$ and $k\in\mathcal{J}\setminus K$.

Adapting the key identification idea to the current context, the identification results for panel multinomial choice models are presented in the following proposition.

Below we show that Proposition (ref) specializes to the corresponding result in pakes2019, who focuses on the static panel multinomial choice model without any endogeneity. Since pakes2019 establishes the sharpness of their identification result under their setup, our Proposition (ref) is also sharp (under their static two-period setting).

However, a key improvement of our result relative to that in pakes2019 is that Proposition (ref) allows for any type of endogeneity including dynamic multinomial models with lagged dependent variable, as well as the inclusion of contemporaneously endogenous variables such as product prices. For example, consider the following dynamic model: \[ u_{ijt}=Z_{ijt}'\beta_{0}+\mathbf{\mathbbm1}\left\{ Y_{i,t-1}=j\right\} \gamma_{0,j}-P_{ijt}\lambda_{0}+\alpha_{ij}+\epsilon_{ijt}. \] where individual $i$'s utility at time $t$ can potentially depend on their choices in the previous period $t-1$ and we allow the dynamic effect $\gamma_{0,j}$ to vary across choices, and $P_{ijt}$ is the price of product $j$ faced by consumer $i$ at time $t$. To the best of our knowledge, no previous work has considered such generalization of pakes2019 that can incorporate price endogeneity and past-choice dependence. Even though Proposition (ref) is presented as a byproduct of our general identification strategy, it nevertheless presents a substantive progress in the related literature on panel multinomial choice models.

Reconciliation with pakes2019

Next, we show that Proposition (ref) specializes to those in pakes2019, who focus on the static panel multinomial choice model without any endogeneity. Since pakes2019 establishes the sharpness of their identification set in a two-period setting and our identification set reproduces theirs, the sharpness of our identification set follows immediately in this setting.

Formally, pakes2019 characterizes the sharp identified set for $\theta_{0}$ under the full stationarity assumption given all covariates: \[ \epsilon_{is}\mid W_{i},\alpha_{i}\stackrel{d}{\sim}\epsilon_{it}\mid W_{i},\alpha_{i}. \] Under this condition, for two periods $\left(t,s\right)$ our identifying condition in (ref) is simplified to

The above equation is only informative when $(w_{js}-w_{ks})'\theta_{0}\leq c_{jk}\leq(w_{jt}-w_{kt})'\theta_{0}$ for any $j\in K,k\in k\in\mathcal{J}\setminus K$; otherwise either the upper bound becomes one or the lower bound becomes zero so that condition (ref) holds for any $\theta$. There exists one value $c_{jk}$ satisfying the condition $(w_{js}-w_{ks})'\theta_{0}\leq c_{jk}\leq(w_{jt}-w_{kt})'\theta_{0}$ is equivalent to $(w_{js}-w_{ks})'\theta_{0}\leq(w_{jt}-w_{kt})'\theta_{0}$, generating the following inequality: for any $K\subset\mathcal{J}$,

which becomes the same result in pakes2019 (Proposition 1, P. 12):

since $Y_{it}^{K}=1$ is equivalent to $Y_{it}\in K$ by the definition.

Simulation

In this section, we focus on the static ordered response model Section (ref) and implement the kernel-based CLR inference approach proposed in the papers by \citet*{chernozhukov2013} and chen2019breaking, which was developed to construct confidence interval based on general conditional moment inequalities.

In Appendix (ref), we also conduct a simulation exercise of a different nature. We numerically compute and visualize the identified set under two DGP configurations in dynamic binary choice setting, but do not implement the finite-sample estimation and inference procedure.

Static Ordered Response Model

This section explores a static ordered choice model with three choices $Y_{it}\in\{1,2,3\}$. We consider the following two-period model with $T=2$, and the latent dependent variable $Y_{it}^{*}$ is generated as: \[ Y_{it}^{*}=Z_{it}^{1}\beta_{01}+Z_{it}^{2}\beta_{02}+\alpha_{i}+\epsilon_{it}, \] where the covariate $Z_{it}^{k}$ satisfies $Z_{it}^{k}\sim\mathcal{N}(0,\sigma_{z})$ for $k\in\{1,2\}$; the fixed effects $\alpha_{i}$ are given as $\alpha_{i}=\sum_{t=1}^{T}(Z_{it}^{1}+Z_{it}^{2})/(4*\sigma_{z}*T)$, so they are correlated with the covariates; the error term $(\epsilon_{i1},\epsilon_{i2})$ follows the normal distribution $\mathcal{N}(\mu,\Sigma)$ with $\mu=(0,0)$ and $\Sigma=(1\ \rho;\rho\ 1)$. The true parameter is $\beta_{0}:=(\beta_{0,1},\beta_{02})'=(1,1)'$, the repetition number is $B=200$, and the sample size is $n=\{2000,8000\}$. We consider three specifications for $\sigma_{z}\in\{1,1.5,2\}$ and $\rho\in\{0,0.25,0.5\}$.

The observed dependent variable $Y_{it}$ is given as \[ Y_{it}=1*(Y_{it}^{*}\leq b_{2})+2*(b_{2}<Y_{it}^{*}\leq b_{3})+3*(Y_{it}^{*}>b_{3}), \] where $b_{2}=-1$ and $b_{3}=1$. We write $Y_{i}:=(Y_{i1},Y_{i2})$ and $Z_{i}:=(Z_{i1},Z_{i2})$,

Proposition (ref) in Appendix (ref), as a corollary of Proposition (ref) with pivotal choices of $c$'s, provides a characterization of the identified set for $\beta_{0}$ under the static ordered choice model via the following conditional moment inequalities: for $s\neq t\leq2$, \[ E[g(Z_{i},Y_{i};\beta_{0})\mid z]\geq0, \] where \[ g(Z_{i},Y_{i};\beta)=\left\{

\right. \] upon which our estimation and inference exercise in this subsection will be based.

The first element $\beta_{01}$ of the parameter $\beta_{0}$ is normalized to one, and we are interested in conducting inference for the parameter $\beta_{02}$ using the CLR approach. Tables (ref) and (ref) report the average confidence interval (CI) for $\beta_{02}$, the coverage probability (CP), the average length of the CI (length), the power of the test at zero (power), and the mean absolute deviation of the lower bound ($l_{MAD}$) and upper bound ($u_{MAD}$) of the CI.

As shown in Tables (ref) and (ref), our approach exhibits robust performance across various specifications of standard deviation $\sigma$ and correlation coefficients $\rho$. The coverage probabilities of the 95% confidence interval (CI) for $\beta_{02}$ are close to the nominal level, the length of the CI is reasonably small, and the CI consistently excludes zero. When the sample size increases, there is a significant decrease in CI length, an improvement in coverage probability, and a reduction of the mean absolute deviation (MAD) for the lower and upper bounds of the CI. Overall, these results demonstrate the good performance of our approach in different DGP designs.

Dynamic Ordered Response Model

In this section, we investigate a dynamic ordered choice model with one lagged dependent variable $Y_{i,t-1}$. The latent dependent variable $Y_{it}^{*}$ is generated as follows: \[ Y_{it}^{*}=Z_{it}\beta_{0}+Y_{i,t-1}\gamma_{0}+\alpha_{i}+\epsilon_{it}. \] where the endogenous variable is the lagged dependent variable $Y_{i,t-1}$. We study three periods $T=3$ to illustrate our approach with multiple periods. The DGP is similar: the exogenous covariate $Z_{it}$ satisfies $Z_{it}\sim\mathcal{N}(0,\sigma_{z})$; the fixed effects $\alpha_{i}$ are given as $\alpha_{i}=\sum_{t=1}^{T}Z_{it}/(4*\sigma_{z}*T)$; the error term $(\epsilon_{i1},\epsilon_{i2},\epsilon_{i3})$ follows the normal distribution $\mathcal{N}(\mu,\Sigma)$ with $\mu=(0,0,0)$ and $\Sigma=(0.5\ c\ c;c\ 0.5\ c;c\ c\ 0.5)$, where $c=0.5*\rho$. The true parameter is $\theta_{0}:=(\beta_{0},\gamma_{0})'=(1,1)'$, the repetition number is $B=200$, and the sample size is $n\in\{2000,8000\}$. We consider three specifications for $\sigma_{z}\in\{1,1.5,2\}$ and $\rho\in\{0,0.25,0.5\}$.

The observed dependent variable $Y_{it}$ is given as \[ Y_{it}=1*(Y_{it}^{*}\leq b_{2})+2*(b_{2}<Y_{it}^{*}\leq b_{3})+3*(Y_{it}^{*}>b_{3}), \] for $1\leq t\leq T$. The initial value $Y_{i0}\in\{1,2,3\}$ is generated independently of all variables and follows the distribution $\mathbb{P}(Y_{i0}=1)=0.6,\mathbb{P}(Y_{i0}=2)=\mathbb{P}(Y_{i0}=3)=0.2$.

In this dynamic model, the covariates $Z_{i}:=(Z_{it})_{t=1}^{T}$ and the initial value $Y_{i0}$ are exogenous, while the lagged variable $Y_{i,t-1}$ is endogenous. Proposition (ref) characterizes the identified set for $\theta_{0}$ with the following conditional moment inequalities:

We normalize the first parameter $\beta_{0}$ to one, and report the performance of the coefficient $\gamma_{0}$ for the lagged dependent variable. Tables (ref) and (ref) illustrate that our approach yields robust and informative results for the dynamic ordered choice model across various DGP specifications. The coverage probability of the CI nearly reaches 95%, and the CI consistently excludes zero, producing significant coefficients. These results remain similar across different values of correlation coefficients. When the standard deviation $\sigma_{z}$ increases, the length of the CI also experiences a slight increase. This phenomenon occurs because, in the dynamic model, only partial identification is achieved, and the bound for $\gamma_{0}$ depends on the variation in $\Delta z'\beta_{0}$. A larger variation in $\Delta z'\beta_{0}$ may result in a wider identified set in this specification, but it still provides informative results. As the sample size increases, the confidence interval shrinks, and concurrently, the coverage probability improves in all specifications.

Empirical Application

In this section, we apply our proposed approach to explore the empirical analysis of income categories using the NLSY79 dataset. The dependent variable is three categories of (log) income, denoted by the three values $\{1,2,3\}$, indicating whether an individual falls within the top 33.3% highest income bracket, the 33.3%-66.6% highest income range, and the lowest 33.3% income tier, respectively. We include two covariates in this analysis: one is tenure, defined as the total duration (in weeks) with the current employer, and the other is a residence indicator for whether one lives in an urban or rural area.\footnote{This dataset also contains other crucial factors for income such as gender and race. However, these variables are time-invariant and cannot be included for panel models with fixed effects.} We use two periods of panel data from the years 1982 and 1983 as well as the income data from 1981 as initial values, and there are $n=5259$ individuals in each period. The following table presents the summary statistics of these variables.

We adopt various ordered response models introduced in Section (ref) to analyze the income category. The first model is the standard static model without any endogeneity. The second is the static model, while treating residence as an endogenous covariate. Residence is potentially endogenous since the choice of living area is typically endogenously determined and may be correlated with individuals' unobserved ability or preference. The last model considers the dynamic model with one lagged dependent variable, allowing people's income in current periods to depend on their income in the last period. All three models allow for individual fixed effects and do not impose any parametric distributions on time-changing shocks. Proposition (ref) characterizes the identified set of the model coefficients for these three models using conditional moment inequalities. Similar to Section (ref), we exploit the kernel-based CLR inference method to construct confidence intervals. The coefficient of the variable “residence” is normalized to one. Table (ref) reports the confidence intervals for the coefficients of the covariate “tenure” and the lagged dependent variable (when applicable).

As shown in Table (ref), tenure exhibits a significantly positive effect on the income category across all specifications. When allowing for the endogeneity of residence, the confidence interval for tenure becomes wider, as we need to account for all possible correlations between residence and unobserved heterogeneity. The results from the dynamic model show that the income category in the current period is also positively affected by last period's income, and this effect is significant. Furthermore, this analysis demonstrates the flexibility of our approach, which can not only allow for endogeneity introduced by dynamics but also account for contemporaneous endogeneity.

Conclusion

We introduce a general method to (partially) identify index parameters in nonlinear panel data models based on a partial stationarity condition. This approach accommodates dynamic models with an arbitrary finite number of lagged outcome variables and other types of endogenous covariates. We demonstrate how our key identification strategy can be applied to obtain informative identifying restrictions in various limited dependent variable models, including binary choice, ordered response, multinomial choice, as well as censored outcome models. Finally, we further extend this approach to study general nonseparable models.

There are some natural directions for follow-up research. In this paper we focus on the identification of model parameters, but it would also be interesting to investigate how our identification strategy can be exploited to obtain informative bounds on average marginal effects and other counterfactual parameters, say, following the approach proposed in botosaru2022identification.\footnote{botosaru2022identification proposes an approach to obtain bounds on counterfactual CCPs in semiparametric dynamic panel data models, assuming that the index parameters are (partially) identified.} Also, our identification strategy should be adaptable to exploit additional restrictions imposed by time-exchangeability assumptions such as in \citet*{mbakop2023identification}, which not only impose homogeneity on per-period marginals of errors but also on their intertemporal dependence structures. Additionally, the idea of bounding an endogenous object (parametric index in our case) by an arbitrary constant so as to obtain an object free of endogeneity issues may have broader applicability beyond the models studied in this work, and it remains to see whether our key identification strategy can be further adapted to other structures.