Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
122,792 characters · 26 sections · 71 citation commands
Moment Conditions for Dynamic Panel Logit Models with Fixed Effects
\thispagestyle{empty} \setcounter{page}{0}
\vskip 3cm
This paper is concerned with estimation of the common parameters in nonlinear panel data models with individual-specific fixed effects in situations where the relevant asymptotics is an increasing number of cross-sectional units observed over a fixed number of time periods. Our contribution is a general approach for constructing conditional moment conditions when the dependent variable can take a finite number of values, and we demonstrate how the approach can be used to construct moment conditions for logit models with strictly exogenous explanatory variables as well as lagged dependent variables.
The economic motivation for the econometric model investigated in this paper is the question of whether persistence in economic data is due to unobserved heterogeneity or state dependence. This question dates back to Heckman78a and can be formulated as a distinction between individual-specific fixed effects and lagged dependent variables. This framework has proven relevant in many areas of economics. For example, PakesPorterShepardCalderWand2022 have employed it to study the importance of switching costs in a model of health insurance plan choice.
Econometrically, this paper makes a contribution to the literature on the estimation of nonlinear econometric models with fixed effects. The challenge is that if the fixed effects enter in a way that is not additive or multiplicative, then one cannot simply difference or quasi-difference it away as one would in a linear or multiplicative model. At the same time, treating the fixed effects as parameters to be estimated in a nonlinear model will generally lead to an inconsistent estimator of the common parameters as the number of cross-sectional units increases with the number of time periods fixed. This is what is known as the incidental parameters problem. See neyman1948consistent. One solution to the incidental parameters problem in parametric models is to look for sufficient statistics for the fixed effects. By definition, the conditional likelihood, conditional on these sufficient statistics, will not depend on the fixed effects, so it can potentially be used for estimation. This approach was pioneered by rasch1960studies and andersen1970asymptotic. Unfortunately, there are relatively few models for which one can find such sufficient statistics, so an alternative approach is to try to construct moment conditions that depend on the parameters of interest, but not on the individual-specific fixed effects. Papers by Honore92, Honore1993, Hu2002, and johnson2004identification are earlier specific examples of this, and bonhomme2012functional developed a general approach for obtaining moment conditions via “functional differencing.”
Our paper operationalizes the proposal in bonhomme2012functional for the case of dependent variables with a finite number of possible values. Specifically, we offer a systematic method for how to first numerically explore the potential for constructing moment conditions and then derive their analytic expressions. We then apply this machinery to create moment conditions for a prominent case: the fixed effects logit model with strictly exogenous explanatory variables and lagged dependent variables. For models with one lag, we give explicit expressions for all available moment conditions when $T\ge 3$, where $T$ is the number of time periods in addition to those that give the initial conditions for the dependent variable. We also provide all the moment conditions for the case with two lagged dependent variables and $T=4$ and $5$, as well as with three lags and $T=5$. Notably, for the case of one lag and three time periods (in addition to the one that delivers the initial), our conditions align with those previously found by kitazawa2013exploration,kitazawa2016root.
Estimation of panel data binary response models dates back to rasch1960studies, who noted that in a logit model with strictly exogenous explanatory variables, one can make inference regarding the remaining parameters by conditioning on the sums of the dependent variable for each individual. Chamberlain1985 and magnac2000subsidised demonstrated that it is also possible to find sufficient statistics for the individual-specific fixed effects in logit models where the only explanatory variables are lagged outcomes, and the common parameters can then be estimated by maximizing the conditional likelihood (conditional on the sufficient statistic). Unfortunately, the conditional likelihood approach referenced above does not generally carry over to logit models that have both lagged dependent variables and strictly exogenous explanatory variables. However, as shown in honore2000panel, this approach does apply if one is also willing to condition on the vector of covariates being equal across certain time periods. This leads to an estimator that is asymptotically normal under suitable regularity conditions, but the rate of convergence is slower than the usual $\sqrt{n}$ when there are continuous covariates. The logit assumption is crucial in the construction of the sufficient statistics above. A number of papers (including Manski87, aristodemou2018semiparametric and khan2019identification) have relaxed the logistic assumption. This literature suggests that point estimation is sometimes possible without the logit assumption, and that informative bounds can be constructed when it is not. On the other hand, chamberlain2010binary showed that in a two-period static threshold-crossing model regular root-$n$ consistent estimation is only possible in a logit setting. This underpins the focus on the logistic assumption throughout this paper.
The paper is organized as follows: Section (ref) details our approach for discerning and constructing moment conditions when the dependent variable assumes a finite number of values, also drawing on insights from dobronyi2021identification regarding the number of available moments. In Section (ref), we showcase this within a panel data logit AR(1) model with \(T=3\) time periods. Section (ref) extends this by exploring various other models, including notably the dynamic ordered logit and dynamic multinomial models. This section also highlights the versatility of our method through examples like the panel data logit AR(1) with a heterogeneous time trend, and a static binary response model leveraging a mixture of logits. Section (ref) discusses conditions under which the moment conditions are guaranteed to identify the common parameters in the AR(1) logit model with strictly exogenous explanatory variables, while Section (ref) demonstrates how to find moment conditions for the AR(1) logit model with strictly exogenous explanatory variables when the number of time periods is not three. Section (ref) illustrates the usefulness of the approach by estimating a simple model for labor force participation, and Section (ref) wraps up the discussion.
In this paper, we consider a panel data setting with $i=1,\ldots,n$ cross-sectional units and $t=1,\ldots,T$ time periods. An econometrician models a sequence of discrete outcomes, $Y_{i}=(Y_{i1},\ldots ,Y_{iT})$, as a function of explanatory variables, $X_{i}=(X_{i1},\ldots ,X_{iT})$, initial conditions, $ Y_{i}^{(0)}=(Y_{it}\,:\,t\leq 0)$, and a time invariant \textquotedblleft fixed effect\textquotedblright , $A_{i}$, as
The function $f$ is assumed to be known up to the finite dimensional parameter ${\Greekmath 0112} $. The variables $Y_i$, $Y_{i}^{(0)}$ and $X_i$ are observed, but $A_i$ is unobserved. The corresponding conditional probabilities that can be identified from the observed data are
where the probability mass or density function, $g\big({\Greekmath 010B} _{i}\,\big| \,\,x_{i},\,y_{i}^{(0)}\big)$, of $A_{i}$ conditional on $X_{i}$ and $ Y_{i}^{(0)}$ is left unspecified. We use $\mathcal{Y}$ to denote the set of possible values of $Y_{i}=(Y_{i1},\ldots ,Y_{iT})$, which will be a finite set in all the models considered in this paper.
Throughout this paper we assume that that $ (Y_{i}^{(0)},Y_{i},X_{i},A_{i})$ are independent and identically distributed across $i=1,\ldots ,n$, and our goal is to estimate the common parameters $ {\Greekmath 0112} $ from the observed data as $n \rightarrow \infty$ and $T$ is fixed. The difficulty in identifying and estimating ${\Greekmath 0112} $ is that the individual specific fixed effects $({\Greekmath 010B} _{1},\ldots ,{\Greekmath 010B} _{n})$, or equivalently their unknown conditional distribution $g\big({\Greekmath 010B} _{i}\,\big| \,\,y_{i}^{(0)},\,x_{i}\big)$, constitute a high-dimensional nuisance parameter, that is, we are faced with a classic neyman1948consistent incidental parameter problem.
The leading example considered throughout most of this paper is the binary choice logit AR(1) model, where $Y_{it}\in \{0,1\}$, $X_{it}\in \mathbb{R} ^{K}$, $A_{i}\in \mathbb{R}$, and the model restriction reads
with $Y_{i}^{t-1}=(Y_{i,t-1},Y_{i,t-2},\ldots )$, ${\Greekmath 010C} \in \mathbb{R}^{K}$ and ${\Greekmath 010D} \in \mathbb{R}$. In this example, we have ${\Greekmath 0112} =({\Greekmath 010C} ,{\Greekmath 010D} )$, $Y_{i}^{(0)}=Y_{i0}$, and
In this binary choice example, the set of possible outcomes $\mathcal{Y} =\{0,1\}^{T}$ has cardinality $|\mathcal{Y}|=2^{T}$.
A very general method to overcome the incidental parameter problem in the models considered here is to find moment functions $ m(y_{i},y_{i}^{(0)},x_{i},{\Greekmath 0112} )$ (different from zero) such that the model restriction (ref) implies that for any true parameter value ${\Greekmath 0112}$,
If we can find such valid moment functions, then they can typically be used to study identification of the parameter ${\Greekmath 0112} $ and to estimate it using generalized method of moments (GMM). The main challenge in this process is to find such valid moment functions for a given panel model of interest.
In the absence of any further restriction on the distribution of $ (Y_{i}^{(0)},X_{i},A_{i})$, the unconditional moment restriction (ref) can only be a consequence of the model (ref) if the conditional moment restriction
holds for all possible realizations $y_{i}^{(0)}$, $x_{i}$, ${\Greekmath 010B} _{i}$. Under weak regularity conditions, (ref) then follows from (ref) by the law of iterated expectations. Furthermore, (ref) can be rewritten as
which shows that knowledge of $f\big(y_{i}\,\big|\,y_{i}^{(0)},\,x_{i},\, {\Greekmath 010B} _{i};\,{\Greekmath 0112} \big)$ is sufficient to verify (ref), and therefore (ref), for a given moment function, $m$.
Consider a single moment function $m(y_{i},y_{i}^{(0)},x_{i},{\Greekmath 0112} )\in \mathbb{R}$ and fixed values of $y_{i}^{(0)}$, $x_{i}$, ${\Greekmath 0112} $. Then, for every value of ${\Greekmath 010B} _{i}$, the condition (ref) constitutes one linear restriction on the vector $ [m(y_{i},y_{i}^{(0)},x_{i},{\Greekmath 0112} )\,:\,y_{i}\in \mathcal{Y}]\in \mathbb{R} ^{|\mathcal{Y}|}$. Finding $m(y_{i},y_{i}^{(0)},x_{i},{\Greekmath 0112} )\in \mathbb{R}$ for fixed $y_{i}^{(0)}$, $x_{i}$, and ${\Greekmath 0112} $ then requires solving an infinite number of linear equations in $|\mathcal{Y}|$ variables. Depending on the choice of $f\big(y_{i}\,\big|\,y_{i}^{(0)},\,x_{i},\,{\Greekmath 010B} _{i};\,{\Greekmath 0112} \big)$ no solution may exist to this infinite dimensional system of equations. The key finding of this paper is that dynamic logit models do generally have solutions to this system, that is, moment conditions of the form (ref) are generally available in such models.
In this subsection, we briefly outline a three-step strategy for obtaining valid moment conditions of the form (ref) for a model $f\big(y\,\big|\,y^{(0)},\,x,\,{\Greekmath 010B} ;\,{\Greekmath 0112} \big)$ with $T$ time periods. We drop all indices $i$ unless they are explicitly required.
The first step is to determine numerically whether it seems likely that one can find moment functions that satisfy (ref). To do this, we choose numerical values for $y^{(0)}$, $x$, and ${\Greekmath 0112} $, and also choose $Q>|\mathcal{Y}|$ different numerical values for the fixed effects $ ({\Greekmath 010B} _{1},\ldots ,{\Greekmath 010B} _{Q})\subset \mathcal{A}^{Q}$. We then check numerically whether for those values, the system
of $Q$ equations in $|\mathcal{Y}|$ unknowns, $[m(y):y\in \mathcal{Y}]\in \mathbb{R}^{|\mathcal{Y}|}$, has a solution other than $m=0$ (and if so, how many). If the $Q$ equations have at least one solution, then one could repeat this exercise for multiple randomly chosen numerical values of $y^{(0)}$, $x$, $ {\Greekmath 0112} $ and of the fixed effects. In this step it is important to use sufficient numerical precision in those calculations, see Appendix (ref) for more details.
If the conclusion of the first step is that moment functions seem to exist, then the next step is to find them. One way to proceed is by choosing specific numerical values for $({\Greekmath 010B} _{1},\ldots ,{\Greekmath 010B} _{Q})$, but now solve the system (ref) analytically for arbitrary values of $y^{(0)}$, $x$, ${\Greekmath 0112} $.\footnote{ Since we consider discrete choice, the initial condition $y^{(0)}$ takes a finite number of discrete values, and we can perform the analysis separately for each of value of $y^{(0)}$. But for $x$ and ${\Greekmath 0112} $ we need to allow for arbitrary general values in this step.} The corresponding solution for $ m(y)$ will depend on $y^{(0)}$, $x$, and ${\Greekmath 0112} $, and we therefore denote the solution by $ m(y,y^{(0)},x,{\Greekmath 0112} )$. The solution will not depend on the specific numeric values of ${\Greekmath 010B} _{1},\ldots ,{\Greekmath 010B} _{Q}$ if we have truly found a valid moment condition for the chosen model. See Section (ref) for a concrete example.
Since the moment functions, $m(y,y^{(0)},x,{\Greekmath 0112} )$, obtained in the second step are obtained using a set of specific numerical values of ${\Greekmath 010B} _{1},\ldots ,{\Greekmath 010B} _{Q}$, the third step is to verify analytically that they satisfy the condition (ref) for all ${\Greekmath 010B} \in \mathbb{R}$.
Once one has constructed moment functions, $m(y,y^{(0)},x,{\Greekmath 0112} )$, using the strategy outlined so far, the next step is to study the implications of those moment functions for identification and estimation of ${\Greekmath 0112} $. It is also useful to study the properties of the moment functions to obtain a better understanding of their structure and origin. In particular, the first two steps can only be implemented for a given number of time periods $T$. However, by studying the moment functions obtained for specific choices of $ T $, one may be able to draw general conclusions that make it possible to write down all moment functions for a given model for all possible values of $T$.
In the next section, we follow the steps outlined above to construct moment conditions for the binary choice logit AR(1) model in equation ((ref) ). However, there are many other interesting semi-parametric discrete choice panel models $f\big(y\,\big|\,y^{(0)},\,x,\,{\Greekmath 010B} ;\,{\Greekmath 0112} \big)$ for which moment conditions of the form (ref) exist, but have not yet been studied systematically --- see Section (ref) below for some concrete examples. The above work program can therefore be seen as a blueprint for an extensive research agenda beyond the current paper. We have recently implemented this blueprint in honore2021dynamic for the case of dynamic ordered choice panel models. davezies2022fixed can be seen as another example that implements the above program for static binary choice panel model with idiosyncratic error distributions that generalize the logistic case in a particular way.
Dano2023arXiv uses “transition functions” to derive moment conditions for dynamic discrete choice logit panel models. This reproduces and extends various results in the current paper, with the advantage that the method more easily generalizes to an arbitrary number of time periods. Dano2023arXiv also works out the semiparametric efficiency bound for the AR(1) panel logit model with regressors.
dobronyi2021identification point out that it is sometimes possible to determine a lower bound on the number of moment conditions that can be derived for a given model. Specifically, for many of the models considered below we have $A \in \mathbb{R}$, and one can write the probability distribution in ((ref)) as
for some $K \in \{1,2,\ldots\}$, some positive function ${\Greekmath 0114}$ of $ a=\exp \left( {\Greekmath 010B} \right) $ that does not depend on $y$, and some functions $c_k$ of $y$ that do not depend on $a$. Here, the functions ${\Greekmath 0114}$ and $c_k$ also depend on $y^{(0)}$, $x$, ${\Greekmath 0112}$, but analogous to our discussion in the last subsection, those arguments are dropped to focus more clearly on the dependence on ${\Greekmath 010B}$ and $y$.
A moment function must then satisfy
which is equivalent to
These are $K$ linear conditions in $\left\vert \mathcal{Y}\right\vert $ unknown parameters $m(y)$. We therefore have at least $\left\vert \mathcal{Y} \right\vert -K$ linearly independent solutions $m(y)$. In other words, the model must have at least $\left\vert \mathcal{Y}\right\vert -K$ conditional moment conditions (conditional on the initial conditions and on the explanatory variables). Of course, there is no guarantee that all of these moment conditions are functions of the common parameter, ${\Greekmath 0112} $.
In this section, we apply the strategy outlined in Section (ref) to construct moment conditions for the binary choice logit AR(1) model in equation ((ref)) when $T$ is three. In most applications, this corresponds to a total of four time periods: three for which the models is assumed to apply, plus one that delivers the initial condition, $y_{0}$.
By numerically evaluating whether solutions to (ref) exist for this model, one finds that $T=3$ is the smallest number of time periods for which non-zero valid moment functions are available. Our discussion in Section (ref) below formally shows that it is indeed not possible to derive moment conditions when $T=2$. This is the reason why we focus on $T=3$ in this section. For $T=3$ and ${\Greekmath 010D} \neq 0$, one furthermore finds by numerical experimentation that for each value of the initial condition $y_{0}$, there exist exactly two linearly independent moment functions that satisfy (ref).
Having verified the existence of moment functions numerically, the next goal is to find analytic formulas for them. That is, we want to find functions $ m(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )$ that satisfy (ref).
Since $T=3$, we have $|\mathcal{Y}|=2^{T}=8$. We define vectors in $\mathbb{R }^{8}$ for the model probabilities and for the candidate moment functions:
For simplicity, we drop the arguments $y_{0}$, $x$, ${\Greekmath 010C} $, and ${\Greekmath 010D} $ for the rest of this subsection. They are all kept fixed in the following derivation, and they are the same in the probability vector $\mathbf{p} ({\Greekmath 010B} )=\mathbf{p}(x,y_{0},{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} )$ and in the moment vector $\mathbf{m}=\mathbf{m}(x,y_{0},{\Greekmath 010C} ,{\Greekmath 010D} )$. The probability vector $\mathbf{p}({\Greekmath 010B} )$ as a function of ${\Greekmath 010B} $ is given by the model specification. A moment vector $\mathbf{m}\in \mathbb{R}^{8}$ with $ \mathbf{m}\neq 0$ is valid if it satisfies $\mathbf{m}^{\prime }\,\mathbf{p} ({\Greekmath 010B} )=0$ for all ${\Greekmath 010B} \in \mathbb{R}$; that is, a valid moment vector needs to be orthogonal to $\mathbf{p}({\Greekmath 010B} )$ for all values of ${\Greekmath 010B} $. If we can find such a valid moment vector, then its entries will provide moment functions that satisfy equation (ref), because $ \mathbf{m}^{\prime }\,\mathbf{p}({\Greekmath 010B} )$ is equal to $\mathbb{E}\left[ m(Y,Y_{0},X,{\Greekmath 010C} _{0},{\Greekmath 010D} _{0})\,\big|\,Y_{0}=y_{0},\,X=x,\,A={\Greekmath 010B} \right] $.
Any valid moment vector also satisfies $\lim_{{\Greekmath 010B} \rightarrow \pm \infty } \mathbf{m}^{\prime }\,\mathbf{p}({\Greekmath 010B} )=0$. Moreover, the model probabilities $\mathbf{p}({\Greekmath 010B} )$ are continuous functions of ${\Greekmath 010B} $ with $\lim_{{\Greekmath 010B} \rightarrow -\infty }\mathbf{p}({\Greekmath 010B} )=\mathbf{e} _{1}=(1,0,0,0,\allowbreak 0,0,0,0)^{\prime }$ and $\lim_{{\Greekmath 010B} \rightarrow +\infty } \mathbf{p}({\Greekmath 010B} )=\mathbf{e}_{8}=(0,0,0,0,0,0,0,1)^{\prime }$, where $ \mathbf{e}_{k}$ denotes the $k$'th standard unit vector in eight dimensions. From this, we conclude:
Furthermore, from our “step 1” analysis with concrete numerical values we already know that:\footnote{ The numerical experiment does not provide a proof of this, but we still take this as an input in our moment condition derivation, with the final justification given by Lemma (ref).}
Motivated by hypothesis (2), the aim is to find two linearly independent moment vectors $\mathbf{m}^{(0)}$ and $\mathbf{m}^{(1)}$ for each $y_{0}\in \{0,1\}$. To distinguish $\mathbf{m}^{(0)}$ and $\mathbf{m}^{(1)}$ from each other, we impose the condition $\mathbf{e}_{7}^{\prime }\,\mathbf{m}^{(0)}=0$ for the first vector and the condition $\mathbf{e}_{2}^{\prime }\,\mathbf{m} ^{(1)}=0$ for the second vector. In addition, we require a normalization for each of these vectors, because an element of the nullspace can be multiplied by an arbitrary nonzero constant to obtain another element of the nullspace. We choose the normalizations $\mathbf{e}_{4}^{\prime }\,\mathbf{m}^{(0)}=-1$ and $\mathbf{e}_{5}^{\prime }\,\mathbf{m}^{(1)}=-1$. Together with the conditions in (1), this specifies four affine restrictions on each of the vectors $\mathbf{m}^{(0)},\mathbf{m}^{(1)}\in \mathbb{R}^{8}$. To define $ \mathbf{m}^{(0)}$ and $\mathbf{m}^{(1)}$ uniquely, we require four more affine conditions for each. We therefore choose four numeric values ${\Greekmath 010B} _{q}$ and impose the orthogonality between $\mathbf{p}({\Greekmath 010B} _{q})$ and $ \mathbf{m}^{(0/1)}$. Thus, motivated by (1) and (2), we need to solve the following two linear systems of equations:
If it is indeed possible to find such moment functions $\mathbf{m}^{(0/1)}$, then it must be possible for the four values of ${\Greekmath 010B} $ to be chosen arbitrarily without affecting the solutions $\mathbf{m}^{(0/1)}$. For example, ${\Greekmath 010B} _{q}=q$ is a valid choice. Note that the two-dimensional span of the vectors $\mathbf{m}^{(0)}$ and $\mathbf{m}^{(1)}$ and the potential of the moment conditions to identify and estimate ${\Greekmath 010C} $ and ${\Greekmath 010D} $ are not affected by the normalizations $\mathbf{e}_{4}^{\prime }\,\mathbf{m}^{(0)}=-1$, $\mathbf{e}_{5}^{\prime }\,\mathbf{m}^{(1)}=-1$, $ \mathbf{e}_{7}^{\prime }\,\mathbf{m}^{(0)}=0$, and $\mathbf{e}_{2}^{\prime }\,\mathbf{m}^{(1)}=0$.
The systems of linear equations (0) and (1) above uniquely determine $ \mathbf{m}^{(0)}$ and $\mathbf{m}^{(1)}$. By defining the $8\times 8$ matrices $\mathbf{B}^{(0)}=[\mathbf{e}_{1},\mathbf{e}_{8},\mathbf{e}_{7}, \mathbf{e}_{4},\allowbreak \mathbf{p}({\Greekmath 010B} _{1}),\mathbf{p}({\Greekmath 010B} _{2}), \mathbf{p}({\Greekmath 010B} _{3}),\mathbf{p}({\Greekmath 010B} _{4})]^{\prime }$, and $\mathbf{B} ^{(1)}=[\mathbf{e}_{1},\mathbf{e}_{8},\mathbf{e}_{2},\mathbf{e}_{5},\mathbf{p }({\Greekmath 010B} _{1}),\mathbf{p}({\Greekmath 010B} _{2}),\mathbf{p}({\Greekmath 010B} _{3}),\mathbf{p} ({\Greekmath 010B} _{4})]^{\prime }$, we can rewrite those systems of equations as $ \mathbf{B}^{(0)}\,\mathbf{m}^{(0)}=-\mathbf{e}_{4}$, and $\mathbf{B}^{(1)}\, \mathbf{m}^{(1)}=-\mathbf{e}_{4}$. Solving this gives
Plugging the analytical expression for $\mathbf{p}({\Greekmath 010B} )=\mathbf{p} (x,y_{0},{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} )$ into the definitions $\mathbf{B}^{(1)}$ and $\mathbf{B}^{(0)}$, we thus obtain analytical expressions for $\mathbf{m} ^{(0)}=\mathbf{m}^{(0)}(x,y_{0},{\Greekmath 010C} ,{\Greekmath 010D} )$ and $\mathbf{m}^{(1)}= \mathbf{m}^{(1)}(x,y_{0},{\Greekmath 010C} ,{\Greekmath 010D} )$.
To report the results, we denote the components of the solutions $\mathbf{m} ^{(0/1)}=\mathbf{m}^{(0/1)}(x,y_0, \allowbreak {\Greekmath 010C} ,{\Greekmath 010D} ) \in \mathbb{R}^8$ by $ m^{(0/1)}(y,x,y_0,{\Greekmath 010C} ,{\Greekmath 010D} ) \in \mathbb{R}$, for $y \in \mathcal{Y}$. Furthermore, let $x_{ts}=x_{t}-x_{s}$. Then, the solutions are
The two solutions in (ref) are closely related: If $ Y_{t} $ is generated according to (ref), then $Z_{t}=1-Y_{t}$ is also generated according to (ref), but with $X_{t}$ replaced by $ -X_{t}$ and $A$ replaced by $A-{\Greekmath 010D} $. The solutions $m^{(0)}$ and $ m^{(1)} $ are symmetric in the sense that $m^{(0)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )=m^{(1)}(1-y,1-y_0,-x,{\Greekmath 010C} ,{\Greekmath 010D} )$.
The following lemma establishes that the moment functions, $ m^{(0/1)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )$, displayed in (ref) are indeed valid. For this, it is not relevant how the moment functions were derived.
This lemma is a special case of Theorem (ref) below. However, one can prove this lemma more easily by direct calculation: just plug-in the definition of the probabilities $p(y,y_{0},x,{\Greekmath 010C} _{0},{\Greekmath 010D} _{0},{\Greekmath 010B} )$ and moments $m^{(0/1)}(y,y_{0},x,{\Greekmath 010C} _{0},{\Greekmath 010D} _{0}$) to show that
The details of this calculation are provided in Appendix B.2.1 of Honore2022moment.
As explained in Section (ref), dobronyi2021identification provide a method for deriving (a lower bound on) the number of moment conditions for a given model. For the panel logit AR(1) model, the probability distribution for $Y_{i}=(Y_{i1},\ldots ,Y_{iT})$ (conditional on $Y_{i0}$, $X_{i}$, $A_{i}$) is given by
With $a=\exp ({\Greekmath 010B} )$ and ${\Greekmath 0119} _{t}(y_{t-1})=\exp [x_{i}^{\prime }\,{\Greekmath 010C} +y_{t-1}\,{\Greekmath 010D} ]$, we then have
where we defined
This has the exact structure of equation ((ref)) with $K=2T$ and $\left\vert \mathcal{Y}\right\vert =2^{T}$, so there must be at least $ 2^{T}-2T$ moment conditions. When $T=3$, the lower bound on the number of conditional moment conditions is $2^{T}-2T=2$, which is exactly the number of moment conditions we found in Lemma (ref) above.
In this section, we briefly discuss some other fixed effects panel data models with discrete outcomes, for which it is possible to use the approach outlined in Section (ref) to derive moment conditions. The goal of this section is to illustrate the broad applicability of the moment condition approach, and it can be skipped by a reader interested in the binary choice AR(1) panel model only.
In a static panel binary response model with strictly exogenous regressors $ X_{i}=(X_{i1}, \ldots , \allowbreak X_{iT})$ and fixed effects $A_{i}$, the conditional distribution of the outcomes $Y_{i}=(Y_{i1},\ldots ,Y_{iT})$ is given by
where $F(\cdot )$ is a cumulative distribution function. The distribution in (ref) is a special case of (ref). For the logistic case, $F({\Greekmath 0122} )=[1+\exp (-{\Greekmath 0122} )]^{-1}$, one can use that $S_{i}=\sum_{t=1}^{T}Y_{it}$ is a sufficient statistic for $A_{i}$ to estimate ${\Greekmath 010C} $ via the conditional maximum likelihood estimator (CMLE) that conditions on $S_{i}$, see rasch1960studies and andersen1970asymptotic. In fact, chamberlain2010binary showed that for $T=2$, and subject to weak regularity conditions, root-$n$-consistent estimation of ${\Greekmath 010C} $ is only possible if $F({\Greekmath 0122} )$ is logistic. \footnote{ This result is for ${\Greekmath 0122} _{it}$ independent across $t$. Generalizations that allow for dependence across $t$ are derived in magnac2004panel.} This implies that non-trivial moment functions $ m(y_{i},x_{i},{\Greekmath 010C} )$ are available for the $T=2$ static panel model if and only if $F({\Greekmath 0122} )=[1+\exp (-{\Greekmath 0114} \,{\Greekmath 0122} +{\Greekmath 0116} )]^{-1}$, for some constants ${\Greekmath 0114} >0$ and ${\Greekmath 0116} \in \mathbb{R}$.
Interestingly, for the static panel model with $T=3$, one can allow for distributions $F(\cdot )$ that are not logistic and still estimate the parameter ${\Greekmath 010C} $ at $\sqrt{n}$ rate, that is, the $T=2$ result of chamberlain2010binary does not apply in that case. In particular, for $T=3$ , johnson2004identification and davezies2022fixed consider distributions of the form $F({\Greekmath 0122} )=\left[ 1+w_{1}\exp (-{\Greekmath 0115} _{1}\,{\Greekmath 0122} )+w_{2}\exp (-{\Greekmath 0115} _{2}\,{\Greekmath 0122} )\right] ^{-1}$, with non-negative real-valued parameters $w_{1}$, $w_{2}$, ${\Greekmath 0115} _{1}$, $ {\Greekmath 0115} _{2}$, and derive moment conditions for (ref). One can use the machinery in Section (ref) to show that the specification considered by johnson2004identification and davezies2022fixed is not the only extension of the logistic distribution that provides moment conditions for the $T=3$ static model. For example, consider the case where $F({\Greekmath 0122} ) $ is a mixture of two logistic distributions with the same variance:
where ${\Greekmath 0121} \in \lbrack 0,1]$ is a mixture weight, ${\Greekmath 0115} >0$ parametrizes the common variance of the logistic components, and ${\Greekmath 0116} _{1},{\Greekmath 0116} _{2}\in \mathbb{R}$ parametrize the mean of the two components. When plugging (ref) into (ref) and then applying the procedure described in Section (ref) to derive valid moment functions, one finds that for ${\Greekmath 0116} _{1}\neq {\Greekmath 0116} _{2}$ and ${\Greekmath 0121} \in (0,1)$ exactly one moment function exists for general values of ${\Greekmath 010C}$ and $x$ when $T=3$. This moment condition is given by
where $\mathcal{P}$ is the set of all six permutations of $(1,2,3)$, and for $(t,s,r)\in \mathcal{P}$ the signature of that permutation is denoted by $ \mathrm{sgn}(t,s,r)$.
In addition, we have found numerically that one can allow for more general finite mixtures of logistic distributions when $T$ exceeds $3$. For example, it appears that one can allow for a mixture of three logistics when $T$ is 4, six when $T=5$, ten when $T=6$, and eighteen when $T$ is 7. Calculations like the ones in Section (ref) suggest that if the number of mixtures is $Q$, then there are $ 2^{T}-TQ-1$ non-trivial conditional moment conditions in this model. For example, if $T$ is 9 then there will be seven moment conditions when $F$ is a mixture of 56 logistic cumulative distribution functions. We leave it to future research to derive these and to investigate the extent to which they identify the common parameters of the model.
The analysis of the dynamic panel data logit model in Section (ref) generalizes to a model with more than one lag. Specifically, consider the model
where ${\Greekmath 010D} =({\Greekmath 010D} _{1},\ldots ,{\Greekmath 010D} _{p})^{\prime }$. We assume that the autoregressive order $p\in \{2,3,4,\ldots \}$ is known, and that outcomes $Y_{it}$ are observed for time periods $t=t_{0},\ldots ,T$, with $ t_{0}=1-p$. Thus, the total number of time periods for which outcomes are observed is $T_{\mathrm{obs}}=T+p$, consisting of $T$ periods for which the model applies and $p$ periods to observe the initial conditions. We maintain the definition $Y_{i}=(Y_{i1},\ldots ,Y_{iT})$, but the initial conditions are now described by the vector $Y_{i}^{(0)}=(Y_{i,t_{0}},\ldots ,Y_{i0})$.
Numeric calculations similar to those for the model with one lag suggest that for a given value of $p$, one requires $T\geq 2+p$ (i.e.\ $T_{\mathrm{ obs}}\geq 2+2p$) time periods to find conditional moment conditions that hold without restrictions on the parameters or on the support of the explanatory variables.\footnote{ In addition to those general moment conditions, there are additional ones that only become available for special values of the parameters and of the regressors.} For example, a model with $p=3$ lags requires a total of eight time periods; three that provide the initial conditions for $Y_{it}$, and five for which the model is assumed to apply. Numerical calculations also suggest that the number of linearly independent moment conditions available for each initial condition, $y^{(0)}$, is equal to\footnote{ We have verified this for $p\in \{0,\ldots ,6\}$ and $T\in \{2+p,\ldots ,8\}$ , but believe that this formula for the number of linearly independent moments holds for all integers $p$, $T$ with $T\geq 2+p$. However, a general proof of this conjecture is beyond the scope of this paper.} $ 2^{T}-(T+1-p)\,2^{p}$. In Appendix (ref), we provide analytic formulas for all the moment functions that can be obtained with $T\leq 5$. Specifically, when $p=2$, we provide four moment functions for $T=4$ and sixteen for $T=5$. For $p=3$, there are eight moment functions, while there are no general moment functions when $p\geq 4$ and $T\leq 5$. Identification of the parameters ${\Greekmath 010C}$ and ${\Greekmath 010D}$ for $p \leq 3$ from the moment conditions is discussed in Appendix Section B.3 of Honore2022moment.
The special case of an AR(2)\ logit model with fixed effects and no explanatory variables was considered in honore2019identification. Numerical calculations in that paper suggested that the common parameters in such a model are point identified for $T=3$ (i.e.\ $T_{\mathrm{obs}}=5$), but no proof of identification was provided. Evaluating the moment functions in Appendix (ref) at ${\Greekmath 010C} =0$ makes it clear why ${\Greekmath 010D} =({\Greekmath 010D} _{1},{\Greekmath 010D} _{2})$ is identified and how one would estimate it. Specifically, if the outcomes $ Y=(Y_{1},Y_{2},Y_{3})$ are generated from the AR(2) panel logit model without explanatory variables, then we have, for all $y^{(0)}\in \{0,1\}^{2}$ and ${\Greekmath 010B} \in \mathbb{R}$, that
with moment functions given by
where $y_{0}\in \{0,1\}$.
The moment functions $m_{(0,0)}$ and $m_{(1,1)}$ are strictly monotone in $ {\Greekmath 010D} _{1}$ and do not depend on ${\Greekmath 010D} _{2}$. Each of them therefore identify the parameter ${\Greekmath 010D} _{1}$. For a given value of ${\Greekmath 010D} _{1}$, the moment function $m_{(0,1)}$ and $m_{(1,0)}$ are strictly monotone in ${\Greekmath 010D} _{2}$, and they therefore each identify the parameter ${\Greekmath 010D} _{2}$ once ${\Greekmath 010D} _{1}$ has been identified. A GMM estimator based on these moment will be root-n consistent under standard regularity conditions.
For arbitrary regressors, no valid moment functions seem to exist when some of the elements of ${\Greekmath 010C} $ are replaced by fixed effects ${\Greekmath 010C}_i$. However, if one of the explanatory variables is a linear time trend, then it is possible to allow for the coefficient on this variable to differ arbitrarily across observations. Specifically, consider the generalization of the model in equation ((ref)) to
By mimicking the calculations in Section (ref), one finds that at least $\ell_{\min} = 2^T - \frac{T}{3}\left( 2T^{2}-3T+7\right) $ moment conditions need to exist in this model. For $T\geq 9$ we have $\ell_{\min}>0$, that is, it must be the case that moment conditions for this model exist. See Appendix Section (ref) for details. However, such calculations only yield a lower bound on the number of moment conditions.\footnote{ Numerically, we find 126 linearly independent moment conditions for the model with additional strictly exogenous explanatory variables when $ T=9$ . This is larger than the lower bound of $\ell_{\min} = 86$ moment conditions derived in Appendix Section (ref). This illustrates that exploring the polynomial structure of the model probabilities (as described in Section (ref)) does not always give the exact number of available moment conditions in binary logit models. } Numerically, we do not find any moment conditions for $T \leq 8$ for general parameter values with $x_{it} \neq 0$, but we do find two valid moment conditions for $T=8$ if $x_{it}=0$ (so there are no additional regressors in the model, and ${\Greekmath 010D}$ is the only common parameter). Both of these moment conditions depend on the parameter ${\Greekmath 010D}$. In Appendix Section (ref), we discuss these moment conditions.
The methods described in this paper can also be applied to dynamic panel data versions of other “textbook” logit models. In particular, honore2021dynamic use the procedure outlined in Section (ref) to find moment conditions for dynamic panel data {\it ordered} logit models, and Dano2023arXiv shows how to obtain moment conditions for dynamic panel data {\it multinomial} logit models.
This section shows that the moment conditions for the panel logit AR(1) model in Lemma (ref) can be used to uniquely identify the parameters ${\Greekmath 010C} $ and ${\Greekmath 010D} $ under appropriate support conditions on the regressor $X$. The following technical lemma turns out to be very useful in showing this.
To explain the lemma, consider the case $K=1$,\footnote{ Note that $K=0$ is trivially allowed in Lemma (ref). We then have $s=\emptyset $ and $g_{\emptyset }:\mathbb{R}\rightarrow \mathbb{R} $ is a single increasing function, implying that $g_{\emptyset }({\Greekmath 010D} )=0$ can at most have one solution.} when we have two scalar parameters ${\Greekmath 010C} ,{\Greekmath 010D} \in \mathbb{R}$. The lemma then requires that the two functions $ g_{+}({\Greekmath 010C} ,{\Greekmath 010D} )$ and $g_{-}({\Greekmath 010C} ,{\Greekmath 010D} )$ are both strictly increasing in ${\Greekmath 010D} $, and $g_{+}$ is also strictly increasing in ${\Greekmath 010C} $ , while $g_{-}$ is strictly decreasing in ${\Greekmath 010C} $. Thus, $g_{+}({\Greekmath 010C} ,{\Greekmath 010D} )=0$ gives a solution for ${\Greekmath 010D}={\Greekmath 010D}({\Greekmath 010C})$ that is strictly decreasing in ${\Greekmath 010C}$, while $g_{-}({\Greekmath 010C} ,{\Greekmath 010D} )=0$ gives a solution ${\Greekmath 010D}({\Greekmath 010C})$ that is strictly increasing, implying that the joint solution must be unique.
Combining the moment conditions in Lemma (ref) with the result of Lemma (ref) allows us to provide sufficient conditions for point identification in panel logit AR(1) models with $T=3$. For that purpose we define the sets
for $k\in \{1,\ldots ,K\}$. The set $\mathcal{X}_{k,+}$ is the set of possible regressor values $x\in \mathbb{R}^{K\times 3}$ such that either $x_{k,1}\leq x_{k,3}<x_{k,2}$ or $x_{k,1}<x_{k,3}\leq x_{k,2}$; that is, the $k$'th regressor takes its smallest value in time period $t=1$ and its largest value in time period $t=2$. Conversely, the set $ \mathcal{X}_{k,-}$ is the set of possible regressor values $x\in \mathbb{R} ^{K\times 3}$ for which the $k$'th regressor takes its largest value in time period $t=1$ and its smallest value in time period $t=2$.
The motivation behind the definition of those sets is that for $x \in \mathcal{X}_{k,\pm}$ our moment functions $m^{(0/1)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )$ defined in Section (ref) have convenient monotonicity properties in the parameters ${\Greekmath 010C}_k$. For example, for $m^{(0)}$ we have
This is because the parameter ${\Greekmath 010C}$ appears in $m^{(0)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )$ only through $\exp \left( x_{23}^{\prime }{\Greekmath 010C} \right)$, $\exp \left( x_{31}^{\prime }{\Greekmath 010C} \right)$ and $\exp \left( x_{21}^{\prime }{\Greekmath 010C} \right) $; $x \in \mathcal{X}_{k,+}$ (or $x \in \mathcal{X}_{k,-}$) guarantees that the differences $x_{k,2}-x_{k,3}$ and $x_{k,3}-x_{k,1}$ and $x_{k,2}-x_{k,1}$ are all $\geq 0$ ($\leq 0$), with some of them strictly positive (negative). The moment function $m^{(1)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )$ has exactly the opposite monotonicity properties in ${\Greekmath 010C}$.
Next, for any vector $s = (s_1,\ldots,s_K) \in \{-,+\}^{K}$ we define the set $\mathcal{X}_{s}=\bigcap_{k\in \{1,\ldots ,K\}}\mathcal{X}_{k,s_{k}}$ and the corresponding expected moment functions, for $q,y_0 \in \{0,1\}$,
Because $\mathcal{X}_{s}$ is the intersection of the sets $\mathcal{X}_{k,\pm}$, the expected moment functions $\overline{m}_{y_{0},s}^{(0/1)}({\Greekmath 010C} ,{\Greekmath 010D} )$ have monotonicity properties with respect to all the elements of ${\Greekmath 010C}$ specified by the sign vector $s = (s_1,\ldots,s_K)$. For example, (ref) implies that $\overline{m}_{y_0,s}^{(0)}({\Greekmath 010C} ,{\Greekmath 010D} )$ is strictly increasing in ${\Greekmath 010C}_k$ if $s_k=+$, and strictly decreasing in ${\Greekmath 010C}_k$ if $s_k=-$, for all $k \in \{1,\ldots,K\}$.
The proof of the theorem is provided in the appendix. Note that only one of the moment functions $m^{(0)}$ or $m^{(1)}$ is required to derive identification in Theorem (ref), and only one of the initial conditions $y_0 \in \{0,1\}$ needs to be observed. The key assumption in Theorem (ref) is that we have enough variation in the observed regressor values $X=(X_1,X_2,X_3)$ to satisfy the condition $\mathrm{Pr}\left( Y_{0}=y_{0},\;X\in \mathcal{X}_{s}\right) >0$, for all $s \in \{+,-\}^K$.
Theorem (ref) achieves identification of ${\Greekmath 010C}$ and ${\Greekmath 010D}$ via conditioning on the sets of regressor values $\mathcal{X}_{s}$, which all have positive Lebesgue measure. By contrast, the conditional likelihood approach in honore2000panel conditions on the set $x_2=x_3$, which has zero Lebesgue measure, and therefore also often zero probability measure, implying that the resulting estimates for ${\Greekmath 010C}$ and ${\Greekmath 010D}$ usually converge at a rate slower than root-$n$. In our approach here, using the sample analogs of the moment conditions $\overline{m}_{y_{0},s}^{(0/1)}({\Greekmath 010C} ,{\Greekmath 010D} ) = 0$ for $s \in \{+,-\}^K$, we immediately obtain GMM estimates for ${\Greekmath 010C}$ and ${\Greekmath 010D}$ that are root-n consistent under standard regularity conditions.
However, in practice, we do not actually recommend estimation via the moment conditions in Theorem (ref), because by conditioning on $X\in \mathcal{X}_{s}$ these moment conditions still only use a small subset of the available information in the data. Instead, many more unconditionally valid moment conditions for ${\Greekmath 010C}$ and ${\Greekmath 010D}$ can be obtained from Lemma (ref) (or from Theorem (ref) below for $T>3$), resulting in potentially much more efficient estimators for ${\Greekmath 010C}$ and ${\Greekmath 010D}$, and Section (ref) describes how we implement such estimators in practice. Nevertheless, from a theoretical perspective, the identification result in Theorem (ref) is important, because it comprises a significant improvement over existing results for dynamic panel logit models with explanatory variables.
In Section (ref) above we already found analytic formulas for valid moment functions that are free of the fixed effects for the panel logit AR(1) model with three time periods. In this section we discuss various generalizations of this result, most importantly to $T>3$ time periods. Before presenting those positive results, we first briefly discuss a negative result for $T=2$ time periods.
Here, we argue that it is not possible to derive moment conditions for model ((ref)) on the basis of two time periods plus the initial condition, $y_{0}$. If one could construct such moment conditions for $T=2$ that hold conditional on the individual specific effects $A$, then the corresponding moment functions $m(y,y_0,x,{\Greekmath 010C},{\Greekmath 010D}) $ would satisfy
for all ${\Greekmath 010B} \in \mathbb{R}$. In the limit ${\Greekmath 010B} \rightarrow \infty$ the model probabilities become zero, except for $p((1,1),y_0,x,{\Greekmath 010C},{\Greekmath 010D},{\Greekmath 010B}) \rightarrow 1$. This implies $m((1,1),y_0,x,{\Greekmath 010C},{\Greekmath 010D}) =0$. Analogously, in the limit ${\Greekmath 010B} \rightarrow -\infty$ we have $p((0,0),y_0,x,{\Greekmath 010C},{\Greekmath 010D},{\Greekmath 010B}) \rightarrow 1$, which implies $m((0,0),y_0,x,{\Greekmath 010C},{\Greekmath 010D}) =0$. Thus, only $m((0,1),y_0,x,{\Greekmath 010C},{\Greekmath 010D})$ and $m((1,0),y_0,x,{\Greekmath 010C},{\Greekmath 010D})$ can be non-zero, and (ref) therefore implies that
Unless ${\Greekmath 010D} =0$, the right hand side of (ref) will always have a non-trivial dependence on ${\Greekmath 010B}$, implying that no moment conditions can be constructed for $T=2$ (that are valid conditional on arbitrary $ A={\Greekmath 010B} $). For ${\Greekmath 010D} =0$ equation ((ref)) yields the moment conditions implied by Rasch60's conditional likelihood.
The fact that there are no moment conditions when $T=2$ is consistent with the non-identification result in Chamberlain2023SERIES.
One can work out analytic moment functions for model ((ref)) with $T=4$ and $T=5$ using the same derivation method described in Section (ref) for $T=3$. These are relatively brute force calculations that only require limited human input and creativity. However, once those analytic moment functions are obtained, one can move on to study their common structure, which leads to the following lemma that allows us to derive all the valid moment conditions for the panel logit AR(1) model for an arbitrary number of time periods.
Before presenting the lemma, we introduce some additional notation: The cumulative distribution function of the logistic distribution is given by $\Lambda({\Greekmath 0118}):=[1+\exp(-{\Greekmath 0118})]^{-1}$. In addition, we define the cyclical decrement function ${\Greekmath 010E} : \{1,2,3\} \rightarrow \{1,2,3\}$ by $$ {\Greekmath 010E}(t) := \left\{
\right. $$
The proof is given in Appendix (ref). Note that the vector of conditioning variables $(X,A)$ is only included in Lemma (ref) to better connect the lemma to our panel AR(1) model, but for the mathematical result of the lemma this vector $(X,A)$ is actually irrelevant (no assumptions are imposed on these conditioning variables, all probability statements are conditional on $(X,A)$), and the lemma may be easier read and understood by initially ignoring all occurrences of $(X,A)$ and $(x,{\Greekmath 010B})$. Furthermore, when applying Lemma (ref) to the $T=3$ panel AR(1) model of Section (ref), we simply have $(\widetilde Y_1, \widetilde Y_2, \widetilde Y_3) =( Y_1, Y_2, Y_3)$ and $(W_1, W_2, W_3) =( Y_0, Y_1, Y_2)$, but the more general notation in the lemma is convenient when generalizing the results to models with $T>3$.
The assumptions imposed in Lemma (ref) are relatively weak. In particular, the outcomes $\widetilde Y_1$, $\widetilde Y_2$, $\widetilde Y_3$ are not assumed to be generated from a logit model. However, the result of Lemma (ref) is in general equally weak, because the moment functions $m^{(q)}(w_1,\widetilde y_1,w_2,\widetilde y_2,w_3,\widetilde y_3,x,{\Greekmath 010B})$ provided by the lemma still depend on the individual specific effects ${\Greekmath 010B}$, that is, the lemma in general does {\it not} deliver the type of moment conditions (ref) that we are interested in this paper. We find it nevertheless useful to state the lemma in this weak form, because it provides some understanding for why the logit assumption is important for obtaining valid moment conditions that are free of the fixed effects.
For a binary choice model with single index $z_t(W_t,X) \in \mathbb{R}$ and additive fixed effects $A \in \mathbb{R}$ we have $\widetilde Y_t = \mathbbm{1}\{z_t(W_t,X) + A + {\Greekmath 0122}_t \geq 0\}$, for $t \in \{1,2,3\}$. If, in addition, we assume a logistic distribution for the random shock ${\Greekmath 0122}_t$, then we obtain, for $\widetilde y \in \{0,1\}$,
which implies that for all $s,t \in \{1,2,3\}$, $$\Lambda^{-1}\Big[ p_{s} \left(\widetilde y \, \big| \,w_{s},x,{\Greekmath 010B} \right) \Big] - \Lambda^{-1}\Big[ p_t \left(\widetilde y \, \big| \,w_t,x,{\Greekmath 010B} \right) \Big] = (2 \widetilde y -1) \, [ z_{s}(w_{s},x) - z_t(w_t,x) ] $$ does not depend on the fixed effects ${\Greekmath 010B}$. For this logistic specification with additive fixed effects we therefore find that the moment functions $m^{(q)}(w_1,\widetilde y_1,w_2,\widetilde y_2,w_3,\widetilde y_3,x,{\Greekmath 010B}) $ in Lemma (ref) do not depend on the fixed effects ${\Greekmath 010B}$, and can be written as\footnote{ Here we also use that $\sum_{t=1}^3 \left[ z_{{\Greekmath 010E}(t)}(w_{{\Greekmath 010E}(t)},x) - z_t(w_t,x) \right] = 0$. }
For the case $T=3$, $(\widetilde Y_1, \widetilde Y_2, \widetilde Y_3) =( Y_1, Y_2, Y_3)$, $(W_1, W_2, W_3) =( Y_0, Y_1, Y_2)$, and $ z_{t}(w_{t},x) = y_{t-1} \, {\Greekmath 010D}_0 + x_t' \, {\Greekmath 010C}_0$ it is easy to verify that $m^{(q)}(w_1,\widetilde y_1,w_2,\widetilde y_2,w_3,\widetilde y_3,x)$ in (ref) is equal to $m^{(q)}(y,y_0,x,{\Greekmath 010C}_0,{\Greekmath 010D}_0)$ in display (ref) above, that is, Lemma (ref) delivers the moment functions derived for the $T=3$ dynamic logit model in Section (ref) as a special case.
We now discuss how the moment functions for $T=3$ generalize to more than three time periods (after the initial $y_{0}$). We have already argued above that Lemma (ref) is useful for our purposes for logit models of the form (ref) where it delivers the moment functions in (ref) that do not depend on the fixed effects. We now apply those results to the fixed effect logit AR(1) model with an arbitrary number of time periods $T \geq 3$ by setting $(\widetilde Y_1, \widetilde Y_2, \widetilde Y_3) =( Y_t, Y_s, Y_r)$ and $(W_1, W_2, W_3) =( Y_{t-1}, Y_{s-1}, Y_{r-1})$, for any triplet of time periods $t,s,r\in \{1,2,\ldots ,T\}$ that satisfy $t<s<r$. Note that for this choice the Markov chain assumption in Lemma (ref) is satisfied, that is, conditional on $(X,A)$, $ Y_{t-1} \rightarrow Y_t \rightarrow Y_{s-1} \rightarrow Y_{s} \rightarrow Y_{r-1} \rightarrow Y_r $ indeed constitutes a Markov chain according to model (ref). Furthermore, in that model, the distribution of $Y_{t}$ conditional on $Y_{t-1}$, $X$, $A$ is indeed of the logistic form (ref) with $z_t(w_t,x) = y_{t-1} \, {\Greekmath 010D}_0 + x_t' \, {\Greekmath 010C}_0$ for all time periods $t$.
Making the unknown parameter dependence explicit, we now define the single index for time period $t$ as $z_{t}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )=x_{t}^{\prime }\,{\Greekmath 010C} +y_{t-1}\,{\Greekmath 010D} $, and we also define the corresponding pairwise differences $z_{ts}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )=z_{t}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )-z_{s}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )$. Then, for triples of time periods $t,s,r\in \{1,2,\ldots ,T\}$ with $t<s<r$, the moment function in (ref) can be written more explicitly as
For $T=3$ and $(t,s,r)=(1,2,3)$, these moment functions are exactly those calculated in Section (ref) above. For general $T \geq 3$ and triplets $(t,s,r)$ we can apply Lemma (ref), conditional also on $Y_0,Y_{1},\ldots ,Y_{t-1}$, to obtain the following theorem.
The proof is given in Appendix (ref), but as argued above, the theorem really is an immediate corollary of Lemma (ref). Instead of conditioning on $Y_{1},\ldots ,Y_{t-1}$, we can also multiply the moment function with an arbitrary function of $Y_{1},\ldots ,Y_{t-1}$. Namely, by applying Theorem (ref) and the law of iterated expectations, we find, for any function $w:\{0,1\}^{t-1}\rightarrow \mathbb{R}$, that
From Section (ref) we know that, for any fixed value of the initial condition $y_0$, there are at least $\ell=2^T-2T$ linearly independent moment conditions available for our AR(1) logit model with $T$ time periods. It turns out for ${\Greekmath 010D} \neq 0$ this is exactly the correct number of linearly independent moment conditions in this model. In the 2020 working paper version of the current paper we conjectured this, and subsequent papers by kruiniger2020further, dobronyi2021identification, and Dano2023arXiv have shown that this is indeed the case.
Equation (ref) provides all of the $\ell=2^T-2T$ available valid moment functions for this model, but not all those moment functions $w(Y_{1},\ldots ,Y_{t-1})\,m^{(q)(t,s,r)}(Y,Y_0,X,{\Greekmath 010C} _{0},{\Greekmath 010D})$ are linearly independent, that is, some of them can be written as linear combinations (with coefficients that depend on $x$, ${\Greekmath 010C}$, ${\Greekmath 010D}$) of the others. However, if we restrict ourselves to $r=T$, then we have verified numerically that a linearly independent basis is obtained. Note that once we fix $r=T$, then, for given $y_0$, we can still choose $q \in \{0,1\}$, $w:\{0,1\}^{t-1}\rightarrow \mathbb{R}$, and $(t,s)$, with $1\leq t<s<T$. The total number of basis elements is therefore equal to
as claimed above.\footnote{ We consider ${\Greekmath 010D} \neq 0$ here. For ${\Greekmath 010D}=0$ we have a static panel logit model, and in that case $T-1$ additional moment conditions become available, bringing the total number of available moments to $2^T - T -1$. The first-order conditions of the conditional likelihood in rasch1960studies and andersen1970asymptotic are linear combinations of these moment functions.}
The only regressor and outcome values that enter into the moment functions $m^{(q)(t,s,r)} $ are $(x_t,x_s,x_r)$ and $(y_{t-1},y_t,y_{s-1},y_s,y_{r-1},y_t)$.\footnote{Of course, $y_t$ coincides with $y_{s-1}$ if $t=s-1$, and $y_s$ coincides with $y_{r-1}$ if $s=r-1$.} Thus, as long as those variables are observed we can evaluate $m^{(q)(t,s,r)} $. The moment conditions for $T>3$ can therefore also be applied to unbalanced panels where regressors and outcomes are not observed in all time periods, provided that the occurrence of missing values is independent of the outcomes $Y$, conditional on the regressors $X$ and the individual-specific effects $A$. The data in our empirical illustration are indeed unbalanced, and in Section (ref) we discuss how to combine the moment functions for unbalanced panels.
The first paper to obtain moment conditions for the dynamic panel logit model without imposing restrictions on the covariate values is the working paper by kitazawa2013exploration, which was recently published (kitazawa2022transformations). That paper defines
where ${\Greekmath 010E}=e^{\Greekmath 010D} -1$ and $\Delta x_t=x_t - x_{t-1}$. The paper then shows that, for $t \in \{2,\ldots,T-1\}$,\footnote{This is written here in our conventions for $t$ and $T$. } the functions $ \hbar U_t$ and $ \hbar \Upsilon_t $ are valid moment functions, in the sense of (ref). kitazawa2016root uses the same moment conditions, but also includes time dummies in the model, which in our notation are included in the parameter vector ${\Greekmath 010C}$ (one just needs to define the regressors $x_t$ as appropriate dummy variables).
Those definitions look quite different to our moment functions above, but one can show that
Thus, apart from a rescaling (with a non-zero function of the parameters and conditioning variables), the moment functions of kitazawa2013exploration coincide with our moment functions for AR(1) models with $T=3$. However, the complete set of moment conditions for $T>3$ in Theorem (ref) is new.
honore2000panel observe that with (ref) and $T=3$, the conditional likelihood function that conditions on $Y=y_{0}$, $Y_{3}=y_{3}$, and $Y_{1}+Y_{2}=1$,
does not depend on ${\Greekmath 010B} $, when $x=(x_{1},x_{2},x_{2})$ (so the explanatory variables are the same in the last two periods). The corresponding scores are
and
where $m^{(0)}$ and $m^{(1)}$ are defined in {(ref). The results for $y_{0}=1$ are analogous. Thus, the score functions of the conditional likelihood in honore2000panel are linear combinations of our moment conditions when $x_{2}=x_{3}$. The conditional likelihood estimation discussed in cox1958regression and Chamberlain1985 are special cases of this without regressors ($x_{1}=x_{2}=x_{3}=0$).
Hahn2001 considers model (ref) with $T=3$, initial condition $ y_{0}=0$, and time dummies as regressors, that is, $x_{t}^{\prime }{\Greekmath 010C} ={\Greekmath 010C} _{t}$, with the normalization ${\Greekmath 010C} _{1}=0$. The common parameters in that model are $({\Greekmath 010C} _{2},{\Greekmath 010C} _{3},{\Greekmath 010D} )$. Hahn shows that these parameters cannot be estimated at root-n-rate. This is not in conflict with our results here, because Lemma (ref) only provides two moment conditions for $y_{0}=0$. However, there are three model parameters in the setup of Hahn2001, so just from counting parameters and moment conditions, we know that our moments cannot identify $({\Greekmath 010C} _{2},{\Greekmath 010C} _{3},{\Greekmath 010D} )$. \footnote{ Including moment conditions that use the initial condition $y_{0}=1$ will give two additional moments. From the point of view of counting moments, this will result in a model which is over-identified.} Thus, our moment conditions cannot be used to estimate the parameters $({\Greekmath 010C} _{2},{\Greekmath 010C} _{3},{\Greekmath 010D} )$ at root-n-rate. This is in agreement with Hahn's calculation of the information bound for this model.
The main reason why we can identify and estimate ${\Greekmath 010C} $ and ${\Greekmath 010D} $ is that we consider non-constant regressors $X=(X_{1},X_{2},X_{3})$, which gives us two moment conditions for each initial condition and each support point of the regressors, and thus many more moment conditions than parameters --- see our formal results on point-identification of ${\Greekmath 010C} $ and ${\Greekmath 010D} $ in Section (ref) above.
As explained in Section (ref), Lemma (ref) delivers valid moment functions for any model with logistic conditional probabilities of the form (ref). This means that the single index of the model need not be of the form $x_{t}^{\prime }\,{\Greekmath 010C} +y_{t-1}\,{\Greekmath 010D}$ that is linear in $x_{t}$ and $y_{t-1}$, but it can actually be any function of the strictly exogenous regressors, lagged dependent variable, and parameters. In particular, if we replace the model specification (ref) by
then the moment functions (ref) and Theorem (ref) remain fully valid, as long as we replace the parameters $({\Greekmath 010C},{\Greekmath 010D})$ by $({\Greekmath 010C}_0,{\Greekmath 010C}_1,{\Greekmath 010D})$, and define the single index by $z_{t}(y,y_{0},x,{\Greekmath 010C}_0,{\Greekmath 010C}_1, ,{\Greekmath 010D} )=(1-y_{t-1}) \,x_{t}^{\prime }\,{\Greekmath 010C}_0 + y_{t-1} \,x_{t}^{\prime }\,{\Greekmath 010C}_1 +y_{t-1}\,{\Greekmath 010D} $.
The generalized model (ref) is interesting, because it allows the effect of the regressors $X_{it}$ on $Y_{it}$ to depend on the current “state” of the process, $Y_{i,t-1}$, with ${\Greekmath 010C}_{0/1}$ measuring the effect of $X_{it}$ on $Y_{it}$ if $Y_{i,t-1} = 0/1$. We do not consider this more general model structure further in this paper, but it is noteworthy that the regressors $(1-Y_{i,t-1}) X_{it}$ and $Y_{i,t-1} X_{it}$ are pre-determined regressors that are more general than just the lagged dependent variable $Y_{i,t-1}$ we have considered so far. Further comments on more general regressors structures are given in Appendix (ref).
Another interesting generalization of this model that still allows for the construction of moment conditions, is to make the AR(1) coefficient ${\Greekmath 010D}$ individual specific. Equation (ref) then reads
We can then treat $(C_i,A_i)$ as a two-dimensional fixed-effect and employ the methods of Section (ref) and (ref) to explore moment conditions for ${\Greekmath 010C}$ that are free of $(C_i,A_i)$. For general covariate and parameter values, we find that no such moment conditions exist for $T=3$, but they do exist for $T \geq 4$. For example, for $T=4$ and $y_0=0$, a valid moment condition in this model (i.e.\ satisfying $\mathbb{E}\left[ m(Y_i,Y_{0,i},X_i,{\Greekmath 010C}_0,{\Greekmath 010C}_1 )\big| \,Y_{0,i}=0,X_i,C_i,A_i\right] =0$) is
where $z_{ts} = z_t - z_s$ as before, and $z_t=z_{t}(y,y_{0},x,{\Greekmath 010C}_0 ,{\Greekmath 010C}_1 ) = (1-y_{t-1}) x_{t}^{\prime }\,{\Greekmath 010C}_0 + y_{t-1} x_{t}^{\prime }\,{\Greekmath 010C}_1$ is the appropriate index function in this model. We have found numerically that there is one additional moment condition for $T=4$, a total of ten for $T=5$, and thirty-two for $T=6$. The moment function in the last display was derived using the ideas described in Section (ref).
In this section, we illustrate how to use the conditional moment functions in this paper to implement a GMM\ approach to estimation. We use data from the National Longitudinal Survey of Youth 1997\footnote{ The analysis is restricted to the years in which the survey was conducted annually, from 1997-2011. For years in which the respondent was not interviewed, all time-varying variables (e.g., employment status, school enrollment status, age, income, marital variables, etc) are marked as missing. Otherwise, unless the raw data was marked as missing in some capacity (e.g., due to non-response, the interviewee not knowing the answer to the question), no other entries had missings imposed upon them.} (NLSY97) covering the years 1997 to 2010, and the dependent variable is a binary variable indicating employment status by whether the respondent reported working $\geq $1000 hours in the past year. We estimate fixed effects logit AR(1) and AR(2) models using the number of biological children the respondent has (Children), a dummy variable for being married (Married), a transformation\footnote{ The spouse's income can be zero or negative. This prevents us from using the logarithm of the income as an explanatory variable. We therefore use the signed fourth root.} of the spouse's income (Sp.Inc.), and a full set of time dummies as the explanatory variables. There are a total of 8,274 individuals aged 16 to 32, resulting in 54,166 observations. For the estimation, we consider the full sample, as well as females and males separately. Figure (ref) displays the number of observations, $T_{i}$, per individual in each of the three samples.
The moment conditions for the fixed effects logit AR(1) in ((ref)) and ((ref)) are all indexed by three time-periods, and they are conditional on the strictly exogenous variables and the initial conditions. One could in principle construct separate conditional moment conditions for each value of $T_{i}$ and each triplet $ 1\leq t<s<r\leq T_{i}$, and then use them to construct efficient unconditional moment functions. See, for example, the discussion in NeweyMcFadden94:HoE. Unfortunately, the construction of these moment functions depends on the conditional expectation of the derivative of the conditional moment function as well as on the conditional variance of the conditional moment function. We therefore pursue a different approach to obtaining unconditional moment functions. We do not claim that the resulting GMM estimator has any optimality properties, but we have found that it performs well in our Monte Carlo simulations even for relatively small sample sizes, see Appendix B.1 of Honore2022moment.
We first normalize all moment functions such that $\sup_{y,x,{\Greekmath 010C} ,{\Greekmath 010D} }\allowbreak \left\vert \widetilde{m}(y,x,{\Greekmath 010C} ,{\Greekmath 010D} )\right\vert <\infty $. For example, the rescaled versions of our $T=3$ moment functions in Section (ref) are given by
Here, each moment function is divided by the sum of the absolute values of all the different positive summands that appear in that moment function. We have found that this rescaling improves the performance of the resulting GMM estimators, particularly for small samples, because it bounds the moment functions and its gradients uniformly over the parameters ${\Greekmath 010C} $ and $ {\Greekmath 010D} $. Interestingly, the score functions of the conditional likelihood in honore2000panel are essentially rescaled in this way.
The rescaled moment functions are valid conditional on any realization of the regressors. We can therefore form unconditional moment functions by multiplying them with arbitrary functions of the regressors and the initial conditions. In our example, we multiply them by 1, the initial condition, and the explanatory variables for the three time periods that index the moment function. For example, for $T=3$, we use
where $\otimes $ is the tensor product.
One could in principle construct a moment function for each $\left( t,s,r\right) $ which indexes a moment function. However, this would create a very large number of moment conditions. For a given individual, we therefore add up all the moment functions over all triplets, $t<s<r$. Observations with $T=T_{i}$ time periods will then contribute ${\binom{T_{i}}{3}}$ terms to the sample analog of the moment. This gives very large weight to observations with large $T_{i}$. We therefore weigh the triplets $ \left( t,s,r\right) $ for an observation with $T_{i}$ time periods by $ \left( T_{i}-1\right) /{\binom{T_{i}}{3}}$. This yields sample moments of the form $\frac{1}{n}\sum_{i=1}^{n}M(Y_{i},Y_{i,0},X_{i},{\Greekmath 010C} ,{\Greekmath 010D} )$, and the corresponding GMM estimator is given by
where $W$ is a symmetric positive-definite weight matrix. We use a diagonal weight matrix with the inverse of the moment variances on the diagonal. The motivation stems from AltonjiSegal1996 who demonstrate that estimating the optimal weighting matrix can result in poor finite sample performance of GMM estimators. They suggest equally weighted moments (i.e., $ W=I$) as an alternative. Of course, using equal weights will not be invariant to changes in units, which explains the practice we have adopted. \footnote{ Our choice of weight matrix is quite common in empirical work. See, for example, GayleShephard2019 for a recent example.}
Under standard regularity conditions we have
with $\Omega =\mathrm{Var}[m(Y_{i},Y_{i,0},X_{i},{\Greekmath 010C} _{0},{\Greekmath 010D} _{0})]$ and $G=\mathbb{E}\left[ \frac{\partial m(Y_{i},Y_{i,0},X_{i},{\Greekmath 010C} _{0},{\Greekmath 010D} _{0})}{\partial {\Greekmath 010C} ^{\prime }},\frac{\partial m(Y_{i},Y_{i,0},X_{i},{\Greekmath 010C} _{0},{\Greekmath 010D} _{0})}{\partial {\Greekmath 010D} ^{\prime }} \right] $.
Table (ref) reports the estimation results. As expected, and consistent with the Monte Carlo results in Appendix B.1 of Honore2022moment, the standard logit maximum likelihood estimator of the coefficient on the lagged dependent variable is much larger than the one that estimates a fixed effect for each individual: the estimated fixed effects will be \textquotedblleft `overfitted\textquotedblright , leading to a downward bias in the estimated state dependence. Moreover, the standard logit estimator that ignores fixed effects will capture the presence of persistent heterogeneity by the lagged dependent variable, leading to an upwards bias if such heterogeneity is present in the data. The GMM\ estimator gives a much smaller coefficient than the standard logit maximum likelihood estimator, suggesting that heterogeneity plays a big role in this application.
To estimate the AR(2)\ version of the model, we apply the moment conditions provided in Appendix (ref) to all consecutive sequences of six outcomes (treating the first two as initial conditions). The moment functions are scaled as described in the Monte Carlo simulations in Appendix B.1 of Honore2022moment. The results are presented in Table (ref). The most interesting finding is that for all three samples, the GMM estimator of $\left({\Greekmath 010D}_1,{\Greekmath 010D}_2 \right)$ is between the maximum likelihood estimator that ignores the fixed effects, and the one that estimates a fixed effect for each individual. This suggests that unobserved individual-specific heterogeneity is important in this example. Economically, it is also interesting that for each estimation method, the estimates of $\left({\Greekmath 010D}_1,{\Greekmath 010D}_2\right)$ are quite similar across the three samples.
bonhomme2012functional proposed a general approach for constructing moment restrictions in nonlinear panel data models that do not depend on individual-specific effects. In this paper, we have operationalized this in models with discrete outcomes by first presenting a blueprint for deciding whether such moment conditions exist, and then an approach for actually finding analytic expressions for the moment conditions.
We have used our approach to derive all the moment conditions for the panel logit AR(1) model that are free of the fixed effects, and we have employed those moment conditions to show identification of the common model parameters and to obtain a GMM estimator that is useful and performs well in practice. The immediate practical relevance of this paper is therefore for the dynamic panel logit model (both AR(1) and AR(2) models are estimated in an empirical application).
While part of this paper emphasises binary logit models, the methods explained in Section (ref) and (ref) for exploring and deriving moment conditions are applicable for more general panel models, as illustrated by the examples provided in Section (ref). Exploring such moment conditions in other interesting models is a research agenda that has only started (e.g.\ honore2021dynamic, davezies2022fixed), and a lot more future work should be done to provide useful new estimation methods in various discrete choice panel models.
\setstretch{1.3} {4pt} \ifx\undefined\leavevmode\rule[.5ex]{3em}{.5pt}\ \fi \ifx\undefined\textsc \let\tmpsmall\tmpsmall\sc \fi