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Moment Conditions for Dynamic Panel Logit Models with Fixed Effects
\title{Moment Conditions for Dynamic Panel Logit Models \\ with Fixed Effects\thanks{
We thank Francesca Molinari and three anonymous referees for constructive feedback
and suggestions.
We are also grateful to St{\'e}phane Bonhomme, Runtong Ding, Geert Dhaene, Luojia Hu, Yoshitsugu Kitazawa, Shakeeb Khan, Chris Muris, Whitney Newey, and numerous seminar
participants for useful
comments and discussions. Sharada
Dharmasankar provided excellent research assistance. This research was
supported by the Gregory C. Chow Econometric Research Program at Princeton
University, by the National Science Foundation (Award Numbers 1824131 and 2116630),
by the Economic and Social Research Council through the ESRC Centre for
Microdata Methods and Practice (grant numbers RES-589-28-0001, RES-589-28-0002 and ES/P008909/1), and by the
European Research Council grants ERC-2014-CoG-646917-ROMIA and
ERC-2018-CoG-819086-PANEDA.}}
\author{\setcounter{footnote}{2}Bo E. Honor{\'e}\thanks{
Princeton University and The Dale T Mortensen
Centre at the University of Aarhus, \texttt{[email removed]} } \and Martin Weidner
\thanks{
University of Oxford, \texttt{[email removed]} } }
\date{December 2023}
\maketitle
\thispagestyle{empty}
\setcounter{page}{0}
\begin{abstract}
\noindent
This paper investigates the construction of moment conditions in discrete choice panel data with individual specific fixed effects. We describe how to systematically explore the existence of moment conditions that do not depend on the fixed effects, and we demonstrate how to construct them when they exist. Our approach is closely related to the numerical “functional differencing” construction in \cite{bonhomme2012functional}, but our emphasis is to find explicit analytic expressions for the moment functions. We first explain the construction and give examples of such moment conditions in various models. Then, we focus on the dynamic binary choice logit model and explore the implications of the moment conditions for identification and estimation of the model parameters that are common to all individuals.
\end{abstract}
\vskip 3cm
\section{Introduction}
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
\cite{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,
\cite{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 \cite{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 \cite{rasch1960studies} and \cite
{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
\cite{Honore92, Honore1993}, \cite{Hu2002}, and \cite{johnson2004identification} are earlier specific examples of this, and \cite{bonhomme2012functional} developed a general approach for obtaining moment conditions
via ``functional
differencing.''
Our paper operationalizes the proposal in \cite{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
\cite{kitazawa2013exploration,kitazawa2016root}.
Estimation of panel data binary response models dates back to \cite
{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. \cite{Chamberlain1985} and \cite{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 \cite{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 \citealt{Manski87}, \citealt{aristodemou2018semiparametric} and \citealt{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,
\cite{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{SEC: Incidental parameter free moment conditions} details our approach for discerning and constructing moment conditions when the dependent variable assumes a finite number of values, also drawing on insights from \cite{dobronyi2021identification} regarding the number of available moments.
In Section~\ref{sec:Derivation}, we showcase this within a panel data logit AR(1) model with \(T=3\) time periods.
Section~\ref{sec:Examples} 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{sec:identitifactionAR1} 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{SEC: Panel logit AR(1) model for general T>3} 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{SEC: Empirical illustration} illustrates the usefulness of the approach by estimating a simple model for labor force participation, and Section~\ref{sec:conc} wraps up the discussion.
\section{Incidental parameter free moment conditions\label{SEC: Incidental parameter free moment conditions}}
\subsection{Model and moment conditions}
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
\begin{equation}
\mathrm{Pr}\left( Y_{i}=y_{i}\,\Big|\,Y_{i}^{(0)}=y_{i}^{(0)},\,X_{i}=x_{i},
\;A_{i}={\Greekmath 010B} _{i}\right) =f\big(y_{i}\,\big|\,y_{i}^{(0)},\,x_{i},\,{\Greekmath 010B}
_{i};\,{\Greekmath 0112} \big). \label{MainModelRestriction}
\end{equation}
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
\begin{equation*}
\mathrm{Pr}\left( Y_{i}=y_{i}\,\Big|\,Y_{i}^{(0)}=y_{i}^{(0)},\,X_{i}=x_{i}
\right) =\int \,f\big(y_{i}\,\big|\,y_{i}^{(0)},\,x_{i},\,{\Greekmath 010B}
_{i};\,{\Greekmath 0112} \big)\;g\big({\Greekmath 010B} _{i}\,\big|\,\,y_{i}^{(0)},\,x_{i}\big)
\;d{\Greekmath 010B} _{i},
\end{equation*}
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 \cite{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
\begin{equation}
\mathrm{Pr}\left( Y_{it}=1\,\big|\,Y_{i}^{t-1},X_{i},A_{i}\right) =\frac{
\exp \big(X_{it}^{\prime }\,{\Greekmath 010C} +Y_{i,t-1}\,{\Greekmath 010D} +A_{i}\big)}{1+\exp
\big(X_{it}^{\prime }\,{\Greekmath 010C} +Y_{i,t-1}\,{\Greekmath 010D} +A_{i}\big)}, \label{model}
\end{equation}
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
\begin{align}
f\big(y_{i}\,\big|\,y_{i}^{(0)},\,x_{i},\,{\Greekmath 010B} _{i};\,{\Greekmath 0112} \big)&
=\prod_{t=1}^{T}\frac{\exp \left(y_{it} \left( x_{it}^{\prime
}\,{\Greekmath 010C} +y_{i,t-1}\,{\Greekmath 010D} +{\Greekmath 010B} _{i}\right) \right)}{1+\exp \left( x_{it}^{\prime
}\,{\Greekmath 010C} +y_{i,t-1}\,{\Greekmath 010D} +{\Greekmath 010B} _{i}\right) } \notag \\[10pt]
& =:p(y_{i}, y_{i}^{(0)},x_{i},{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} _{i}).
\label{DefProb}
\end{align}
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
\eqref{MainModelRestriction} implies that for any true
parameter value ${\Greekmath 0112}$,
\begin{equation}
\mathbb{E}\left[ m(Y_{i},Y_{i}^{(0)},X_{i},{\Greekmath 0112} )\right] = 0.
\label{MomentsUnconditional}
\end{equation}
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
\eqref{MomentsUnconditional} can only be a consequence of the model
\eqref{MainModelRestriction} if the conditional moment restriction
\begin{equation}
\mathbb{E}\left[ m(Y_{i},Y_{i}^{(0)},X_{i},{\Greekmath 0112} )\Big|
\,Y_{i}^{(0)}=y_{i}^{(0)},\,X_{i}=x_{i},\;A_{i}={\Greekmath 010B} _{i}\right] =0
\label{MomentsConditional}
\end{equation}
holds for all possible realizations $y_{i}^{(0)}$, $x_{i}$, ${\Greekmath 010B} _{i}$.
Under weak regularity conditions, \eqref{MomentsUnconditional} then follows
from \eqref{MomentsConditional} by the law of iterated expectations.
Furthermore, \eqref{MomentsConditional} can be rewritten as
\begin{equation}
\sum_{y_{i}\in \mathcal{Y}}m(y_{i},y_{i}^{(0)},x_{i},{\Greekmath 0112} )\,f\big(y_{i}\,
\big|\,y_{i}^{(0)},\,x_{i},\,{\Greekmath 010B} _{i};\,{\Greekmath 0112} \big)=0,
\label{MomentsConditional2}
\end{equation}
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
\eqref{MomentsConditional}, and therefore \eqref{MomentsUnconditional}, 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 \eqref{MomentsConditional2}
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 \eqref{MomentsUnconditional} are generally available in such models.
\subsection{Strategy for exploring and using such moment conditions\label
{Sec: Strategy for exploring and using such moment conditions}}
In this subsection, we briefly outline a three-step strategy for obtaining
valid moment conditions of the form \eqref{MomentsUnconditional} 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 \eqref{MomentsConditional}. 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
\begin{equation}
\sum_{y\in \mathcal{Y}}m(y)\,f\big(y\,\big|\,y^{(0)},\,x,\,{\Greekmath 010B}
_{q};\,{\Greekmath 0112} \big)=0,\qquad q=1,\ldots ,Q, \label{MomentsConditional3}
\end{equation}
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{sec:Computation} 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 \eqref{MomentsConditional3} 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{sec:Derivation} 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 \eqref{MomentsConditional2} 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{model}
). 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 \eqref{MomentsUnconditional} exist, but
have not yet been studied systematically --- see Section~\ref{sec:Examples}
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 \cite{honore2021dynamic} for the
case of dynamic ordered choice panel models. \cite{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.
\cite{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. \cite{Dano2023arXiv} also works out the semiparametric efficiency bound for the AR(1) panel logit model with regressors.
\subsection{Lower bound on the number of moment conditions\label{SEC: Lower
bound on number of moment conditions}}
\cite{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{MainModelRestriction}) as
\begin{equation*}
f\big(y\,\big|\,y^{(0)},\,x,\,{\Greekmath 010B} ;\,{\Greekmath 0112} \big)={\Greekmath 0114} \left( a \right) \,
\mathop{\displaystyle \sum }\limits_{k=1}^{K}a^{k-1}c_{k}\left( y \right)
\end{equation*}
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
\begin{equation}
\sum_{y\in \mathcal{Y}}m(y)\,\mathop{\displaystyle \sum }\limits_{k=1}^{K}a^{k-1}c_{k}\left(
y\right) =0,\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all }a\in \left( 0,\infty \right)
\label{EQ: Polynomial}
\end{equation}
which is equivalent to
\begin{equation*}
\sum_{y\in \mathcal{Y}}\,m(y)\,c_{k}(y)=0,\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all $k\in
\{1,\ldots ,K\}$.}
\end{equation*}
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} $.
\section{Moment functions for the $T=3$ logit AR(1) model}
\label{sec:Derivation}
In this section, we apply the strategy outlined in Section \ref{Sec:
Strategy for exploring and using such moment conditions} to construct moment
conditions for the binary choice logit AR(1) model in equation (\ref{model})
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, $y_{0}$.
\subsection{Verifying existence of moment functions numerically}
By numerically evaluating whether solutions to \eqref{MomentsConditional3}
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{Moments when T=2} 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 \eqref{MomentsConditional3}.
\subsection{Finding analytical solutions for the moment functions}
\label{sec:FindSolutions}
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 \eqref{MomentsConditional2}.
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:
\begin{equation*}
\hspace{-3pt}
\mathbf{p}(y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} )
\hspace{-3pt}
=
\hspace{-3pt}
\left(
\begin{array}{@{}c@{}}
p((0,0,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((0,0,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((0,1,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((0,1,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((1,0,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((1,0,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((1,1,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} ) \\
p((1,1,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 010B} )
\end{array}
\right) \hspace{-3pt},\, \, \, \mathbf{m}(x,y_{0},{\Greekmath 010C} ,{\Greekmath 010D} )
\hspace{-3pt}
=
\hspace{-3pt}
\left(
\begin{array}{@{}c@{}}
m((0,0,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((0,0,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((0,1,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((0,1,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((1,0,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((1,0,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((1,1,0),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
m((1,1,1),y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )
\end{array}
\right) \hspace{-3pt} .
\end{equation*}
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 \eqref{MomentsConditional}, 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:
\begin{itemize}
\item[(1)] Any valid moment vector $\mathbf{m}$ satisfies $\mathbf{e}
_1^{\prime }\, \mathbf{m} = 0$ and $\mathbf{e}_8^{\prime }\, \mathbf{m} = 0$.
\end{itemize}
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{lemma:moments_p1T3}.}
\begin{itemize}
\item[(2)] There are two linearly independent solutions to the equations $
\mathbf{m}^{\prime }\,\mathbf{p}({\Greekmath 010B} )=0$ for all ${\Greekmath 010B} \in \mathbb{R}$
. This implies that the set of valid moment vectors $\mathbf{m}$ is
two-dimensional.
\end{itemize}
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:
\begin{itemize}
\item[(0)] $\mathbf{e}_1^{\prime }\, \mathbf{m}^{(0)}=0$, \; \; $\mathbf{e}
_8^{\prime }\, \mathbf{m}^{(0)}=0$, \; \; $\mathbf{e}_7^{\prime }\, \mathbf{m
}^{(0)}=0$, \; \; $\mathbf{e}_4^{\prime }\, \mathbf{m}^{(0)}=-1$, \newline
$\mathbf{p}^{\prime }({\Greekmath 010B}_q) \, \mathbf{m}^{(0)}=0$, \;\; for $q=1,2,3,4$.
\item[(1)] $\mathbf{e}_1^{\prime }\, \mathbf{m}^{(1)}=0$, \; \; $\mathbf{e}
_8^{\prime }\, \mathbf{m}^{(1)}=0$, \; \; $\mathbf{e}_2^{\prime }\, \mathbf{m
}^{(1)}=0$, \; \; $\mathbf{e}_5^{\prime }\, \mathbf{m}^{(1)}=-1$, \newline
$\mathbf{p}^{\prime }({\Greekmath 010B}_q) \, \mathbf{m}^{(1)}=0$, \;\; for $q=1,2,3,4$.
\end{itemize}
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
\begin{eqnarray}
\mathbf{m}^{(0)}\, &=&\,-\,\left( \mathbf{B}^{(0)}\right) ^{-1}\,\mathbf{e}
_{4} \, , \nonumber \\
\mathbf{m}^{(1)}\, &=&\,-\,\left( \mathbf{B}^{(1)}\right) ^{-1}\,\mathbf{e}
_{4} \, .
\label{FindMomentsAnalytical}
\end{eqnarray}
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
\begin{align}
m^{(0)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )& =\left\{
\begin{array}{ll}
\exp \left( x_{23}^{\prime }{\Greekmath 010C} \right) -1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(0,0,1), \\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }(y_{1},y_{2})=(0,1), \\
\exp \left( x_{31}^{\prime }{\Greekmath 010C} - y_0 \, {\Greekmath 010D} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
y=(1,0,0), \\
\exp \left( x_{21}^{\prime }{\Greekmath 010C} + (1-y_0) \, {\Greekmath 010D} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
y=(1,0,1), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise},
\end{array}
\right. \notag \\[5pt]
m^{(1)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )& =\left\{
\begin{array}{ll}
\exp \left( x_{12}^{\prime }{\Greekmath 010C} + y_0 \, {\Greekmath 010D} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
y=(0,1,0), \\
\exp \left( x_{13}^{\prime }{\Greekmath 010C} - (1-y_0) \,{\Greekmath 010D} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
y=(0,1,1), \\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }(y_{1},y_{2})=(1,0), \\
\exp \left( x_{32}^{\prime }{\Greekmath 010C} \right) -1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(1,1,0), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array}
\right. \label{SolutionMomentsT3}
\end{align}
The two solutions in \eqref{SolutionMomentsT3} are closely related: If $
Y_{t} $ is generated according to \eqref{model}, then $Z_{t}=1-Y_{t}$ is
also generated according to \eqref{model}, 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 $m^{(0)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D}
)=m^{(1)}(1-y,1-y_0,-x,{\Greekmath 010C} ,{\Greekmath 010D} )$.
\subsection{Verifying that the moment functions are valid}
The following lemma establishes that the moment functions, $
m^{(0/1)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )$, displayed in \eqref{SolutionMomentsT3}
are indeed valid. For this, it is not relevant how the moment functions were
derived.
\begin{lemma}
\label{lemma:moments_p1T3} If the outcomes $Y=(Y_1,Y_2,Y_3)$ are generated
from model \eqref{model} with $T=3$ and true parameters ${\Greekmath 010C}_0$ and $
{\Greekmath 010D}_0$, then we have for all $q \in \{0,1\}$, $y_0 \in \{0,1\}$, $x \in
\mathbb{R}^{K \times 3}$, ${\Greekmath 010B} \in \mathbb{R}$ that
\begin{align*}
\mathbb{E} \left[ m^{(q)}(Y,Y_0,X,{\Greekmath 010C}_0,{\Greekmath 010D}_0) \, \big| \, Y_0=y_0, \,
X=x, \, A={\Greekmath 010B} \right] &= 0 .
\end{align*}
\end{lemma}
This lemma is a special case of Theorem~\ref{th:AR1moments} 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
\begin{equation*}
\sum_{y\in \{0,1\}^{3}}\,p(y,y_{0},x,{\Greekmath 010C} _{0},{\Greekmath 010D} _{0},{\Greekmath 010B}
)\;m^{(0/1)}(y,y_{0},x,{\Greekmath 010C} _{0},{\Greekmath 010D} _{0})=0.
\end{equation*}
The details of this calculation are provided in Appendix~B.2.1 of \cite{Honore2022moment}.
\subsection{On the number of moment conditions\label{SEC: On the number of moment conditions}}
As explained in Section \ref{SEC: Lower bound on number of moment
conditions}, \cite{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
\begin{equation*}
f\big(y\,\big|\,y^{(0)},\,x,\,{\Greekmath 010B} ;\,{\Greekmath 0112} \big)\ =\prod_{t=1}^{T}\frac{
\left[\exp \left( x_{t}^{\prime }\,{\Greekmath 010C} +y_{t-1}\,{\Greekmath 010D} +{\Greekmath 010B} \right)
\right]^{y_{it}}}{1+\exp \left( x_{t}^{\prime }\,{\Greekmath 010C} +y_{t-1}\,{\Greekmath 010D} +{\Greekmath 010B}
\right) }.
\end{equation*}
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
\begin{align*}
f\big(y\,\big|\,y^{(0)},\,x,\,{\Greekmath 010B} ;\,{\Greekmath 0112} \big)\ & =\prod_{t=1}^{T}
\frac{\left[ a\,{\Greekmath 0119} _{t}(y_{t-1})\right] ^{y_{t}}}{1+a\,{\Greekmath 0119} _{t}(y_{t-1})}=
\frac{\left[ a\,{\Greekmath 0119} _{1}(y_{0})\right] ^{y_{1}}}{1+a\,{\Greekmath 0119} _{1}(y_{0})}
\,\prod_{t=2}^{T}\frac{\left[ a\,{\Greekmath 0119} _{t}(y_{t-1})\right] ^{y_{t}}}{1+a\,{\Greekmath 0119}
_{t}(y_{t-1})} \\
& =\frac{\left[ a\,{\Greekmath 0119} _{1}(y_{0})\right] ^{y_{1}}}{1+a\,{\Greekmath 0119} _{1}(y_{0})}
\,\prod_{t=2}^{T}\frac{[1+a\,{\Greekmath 0119} _{t}(1-y_{t-1})]\,\left[ a\,{\Greekmath 0119}
_{t}(y_{t-1})\right] ^{y_{t}}}{[1+a\,{\Greekmath 0119} _{t}(0)][1+a\,{\Greekmath 0119} _{t}(1)]}={\Greekmath 0114}
(a)\,\cdot \widetilde{p}(y,a),
\end{align*}
where we defined
\begin{align}
{\Greekmath 0114} (a)& =\frac{1}{1+a\,{\Greekmath 0119} _{1}(y_{0})}\,\cdot \,\prod_{t=2}^{T}\,\frac{1
}{[1+a\,{\Greekmath 0119} _{t}(0)][1+a\,{\Greekmath 0119} _{t}(1)]}, \notag \\
\widetilde{p}(y,a)& =\left[ a\,{\Greekmath 0119} _{1}(y_{0})\right] ^{y_{1}}
\prod_{t=2}^{T}\left\{ [1+a\,{\Greekmath 0119} _{t}(1-y_{t-1})]\,\left[ a\,{\Greekmath 0119}
_{t}(y_{t-1})\right] ^{y_{t}}\right\} =\sum_{k=1}^{2T}\,a^{k-1}\,c_{k}(y).
\notag
\end{align}
This has the exact structure of equation (\ref{EQ: Polynomial}) 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{lemma:moments_p1T3} above.
\section{Examples of moment functions in other models}
\label{sec:Examples}
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{Sec: Strategy for exploring and using such moment
conditions} 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.
\subsection{Static binary choice models}
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
\begin{equation}
f\big(y_{i}\,\big|\,\,x_{i},\,{\Greekmath 010B} _{i};\,{\Greekmath 010C} \big)=\prod_{t=1}^{T}\left[
F\left( x_{it}^{\prime }\,{\Greekmath 010C} +{\Greekmath 010B} _{i}\right) \right] ^{y_{it}}\left[
1-F\left( x_{it}^{\prime }\,{\Greekmath 010C} +{\Greekmath 010B} _{i}\right) \right] ^{1-y_{it}},
\label{StaticModel}
\end{equation}
where $F(\cdot )$ is a cumulative distribution function. The distribution in
\eqref{StaticModel} is a special case of \eqref{MainModelRestriction}. 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 \cite{rasch1960studies} and \cite
{andersen1970asymptotic}. In fact, \cite{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 \cite
{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 \cite
{chamberlain2010binary} does not apply in that case. In particular, for $T=3$
, \cite{johnson2004identification} and \cite{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 \eqref{StaticModel}. One can use
the machinery in Section \ref{SEC: Incidental parameter free moment
conditions} to show that the specification considered by \cite
{johnson2004identification} and \cite{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:
\begin{equation}
F({\Greekmath 0122} )=\frac{{\Greekmath 0121} }{1+\exp (-{\Greekmath 0115} \,{\Greekmath 0122} +{\Greekmath 0116} _{1})}+
\frac{1-{\Greekmath 0121} }{1+\exp (-{\Greekmath 0115} \,{\Greekmath 0122} +{\Greekmath 0116} _{2})},
\label{LogitMixture}
\end{equation}
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 \eqref{LogitMixture} into \eqref{StaticModel} and then
applying the procedure described in
Section \ref{Sec: Strategy for exploring and using such moment conditions}
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
\begin{align}
m(y,x,{\Greekmath 010C} )& =\sum_{(t,s,r)\in \mathcal{P}}\mathbbm{1}\left\{
(y_{t},y_{s})=(0,1)\right\} \;(1-2\,y_{r})\;\mathrm{sgn}(t,s,r)\;\exp
[{\Greekmath 0115} \,(x_{t}-x_{s})^{\prime }{\Greekmath 010C} ] \notag \\
& \qquad \qquad \times \Big\{{\Greekmath 0121} \,\exp [(1-y_{r})({\Greekmath 0116} _{2}-{\Greekmath 0116}
_{1})]+(1-{\Greekmath 0121} )\,\exp [y_{r}\,({\Greekmath 0116} _{2}-{\Greekmath 0116} _{1})]\Big\},
\label{MomentFunctionStaticLogitMixtureT3}
\end{align}
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{SEC: Lower bound on number of moment
conditions} 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.
\subsection{Fixed Effect Logit AR($p$) Models With $p>1$}
\label{subsecARp}
The analysis of the dynamic panel data logit model in
Section~\ref{sec:Derivation} generalizes
to a model with more than one lag. Specifically, consider the model
\begin{equation}
\mathrm{Pr}\left( Y_{it}=1\,\big|\,Y_{i}^{t-1},X_{i},A_{i},{\Greekmath 010C} ,{\Greekmath 010D}
\right) =\frac{\exp \big(X_{it}^{\prime }\,{\Greekmath 010C} +\sum_{\ell
=1}^{p}\,Y_{i,t-\ell }\,{\Greekmath 010D} _{\ell }+A_{i}\big)}{1+\exp \big(
X_{it}^{\prime }\,{\Greekmath 010C} +\sum_{\ell =1}^{p}\,Y_{i,t-\ell }\,{\Greekmath 010D} _{\ell
}+A_{i}\big)}, \label{modelARp}
\end{equation}
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{sec:ARp}, 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 \cite{Honore2022moment}.
The special case of an AR(2)\ logit model with fixed effects and no
explanatory variables was considered in \cite{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{sec:ARp} 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
\begin{equation*}
\mathbb{E}\left[ m_{y^{(0)}}(Y,{\Greekmath 010D} _{0})\,\big|\,Y^{(0)}=y^{(0)},\,A=
{\Greekmath 010B} \right] =0,
\end{equation*}
with moment functions given by
\begin{align*}
m_{(y_0,y_0)}(y,\!{\Greekmath 010D} )&\!=\!\!\left\{
\begin{array}{@{}l@{\,}l}
1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(y_0,\!1\!\!-\!\!y_0,\!y_0), \\
e^{-{\Greekmath 010D} _{1}} & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(y_0,\!1\!\!-\!\!y_0,\!1\!\!-\!\!y_0), \\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }(y_{1},\!y_{2})=(1\!\!-\!\!y_0,\!y_0), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise},
\end{array}
\right. \!\! & m_{(1\!-y_0,y_0)}(y,\!{\Greekmath 010D} )&\!=\!\!\left\{
\begin{array}{@{}l@{\,}l}
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }(y_{1},\!y_{2})=(1\!\!-\!\!y_0,\!y_0), \\
e^{{\Greekmath 010D} _{2}-{\Greekmath 010D} _{1}} & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(y_0,\!1\!\!-\!\!y_0,\!1\!\!-\!\!y_0), \\
e^{{\Greekmath 010D} _{2}} & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(y_0,\!1-y_0,\!y_0), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise},
\end{array}
\right.
\end{align*}
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.
\subsection{Panel logit AR(1) with heterogeneous time trends\label{Panel logit AR(1) with heterogeneous time trends}}
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{model}) to
\begin{equation*}
\mathrm{Pr}\left( Y_{it}=1\,\big|\,Y_{i}^{t-1},X_{i},A_{i}\right) =\frac{
\exp \big(X_{it}^{\prime }\,{\Greekmath 010C} +Y_{i,t-1}\,{\Greekmath 010D} +tD_{i}+A_{i}\big)}{
1+\exp \big(X_{it}^{\prime }\,{\Greekmath 010C} +Y_{i,t-1}\,{\Greekmath 010D} +tD_{i}+A_{i}\big)}.
\end{equation*}
By mimicking the calculations in Section \ref{SEC: On the number of moment conditions}, 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{Additional Calculations for Time Trends}
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{Additional Calculations for Time Trends}.
This illustrates that exploring the polynomial structure of the model probabilities
(as described in Section \ref{SEC: On the number of moment conditions})
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{Additional Calculations for Time Trends}, we discuss these moment conditions.
\subsection{Extensions to dynamic ordered logit model and dynamic
multinomial logit models}
The methods described in this paper can also be applied to dynamic panel data versions of other ``textbook'' logit models. In particular, \cite
{honore2021dynamic}
use the procedure outlined in Section \ref{Sec: Strategy
for exploring and using such moment conditions} to find moment
conditions for dynamic panel data {\it ordered} logit models,
and \cite{Dano2023arXiv} shows how to obtain moment conditions for dynamic panel data
{\it multinomial} logit models.
\section{Identification}
\label{sec:identitifactionAR1}
This section shows that the moment conditions for the panel logit AR(1)
model in Lemma~\ref{lemma:moments_p1T3}
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.
\begin{lemma}
\label{lemma:INVERSION} Let $K \in \mathbb{N}_0$. For every $s =
(s_1,\ldots,s_K) \in \{-,+\}^K$ let $g_s : \mathbb{R}^{K} \times \mathbb{R}
\rightarrow \mathbb{R}$ be a continuous function such that for all $
({\Greekmath 010C},{\Greekmath 010D}) \in \mathbb{R}^{K} \times \mathbb{R} $ we have
\begin{itemize}
\item[(i)] $g_s({\Greekmath 010C},{\Greekmath 010D})$ is strictly increasing in ${\Greekmath 010D}$.
\item[(ii)] For all $k \in \{1,\ldots,K\}$: If $s_k = +$, then $
g_s({\Greekmath 010C},{\Greekmath 010D})$ is strictly increasing in ${\Greekmath 010C}_k$.
\item[(iii)] For all $k \in \{1,\ldots,K\}$: If $s_k = -$, then $
g_s({\Greekmath 010C},{\Greekmath 010D})$ is strictly decreasing in ${\Greekmath 010C}_k$.
\end{itemize}
Then, the system of $2^K$ equations in $K+1$ variables
\begin{align}
g_s({\Greekmath 010C},{\Greekmath 010D})=0 , \qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all} \; s \in \{-,+\}^K ,
\label{systemEQ}
\end{align}
has at most one solution.
\end{lemma}
To explain the lemma, consider the case $K=1$,\footnote{
Note that $K=0$ is trivially allowed in Lemma~\ref{lemma:INVERSION}. 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{lemma:moments_p1T3}
with the result of
Lemma \ref{lemma:INVERSION} 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
\begin{align*}
\mathcal{X}_{k,+}& =\{x\in \mathbb{R}^{K\times 3}\,:\,x_{k,1}\leq
x_{k,3}<x_{k,2}\;\;\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{or}\;\;x_{k,1}<x_{k,3}\leq x_{k,2}\}, \\
\mathcal{X}_{k,-}& =\{x\in \mathbb{R}^{K\times 3}\,:\,x_{k,1}\geq
x_{k,3}>x_{k,2}\;\;\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{or}\;\;x_{k,1}>x_{k,3}\geq x_{k,2}\},
\end{align*}
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{sec:FindSolutions}
have convenient monotonicity properties in the parameters ${\Greekmath 010C}_k$.
For example, for $m^{(0)}$ we have
\begin{align}
\frac{\partial \, \mathbb{E}\left[ m^{(0)}(Y,Y_0,X,{\Greekmath 010C} ,{\Greekmath 010D} )\,\Big|\,Y_{0}=y_0,\; X=x \right] }
{\partial {\Greekmath 010C}_k}
&>0, \;\; \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for $x \in \mathcal{X}_{k,+}$} \; ,
\nonumber \\
&<0 , \;\; \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for $x \in \mathcal{X}_{k,-}$} \;.
\label{MonotonicityEmoments}
\end{align}
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 $q,y_0 \in \{0,1\}$,
\begin{align*}
\overline{m}_{y_{0},s}^{(q)}({\Greekmath 010C} ,{\Greekmath 010D} )& =\mathbb{E}\left[
m^{(q)}(Y,Y_0,X,{\Greekmath 010C} ,{\Greekmath 010D} )\,\Big|\,Y_{0}=y_{0},\;X\in \mathcal{X}_{s}\right] .
\end{align*}
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, \eqref{MonotonicityEmoments} 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 $k \in \{1,\ldots,K\}$.
\begin{theorem}
\label{th:id1} Let $q,y_{0}\in \{0,1\}$. Let the
outcomes $Y=(Y_{1},Y_{2},Y_{3})$ be generated from model \eqref{model} with $
T=3$ and true parameters ${\Greekmath 010C} _{0}$ and ${\Greekmath 010D} _{0}$. Furthermore, for
all $s\in \{-,+\}^{K}$ assume that
\begin{equation*}
\mathrm{Pr}\left( Y_{0}=y_{0},\;X\in \mathcal{X}_{s}\right) >0,
\end{equation*}
and that the expected moment function $\overline{m}_{y_{0},s}^{(q)}({\Greekmath 010C} ,{\Greekmath 010D} )$ is well-defined.\footnote{
We could always guarantee $\overline{m}_{y_{0},s}^{(q )}({\Greekmath 010C} ,{\Greekmath 010D} )$ to be well-defined
by modifying the definition of the set $\mathcal{X}_{s}$ to only contain bounded regressor values.
}
Then, the solution to
\begin{equation}
\overline{m}_{y_{0},s}^{(q )}({\Greekmath 010C} ,{\Greekmath 010D} )=0 \qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all}
\;s\in \{-,+\}^{K}
\label{SystemTheoremID}
\end{equation}
is unique and given by $({\Greekmath 010C} _{0},{\Greekmath 010D} _{0})$. Thus, the parameters ${\Greekmath 010C}_0$ and ${\Greekmath 010D}_0$ are point-identified
\end{theorem}
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{th:id1},
and only one of the initial conditions $y_0 \in \{0,1\}$ needs to be observed.
The key assumption in Theorem~\ref{th:id1}
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{th:id1} 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
\cite{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{th:id1}, 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{lemma:moments_p1T3} (or from Theorem~\ref{th:AR1moments}
below for $T>3$), resulting in potentially much more
efficient estimators for ${\Greekmath 010C}$ and ${\Greekmath 010D}$, and Section~\ref{sec:Emp} describes how we implement such estimators in practice.
Nevertheless, from a theoretical perspective, the identification result in
Theorem~\ref{th:id1} is important, because it comprises a significant improvement over
existing results for dynamic panel logit models with explanatory variables.
\section{Panel logit AR(1) model for general $T \geq 3$}
\label{SEC: Panel logit AR(1) model for general T>3}
In Section~\ref{sec:FindSolutions} 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.
\subsection{Impossibility of moment conditions when $T=2\label{Moments when T=2}$}
Here, we argue that it is not possible to derive moment conditions for model (\ref{model}) 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
\begin{align}
\sum_{y \in \{ 0,1\}^2} \, p(y,y_0,x,{\Greekmath 010C},{\Greekmath 010D},{\Greekmath 010B}) \;
m(y,y_0,x,{\Greekmath 010C},{\Greekmath 010D}) = 0,
\label{MomentT2}
\end{align}
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 \eqref{MomentT2} therefore implies that
\begin{align}
\frac{m((0,1),y_0,x,{\Greekmath 010C},{\Greekmath 010D})} {m((1,0),y_0,x,{\Greekmath 010C},{\Greekmath 010D})}
&= - \frac{p((1,0),y_0,x,{\Greekmath 010C},{\Greekmath 010D},{\Greekmath 010B})} {p((0,1),y_0,x,{\Greekmath 010C},{\Greekmath 010D},{\Greekmath 010B})}
\nonumber \\
&=- \exp
\left( \left( x_{1}-x_{2}\right) ^{\prime }{\Greekmath 010C} +{\Greekmath 010D} y_{0}\right)
\frac{1+\exp \left( x_{2}^{\prime }{\Greekmath 010C} +{\Greekmath 010B} \right) }
{1+\exp \left( x_{2}^{\prime }{\Greekmath 010C} +{\Greekmath 010D} +{\Greekmath 010B} \right) } .
\label{MM T=2}
\end{align}
Unless ${\Greekmath 010D} =0$, the right hand side of \eqref{MM T=2}
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{MM T=2}) yields the moment
conditions implied by \cite{Rasch60}'s conditional likelihood.
The fact that there are no moment conditions when $T=2$ is consistent with the non-identification result in \cite{Chamberlain2023SERIES}.
\subsection{Master lemma for obtaining all moment conditions}
\label{subsec:MasterLemma}
One can work out analytic moment functions for
model (\ref{model}) with $T=4$
and $T=5$ using the same derivation method described in
Section~\ref{sec:FindSolutions} 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
$$
{\Greekmath 010E}(t) := \left\{ \begin{array}{ll}
3 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for } t=1, \\
1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for } t=2, \\
2 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for } t=3.
\end{array} \right.
$$
\begin{lemma}
\label{lemma:Markov}
Let $\widetilde Y_1, \widetilde Y_2, \widetilde Y_3 \in \{0,1\}$
be binary random variables, and $W_1, W_2, W_3$
be random variables (or vectors) such that
$
W_1
\rightarrow \widetilde Y_1
\rightarrow W_2
\rightarrow \widetilde Y_2
\rightarrow W_3
\rightarrow \widetilde Y_3
$
is a Markov chain,
conditional on the random vector $(X,A)$.\footnote{
The Markov chain assumption means that the
density of
$(W_1,\widetilde Y_1, W_2, \widetilde Y_2, W_3, \widetilde Y_3)$, conditional on $(X,A)$,
can be written as a product
$ f_{\widetilde Y_3|W_3,X,A} \,
f_{W_3|\widetilde Y_2,X,A} \,
f_{\widetilde Y_2|W_2,X,A} \,
f_{W_2|\widetilde Y_1,X,A}
f_{\widetilde Y_1|W_1,X,A} \,
f_{W_1|X,A} $.
}
Assume furthermore that
$p_t(\widetilde y_t \, |\, w_t,x,{\Greekmath 010B}):={\rm Pr}\big( \widetilde Y_t = \widetilde y_t \, \big| \, W_t=w_t,
\, X=x,
\, A={\Greekmath 010B} \big)$
satisfies
$0<p_t(\widetilde y_t \, |\,w_t, x,{\Greekmath 010B})<1$,
for all $\widetilde y_t$, $w_t$, $x$, ${\Greekmath 010B}$,
and
$t \in \{1,2,3\}$.
Then, for $q \in \{0,1\}$, the function
\begin{align*}
& m^{(q)}(w_1,\widetilde y_1,w_2,\widetilde y_2,w_3,\widetilde y_3,x,{\Greekmath 010B})
:= - \mathbbm{1}\left\{ \widetilde y_1 = q \right\}
\\ & \qquad
+
\mathbbm{1}\left\{ \widetilde y_2 = q \right\} \,
\exp \left( \frac 1 2
\sum_{t=1}^3 \bigg\{
\Lambda^{-1}\Big[
p_{{\Greekmath 010E}(t)} \left(\widetilde y_t \, \big| \,w_{{\Greekmath 010E}(t)},x,{\Greekmath 010B} \right) \Big]
-
\Lambda^{-1}\Big[ p_t \left(\widetilde y_{t} \, \big| \,w_t,x,{\Greekmath 010B} \right) \Big]
\bigg\}
\right)
\end{align*}
satisfies
$
\mathbb{E}\left[ m^{(q)}(W_1,\widetilde Y_1,W_2,\widetilde Y_2,W_3,\widetilde Y_3,X,A) \, \Big| \, W_1, \, X, \, A \right] = 0.
$
\end{lemma}
The proof is given in Appendix~\ref{app:proofs}.
Note that the vector of conditioning variables $(X,A)$
is only included in Lemma~\ref{lemma:Markov} 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{lemma:Markov} to the $T=3$ panel AR(1) model of Section~\ref{sec:Derivation}, 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{lemma:Markov}
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{lemma:Markov}
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
\eqref{MomentsConditional} 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 $\widetilde y \in \{0,1\}$,
\begin{align}
p_t(\widetilde y \, |\, w_t,x,{\Greekmath 010B})
= \Lambda\big\{ (2 \widetilde y -1) \, [ z_t(w_t,x) + {\Greekmath 010B} ] \big\},
\label{GeneralLogitBinaryChoice}
\end{align}
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{lemma:Markov} do not depend on the
fixed effects ${\Greekmath 010B}$, and can be written as\footnote{
Here we also use that
$\sum_{t=1}^3 \left[ z_{{\Greekmath 010E}(t)}(w_{{\Greekmath 010E}(t)},x) - z_t(w_t,x) \right] = 0$.
}
\begin{align}
& m^{(q)}(w_1,\widetilde y_1,w_2,\widetilde y_2,w_3,\widetilde y_3,x)
\nonumber \\ & \qquad
= - \mathbbm{1}\left\{ \widetilde y_1 = q \right\} +
\mathbbm{1}\left\{ \widetilde y_2 = q \right\} \,
\exp \left\{
\sum_{t=1}^3 \widetilde y_t \, \Big[ z_{{\Greekmath 010E}(t)}(w_{{\Greekmath 010E}(t)},x) - z_t(w_t,x) \Big]
\right\}.
\label{MomentFunctionLogitGeneral}
\end{align}
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
\eqref{MomentFunctionLogitGeneral}
is equal to $m^{(q)}(y,y_0,x,{\Greekmath 010C}_0,{\Greekmath 010D}_0)$
in display~\eqref{SolutionMomentsT3} above, that is,
Lemma~\ref{lemma:Markov} delivers
the moment functions derived
for the $T=3$ dynamic logit model
in Section~\ref{sec:FindSolutions} as a special case.
\subsection{Moment conditions for $T \geq 3$}
\label{Section: T greater that 3}
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{lemma:Markov} is useful for our purposes for logit models of the form \eqref{GeneralLogitBinaryChoice} where it
delivers the moment functions in
\eqref{MomentFunctionLogitGeneral} 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{lemma:Markov}
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 \eqref{model}.
Furthermore, in that model, the distribution
of $Y_{t}$ conditional on
$Y_{t-1}$, $X$, $A$ is indeed of
the logistic form \eqref{GeneralLogitBinaryChoice}
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 \eqref{MomentFunctionLogitGeneral}
can be written more explicitly as
\begin{align}
m^{(0)(t,s,r)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )& =\left\{
\begin{array}{ll}
\exp \left[ z_{sr}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )\right] -1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
(y_{t},y_{s},y_{r})=(0,0,1), \\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }(y_{t},y_{s})=(0,1), \\
\exp \left[ z_{rt}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )\right] & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
(y_{t},y_{s},y_{r})=(1,0,0), \\
\exp \left[ z_{st}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )\right] & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
(y_{t},y_{s},y_{r})=(1,0,1), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise},
\end{array}
\right.
\nonumber \\[5pt]
m^{(1)(t,s,r)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} )& =\left\{
\begin{array}{ll}
\exp \left[ z_{ts}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )\right] & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
(y_{t},y_{s},y_{r})=(0,1,0), \\
\exp \left[ z_{tr}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )\right] & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
(y_{t},y_{s},y_{r})=(0,1,1), \\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }(y_{t},y_{s})=(1,0), \\
\exp \left[ z_{rs}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )\right] -1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }
(y_{t},y_{s},y_{r})=(1,1,0), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array}
\right.
\label{MomentsGeneralT}
\end{align}
For $T=3$ and $(t,s,r)=(1,2,3)$, these moment functions are exactly those
calculated in Section~\ref{sec:FindSolutions} above.
For general $T \geq 3$
and triplets $(t,s,r)$
we can apply Lemma~\ref{lemma:Markov}, conditional also on $Y_0,Y_{1},\ldots ,Y_{t-1}$,
to obtain the following theorem.
\begin{theorem}
\label{th:AR1moments} If the outcomes $Y$ are generated from the panel
logit AR(1) model with $T\geq 3$ and true parameters ${\Greekmath 010C} _{0}$ and $
{\Greekmath 010D} _{0}$, then we have for all $t,s,r\in \{1,2,\ldots ,T\}$ with $t<s<r$
, and for all $q \in \{0,1\}$, $y^{(t)} \in \{0,1\}^t$, $x\in \mathbb{R}^{K\times T}$,
${\Greekmath 010B} \in \mathbb{R}$, that
\begin{align*}
\mathbb{E}\left[ m^{(q)(t,s,r)}(Y,Y_0,X,{\Greekmath 010C}
_{0},{\Greekmath 010D} _{0})\,\big|\,(Y_{0},\, Y_{1},\ldots ,Y_{t-1})=y^{(t)}, \,X=x,\,A={\Greekmath 010B} \right] & =0.
\end{align*}
\end{theorem}
The proof is given in Appendix~\ref{app:proofs}, but as argued above, the theorem
really is an immediate corollary of Lemma~\ref{lemma:Markov}.
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{th:AR1moments}
and the law of iterated expectations, we find, for
any function $w:\{0,1\}^{t-1}\rightarrow \mathbb{R}$, that
\begin{align}
\mathbb{E}\left[ w(Y_{1},\ldots ,Y_{t-1}) \; m^{(q)(t,s,r)}(Y,Y_0,X,{\Greekmath 010C}
_{0},{\Greekmath 010D} _{0})\,\big|\,Y_{0}=y_{0},\,X=x,\,A={\Greekmath 010B} \right] & =0.
\label{allAR1moments}
\end{align}
From Section~\ref{SEC: On the number of moment conditions} 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
\cite{kruiniger2020further},
\cite{dobronyi2021identification},
and \cite{Dano2023arXiv}
have shown that this is indeed the case.
Equation \eqref{allAR1moments}
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
\begin{align*}
\ell = \underbrace{ 2 }_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{$q \in \{0,1\}$}} \times
\underbrace{ \sum_{t=1}^{T-2} \sum_{s=t+1}^{T-1} }_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{allowed $t,s$ values}}
\underbrace{ 2^{t-1} }_{\begin{minipage}{3.2cm} \center \scriptsize
\setstretch{1.0}
number of linearly independent functions $w(y_{1},\ldots ,y_{t-1})$
\end{minipage}}
= \sum_{t=1}^{T-2} \, 2^t \, (T-t-1) = 2^T - 2T,
\end{align*}
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 \cite{rasch1960studies} and \cite{andersen1970asymptotic} are linear combinations of these moment functions.}
\subsubsection{Unbalanced panels and missing time periods}
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{sec:Emp} we discuss how to combine the moment functions for unbalanced panels.
\subsubsection{Relation to \cite{kitazawa2013exploration,kitazawa2016root}}
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
\cite{kitazawa2013exploration}, which was recently published
(\citealt{kitazawa2022transformations}).
That paper defines
\begin{align}
U_t &= y_t + (1-y_t) y_{t+1} - (1-y_t) y_{t+1} \exp(-{\Greekmath 010C} \Delta x_{t+1}) - {\Greekmath 010E} y_{t-1} (1-y_t)y_{t+1} \exp(-{\Greekmath 010C} \Delta x_{t+1}),
\nonumber \\
\hbar U_t &= U_t - y_{t-1} - \tanh\left[ \frac{ -{\Greekmath 010D} y_{t-2} + {\Greekmath 010C} (\Delta x_t + \Delta x_{t+1}) } 2 \right]
(U_t + y_{t-1} - 2U_t y_{t-1}) ,
\nonumber \\
\Upsilon_t &= y_t y_{t+1} + y_t (1-y_{t+1}) \exp({\Greekmath 010C} \Delta x_{t+1}) + {\Greekmath 010E} (1-y_{t-1}) y_t (1-y_{t+1}) \exp({\Greekmath 010C} \Delta x_{t+1}) ,
\nonumber \\
\hbar \Upsilon_t &= \Upsilon_t - y_{t-1} - \tanh\left[ \frac{ {\Greekmath 010D} (1-y_{t-2}) + {\Greekmath 010C} (\Delta x_t + \Delta x_{t+1}) } 2 \right]
(\Upsilon_t + y_{t-1} - 2\Upsilon_t y_{t-1}) ,
\label{KitazawaMomentFunctions}
\end{align}
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 \eqref{MomentsConditional}.
\cite{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
\begin{align*}
\hbar U_2 &= \left\{ \tanh\left[ \frac{ -{\Greekmath 010D} y_{0} + {\Greekmath 010C} (\Delta x_2 + \Delta x_{3}) } 2 \right] -1 \right\} \, m^{(0)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} ) ,
\\
\hbar \Upsilon_2 &= \left\{ \tanh\left[ \frac{ {\Greekmath 010D} (1-y_{0}) + {\Greekmath 010C} (\Delta x_2 + \Delta x_{3}) } 2 \right] +1 \right\} \, m^{(1)}(y,y_0,x,{\Greekmath 010C} ,{\Greekmath 010D} ) .
\end{align*}
Thus, apart from a rescaling (with a non-zero function of the parameters and conditioning variables), the moment functions of
\cite{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{th:AR1moments} is new.
\subsubsection{Relation to other existing results}
\cite{honore2000panel} observe that with \eqref{model} and $T=3$, the
conditional likelihood function that conditions on $Y=y_{0}$, $Y_{3}=y_{3}$,
and $Y_{1}+Y_{2}=1$,
\begin{equation*}
\ell _{y_{0},y_{3}}(y,x,{\Greekmath 010C} ,{\Greekmath 010D} )= \,\mathrm{Pr}\left( Y=y\,\big|
\,Y_{0}=y_{0},\,Y_{1}+Y_{2}=1,\,Y_{3}=y_{3},\,X=x,{\Greekmath 010C} ,{\Greekmath 010D} \right) ,
\end{equation*}
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
\begin{align}
\frac{\partial \ell _{0,0}(y,x,{\Greekmath 010C} ,{\Greekmath 010D} )}{\partial {\Greekmath 010D} }& =0,
\notag \\
\frac{\partial \ell _{0,0}(y,x,{\Greekmath 010C} ,{\Greekmath 010D} )}{\partial {\Greekmath 010C} }& =\frac{
x_{12}}{1+\exp \left( x_{12}^{\prime }{\Greekmath 010C} \right) }\left[ \frac{
m^{(1)}(y,0,x,{\Greekmath 010C} ,{\Greekmath 010D} )+\exp \left( x_{12}^{\prime }{\Greekmath 010C} -{\Greekmath 010D}
\right) \,m^{(0)}(y,0,x,{\Greekmath 010C} ,{\Greekmath 010D} )}{\exp \left( -{\Greekmath 010D} \right) -1}
\right] , \label{FOClikelihood1}
\end{align}
and
\begin{align}
\left(
\begin{array}{c}
\displaystyle\frac{\partial \ell _{0,1}(y,x,{\Greekmath 010C} ,{\Greekmath 010D} )}{\partial {\Greekmath 010D}
} \\[3pt]
\displaystyle\frac{\partial \ell _{0,1}(y,x,{\Greekmath 010C} ,{\Greekmath 010D} )}{\partial {\Greekmath 010C} }
\end{array}
\right) & =\left(
\begin{array}{c}
\displaystyle-1\nonumber \\[3pt]
\displaystyle x_{12}
\end{array}
\right) \frac{1}{1+\exp \left( x_{12}^{\prime }{\Greekmath 010C} -{\Greekmath 010D} \right) }
\notag \\
& \qquad \qquad \times \left[ \frac{m^{(1)}(y,0,x,{\Greekmath 010C} ,{\Greekmath 010D} )+\exp
\left( x_{12}^{\prime }{\Greekmath 010C} \right) \,m^{(0)}(y,0,x,{\Greekmath 010C} ,{\Greekmath 010D} )}{\exp
({\Greekmath 010D} )-1}\right] , \label{FOClikelihood2}
\end{align}
where $m^{(0)}$ and $m^{(1)}$ are defined in {\eqref{SolutionMomentsT3}. The results for $y_{0}=1$
are analogous. Thus, the score functions of the conditional likelihood in
\cite{honore2000panel} are linear combinations of our moment conditions when
$x_{2}=x_{3}$. The conditional likelihood estimation discussed in \cite
{cox1958regression} and \cite{Chamberlain1985} are special cases of this
without regressors ($x_{1}=x_{2}=x_{3}=0$).
\cite{Hahn2001} considers model \eqref{model} 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{lemma:moments_p1T3} only provides
two moment conditions for $y_{0}=0$. However, there are three model
parameters in the setup of \cite{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{sec:identitifactionAR1} above.
\subsection{More general dynamic panel models}
As explained in Section~\ref{subsec:MasterLemma},
Lemma~\ref{lemma:Markov}
delivers valid moment functions
for any model with logistic
conditional
probabilities of the form
\eqref{GeneralLogitBinaryChoice}.
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
\eqref{model} by
\begin{equation}
\mathrm{Pr}\left( Y_{it}=1\,\big|\,Y_{i}^{t-1},X_{i},A_{i}\right) =\frac{
\exp \big[
(1-Y_{i,t-1}) X_{it}^{\prime }\,{\Greekmath 010C}_0
+
Y_{i,t-1} X_{it}^{\prime }\,{\Greekmath 010C}_1 +Y_{i,t-1}\,{\Greekmath 010D} +A_{i}\big)}{1+\exp
\big((1-Y_{i,t-1}) X_{it}^{\prime }\,{\Greekmath 010C}_0
+
Y_{i,t-1} X_{it}^{\prime }\,{\Greekmath 010C}_1 +Y_{i,t-1}\,{\Greekmath 010D} +A_{i}\big]}, \label{modelGENERALIZED}
\end{equation}
then the moment functions \eqref{MomentsGeneralT}
and Theorem~\ref{th:AR1moments}
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 \eqref{modelGENERALIZED} 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{app:MoreGeneralPredetermined}.
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 \eqref{modelGENERALIZED} then reads
\begin{align}
&\mathrm{Pr}\left( Y_{it}=1\,\big|\,Y_{i}^{t-1},X_{i},A_{i},C_i\right)
\nonumber \\ & \qquad \qquad \qquad
=\frac{
\exp \big[(1-Y_{i,t-1}) X_{it}^{\prime }\,{\Greekmath 010C}_0
+
Y_{i,t-1} X_{it}^{\prime }\,{\Greekmath 010C}_1 +Y_{i,t-1}\,C_i +A_{i}\big]}{1+\exp
\big[(1-Y_{i,t-1}) X_{it}^{\prime }\,{\Greekmath 010C}_0
+
Y_{i,t-1} X_{it}^{\prime }\,{\Greekmath 010C}_1 +Y_{i,t-1}\,C_i +A_{i}\big]}.
\label{modelGammaHeterogeneous}
\end{align}
We can then treat $(C_i,A_i)$ as a two-dimensional fixed-effect and employ the
methods of Section~\ref{SEC: Incidental parameter free moment conditions} and \ref{sec:Derivation} 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
\begin{align}
m(y,0,x,{\Greekmath 010C}_0,{\Greekmath 010C}_1 )& =
- \mathbbm{1}\Big\{ (y_1,y_3,y_4) = (1,0,0) \Big\}
+
\left\{
\begin{array}{ll}
\exp \left( z_{12} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(0,1,0,0), \\
\exp \left( z_{14} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(0,1,0,1), \\
- \exp \left( z_{34} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(1,0,0,1), \\
\exp \left( z_{32} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }y=(1,1,0,0), \\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise},
\end{array}
\right.
\label{momentGammaHeterogeneous}
\end{align}
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{sec:Derivation}.
\section{Empirical illustration\label{SEC: Empirical illustration}}
\label{sec:Emp}
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{Figure: Histograms} displays the number of
observations, $T_{i}$, per individual in each of the three samples.
\begin{figure}[h]
\caption{Histogram of Number of Observations Per Individual.}
\label{Figure: Histograms}\centering
\begin{subfigure}[t]{0.30\textwidth}
\includegraphics[scale=0.35]{HistoAll.png}
\caption{All}
\end{subfigure} \quad
\begin{subfigure}[t]{0.30\textwidth}
\includegraphics[scale=0.35]{HistoFemales.png}
\caption{Females}
\end{subfigure} \quad
\begin{subfigure}[t]{0.30\textwidth}
\includegraphics[scale=0.35]{HistoMales.png}
\caption{Males}
\end{subfigure}
\end{figure}
The moment conditions for the fixed effects logit AR(1) in (\ref
{SolutionMomentsT3}) and (\ref{MomentsGeneralT}) 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 \cite{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 \cite{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{sec:FindSolutions} are given by
\begin{eqnarray*}
\widetilde{m}^{(0)}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) &=&\frac{m^{(0)}(y,y_{0},x,
{\Greekmath 010C} ,{\Greekmath 010D} )}{1+\exp \left( x_{23}^{\prime }{\Greekmath 010C} \right) +\exp \left(
x_{31}^{\prime }{\Greekmath 010C} -y_{0}\,{\Greekmath 010D} \right) +\exp \left( x_{21}^{\prime
}{\Greekmath 010C} +(1-y_{0})\,{\Greekmath 010D} \right) } \\
\widetilde{m}^{(1)}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) &=&\frac{m^{(2)}(y,y_{0},x,
{\Greekmath 010C} ,{\Greekmath 010D} )}{1+\exp \left( x_{12}^{\prime }{\Greekmath 010C} +y_{0}\,{\Greekmath 010D} \right)
+\exp \left( x_{13}^{\prime }{\Greekmath 010C} -(1-y_{0})\,{\Greekmath 010D} \right) +\exp \left(
x_{32}^{\prime }{\Greekmath 010C} \right) }
\end{eqnarray*}
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 \cite{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
\begin{equation*}
M(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )=\big(1,\;y_{0},\;x_{1}^{\prime },\;x_{2}^{\prime
},\;x_{3}^{\prime }\big)^{\prime }\otimes \left(
\begin{array}{c}
\widetilde{m}^{(1)}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} ) \\
\widetilde{m}^{(0)}(y,y_{0},x,{\Greekmath 010C} ,{\Greekmath 010D} )
\end{array}
\right) ,
\end{equation*}
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
\begin{equation*}
{\binom{\widehat{{\Greekmath 010C} }}{\widehat{{\Greekmath 010D} }}}=\limfunc{argmin}_{{\Greekmath 010C} \in
\mathbb{R}^{K},\,{\Greekmath 010D} \in \mathbb{R}}\left(
\sum_{i=1}^{n}M(Y_{i},Y_{i,0},X_{i},{\Greekmath 010C} ,{\Greekmath 010D} )\right) ^{\prime }W\left(
\sum_{i=1}^{n}M(Y_{i},Y_{i,0},X_{i},{\Greekmath 010C} ,{\Greekmath 010D} )\right) ,
\end{equation*}
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 \cite{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, \cite{GayleShephard2019} for a recent example.}
Under standard regularity conditions we have
\begin{equation*}
\sqrt{n}\left[ {\binom{\widehat{{\Greekmath 010C} }}{\widehat{{\Greekmath 010D} }}}-{\binom{{\Greekmath 010C}
_{0}}{{\Greekmath 010D} _{0}}}\right] \Rightarrow \mathcal{N}\left(0, \,
(G'WG)^{-1} \,
G^{\prime }W\,\Omega \,WG\,
(G'WG)^{-1}
\right),
\end{equation*}
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{Table: Empirical} reports the estimation results. As expected,
and consistent with the Monte Carlo results in Appendix~B.1 of \cite{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.
\begin{table}[tbp!]
\caption{Empirical Results (AR(1)).}
\label{Table: Empirical}\centering{\hspace{-0.3cm} {\footnotesize
\begin{tabular}{l@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r}
& & & & & & & & & & & \\
& \multicolumn{3}{c}{Females} & & \multicolumn{3}{c}{Males} & &
\multicolumn{3}{c}{All} \\
& & & & & & & & & & & \\
& \multicolumn{1}{c}{Logit} & \multicolumn{1}{c}{
\begin{tabular}{@{}r}
Logit \\
w FE
\end{tabular}
} & \multicolumn{1}{c}{GMM} & & \multicolumn{1}{c}{Logit} &
\multicolumn{1}{c}{
\begin{tabular}{@{}r}
Logit \\
w FE
\end{tabular}
} & \multicolumn{1}{c}{GMM} & & \multicolumn{1}{c}{Logit} &
\multicolumn{1}{c}{
\begin{tabular}{@{}r}
Logit \\
w FE
\end{tabular}
} & \multicolumn{1}{c}{GMM} \\ \cline{2-4}\cline{6-8}\cline{10-12}
& & & & & & & & & & & \\
Lagged $y$ & $2.585 $ & $0.780 $ & $1.512 $ & & $2.947 $ & $0.709 $ & $
1.454 $ & & $2.797 $ & $0.768 $ & $1.417 $ \\
& $( 0.038 \rlap{)}$ & $( 0.050 \rlap{)}$ & $( 0.076 \rlap{)}$ & & $( 0.040
\rlap{)}$ & $( 0.063 \rlap{)}$ & $( 0.088 \rlap{)}$ & & $( 0.027 \rlap{)}$
& $( 0.039 \rlap{)}$ & $( 0.060 \rlap{)}$ \\
& & & & & & & & & & & \\
Children & $-0.335 $ & $-0.444 $ & $-0.244 $ & & $-0.153 $ & $0.018 $ & $
-0.275 $ & & $-0.278 $ & $-0.252 $ & $-0.214 $ \\
& $( 0.016 \rlap{)}$ & $( 0.052 \rlap{)}$ & $( 0.196 \rlap{)}$ & & $( 0.021
\rlap{)}$ & $( 0.067 \rlap{)}$ & $( 0.133 \rlap{)}$ & & $( 0.012 \rlap{)}$
& $( 0.043 \rlap{)}$ & $( 0.102 \rlap{)}$ \\
& & & & & & & & & & & \\
Married & $0.082 $ & $-0.044 $ & $0.637 $ & & $0.335 $ & $0.332 $ & $0.038 $
& & $0.349 $ & $0.173 $ & $0.707 $ \\
& $( 0.084 \rlap{)}$ & $( 0.159 \rlap{)}$ & $( 0.890 \rlap{)}$ & & $( 0.071
\rlap{)}$ & $( 0.171 \rlap{)}$ & $( 0.295 \rlap{)}$ & & $( 0.053 \rlap{)}$
& $( 0.111 \rlap{)}$ & $( 0.397 \rlap{)}$ \\
& & & & & & & & & & & \\
SP.Inc. & $-0.010 $ & $-0.050 $ & $-0.104 $ & & $0.033 $ & $0.003 $ & $
0.019 $ & & $-0.017 $ & $-0.044 $ & $-0.089 $ \\
& $( 0.006 \rlap{)}$ & $( 0.011 \rlap{)}$ & $( 0.068 \rlap{)}$ & & $( 0.007
\rlap{)}$ & $( 0.016 \rlap{)}$ & $( 0.026 \rlap{)}$ & & $( 0.004 \rlap{)}$
& $( 0.009 \rlap{)}$ & $( 0.033 \rlap{)}$ \\
& & & & & & & & & & & \\
& & & & & & & & & & &
\end{tabular}
}}
\begin{tablenotes}
\tmpsmall\sc
\item The estimation also includes 12 time dummies. Standard error for the GMM and Logit Fixed Effects Estimators are calculated as the interquartile range of 1,000 bootstrap replications divided by 1.35.
\end{tablenotes}
\end{table}
\begin{table}[tbp!]
\caption{Empirical Results (AR(2)).}
\label{Table: Empirical AR2}\centering{\hspace{-0.3cm} {\footnotesize
\begin{tabular}{l@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r@{\;\;\;\;}r}
& & & & & & & & & & & \\
& \multicolumn{3}{c}{Females} & & \multicolumn{3}{c}{Males} & &
\multicolumn{3}{c}{All} \\
& & & & & & & & & & & \\
& \multicolumn{1}{c}{Logit} & \multicolumn{1}{c}{
\begin{tabular}{@{}r}
Logit \\
w FE
\end{tabular}
} & \multicolumn{1}{c}{GMM} & & \multicolumn{1}{c}{Logit} &
\multicolumn{1}{c}{
\begin{tabular}{@{}r}
Logit \\
w FE
\end{tabular}
} & \multicolumn{1}{c}{GMM} & & \multicolumn{1}{c}{Logit} &
\multicolumn{1}{c}{
\begin{tabular}{@{}r}
Logit \\
w FE
\end{tabular}
} & \multicolumn{1}{c}{GMM} \\ \cline{2-4}\cline{6-8}\cline{10-12}
& & & & & & & & & & & \\
$y_{t-1}$ & $2.259 $ & $0.742 $ & $1.356 $ & & $2.422 $ & $0.514 $ & $1.116
$ & & $2.361 $ & $0.665 $ & $1.297 $ \\
& $( 0.047 \rlap{)}$ & $( 0.069 \rlap{)}$ & $( 0.162 \rlap{)}$ & & $( 0.052
\rlap{)}$ & $( 0.083 \rlap{)}$ & $( 0.131 \rlap{)}$ & & $( 0.035 \rlap{)}$
& $( 0.053 \rlap{)}$ & $( 0.092 \rlap{)}$ \\
& & & & & & & & & & & \\
$y_{t-2}$ & $0.917 $ & $-0.379 $ & $0.678 $ & & $1.332 $ & $-0.286 $ & $
0.558 $ & & $1.137 $ & $-0.319 $ & $0.648 $ \\
& $( 0.048 \rlap{)}$ & $( 0.072 \rlap{)}$ & $( 0.081 \rlap{)}$ & & $( 0.053
\rlap{)}$ & $( 0.080 \rlap{)}$ & $( 0.066 \rlap{)}$ & & $( 0.036 \rlap{)}$
& $( 0.054 \rlap{)}$ & $( 0.046 \rlap{)}$ \\
& & & & & & & & & & & \\
Children & $-0.260 $ & $-0.410 $ & $-1.926 $ & & $-0.143 $ & $0.112 $ & $
-0.188 $ & & $-0.223 $ & $-0.192 $ & $-1.209 $ \\
& $( 0.018 \rlap{)}$ & $( 0.069 \rlap{)}$ & $( 0.282 \rlap{)}$ & & $( 0.024
\rlap{)}$ & $( 0.100 \rlap{)}$ & $( 0.251 \rlap{)}$ & & $( 0.014 \rlap{)}$
& $( 0.051 \rlap{)}$ & $( 0.219 \rlap{)}$ \\
& & & & & & & & & & & \\
Married & $0.136 $ & $0.022 $ & $-0.193 $ & & $0.411 $ & $0.534 $ & $-0.019
$ & & $0.393 $ & $0.269 $ & $-0.121 $ \\
& $( 0.095 \rlap{)}$ & $( 0.184 \rlap{)}$ & $( 0.028 \rlap{)}$ & & $( 0.083
\rlap{)}$ & $( 0.203 \rlap{)}$ & $( 0.025 \rlap{)}$ & & $( 0.061 \rlap{)}$
& $( 0.145 \rlap{)}$ & $( 0.022 \rlap{)}$ \\
& & & & & & & & & & & \\
Sp.Inc & $-0.015 $ & $-0.050 $ & $0.255 $ & & $0.023 $ & $-0.008 $ & $0.179
$ & & $-0.022 $ & $-0.046 $ & $0.093 $ \\
& $( 0.007 \rlap{)}$ & $( 0.014 \rlap{)}$ & $( 0.210 \rlap{)}$ & & $( 0.008
\rlap{)}$ & $( 0.019 \rlap{)}$ & $( 0.316 \rlap{)}$ & & $( 0.005 \rlap{)}$
& $( 0.011 \rlap{)}$ & $( 0.183 \rlap{)}$ \\
& & & & & & & & & & & \\
& & & & & & & & & & &
\end{tabular}
}}
\begin{tablenotes}
\tmpsmall\sc
\item The estimation also includes 11 time dummies. Standard error for the GMM and Logit Fixed Effects Estimators are calculated as the interquartile range of 1,000 bootstrap replications divided by 1.35.
\end{tablenotes}
\end{table}
To estimate the AR(2)\ version of the model, we apply the moment conditions provided
in Appendix~\ref{sec:moments_p2T4} 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 \cite{Honore2022moment}.
The results are
presented in Table \ref{Table: Empirical AR2}. 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.
\section{Conclusion}
\label{sec:conc}
\cite{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{SEC: Incidental parameter free moment conditions} and \ref{sec:Derivation}
for exploring and deriving
moment conditions are applicable for more general panel models,
as illustrated by the examples provided in Section~\ref{sec:Examples}.
Exploring such moment conditions in other interesting models is a research agenda that has only started (e.g.\ \citealt{honore2021dynamic}, \citealt{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}
\setlength{\bibsep}{4pt}
\ifx\undefined\leavevmode\rule[.5ex]{3em}{.5pt}\
\fi
\ifx\undefined\textsc
\let\tmpsmall\tmpsmall\sc
\fi
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