The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
147,499 characters
Dynamic Ordered Panel Logit Models
\title{{\bf Dynamic Ordered Panel Logit Models}\thanks{
This research was
supported by the Gregory C. Chow Econometric Research Program at Princeton
University, by the National Science Foundation (Grant Number SES-1530741),
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),
by the Social Sciences and Humanities Research Council of Canada (grant number IG 435-2021-0778),
and by the European Research Council grants ERC-2014-CoG-646917-ROMIA and
ERC-2018-CoG-819086-PANEDA. We are grateful to seminar participants at the Universities of Leuven and Toronto, to Paul Contoyannis for helpful conversations about the BHPS data used in \citet{contoyannis_dynamics_2004}, and to Riccardo D'Adamo, Geert Dhaene, Sharada Dharmasankar and Runtong Ding for useful comments and discussion. We also thank Limor Golan and four anonymous referees for constructive feedback and suggestions.}
}
\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
Chris Muris
\thanks{
McMaster University, \texttt{[email removed]}}
\and
Martin Weidner
\thanks{
University of Oxford, \texttt{[email removed]} } }
\date{April 2024}
\maketitle
\thispagestyle{empty}
\setcounter{page}{0}
\begin{abstract}
\noindent
This paper studies a dynamic ordered logit model for panel data with fixed effects. The main contribution of the paper is to construct a set of valid moment conditions that are free of the fixed effects. The moment functions can be computed using four or more periods of data, and the paper presents sufficient conditions for the moment conditions to identify the common parameters of the model, namely the regression coefficients, the autoregressive parameters, and the threshold parameters. The availability of moment conditions suggests that these common parameters can be estimated using the generalized method of moments, and the paper documents the performance of this estimator using Monte Carlo simulations and an empirical illustration to self-reported health status using the British Household Panel Survey.
\end{abstract}
\newpage
\section{Introduction}
Panel surveys routinely collect data on an ordinal scale. For example, many nationally representative surveys ask respondents to rate their health or life satisfaction on an ordinal scale.\footnote{One example is the British Household Panel Survey in our empirical application. Others include the U.S. Health and Retirement Study and Medical Expenditure Panel Survey, the Canadian Longitudinal Study on Ageing and the National Longitudinal Survey of Children and Youth, the Australian Longitudinal Study on Women's Health, the European Union Statistics on Income and Living Conditions, the Survey on Health, Ageing, and Retirement in Europe, among many others.} Other examples include test results in longitudinal data sets gathered for studying education.
We are interested in regression models for ordinal outcomes that allow for lagged dependent variables as well as fixed effects. In the model that we propose, the ordered outcome depends on a fixed effect, a lagged dependent variable, regressors, and a logistic error term. We study identification and estimation of the finite-dimensional parameters in this model when only a small number ($\geq 4$) of time periods is available.
For other types of outcome variables (continuous outcomes in linear models, binary and multinomial outcomes), results for regression models with fixed effects and lagged dependent variables are already available. Such results are of great importance for applied practice, as they allow researchers to distinguish unobserved heterogeneity from state dependence, and to control for both when estimating the effect of regressors. The demand for such methods is evidenced by the popularity of existing approaches for the linear model, such as those proposed by \cite{arellano_tests_1991} and \cite{blundell_initial_1998}. In contrast, for ordinal outcomes, almost no results are available.
The challenge of accommodating unobserved heterogeneity in nonlinear models is well understood, especially when the researcher also wants to allow for lagged dependent variables. For example, while recent developments (\citealt{kitazawa2021transformations} and \citealt{honore2020dynamic}) relax these requirements, early work on the dynamic binary logit model with fixed effects either assumed no regressors, or restricted their joint distribution (cf. \citealt{Chamberlain1985} and \citealt{honore2000panel}). The challenge of accommodating unobserved heterogeneity in the ordered logit model seems even greater than in the binary model. The reason is that even the static version of the model is not in the exponential family (\citealt{hahn_note_1997}). As a result, one cannot directly appeal to a sufficient statistic approach. An alternative approach in the static ordered logit model is to reduce it to a set of binary choice models (cf. \citealt{das_panel_1999}, \citealt{JohnsonThesis2004}, \citealt{Baetschmann2015}, \citealt{Muris2017}, and \citealt{bmp2021}). Unfortunately, the dynamic ordered logit model cannot be similarly reduced to a dynamic binary choice model (see \citealt{muris2020dynamic}). Therefore, a new approach is needed. The contribution of this paper is to develop such an approach.
To do this, we follow the functional differencing approach in \cite{bonhomme2012functional} to obtain moment conditions for the finite-dimensional parameters in this model, namely the autoregressive parameters (one for each level of the lagged dependent variable), the threshold parameters in the underlying latent variable formulation, and the regression coefficients. Our approach is closely related to \cite{honore2020dynamic}, and can be seen as the extension of their method to the case of an ordered response variable.
This paper contributes to the literature on dynamic ordered logit models. We are aware of only one paper that studies a fixed-$T$ version of this model while allowing for fixed effects. The approach in \cite{muris2020dynamic} builds on methods for dynamic binary choice models in \cite{honore2000panel} by restricting how past values of the dependent variable enter the model. In particular, in \cite{muris2020dynamic}, the lagged dependent variable $Y_{i,t-1}$ enters the model only via $\mathbbm{1}\{Y_{i,t-1}\geq k\}$ for some known $k$. We do not impose such a restriction, and allow the effect of $Y_{i,t-1}$ to vary freely with its level.
Other existing work on dynamic panel models for ordered outcomes uses a random effects approach (\citealt{contoyannis_dynamics_2004}, \citealt{albarran_estimation_2019}) or requires a large number of time periods for consistency (\citealt{carro_state_2014}, \citealt{fernandez-val_evaluating_2017}). An earlier version of \cite{aristodemou2018semiparametric} contained partial identification results for a dynamic ordered choice model without logistic errors. Our approach places no restrictions on the dependence between fixed effects and regressors, requires only four periods of data for consistency, and delivers point identification and estimates.
More broadly, this paper contributes to the literature on fixed-$T$ identification and estimation in nonlinear panel models with fixed effects (see \citealt{honore_nonlinear_2002}, \citealt{arellano2003discrete}, and \citealt{ArellanoBonhomme2011} for overviews). The literature contains results for several models adjacent to ours. For example, the static panel ordered logit model with fixed effects was studied by \cite{das_panel_1999}, \cite{JohnsonThesis2004}, \cite{Baetschmann2015}, and \cite{Muris2017}; results for static and dynamic binomial and multinomial choice models are in \cite{chamberlain_analysis_1980}, \cite{honore2000panel}, \cite{magnac2000subsidised}, \cite{shi_estimating_2018}, \cite{aguirregabiriaSufficientStatisticsUnobserved2021},
\cite{aguirregabiriaIdentificationAverageMarginal},
\cite{PakesPorterShepardCalderWang2022} and \cite{khan_inference_2021}.
Our main contribution is to obtain novel moment conditions for the common parameters in the dynamic ordered logit model with fixed effects. Additionally, we obtain conditions under which these moment conditions identify those parameters. Finally, we discuss the implied generalized method of moments (GMM) estimator and demonstrate its performance in both a Monte Carlo study and an empirical application to self-reported health status in the British Household Panel Study.
The remainder of this paper is organized as follows. Section~\ref{sec:model} introduces the model and the moment conditions that are free of fixed effects. Section~\ref{sec:Identification} presents identification results for the common model parameters based on those moment conditions.
Section~\ref{Section: Implication for estimation} discusses how to use the moment conditions for estimation and inference.
Section~\ref{PracticalPerformance} explores the practical performance of the resulting estimation method through Monte Carlo simulations, including comparison with a correlated random effects approach.
That section also provides an empirical illustration to health data. Section~\ref{Conclusions} concludes the paper. The appendix provides proofs and further computational details.
\section{Model and moment conditions}
\label{sec:model}
In this section, we first describe the panel ordered logit model that
is used throughout the paper, and then present moment
conditions for the model that can be used to estimate the common
parameters of the model without imposing any knowledge of the
individual-specific effects.
\subsection{Model and notation}
We consider panel data with cross-sectional units
$i=1,\ldots,n$ and time periods $t=0,\ldots,T$.
For each pair $(i,t)$,
we observe the discrete outcome $Y_{it}\in \{1,2,\ldots,Q\}$,
which can take $Q \in \{2,3,4,\ldots\}$ different values,
and the strictly exogenous regressors $X_{it}\in \mathbb{R}^{K}$.
We discuss unbalanced panels in Section~\ref{sec:MomentsTlarger3}, but for now, we assume a balanced panel where outcomes
are observed for all $t \geq 0$ and regressors for all $t \geq 1$.
Thus, the total number of time periods for which outcomes are
observed is $T+1$. For $t \geq 1$, the observed discrete outcomes depend on
an unobserved latent variable $Y^*_{it} \in \mathbb{R}$ as follows:
\begin{align}
Y_{it} = \left\{
\begin{array}{ll}
1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if \; $ Y^*_{it} \; \in \; (-\infty \,,{\Greekmath 0115}_{1}]$,}
\\
2 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if \; $ Y^*_{it} \; \in \; ({\Greekmath 0115}_{1} \, ,{\Greekmath 0115}_{2}]$,}
\\
\vdots
\\
Q & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if \; $ Y^*_{it} \; \in \; ({\Greekmath 0115}_{Q-1} \, , \infty)$,}
\end{array}
\right.
\label{ModelOutcomes}
\end{align}
where ${\Greekmath 0115}=({\Greekmath 0115}_1,\ldots,{\Greekmath 0115}_{Q-1}) \in \mathbb{R}^{Q-1}$ is a vector of unknown parameters with ${\Greekmath 0115}_1 < {\Greekmath 0115}_2 < \ldots < {\Greekmath 0115}_{Q-1}$.
The latent variable is generated by the model
\begin{align}
Y^*_{it} = X_{it}^{\prime }\,{\Greekmath 010C} + \sum_{q=1}^{Q} {\Greekmath 010D}_q \, \mathbbm{1}\left\{ Y_{i,t-1} = q \right\} + A_i + {\Greekmath 0122}_{it} ,
\label{ModelYstar}
\end{align}
with unknown parameters ${\Greekmath 010C} \in \mathbb{R}^K$ and ${\Greekmath 010D} = ({\Greekmath 010D}_1,\ldots,{\Greekmath 010D}_Q) \in \mathbb{R}^{Q}$. Here, $A_i \in \mathbb{R} \cup \{\pm \infty\}$ is an unobserved individual-specific effect
whose distribution is not specified, and $A_i$ is allowed to be arbitrarily correlated with the regressors $X_{it}$ and the initial conditions $Y_{i0}$. Let $X_{i}:=(X_{i1},\ldots ,X_{iT})$.
Conditional on $Y_{i0}$, $X_i$,
and $A_i$,
the idiosyncratic error term $ {\Greekmath 0122}_{it}$ is assumed to
be independent and identically distributed over $t$ with
cumulative distribution function $ \Lambda({\Greekmath 0122}) := [1+\exp(-{\Greekmath 0122})]^{-1}$.
Thus, $ {\Greekmath 0122}_{it}$ is a logistic error term.
For the cross-sectional sampling, we assume that
$(Y_{i0}, X_{i1},\ldots ,X_{iT}, A_{i}, {\Greekmath 0122}_{i1},\ldots ,{\Greekmath 0122}_{iT})$ are independent and identically distributed across $i$.
The model described by \eqref{ModelOutcomes} and \eqref{ModelYstar} is a dynamic ordered panel logit model, where an arbitrary function ${\Greekmath 010D}_{Y_{i,t-1}}$ of the lagged dependent variable $Y_{i,t-1}$ is allowed to enter additively into the latent variable
$Y^*_{it}$.
This model strikes a balance between
a general functional form and a parsimonious parameter structure.
We discuss possible generalizations of the
model for $Y^*_{it}$ in Section~\ref{sec:GeneralModel}, but
otherwise impose \eqref{ModelYstar} throughout the paper.\footnote{If the observed
$Y_{it}$ is a discretized version of a continuous variable with a natural economic interpretation, then it would be more natural to model the state dependence in \eqref{ModelYstar} as $Y_{i,t-1}^{\star}{\Greekmath 010D}$. A numerical investigation suggests that it is not possible to develop moment conditions for such a model. This suggests that the common parameters in this model are not point-identified, or are only point-identified under strong support assumptions on the covariates and hence not generally $\sqrt{n}$ estimable.
We explore this alternative model in Appendix \ref{app:alternative_specification}.}
Our ultimate goal is to estimate the unknown parameters ${\Greekmath 0112} = ({\Greekmath 010C},{\Greekmath 010D},{\Greekmath 0115}) \in \Theta := \mathbb{R}^{K+2Q-1}$ without imposing any assumptions on the individual-specific effect $A_i$. This requires two normalizations, because
common additive shifts of all the parameters ${\Greekmath 010D}_q$ or of all the parameters ${\Greekmath 0115}_q$ can
be absorbed into $A_i$. For example, we could impose
the normalizations ${\Greekmath 010D}_1=0$ and ${\Greekmath 0115}_1=0$, but in this section there is no need to specify such normalizations.
It is convenient to define ${\Greekmath 0115}_{0} := -\infty$, and ${\Greekmath 0115}_{Q} := \infty$,
and
\begin{align}
z(Y_{i,t-1},X_{it},{\Greekmath 0112}) := X_{it}^{\prime }\,{\Greekmath 010C} + \sum_{q=1}^{Q} {\Greekmath 010D}_q \, \mathbbm{1}\left\{ Y_{i,t-1} = q \right\} .
\label{DefSingleIndex}
\end{align}
With this notation, the model assumptions imposed so far imply that the distribution of $Y_{it}$ conditional on the regressors $X_{i}$,
past outcomes $Y_{i}^{t-1}=(Y_{i,t-1},Y_{i,t-2},\ldots )$, and fixed effects $A_i$,
is given by
\begin{equation}
\mathrm{Pr}\left( Y_{it}=q \, \Big| \, Y_{i}^{t-1},X_{i},A_{i}, {\Greekmath 0112}
\right)
= \Lambda\Big[ z(Y_{i,t-1},X_{it},{\Greekmath 0112}) + A_{i} - {\Greekmath 0115}_{q-1} \Big] - \Lambda\Big[ z(Y_{i,t-1},X_{it},{\Greekmath 0112}) + A_{i} - {\Greekmath 0115}_{q} \Big]
\label{model}
\end{equation}
for all $i\in \{1,\ldots ,n\}$, $t\in \{1,2,\ldots ,T\}$, and $q \in \{1,2,\ldots,Q\}$.
Let $Y_{i}=(Y_{i1},\ldots ,Y_{iT})$, and let the true model parameters be denoted by ${\Greekmath 0112}^0 = ({\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}^0)$. In the following, all probabilistic statements are for the model distribution generated under ${\Greekmath 0112}^0$. For
example, we have $\mathrm{Pr}\big(Y_{i}=y_{i}\,\big|\,Y_{i0}=y_{i0},\,X_{i}=x_{i},
\,A_{i}={\Greekmath 010B} _{i}\big)=p_{y_{i0}}(y_{i},x_{i},{\Greekmath 0112}^0,{\Greekmath 010B} _{i})$, where
\begin{equation}
p_{y_{i0}}(y_{i},x_{i},{\Greekmath 0112} ,{\Greekmath 010B} _{i}) :=\prod_{t=1}^{T}
\left\{ \Lambda\Big[ z(y_{i,t-1},x_{it},{\Greekmath 0112}) + {\Greekmath 010B}_{i} - {\Greekmath 0115}_{y_{it}-1} \Big] - \Lambda\Big[ z(y_{i,t-1},x_{it},{\Greekmath 0112}) + {\Greekmath 010B}_{i} - {\Greekmath 0115}_{y_{it}} \Big]
\right\} .
\label{DefProb}
\end{equation}
Below, we drop the index $i$ until we discuss estimation;
instead of $Y_{i0}$, $Y_i$, $X_i$, $A_i$, we just write $Y_0$, $Y$, $X$, $A$ for those random variables and random vectors.
\subsection{Moment condition approach}
\label{subsec:momentApproach}
In the next subsection, we present moment functions for the ordered logit model discussed above. These are functions
$m:\{1,\ldots,Q\} \times \{1,\ldots,Q\}^T \times \mathbb{R}^{T \times K} \times \Theta
\rightarrow \mathbb{R}$ such that
\begin{align}
\mathbb{E} \left[ m_{Y_0}(Y,X,{\Greekmath 0112}^0) \, \Big| \, Y_0=y_0,
\, X=x, \, A={\Greekmath 010B} \right] &= 0
\label{MomentFunctionIdea}
\end{align}
for all $y_0 \in \{1,\ldots,Q\}$, $x \in \mathbb{R}^{T \times K}$,
and ${\Greekmath 010B} \in \mathbb{R} \cup \{\pm \infty\}$. We write the first argument $y_0$ of the moment function as an index, but that is purely
for notational convenience.
Conditional on
$Y_0=y_0$, $X=x$, and $A={\Greekmath 010B}$, the distribution of
$Y=(Y_1,\ldots,Y_T) \in \{1,\ldots,Q\}^T$ is given by
\eqref{DefProb}. The model assumptions in the last subsection are therefore sufficient to evaluate the conditional expectation
in \eqref{MomentFunctionIdea}.
If we can establish
the conditional moment condition \eqref{MomentFunctionIdea} then,
by the law of iterated expectations, we also have
the unconditional moment conditions
\begin{align}
\mathbb{E} \left[ h(Y_0,X,{\Greekmath 0112}^0) \, m_{Y_0}(Y,X,{\Greekmath 0112}^0) \right] &= 0
\label{MomentFunctionUnconditional}
\end{align}
for any function $h : \{1,\ldots,Q\} \times \mathbb{R}^{T \times K} \times \Theta
\rightarrow \mathbb{R}$ such that the expectation is well-defined.
Those unconditional moment conditions can then be used to estimate
the model parameters ${\Greekmath 0112}^0$ by the generalized method of moments (GMM).
Such an estimation approach solves the incidental parameter
problem (\citealt{neyman1948consistent}), because the moment condition
\eqref{MomentFunctionUnconditional} does not feature the
individual-specific effect $A$ at all. No
assumptions are imposed on the distribution of those nuisance parameters,
and they need not be estimated. On the flip-side, this implies
that we do not learn anything about the distribution of $A$. Notice, however, that if one is interested in (functions of)
the individual-specific effects such as average partial effects, then the estimation of the common parameters ${\Greekmath 0112}$ will always be
a key first step in any inference procedure.
The moment condition approach just described eliminates
the individual-specific effect $A$ from the estimation, because
\eqref{MomentFunctionIdea} is assumed to hold for all ${\Greekmath 010B} \in \mathbb{R} \cup \{\pm \infty\}$, but the moment function
$m_{Y_0}(Y,X,{\Greekmath 0112}^0)$ does not depend on $A$ at all.
The existence of moment functions with this property is quite
miraculous:
for any given values of $Y_0=y_0$, $X=x$, and ${\Greekmath 0112}^0$, the moment function
$m_{y_0}(\cdot,x,{\Greekmath 0112}^0) : \{1,\ldots,Q\}^T \rightarrow \mathbb{R}$
can be viewed as a finite-dimensional vector (with $Q^T$ real numbers),
but \eqref{MomentFunctionIdea} imposes an infinite number of linear constraints -- one for each ${\Greekmath 010B} \in \mathbb{R} \cup \{\pm \infty\}$.
The logistic assumption on ${\Greekmath 0122}_{it}$ is important for finding
solutions
of this infinite-dimensional linear system in a finite number
of variables, and for most choices of error distributions (e.g.\
normally distributed error),
we do not expect such solutions to exist. It seems
likely that for the error distributions in \cite{johnson2004identification} and \cite{davezies2022fixed},
and also for a mixture of logistics (briefly discussed in
\citealt{honore2020dynamic}), one could also find
valid moment conditions, if a sufficient number of
time periods are available, but we focus purely
on logistic errors in this paper.
In the following, we present moment functions $m_{Y_0}(Y,X,{\Greekmath 0112}^0)$
that satisfy \eqref{MomentFunctionIdea}.
We derive those moment functions for the dynamic panel ordered
logit model analogously to the results for the dynamic panel
binary choice logit model in \cite{honore2020dynamic}.
Indeed, for the binary choice case ($Q=2$), our moment functions below
exactly coincide with those in \cite{honore2020dynamic}, and
we refer to that paper for more details on the derivation, which is closely related to the functional differencing method in
\cite{bonhomme2012functional}.
Once we have obtained expressions for the moment functions, their derivation is no longer relevant and we can focus on showing that they are valid -- i.e. that \eqref{MomentFunctionIdea} holds -- and on their implications for the identification and estimation of ${\Greekmath 0112}$.
\subsection{Moment conditions for $T=3$}
We first introduce our moment functions for $T=3$. In our convention, this means that outcomes $Y_t$ are observed
for the four time periods $t =0,1,2,3$ (including the initial conditions $Y_0$).
We have verified numerically
that no moment functions satisfying \eqref{MomentFunctionIdea}
for general parameter and regressor values
exist for $T<3$,
and for the binary choice case ($Q=2$) a proof of this non-existence is given in \cite{honore2020dynamic}. Thus, $T=3$ is the smallest
number of time periods that we can consider.
We use lower case letters for generic arguments (as opposed to
random variables) of the moment function
$m_{y_0}(y,x,{\Greekmath 0112})$, where $y_0 \in \{1,\ldots,Q\}$,
$y \in \{1,\ldots,Q\}^T$, $x \in \mathbb{R}^{T \times K}$
and ${\Greekmath 0112} = ({\Greekmath 010C},{\Greekmath 010D},{\Greekmath 0115}) \in \Theta$.
The
$t$'th row of $x$ is denoted by $x_t' \in \mathbb{R}^K$,
and we define $x_{ts} := x_t - x_s$, ${\Greekmath 010D}_{qr} := {\Greekmath 010D}_q - {\Greekmath 010D}_r$
and ${\Greekmath 0115}_{qr} := {\Greekmath 0115}_q - {\Greekmath 0115}_r$.
We find multiple moment functions
$m_{y_0,q_1,q_2,q_3} \allowbreak (y,x,{\Greekmath 0112})$
which are distinguished by the additional indices $q_1 \in \{1,\ldots,Q-1\}$, $q_2 \in \{1,\ldots,Q\}$, $q_3 \in \{1,\ldots,Q-1\}$.
For the moment function labelled by $q_1$, $q_2$, $q_3$, the dependence
on $y$ is only through the coarser outcome $\widetilde y_{q_1,q_2,q_3}(y)
\in \{0,1\} \times \{1,2,3\} \times \{0,1\}$, which is a vector
with three components $ \widetilde y_{t,q_1,q_2,q_3}(y) $, $t \in \{1,2,3\}$,
given by
\begin{align*}
\widetilde y_{2,q_1,q_2,q_3}(y) &:= \left\{
\begin{array}{ll} 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_2 < q_2 ,
\\
2 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_2 = q_2 ,
\\
3 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_2 > q_2 .
\end{array} \right.
&
\widetilde y_{t,q_1,q_2,q_3}(y) &:= \mathbbm{1}\left\{ y_r > q_r \right\} ,
\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for $t \in \{1,3\}$}.
\end{align*}
The moment functions presented below have the property that
\begin{align}
m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}) = m_{y_0,q_1,q_2,q_3}(\widetilde y_{q_1,q_2,q_3}(y),x,{\Greekmath 0112}) .
\label{BinarizationTrinarization}
\end{align}
Thus, for given $q_1$, $q_2$, $q_3$, it would be sufficient to observe the outcome
$\widetilde Y = \widetilde y_{q_1,q_2,q_3}(Y)$ to implement this moment function.
Notice that $ \widetilde y_{t,q_1,q_2,q_3}(y)$, for time periods
$t=1$ and $t=3$ is just a binarization, as in
\cite{Muris2017} and \cite{muris2020dynamic}, but for
$t=2$ we crucially deviate from those existing papers, because
for $q_2 \in \{2,\ldots,Q-1\}$ the coarser outcome
$\widetilde y_{2,q_1,q_2,q_3}(y)$ is a trinarization of the second
period outcome, not a binarization. It turns out that this is a crucial
extension to obtain all the valid moment conditions in our model.
The moment functions presented below were obtained
using Mathematica by following
the methods described in Section~2 of
\cite{honore2020dynamic}, which builds on
ideas in \cite{bonhomme2012functional}.
Once derived, we can prove by hand that the moment functions are valid (proof of Theorem~\ref{th:momentsT3} in the appendix),
but we do not have
any useful explanation or intuition for the detailed functional
form of these moment functions.
However,
the binarized/trinarized outcome $\widetilde Y$
and equation \eqref{BinarizationTrinarization}
help to appreciate some aspects of the structure
of the moment functions (see also Lemma~\ref{lemma:MASTER} in the appendix).
Furthermore, while the functional form of the
moment functions is mysterious, one can show
the existence of valid moment functions much more easily,
see Appendix~\ref{app:LowerBound}.
For $q_1,q_3 \in \{1,\ldots,Q-1\}$ and $q_2 \in \{2,\ldots,Q-1\}$, we define
\begin{align}
&m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112})
\nonumber \\[10pt]
&:=
\left\{ \begin{array}{ll}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ \exp\left( x_{13}^{\prime }\,{\Greekmath 010C} + {\Greekmath 010D}_{y_0,q_2} + {\Greekmath 0115}_{q_3,q_1} \right)
\frac {\displaystyle \exp(x_{32}' \, {\Greekmath 010C} + {\Greekmath 010D}_{q_2,y_1} + {\Greekmath 0115}_{q_2,q_3}) - 1 }
{\displaystyle \exp\left( {\Greekmath 0115}_{q_2,q_2-1} \right) - 1 }
$}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if $ y_1 \leq q_1, \, y_2=q_2 , y_3 \leq q_3$},
\\[6pt]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ \exp\left( x_{13}^{\prime }\,{\Greekmath 010C} + {\Greekmath 010D}_{y_0,q_2} + {\Greekmath 0115}_{q_3,q_1} \right)
\frac {\displaystyle 1 - \exp(x_{23}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1,q_2} + {\Greekmath 0115}_{q_3,q_2}) }
{\displaystyle 1 - \exp\left( {\Greekmath 0115}_{q_2-1,q_2} \right) }
$}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2=q_2 , \, y_3 > q_3,
\\[6pt]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ \exp\left( x_{13}^{\prime }\,{\Greekmath 010C} + {\Greekmath 010D}_{y_0,q_2} + {\Greekmath 0115}_{q_3,q_1} \right)
$}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2>q_2 ,
\\[6pt]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ -1 $}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2<q_2 ,
\\[6pt]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ - \, \frac{\displaystyle 1 - \exp(x_{32}' \, {\Greekmath 010C} + {\Greekmath 010D}_{q_2,y_1} + {\Greekmath 0115}_{q_2-1,q_3}) }
{\displaystyle 1 - \exp\left( {\Greekmath 0115}_{q_2-1,q_2} \right) }
$}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=q_2 , \, y_3 \leq q_3,
\\[6pt]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ - \, \frac{ \displaystyle \exp( x_{23}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1,q_2} + {\Greekmath 0115}_{q_3,q_2-1}) - 1 }
{\displaystyle \exp\left( {\Greekmath 0115}_{q_2,q_2-1} \right) - 1 }
$}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=q_2 , \,y_3 > q_3,
\\[6pt]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\footnotesize $ 0 $}
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{MomentFunctionsT3}
\end{align}
Any valid moment function satisfying \eqref{MomentFunctionIdea}
can be multiplied by an arbitrary constant and remain a valid
moment function. In \eqref{MomentFunctionsT3}
we used that rescaling freedom to
normalize the entry for the case
$(y_1 > q_1, y_2<q_2)$ to be equal to $-1$.
If, alternatively, we normalize the entry for $(y_1 \leq q_1, y_2>q_2)$ to be equal to $-1$, then we obtain the equally valid moment function
\begin{align*}
\widetilde m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}) &= - \frac{ m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112})} { \exp\left( x_{13}^{\prime }\,{\Greekmath 010C} + {\Greekmath 010D}_{y_0,q_2} + {\Greekmath 0115}_{q_3,q_1} \right) } .
\end{align*}
This rescaling is interesting, because
if we reverse the order of the outcome labels (i.e.\ $Y_t \mapsto Q+1-Y_{t}$), the model remains unchanged except for the parameter
transformations ${\Greekmath 010C} \mapsto - {\Greekmath 010C}$, ${\Greekmath 010D}_q \mapsto - {\Greekmath 010D}_{Q+1-q}$, and ${\Greekmath 0115}_q \mapsto - {\Greekmath 0115}_{Q-q}$.
Under this transformation, the moment function $m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}) $
becomes $ \widetilde m_{\widetilde y_0,\widetilde q_1,\widetilde q_2,\widetilde q_3}(y,x,{\Greekmath 0112}) $
with $\widetilde y_0 = Q+1- y_0$ and
$(\widetilde q_1,\widetilde q_2,\widetilde q_3) = (Q-q_1, Q+1-q_2, Q-q_3)$. This transformation therefore does not deliver any new moment functions,
which are not already (up to rescaling) given by~\eqref{MomentFunctionsT3}.
Equation \eqref{MomentFunctionsT3} does not define $m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112})$ for $q_2=1$ and $q_2 = Q$.
If we plug those values of $q_2$ into \eqref{MomentFunctionsT3}, then various undefined terms appear
since ${\Greekmath 0115}_{0} = -\infty$ and ${\Greekmath 0115}_{Q} = \infty$. However,
if for $q_2=1$ we properly evaluate the limit of
$\widetilde m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112})$ as
${\Greekmath 0115}_{0} \rightarrow -\infty$,
then we obtain
\begin{align}
m_{y_0,q_1,1,q_3}(y,x,{\Greekmath 0112}) &:=
\left\{ \begin{array}{ll}
\exp\left( x_{23}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1,1} + {\Greekmath 0115}_{q_3,1} \right) - 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2=1 , \, y_3>q_3, \\
- 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2>1,\\
\exp\left( x_{31}' \, {\Greekmath 010C} + {\Greekmath 010D}_{1,y_0} + {\Greekmath 0115}_{q_1,q_3} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=1 , \,y_3\leq q_3,\\
\exp\left( x_{21}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1,y_0} + {\Greekmath 0115}_{q_1,1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=1 , \, y_3>q_3,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{MomentFunctionsT3b}
\end{align}
Similarly, if for $q_2 = Q$ we properly evaluate the limit of $m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112})$ as ${\Greekmath 0115}_{Q} \rightarrow \infty$,
then we obtain
\begin{align}
m_{y_0,q_1,Q,q_3}(y,x,{\Greekmath 0112}) &:=
\left\{ \begin{array}{ll}
\exp\left( x_{12}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_0,y_1} + {\Greekmath 0115}_{Q-1,q_1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2=Q , \, y_3\leq q_3, \\
\exp\left( x_{13}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_0,Q} + {\Greekmath 0115}_{q_3,q_1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2=Q , \, y_3>q_3,\\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2<Q,\\
\exp\left( x_{32}' \, {\Greekmath 010C} + {\Greekmath 010D}_{Q,y_1} + {\Greekmath 0115}_{Q-1,q_3} \right) -1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=Q , \, y_3\leq q_3,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{MomentFunctionsT3c}
\end{align}
Notice that the moment functions for
$q_2=1$ and $q_2=Q$ also satisfy \eqref{BinarizationTrinarization}, but
here $\widetilde y_{q_1,q_2,q_3}(y)$ corresponds to a binarization of
the outcome in all time periods, because $\widetilde y_{t,q_1,q_2,q_3}(y)$
also only takes two values for those values of $q_2$. Those moment
functions are therefore conceptually much closer
to \cite{Muris2017} and \cite{muris2020dynamic}, and they also
incorporate the moment functions for the dynamic binary choice model
in \cite{honore2020dynamic} as special cases.
Together, the formulas \eqref{MomentFunctionsT3}, \eqref{MomentFunctionsT3b}, and \eqref{MomentFunctionsT3c}
provide one moment function for every value of $(y_0,q_1,q_2,q_3) \in \{1,\ldots,Q\} \times \{1,\ldots,Q-1\} \times \{1,\ldots,Q\} \times \{1,\ldots,Q-1\} $, and these constitute all our moment
functions for the dynamic ordered logit model with $T=3$.
\footnote{By the limiting arguments (${\Greekmath 0115}_{0} \rightarrow -\infty$ and ${\Greekmath 0115}_{Q} \rightarrow \infty$) described
above, all of those moment functions are already implicitly defined
via \eqref{MomentFunctionsT3} alone.}
The following theorem states that these are indeed valid moment functions for the dynamic panel ordered logit model, independent of the value
of the fixed effect $A$.
\begin{theorem}
\label{th:momentsT3}
If the outcomes $Y=(Y_1,Y_2,Y_3)$ are generated
from model \eqref{model} with $Q \geq 2$, $T=3$ and true parameters ${\Greekmath 0112}^0 = ({\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}^0)$, then we have for all
$y_0 \in \{1,\ldots,Q\}$, $q_1,q_3 \in \{1,\ldots,Q-1\}$, $q_2 \in \{1,\ldots,Q\}$, $x \in \mathbb{R}^{K
\times 3}$, and ${\Greekmath 010B} \in \mathbb{R} \cup \{\pm \infty\}$ that
\begin{align*}
\mathbb{E} \left[ m_{y_0,q_1,q_2,q_3}(Y,X,{\Greekmath 0112}^0) \, \big| \, Y_0=y_0,
\, X=x, \, A={\Greekmath 010B} \right] &= 0 .
\end{align*}
\end{theorem}
The proof of the theorem is given in the appendix.
For any fixed value of $Q$ one could, in principle, show by direct calculation that
\begin{align*}
\sum_{y \in \{ 1,2,\ldots,Q\}^3} \, p_{y_0}(y,x,{\Greekmath 0112}^0,{\Greekmath 010B}) \;
m_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}^0) = 0
\end{align*}
for the model probabilities $p_{y_0}(y,x,{\Greekmath 0112}^0,{\Greekmath 010B})$ given by \eqref{DefProb},
but our proof in the appendix does not rely on such a brute force calculation and is valid for any $Q \geq 2$.
For each initial condition $y_0$ we thus have $\ell = Q (Q-1)^2$ available moment conditions.
For example, for $Q=2,3,4,5$ there are respectively $\ell=2$, $12$, $36$, $80$ available moment conditions for each initial condition.
For those values of $Q$ we have verified numerically that our $\ell$ moment conditions are linearly independent, and that they constitute all the valid moment conditions that are available for the dynamic panel ordered logit model with $T=3$, for generic values of ${\Greekmath 010D}$.\footnote{If some of the parameters ${\Greekmath 010D}_q$ are equal to each other, then additional moment conditions become available.}
We believe that this is true for all $Q \geq 2$, but a proof of this completeness result is beyond the scope of this paper.
For the special case of dynamic binary choice ($Q=2$), the moment conditions here are identical to those in \cite{honore2020dynamic}
and \cite{kitazawa2021transformations},
and the completeness of those binary choice moment conditions is discussed in \cite{kruiniger2020further}
and \cite{dobronyi2021identification}.
\subsection{Moment conditions for $T>3$}
\label{sec:MomentsTlarger3}
We now consider the case where the econometrician has data for more than three time periods (in addition to the period that gives the initial condition).
Obviously, all the moment conditions above for $T=3$ are still
valid when applied to three consecutive periods, but additional moment conditions become available
for $T>3$. We first consider moment conditions that are based on the outcome in three periods, where the last two are consecutive.
Let $z_t := z(y_{t-1},x_{t},{\Greekmath 0112})$, with $z(\cdot,\cdot,\cdot)$
defined in \eqref{DefSingleIndex},
and define $z_{ts} := z_t - z_s$.
For $y_0 \in \{1,\ldots,Q\}$, $q_1,q_3 \in \{1,\ldots,Q-1\}$, $q_2 \in \{2,\ldots,Q-1\}$,
and $t,s \in \{1,2,\ldots,T-1\}$ with $t<s$ we define
\begin{align}
m^{(t,s,s+1)}_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}) &:=
\left\{ \begin{array}{ll}
\exp\left( z_{t,s+1} + {\Greekmath 0115}_{q_3,q_1} \right) \frac {\displaystyle \exp(z_{s+1,s} + {\Greekmath 0115}_{q_2,q_3}) - 1 }
{\displaystyle \exp\left( {\Greekmath 0115}_{q_2,q_2-1} \right) - 1 }
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s=q_2 , y_{s+1} \leq q_3,
\\[10pt]
\exp\left( z_{t,s+1} + {\Greekmath 0115}_{q_3,q_1} \right) \frac {\displaystyle 1 - \exp(z_{s,s+1} + {\Greekmath 0115}_{q_3,q_2}) }
{\displaystyle 1 - \exp\left( {\Greekmath 0115}_{q_2-1,q_2} \right) }
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s=q_2 , \, y_{s+1} > q_3,
\\[10pt]
\exp\left( z_{t,s+1} + {\Greekmath 010D}_{y_s,q_2} + {\Greekmath 0115}_{q_3,q_1} \right)
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s>q_2 ,
\\[10pt]
-1
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s<q_2 ,
\\[10pt]
- \, \frac{\displaystyle 1 - \exp(z_{s+1,s} + {\Greekmath 0115}_{q_2-1,q_3}) }
{\displaystyle 1 - \exp\left( {\Greekmath 0115}_{q_2-1,q_2} \right) }
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s=q_2 , \, y_{s+1} \leq q_3,
\\[10pt]
- \, \frac{ \displaystyle \exp( z_{s,s+1} + {\Greekmath 0115}_{q_3,q_2-1}) - 1 }
{\displaystyle \exp\left( {\Greekmath 0115}_{q_2,q_2-1} \right) - 1 }
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s=q_2 , \,y_{s+1} > q_3,
\\[10pt] 0
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{MomentFunctionsTgeneral}
\end{align}
For $T=3$, $t=1$, and $s=2$, it is straightforward to verify that $ m^{(t,s,s+1)}_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}) $ in equation \eqref{MomentFunctionsTgeneral}
equals
the moment function in equation \eqref{MomentFunctionsT3}.
For larger values of $T$,
the moment function in \eqref{MomentFunctionsTgeneral}
can be implemented as long as outcomes
are observed for the time periods $\{t-1,t,s-1,s,s+1\}$
and covariates are observed for time periods
$\{t,s,s+1\}$.
Since ${\Greekmath 0115}_0= -\infty$ and ${\Greekmath 0115}_Q=\infty$, equation \eqref{MomentFunctionsTgeneral} can not be used to define a moment function when $q_2$ equals 1 or $Q$. We next define moment functions for these cases.
For $y_0 \in \{1,\ldots,Q\}$, $q_1,q_3 \in \{1,\ldots,Q-1\}$, $t,s,r\in \{1,2,\ldots ,T\}$, and $t<s<r$, we
define
\begin{align}
m^{(t,s,r)}_{y_0,q_1,1,q_3}(y,x,{\Greekmath 0112}) &:=
\left\{ \begin{array}{ll}
\exp\left( z_{sr} + {\Greekmath 0115}_{q_3,1} \right) - 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s=1 , \, y_r>q_3, \\
- 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s>1,\\
\exp\left( z_{rt} + {\Greekmath 0115}_{q_1,q_3} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s=1 , \,y_r\leq q_3,\\
\exp\left( z_{st} + {\Greekmath 0115}_{q_1,1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s=1 , \, y_r>q_3,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise},
\end{array} \right.
\nonumber \\[10pt]
m^{(t,s,r)}_{y_0,q_1,Q,q_3}(y,x,{\Greekmath 0112}) &:=
\left\{ \begin{array}{ll}
\exp\left( z_{ts} + {\Greekmath 0115}_{Q-1,q_1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s=Q , \, y_r\leq q_3, \\
\exp\left( z_{tr} + {\Greekmath 0115}_{q_3,q_1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t \leq q_1, \, y_s=Q , \, y_r>q_3,\\
-1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s<Q,\\
\exp\left( z_{rs} + {\Greekmath 0115}_{Q-1,q_3} \right) -1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_t > q_1, \, y_s=Q , \, y_r\leq q_3,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{MomentFunctionsTgeneralBC}
\end{align}
When $T$ equals 3 and $(t,s,r)=(1,2,3)$, these moment functions agree with the ones in equations \eqref{MomentFunctionsT3b} and \eqref{MomentFunctionsT3c}, where all the arguments were made explicit.
For $r=s+1$,
analogous to \eqref{MomentFunctionsT3b} and \eqref{MomentFunctionsT3c} for $T=3$, the two moment conditions in \eqref{MomentFunctionsTgeneralBC} for $T \geq 3$ can be obtained from
\eqref{MomentFunctionsTgeneral} by setting $q_2=1$ and carefully evaluating the limit ${\Greekmath 0115}_{0} \rightarrow -\infty$
(after normalizing the value for $y_t \leq q_1$, $y_s>1$ to be $-1$),
or setting $q_2=Q$ and taking the limit ${\Greekmath 0115}_{Q} \rightarrow \infty$. It is therefore
appropriate to think of \eqref{MomentFunctionsTgeneral} as our master equation, which summarizes all
the moment conditions provided in this paper.
In \eqref{MomentFunctionsTgeneralBC} we can choose more general $r\geq s+1$,
but otherwise the structure of \eqref{MomentFunctionsTgeneralBC} can be derived from \eqref{MomentFunctionsTgeneral}.
It turns out that the moment functions with $r > s+1$ are not actually needed to span all possible valid moment functions of the dynamic ordered choice logit model (see our discussion of independence and completeness below). However, since implementation of these moment functions requires only that we observe three pairs $(y_{t-1},y_t)$, $(y_{s-1},y_s)$, $(y_{r-1},y_r)$ of consecutive outcomes, they may be empirically relevant for the case where observations for some time periods are (exogenously) missing.
\footnote{For example, an estimator that allows for selection to be correlated with $(Y_{i0},X_i,A_i)$ can be constructed using the results on GMM estimation with incomplete data in \cite{muris_incomplete}.}
We also include $r > s+1$ in our discussion
here to ensure that our results in this paper
contain those for the dynamic binary choice logit model studied in \cite{honore2020dynamic} as a special case
--- notice that for $Q=2$ we always have $q_2=1$
or $q_2=Q$, that is, for the binary choice case all available moment functions are stated in \eqref{MomentFunctionsTgeneralBC}.
The following theorem establishes that the moment functions in \eqref{MomentFunctionsTgeneral} and \eqref{MomentFunctionsTgeneralBC} do indeed deliver valid moment conditions.
\begin{theorem}
\label{th:momentsTgeneralA}
If the outcomes $Y=(Y_1,\ldots,Y_T)$ are generated
from model \eqref{model} with $Q \geq 2$, $T\geq 3$ and true parameters ${\Greekmath 0112}^0 = ({\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}^0)$, then we have for all
$t,s,r \in \{1,2,\ldots,T\}$ with $t<s<r$,
$y_0 \in \{1,\ldots,Q\}$,
$q_1,q_3 \in \{1,\ldots,Q-1\}$, $x\in \mathbb{R}^{K\times T}$, ${\Greekmath 010B}
\in \mathbb{R} \cup \{\pm \infty\}$, and $w:\{1,\ldots,Q\}^{t-1}\rightarrow \mathbb{R}$ that
\begin{align*}
\mathbb{E}\left[ w(Y_{1},\ldots ,Y_{t-1})\,m^{(t,s,s+1)}_{y_0,q_1,q_2,q_3}(Y,X,{\Greekmath 0112}^0)\,\big|\,Y_{0}=y_{0},\,X=x,\,A={\Greekmath 010B} \right] & =0,
\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for $q_2 \in \{2,\ldots,Q-1\}$,}
\\
\mathbb{E}\left[ w(Y_{1},\ldots ,Y_{t-1})\,m^{(t,s,r)}_{y_0,q_1,q_2,q_3}(Y,X,{\Greekmath 0112})\,\big|\,Y_{0}=y_{0},\,X=x,\,A={\Greekmath 010B} \right] & =0,
\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for $q_2 \in \{1,Q\}$.}
\end{align*}
\end{theorem}
The proof is provided in the appendix. Notice that for $q_2 \in \{1,Q\}$ we can choose the time indices $t<s<r$ freely.
By contrast, for $q_2 \in \{2,\ldots,Q-1\}$ we can only choose $t<s$ freely, but the third time index that appears in the definition
of the moment function needs to be equal to $s+1$, otherwise we do not obtain a valid moment function for those values of $q_2$.
This distinction between $q_2 \in \{1,Q\}$ and $q_2 \in \{2,\ldots,Q-1\}$ is also reflected in the proof of Theorem~\ref{th:momentsTgeneralA}.
The moment functions in \eqref{MomentFunctionsTgeneralBC} for $q_2 \in \{1,Q\}$
only depend on $Y_1$, $Y_2$, $Y_3$ through the binarized variables
$\widetilde Y_1 = \mathbbm{1}\left\{ Y_1 > q_1 \right\}$,
$\widetilde Y_2 = \mathbbm{1}\left\{ Y_2 = q_2 \right\}$,
$\widetilde Y_3 = \mathbbm{1}\left\{ Y_3 > q_3 \right\}$,
and the proof relies on Lemma~\ref{lemma:MASTER2} in the appendix, which
provides a general set of valid moment functions for such binary variables,
very closely related to the dynamic binary choice results in
\cite{honore2020dynamic}.
By contrast, the moment functions in \eqref{MomentFunctionsTgeneral}
for $q_2 \in \{2,\ldots,Q-1\}$ cannot be expressed through binarized variables only,
because there the dependence on $Y_2$ requires distinguishing three cases
($Y_s < q_2 $, $Y_s = q_2$, $Y_s > q_2$). The proof, in this case, relies on
Lemma~\ref{lemma:MASTER} in the appendix which is completely novel to the
current paper. However, that proof strategy for $q_2 \in \{2,\ldots,Q-1\}$
does not work for $s>r+1$, and we have also numerically verified that our
moment conditions for $q_2 \in \{2,\ldots,Q-1\}$ indeed do not
generalize to $s>r+1$.
\subsubsection*{Conjecture on the completeness of the moment conditions}
Theorem~\ref{th:momentsTgeneralA} states that the moment functions
in \eqref{MomentFunctionsTgeneral} and \eqref{MomentFunctionsTgeneralBC}
are valid, but it is natural to ask whether they are also linearly
independent, and whether they constitute all possible valid moment
functions of the dynamic panel ordered logit model.
We do not aim to formally prove such a linear independence and completeness result in this paper, and the
following statement should accordingly be understood as a conjecture, which we have numerically confirmed for various combinations of $Q$ and $T$ and for
many different numerical values of the regressors and model parameters:
Let the outcomes $Y=(Y_1,\ldots,Y_T)$ be generated
from model \eqref{model} with $Q \geq 2$, $T\geq 3$, and let the true parameters
${\Greekmath 0112}^0 = ({\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}^0)$ be
such that ${\Greekmath 010D}^0_{q_1} \neq {\Greekmath 010D}^0_{q_2}$ for
all $q_1 \neq q_2$. For given $y_0 \in \{1,\ldots,Q\}$ and
$x\in \mathbb{R}^{K\times T}$, let $m_{y_0}(y,x,{\Greekmath 0112}^0) \in \mathbb{R}$ be a
moment function
that satisfies \eqref{MomentFunctionIdea} for all ${\Greekmath 010B} \in \mathbb{R} \cup \{\pm \infty\}$. Our calculations suggest that there exist unique
weights $w_{y_{0}}(q_1,q_2,q_3,s,y_{1},\ldots ,y_{t-1},x,{\Greekmath 0112}^0) \in \mathbb{R}$ such that for all $y \in \{1,\ldots,Q\}^T$ we have
\begin{align}
m_{y_0}(y,x,{\Greekmath 0112}^0) =
\sum_{q_1=1}^{Q-1}
\sum_{q_2=1}^{Q}
\sum_{q_3=1}^{Q-1}
\sum_{t=1}^{T-2} \sum_{s=t+1}^{T-1}
w_{y_{0}}(q_1,q_2,q_3,t,s,y_{1},\ldots ,y_{t-1},x,{\Greekmath 0112}^0) \;
m^{(t,s,s+1)}_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}^0),
\label{completeness}
\end{align}
where $m^{(t,s,s+1)}_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}^0)$
are the moment functions defined in \eqref{MomentFunctionsTgeneral} and \eqref{MomentFunctionsTgeneralBC}. In other words,
we conjecture that
every valid
moment condition in this model is a unique linear combination of the moment conditions in
Theorem~\ref{th:momentsTgeneralA} with $r=s+1$. Notice that
the uniqueness of the linear combination implies that the moment
functions involved in this linear combination are linearly independent.
In equation \eqref{completeness}, the function
$m^{(t,s,s+1)}_{y_0,q_1,q_2,q_3}(y,x,{\Greekmath 0112}^0)$ is
multiplied with an arbitrary function of $y_1,\ldots,y_{t-1}$.
Those functions of $y_1,\ldots,y_{t-1}$ constitute a
$Q^{t-1}$ dimensional space. Thus, \eqref{completeness} suggests that the total number of available moment conditions for each value of the covariates $x$ and initial conditions $y_0$ is equal to
\begin{align}
\ell &= \sum_{q_1=1}^{Q-1}
\sum_{q_2=1}^{Q}
\sum_{q_3=1}^{Q-1}
\sum_{t=1}^{T-2} \sum_{s=t+1}^{T-1} Q^{t-1}
= (Q-1) \, Q\, (Q-1) \sum_{t=1}^{T-2} \, (T-t-1) \, Q^{t-1}
\nonumber \\
&= Q^T - (T-1) \, Q^2 + (T-2) \, Q .
\label{NumberMoment}
\end{align}
As explained in Section~\ref{subsec:momentApproach}, the function $m_{y_0}(\cdot,x,{\Greekmath 0112}^0) : \{1,\ldots,Q\}^T \rightarrow \mathbb{R}$ is a vector in a $Q^T$ dimensional space.
The condition \eqref{MomentFunctionIdea}, for all ${\Greekmath 010B}$,
then imposes $Q^T - \ell = (T-1) \, Q^2 + (T-2) \, Q$ linear restrictions on this vector, leaving an $\ell$-dimensional linear subspace
of valid moment functions, a basis representation
of which is given by
\eqref{completeness}. For fixed values
of $y_0$, $x$, ${\Greekmath 0112}_0$, $T$, $Q$, one can numerically verify
the dimension of the solution space of the system of linear equations \eqref{MomentFunctionIdea}, and thereby check
\eqref{NumberMoment} numerically. In Appendix~\ref{app:LowerBound} we furthermore show that
the total number of linearly independent
conditional moment conditions for our model
is at least
the number obtained in \eqref{app:LowerBound},
but that argument in the appendix still allows for the possibility
that there could be more, although we do not believe that there are.
The condition
${\Greekmath 010D}^0_{q_1} \neq {\Greekmath 010D}^0_{q_2}$ for
all $q_1 \neq q_2$ is important for this result. For example,
if all the ${\Greekmath 010D}^0_q$ are the same, then the parameter ${\Greekmath 010D}^0$ can be absorbed into
the fixed effects, and we are left with a static ordered logit model
as in \cite{Muris2017}, for which
one finds an additional $(T-1) (Q-1)^2$ moment conditions to be available, bringing the total
number of linearly independent valid moment conditions (for each value of covariates and parameters) in the static model
to
$\ell = Q^T -T (Q-1) - 1$.
We reiterate that the discussion of linear independence and completeness
of the moment functions presented above are conjectures which we
do not aim to prove in this paper. A proof for the special case $Q=2$ (dynamic binary choice logit models) is provided in \cite{kruiniger2020further}
and \cite{dobronyi2021identification}.
We also note that the counting of moment conditions as above does not consider whether the resulting moment conditions actually contain
information about (all) the parameters ${\Greekmath 0112}$. Some of the valid moment
functions may not depend on (all of) those model parameters. Identification of the model
parameters through the moment conditions is discussed in Section~\ref{sec:Identification}.
\subsection{More general regressors}
\label{sec:GeneralModel}
The model probabilities in \eqref{DefProb} and the moment functions in \eqref{MomentFunctionsTgeneral}
and \eqref{MomentFunctionsTgeneralBC}
only depend on the regressors and the parameters ${\Greekmath 010C}$ and ${\Greekmath 010D}$
through the single index $z_t=z(y_{t-1},x_{t},{\Greekmath 0112})$.\footnote{
As written, the moment condition in \eqref{MomentFunctionsTgeneral}
depends explicitly on the model parameter ${\Greekmath 010D}$ for
the case that $y_t \leq q_1$ and $y_s>q_2$. However, that is a notational
artefact, because in that line of the moment condition we could have
written
$ \exp\left[ z(y_{t-1},x_{t},{\Greekmath 0112}) - z(q_2,x_{s+1},{\Greekmath 0112}) + {\Greekmath 0115}_{q_3,q_1} \right]$
instead of
$ \exp\left( z_{t,s+1} + {\Greekmath 010D}_{y_s,q_2} + {\Greekmath 0115}_{q_3,q_1} \right)$; that is, the explicit
dependence on ${\Greekmath 010D}$ can be fully absorbed into the single index, but one needs to evaluate $z_{s+1}=z(y_s,x_{s+1},{\Greekmath 0112})$ at $q_2$ instead of $y_s$.
}
So far, we have only explicitly discussed the linear specification in \eqref{DefSingleIndex} for this single index,
but Theorem~\ref{th:momentsTgeneralA} is valid independently of the functional form
of $z(y_{t-1},x_{t},{\Greekmath 0112})$.\footnote{
The parameters ${\Greekmath 0115}$ can also be absorbed into the single index. One just needs to
define $\widetilde z_q(y_{t-1},x_{t},{\Greekmath 0112}) := z(y_{t-1},x_{t},{\Greekmath 0112}) - {\Greekmath 0115}_{q} $
and rewrite \eqref{DefProb} as
\begin{equation*}
p_{y_{0}}(y_{},x_{},{\Greekmath 0112} ,{\Greekmath 010B} _{}) =\prod_{t=1}^{T}
\left\{ \Lambda\Big[ \widetilde{z}_{y_{t}-1}(y_{t-1},x_{t},{\Greekmath 0112}) + {\Greekmath 010B} \Big] - \Lambda\Big[ \widetilde{z}_{y_{t}}(y_{t-1},x_{t},{\Greekmath 0112}) + {\Greekmath 010B} \Big]
\right\} .
\end{equation*}
The moment functions in \eqref{MomentFunctionsTgeneral}
and \eqref{MomentFunctionsTgeneralBC} then remain valid for arbitrary functional forms
of $\widetilde z_q(y_{t-1},x_{t},{\Greekmath 0112})$. We just need to replace $z_t - {\Greekmath 0115}_{q_1}$, $z_s - {\Greekmath 0115}_{q_2}$,
and $z_r - {\Greekmath 0115}_{q_3}$ (with $r=s+1$ in \eqref{MomentFunctionsTgeneral}) by
$\widetilde z_{q_1}(y_{t-1},x_{t},{\Greekmath 0112})$, $\widetilde z_{q_2}(y_{s-1},x_{s},{\Greekmath 0112})$,
and $\widetilde z_{q_3}(y_{r-1},x_{r},{\Greekmath 0112})$, respectively. The proof of Theorem~\ref{th:momentsTgeneralA} remains
valid under that replacement.
}
In other words, if we replace the latent variable specification
in \eqref{ModelYstar} by
\begin{align*}
Y^*_{it} = z \left( Y_{i,t-1},X_{it},{\Greekmath 0112} \right)
+ A_i + {\Greekmath 0122}_{it}
\end{align*}
for an arbitrary function $z(\cdot,\cdot,\cdot)$, then
the moment functions~\eqref{MomentFunctionsT3}, \eqref{MomentFunctionsT3b},
\eqref{MomentFunctionsT3c}, and Theorem~\ref{th:momentsTgeneralA} remain
fully valid.
We believe that the linear specification in \eqref{ModelYstar} is the most relevant in practice, but one could certainly consider other
specifications as well. In particular, it is possible to include regressors that are interactions between the
observed regressors
and the lagged dependent variable:
\begin{align}
z(Y_{i,t-1},X_{it},{\Greekmath 0112}) := X_{it}^{\prime }\,{\Greekmath 010C} + \sum_{q=1}^{Q} {\Greekmath 010D}_q
\left[ \mathbbm{1}\left\{ Y_{i,t-1} = q \right\}
+ \mathbbm{1}\left\{ Y_{i,t-1} = q \right\} X_{it}^{\prime }\,{\Greekmath 010E}_q \right],
\label{AlternativeSpecification}
\end{align}
where ${\Greekmath 010E}_q \in \mathbb{R}^K$ are the additional unknown parameters to be included in ${\Greekmath 0112}$. This specification allows
the effect of the regressors $X_{it}$ on the outcome $Y_{it}$ to be arbitrarily dependent on the current state $Y_{i,t-1}$.
While a GMM estimator based on moment functions developed in this paper could be employed in applications with the more general state dependence as in \eqref{AlternativeSpecification}, we do not consider these more general models further.
\section{Identification}
\label{sec:Identification}
This section presents identification results for the parameters ${\Greekmath 0112} = ({\Greekmath 010C},{\Greekmath 010D},{\Greekmath 0115})$ based on the moment conditions for $T=3$
in Theorem~\ref{th:momentsT3}.
All results in this section impose the following model assumption.
\newtheorem{IDassumption}{Assumption}
\begin{IDassumption}
\label{ass:ID}
The outcomes $Y=(Y_1,Y_2,Y_3)$ are generated
from model \eqref{model} with $z(\cdot,\cdot,\cdot)$
defined in \eqref{DefSingleIndex}, $Q \geq 2$,
$T=3$, and true parameters ${\Greekmath 0112}^0 = ({\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}^0)$.
Furthermore, for all $y_0 \in \{1,\ldots,Q\}$,
there exists a non-empty set ${\cal X}^{\rm reg}_{y_0} \subset \mathbb{R}^{K\times 3}$ such that
for all $x \in {\cal X}^{\rm reg}_{y_0}$, the conditional probability
$ {\rm Pr}(A \in \{ \pm \infty\} \mid Y_0=y_0, \, X=x )$ is well-defined and smaller than one.
\end{IDassumption}
We impose the assumption $ {\rm Pr}(A \in \{ \pm \infty\} \mid Y_0=y_0, \, X=x ) < 1$ for some $x$ in order to ensure that
the model probabilities in \eqref{DefProb} are strictly positive for all possible outcomes. If $ {\rm Pr}(A \in \{ \pm \infty\} \mid Y_0=y_0, \, X=x ) = 1$
for all $x$,
then only the outcomes $Y_t=1$ and $Y_t=Q$ would be generated by the model. A violation of this assumption on the fixed effects $A$ would
therefore be readily observable from the data.
All the propositions below also impose that $X \in {\cal X}^{\rm reg}_{y_0}$
occurs with non-zero probability.
The aim is to identify the parameter vector ${\Greekmath 0112}^0$ from the distribution of $Y$ conditional on $Y_0$ and $X$ under Assumption~\ref{ass:ID}.
The model for that conditional distribution is semi-parametric:
The distribution of $Y$ conditional on $Y_0$, $X$,
and $A$ is specified parametrically, but only weak regularity conditions are
imposed on the unknown distribution of $A$ conditional on $Y_0$ and $X$.
The main challenge in the identification problem is how
to deal with the unspecified conditional distribution of $A$, which is
an infinite-dimensional component of the parameter space of the model. Fortunately, the moment conditions in Theorem~\ref{th:momentsT3} already partly
solve this challenge, because they give us implications of the model
that do not depend on $A$. The remaining question is whether ${\Greekmath 0112}^0$ is point-identified from those moment conditions.
\subsubsection*{Identification of ${\Greekmath 010D}$}
In order to identify the parameters ${\Greekmath 010D}=({\Greekmath 010D}_1,\ldots,{\Greekmath 010D}_Q)$ up to normalization, we condition on the event $X_1=X_2=X_3$.
For $x=(x_1,x_1,x_1)$ and $q_1=q_2=q_3=1$, the moment function in \eqref{MomentFunctionsT3b} reads
\begin{align}
m_{y_0}(y,{\Greekmath 010D}) := \exp({\Greekmath 010D}_{y_0} ) \, m_{y_0,1,1,1}(y,x,{\Greekmath 0112}) &=
\left\{ \begin{array}{ll}
- \exp({\Greekmath 010D}_{y_0}) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 = 1, \, y_2>1,\\
\exp\left( {\Greekmath 010D}_{1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > 1, \, y_2=1 , \,y_3= 1,\\
\exp\left( {\Greekmath 010D}_{y_1} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > 1, \, y_2=1 , \, y_3>1,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{MomentFunctionID}
\end{align}
Theorem~\ref{th:momentsT3} implies that
$\mathbb{E} \left[ m_{y_0}(Y,{\Greekmath 010D}^0) \, \big| \, Y_0=y_0, \, X=(x_1,x_1,x_1) \right] = 0$.
The following lemma states that these moment conditions are sufficient to uniquely identify ${\Greekmath 010D}$ up to a normalization.
\begin{proposition}
\label{prop:IDgamma}
Let Assumption~\ref{ass:ID} hold, and let $x_1 \in \mathbb{R}$ be such that
$${\rm Pr}\left(
Y_0=y_0
\; \& \;
X \in {\cal X}^{\rm reg}_{y_0}
\; \& \;
\left\| X- (x_1,x_1,x_1) \right\| \leq {\Greekmath 010F} \right) >0
\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all $y_0 \in \{1,\ldots,Q\}$ and ${\Greekmath 010F}>0$.}
$$
Then, if ${\Greekmath 010D} \in \mathbb{R}^Q$ satisfies\footnote{
Here, we implicitly assume that
$ \mathbb{E} \left[ m_{y_0}(Y,{\Greekmath 010D}) \, \big| \, Y_0=y_0, \, X=(x_1,x_1,x_1) \right]$ is uniquely
defined.
This can be guaranteed, for example,
by demanding that this conditional expectation is
continuous in $x_1$.
}
\begin{align}
\mathbb{E} \left[ m_{y_0}(Y,{\Greekmath 010D}) \, \big| \, Y_0=y_0, \, X=(x_1,x_1,x_1) \right] = 0
\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all $y_0 \in \{1,\ldots,Q\}$},
\label{MomentConditionIDprop1}
\end{align}
for $m_{y_0}(y,{\Greekmath 010D})$ as defined in \eqref{MomentFunctionID},
we have ${\Greekmath 010D} = {\Greekmath 010D}^0 + c$ for some $c \in \mathbb{R}$.
Thus,
if we normalize ${\Greekmath 010D}^0_1=0$, then $ {\Greekmath 010D}^0$ is uniquely identified from the data.
\end{proposition}
The proof is given in the appendix. This identification result requires observed data for every initial condition $y_0 \in \{1,\ldots,Q\}$.
If this is not available, but we observe $T=4$ time periods of data after the initial condition, then we can instead apply
Proposition~\ref{prop:IDgamma} to the data shifted by one time period.
In addition to Assumption~\ref{ass:ID}, the proposition
demands that covariate values $X \in {\cal X}^{\rm reg}_{y_0}$
in any ${\Greekmath 010F}$-ball around $(x_1,x_1,x_1)$ occur with positive probability. This condition, in particular, guarantees that
the conditional expectation in \eqref{MomentConditionIDprop1}
is well-defined, and that conditional on
$X=(x_1,x_1,x_1)$ the event $A \in \{ \pm \infty\}$ occurs
with probability less than one for every value of the initial condition
$Y_0$.
\subsubsection*{Identification of ${\Greekmath 010C}$}
Taking the identification result for ${\Greekmath 010D}$ as given, we now turn to the problem of identifying ${\Greekmath 010C}$.
We again
consider the moment function in \eqref{MomentFunctionsT3b}
with $q_1=q_2=q_3=1$, but now for general regressor values
\begin{align}
m_{y_0,1,1,1}(y,x,{\Greekmath 010C},{\Greekmath 010D})
:=
m_{y_0,1,1,1}(y,x,{\Greekmath 0112}) & =
\left\{ \begin{array}{ll}
\exp\left( x_{23}' \, {\Greekmath 010C} \right) - 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 = 1, \, y_2=1 , \, y_3>1, \\
- 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 = 1, \, y_2>1,\\
\exp\left( x_{31}' \, {\Greekmath 010C} + {\Greekmath 010D}_1 - {\Greekmath 010D}_{y_0} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > 1, \, y_2=1 , \,y_3 = 1,\\
\exp\left( x_{21}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1} - {\Greekmath 010D}_{y_0} \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > 1, \, y_2=1 , \, y_3>1,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{momIdentify}
\end{align}
For $k\in \{1,\ldots ,K\}$ we define
\begin{align*}
\mathcal{X}_{k,+}& :=\{x\in {\cal X}^{\rm reg}_{y_0} \,:\,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 {\cal X}^{\rm reg}_{y_0}\,:\,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*}
Here, the set $\mathcal{X}_{k,+}$ is the
set of possible regressor values $x\in \mathbb{R}^{K\times 3}$ such that
$x_{k,1}\leq x_{k,3}\leq x_{k,2}$ with at least one of the inequalities being strict.
For the set $\mathcal{X}_{k,-}$ those inequalities are reversed.
Therefore, if $x \in \mathcal{X}_{k,+}$, then $m_{y_0,1,1,1}(y,x,{\Greekmath 010C},{\Greekmath 010D})$ is strictly
increasing in ${\Greekmath 010C}_k$,
and if $x \in \mathcal{X}_{k,-}$, then $m_{y_0,1,1,1}(y,x,{\Greekmath 010C},{\Greekmath 010D})$ is strictly
decreasing in ${\Greekmath 010C}_k$.
For any vector $s\in \{-,+\}^{K}$, we furthermore define the set $\mathcal{X}_{s}=\bigcap_{k\in
\{1,\ldots ,K\}}\mathcal{X}_{k,s_{k}}$. If $x \in \mathcal{X}_{s}$, then for all $k\in
\{1,\ldots ,K\}$ we have that ${\Greekmath 010C}_k$ is strictly increasing (or strictly decreasing) in $m_{y_0,1,1,1}(y,x,{\Greekmath 010C},{\Greekmath 010D})$
if $s_k=+$ (or $s_k=-$). These monotonicity properties allow us to uniquely identify ${\Greekmath 010C}$
from the moment conditions $\mathbb{E}\left[ m_{y_0,1,1,1}(Y,X,{\Greekmath 010C}^0,{\Greekmath 010D}^0)\,\Big|\,Y_{0}=y_{0},\;X\in \mathcal{X}_{s}\right] = 0$,
which are valid moment conditions according to Theorem~\ref{th:momentsT3}. The following proposition formalizes this.
\begin{proposition}
\label{prop:IDbeta}
Let Assumption~\ref{ass:ID} hold and let $y_0 \in \{1,\ldots,Q\}$ be such that
$${\rm Pr}\left(
Y_0=y_0
\; \& \;
X\in \mathcal{X}_{s} \right) >0
\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all $s \in \{-,+\}^{K}$ with $s_K=+$.}
$$
Then, if ${\Greekmath 010C} \in \mathbb{R}^K$ satisfies
\begin{align}
\mathbb{E}\left[ m_{y_0,1,1,1}(Y,X,{\Greekmath 010C},{\Greekmath 010D}^0)\,\Big|\,Y_{0}=y_{0},\;X\in \mathcal{X}_{s}\right] = 0
\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all $s \in \{-,+\}^{K}$ with $s_K=+$,}
\label{Condition:prop:IDbeta}
\end{align}
we have ${\Greekmath 010C}= {\Greekmath 010C}^0$.
Thus, since ${\Greekmath 010D}^0$ is already identified from Proposition~\ref{prop:IDgamma},
we find that ${\Greekmath 010C}^0$ is also uniquely identified from the data.
\end{proposition}
The proof is given in the appendix.
Again, in addition
to Assumption~\ref{ass:ID}, the additional
condition in Proposition~\ref{prop:IDbeta}
simply guarantees that
the conditional expectation
in \eqref{Condition:prop:IDbeta} is well-defined.
\subsubsection*{Identification of ${\Greekmath 0115}$}
Having identified ${\Greekmath 010D}$ and ${\Greekmath 010C}$, we now turn to the problem of identifying ${\Greekmath 0115}$ up to a normalization.
The moment function in \eqref{MomentFunctionsT3b}
with $q_2=q_3=1$ and $q_1 \in \{2,\ldots,Q-1\}$ can be written as
\begin{align}
m_{y_0,q_1,1,1}(y,x,{\Greekmath 010C},{\Greekmath 010D},{\Greekmath 0115}) =
\left\{ \begin{array}{ll}
\exp\left( x_{23}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1,1} \right) - 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2=1 , \, y_3>1, \\
- 1 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 \leq q_1, \, y_2>1,\\
\exp\left( x_{31}' \, {\Greekmath 010C} + {\Greekmath 010D}_{1,y_0} + {\Greekmath 0115}_{q_1} - {\Greekmath 0115}_1 \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=1 , \,y_3 =1,\\
\exp\left( x_{21}' \, {\Greekmath 010C} + {\Greekmath 010D}_{y_1,y_0} + {\Greekmath 0115}_{q_1} - {\Greekmath 0115}_1 \right) & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if } y_1 > q_1, \, y_2=1 , \, y_3>1,\\
0 & \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{otherwise}.
\end{array} \right.
\label{Moments:prop:IDlambda}
\end{align}
The expected value of this moment function
only depends on ${\Greekmath 0115}$ through ${\Greekmath 0115}_{q_1} - {\Greekmath 0115}_1$,
and is strictly increasing in ${\Greekmath 0115}_{q_1} - {\Greekmath 0115}_1$.
This implies that this moment function identifies ${\Greekmath 0115}_{q_1} - {\Greekmath 0115}_1$ uniquely.
By applying this argument to all $q_1 \in \{2,\ldots,Q-1\}$,
we can therefore identify ${\Greekmath 0115}$ up to an additive constant.
This is summarized in the following proposition.
\begin{proposition}
\label{prop:IDlambda}
Let Assumption~\ref{ass:ID} hold. Let $y_0 \in \{1,\ldots,Q\}$ be such that
${\rm Pr}\big(
Y_0=y_0 \; \& \;
X\in {\cal X}^{\rm reg}_{y_0} \big) > 0$.
Then, if ${\Greekmath 0115}$ satisfies
$$
\mathbb{E}\left[ m_{y_0,q_1,1,1}(Y,X,{\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}) \,\Big|\,Y_{0}=y_{0} \right] = 0
\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for all $q_1 \in \{2,\ldots,Q-1\}$,}
$$
we have ${\Greekmath 0115} = {\Greekmath 0115}^0 + c$ for some $c \in \mathbb{R}$.
Thus, if we normalize ${\Greekmath 0115}^0_1=0$,
and
since ${\Greekmath 010D}^0$ and ${\Greekmath 010C}^0$ are already identified from Proposition~\ref{prop:IDgamma} and \ref{prop:IDbeta},
we find that ${\Greekmath 0115}^0$ is also uniquely identified from the data.
\end{proposition}
The proof is given in the appendix.
Combining Proposition~\ref{prop:IDgamma}, \ref{prop:IDbeta},
and \ref{prop:IDlambda}, we find that ${\Greekmath 0112}^0$ is uniquely identified
from the data. Under the regularity conditions
of those propositions, we can recover ${\Greekmath 0112}^0 = ({\Greekmath 010C}^0,{\Greekmath 010D}^0,{\Greekmath 0115}^0)$ uniquely from the distribution of $Y$ conditional on $Y_0$ and $X$.
Our identification arguments in this section are constructive.
However, they condition on special values of the regressors. In particular,
Proposition~\ref{prop:IDgamma} conditions on the event $X_1=X_2=X_3$,
which is a zero-probability event if $X$ is continuously distributed
(and may happen rarely even for discrete $X$). An estimator based on
the identification strategy in this section would therefore in general
be quite inefficient. Hence, in our Monte Carlo simulations and empirical
application, we construct more general GMM estimators based
on our moment conditions.
\section{Implication for estimation and specification testing\label{Section:
Implication for estimation}}
The moment conditions in Section\ \ref{sec:model} are conditional on the
initial condition $Y_{i0}$ and the strictly exogenous explanatory variables $
X_{i}$. It is tempting to try to mimic the identification argument in
Section \ref{sec:Identification} in order to turn these moment conditions
into an estimator. The problem with such an approach is that the
conditioning set in Proposition \ref{prop:IDgamma} will often have
probability 0. Alternatively, one can form a set of unconditional moment
functions by constructing
\begin{equation*}
M(Y_{i0},Y_{i},X_{i},{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 0115} )=g\left( Y_{i0},X_{i}\right)
\otimes m_{Y_{i0}}\left( Y_{i},X_{i},{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 0115} \right)
\end{equation*}
where the vector-valued function, $m_{Y_{i0}}$, is composed of linear
combinations of the moment functions in \eqref{MomentFunctionsT3},
\eqref{MomentFunctionsT3b}, and \eqref{MomentFunctionsT3c}, and $g$ is a
vector-valued function of the initial condition $Y_{i0}$ and the strictly
exogenous $X_{i}$. Let ${\Greekmath 0112} =\left( {\Greekmath 010C} ^{\prime },{\Greekmath 010D} ^{\prime
},{\Greekmath 0115} ^{\prime }\right) ^{\prime }$. A generalized method of moments
(GMM) estimator can then be defined by\footnote{
As mentioned in Section \ref{sec:model}, it is necessary to normalize one of
the $Q$ elements of ${\Greekmath 010D} $ and one of the $Q-1$ elements of ${\Greekmath 0115} $.}
\begin{multline*}
\widehat{{\Greekmath 0112} }=
\begin{pmatrix}
\widehat{{\Greekmath 010C} } \\
\widehat{{\Greekmath 010D} } \\
\widehat{{\Greekmath 0115} }
\end{pmatrix}
=\limfunc{argmin}_{{\Greekmath 010C} \in \mathbb{R}^{K},\,{\Greekmath 010D} \in \mathbb{R}
^{Q-1},\,{\Greekmath 0115} \in \mathbb{R}^{Q-2}}\left(
\sum_{i=1}^{n}M(Y_{i0},Y_{i},X_{i},{\Greekmath 010C} ,{\Greekmath 010D} ,{\Greekmath 0115} )\right) ^{\prime }
\\
\widehat{W}_{n}\left( \sum_{i=1}^{n}M(Y_{i0},Y_{i},X_{i},{\Greekmath 010C} ,{\Greekmath 010D}
,{\Greekmath 0115} )\right) ,
\end{multline*}
where the weighting matrix $\widehat{W}_{n}$ converges to a positive
definite matrix, $W_{0}$. Assuming that $\mathbb{E}\left[
M(Y_{i0},Y_{i},X_{i},{\Greekmath 0112} )\right] =0$ is \textsl{uniquely} satisfied at $
{\Greekmath 0112} ={\Greekmath 0112} ^{0}$, and that mild regularity conditions (see
\citealt{Hansen1982}) are satisfied, $\widehat{{\Greekmath 0112} }$ will be consistent
and asymptotically normally distributed.
One limitation of the GMM\ approach is that it is often difficult to know
whether the moment condition $\mathbb{E}\left[ M(Y_{i0},Y_{i},X_{i},{\Greekmath 0112} )
\right] =0$ is uniquely satisfied at the true parameter value. When the
strictly exogenous explanatory variables, $X_{i}$, are discrete, sufficient
conditions for this can be obtained from the identification results in
Section~\ref{sec:Identification} by defining $g\left( Y_{i0},X_{i}\right) $
to be a vector of indicator functions for values in the support of $\left(
Y_{i0},X_{i}\right) $. If $X_{i}$ is not discrete, it may be possible to
define a root-$n$ consistent estimator by combining nonparametrically
estimated conditional moment conditions with the unconditional moment
conditions. See, for example, \cite{HonoreHu2004a} for such an approach.
Whether or not $\mathbb{E}\left[ M(Y_{i0},Y_{i},X_{i},{\Greekmath 0112} )\right] =0$ is
uniquely satisfied at the true parameter value, one can calculate valid
confidence sets for ${\Greekmath 0112} _{0}$ based on moment conditions like $\mathbb{E}
\left[ M(Y_{i0},Y_{i},X_{i},{\Greekmath 0112} )\right] =0$. See, for example, \cite
{ChenChristensenTamer2018}.
A second limitation of the GMM\ approach is that even if one ignores the
issue of identification, there are many ways to form a finite set of
unconditional moment conditions from the expressions in
\eqref{MomentFunctionsT3}, \eqref{MomentFunctionsT3b}, and
\eqref{MomentFunctionsT3c}. Moreover, the most natural ad hoc ways to do this, such as
considering all interactions between the conditional moment and the explanatory
variables and the initial condition, can lead to a very large number of moment conditions,
which in turn can result in poor small sample performance.
It is in principle known how to most efficiently
turn a set of conditional moment conditions into a set of moment conditions
of the same dimensionality as the parameter to be estimated. See, for
example, the discussion in \cite{NeweyMcFadden94:HoE}. Specifically, with a
conditional moment condition $\mathbb{E}\left[ \left. m_{Y_{0}}\left(
Y,X,{\Greekmath 0112} \right) \right\vert X,Y_{0}\right] =0$ when ${\Greekmath 0112} $ takes its
true value, ${\Greekmath 0112} _{0}$, the optimal unconditional moment function is $
A\left( X,Y_{0}\right) m_{Y_{0}}\left( Y,X,{\Greekmath 0112} \right) $, where $A\left(
X,Y_{0}\right) =\mathbb{E}\left[ \left. {\Greekmath 0272} _{{\Greekmath 0112} }m_{Y_{0}}\left(
Y,X,{\Greekmath 0112} _{0}\right) \right\vert X,Y_{0}\right] ^{\prime }V\left[ \left.
m_{Y_{0}}\left( Y,X,{\Greekmath 0112} _{0}\right) \right\vert X,Y_{0}\right] ^{-1}$.
Unfortunately, the construction of estimators of these efficient moments
depends heavily on the distribution of $Y$ given $\left( X,Y_{0}\right) $.
On the other hand, the moment conditions are still valid if $A\left(
X,Y_{0}\right) $ is misspecified. One approach therefore is to estimate a
flexible reduced form model for the distribution of $Y$ given $\left(
X,Y_{0}\right) $, and then use this reduced form for the distribution of $Y$
given $X$ to construct an estimate of $A\left( X,Y_{0}\right) $. In the
simulations and the empirical illustration below, we take this approach
using a correlated random effects approach to obtain the reduced form for
the distribution of $Y$ given $\left( X,Y_{0}\right) $. The dimensionality
of the moment function $A\left( X,Y_{0}\right) m_{Y_{0}}\left( Y,X,{\Greekmath 0112}
\right) $ is the same as that of the parameter vector, and the asymptotic
distribution of the estimator therefore follows from the theory of nonlinear
method of moments estimators.
The moment conditions derived in this paper can also be used for
specification testing. Suppose that a researcher has estimated the
parameters of interest, ${\Greekmath 0112} _{0}=\left( {\Greekmath 010C} _{0},{\Greekmath 010D} _{0},{\Greekmath 0115}
_{0}\right) $, by an estimator, $\widehat{{\Greekmath 0112} }$, that solves a moment
condition of the type $\frac{1}{n}\sum_{i=1}^{n}{\Greekmath 0120} \left( Y_{i},X_{i},
\widehat{{\Greekmath 0112} }\right) =0$. For example, she might have estimated a model
without individual-specific heterogeneity or a model in which the
heterogeneity is captured parametrically by a random effects approach, and
she might be interested in testing her parametric assumptions against the
less parametric fixed effects model. Let $\widehat{M}=\frac{1}{n}
\sum_{i=1}^{n}M\left( Y_{i},X_{i},\widehat{{\Greekmath 0112} }\right) $ where $M$ is
defined as above. $\widehat{M}$ is then a standard two-step estimator, and
it is straightforward to test whether $\widehat{M}$ is statistically
different from 0.
\section{Practical performance of a method of moments estimator}
\label{PracticalPerformance}
In the next subsection, we present the results from a small Monte Carlo
experiment designed to illustrate the performance of the method of moments
estimator based on the discussion in Section \ref{Section: Implication for
estimation}, and we compare the performance of the estimator to its
asymptotic distribution as well as to a correlated random effects estimator.
We then illustrate the use of the method of moments estimator in an
empirical example.
\subsection{Monte Carlo illustration\label{Section: Monte Carlo Results}}
We illustrate the performance of the GMM\ estimator described above through
a Monte Carlo study that considers three data generating processes,
two with a fixed effect and one without a fixed effect. The two data
generating processes that include a fixed effect are chosen such that one
satisfies the assumptions underpinning the correlated random effects
estimator proposed by \cite{wooldridge2005}, while the other does not. We consider
sample sizes of $N=500$, $1000$, and $2000$ with five time periods for each
individual. This includes the initial observations, so $T=4$ using the
notation above. There are $k=3$ explanatory variables and the dependent
variable can take $Q=4$ values. The true parameters are ${\Greekmath 010C} =\left(
1,0,0\right) ^{\prime },$ ${\Greekmath 010D} =\left( -1,0,0,1\right) ^{\prime }$ and $
{\Greekmath 0115} =\left( -2,0,2\right) ^{\prime }$ and we normalize ${\Greekmath 010D} _{2}={\Greekmath 0115}
_{2}=0$.
The explanatory variables are drawn as follows. First, let $\tilde{A}_{i}$
be a discrete random variable with $E\left[ \tilde{A}_{i}\right] =0$ and $V
\left[ \tilde{A}_{i}\right] =3$. The exact distribution of $\tilde{A}_{i}$
differs across specifications.\ Secondly, let $Z_{ijt}$ ($j=1,...,k$, $
t=0,...,4$) be independent normal random variables with mean 0 and variance
3, and let the first explanatory variable be $X_{i1t}=\left( Z_{i1t}+\tilde{A
}_{i}\right) /\sqrt{2}$. The second through $k$'th explanatory variables are
given by $X_{ijt}=\left. \left( Z_{ijt}+X_{i1t}\right) \right/ \sqrt{2}$.
This implies that all the explanatory variables and $
X_{it}^{\prime }{\Greekmath 010C} $ have mean 0 and variance 3. This is
comparable to the magnitude of the logistic distribution, which has mean 0
and variance $\left. {\Greekmath 0119} ^{2}\right/ 3$. \
For the two data generating
processes with a fixed effect, one (Design B) has $\tilde{A}_{i}$ normally
distributed while the other\ (Design C) has $P\left( \tilde{A}_{i}=\sqrt{6}
\right) =\frac{1}{3}$ and $P\left( \tilde{A}_{i}=-\left. \sqrt{6}\right/
2\right) =\frac{2}{3}$. For both of these specifications, the fixed effect, $
A_{i}$, equals $\tilde{A}_{i}$. The data generating process without fixed
effects (Design A) has the same distribution of $\tilde{A}_{i}$ as Design C,
but here $A_{i}=0$.
For Design C, the initial dependent variable, $Y_{i0}$
is generated from the ordered logit model where the only explanatory
variable is $A_{i}$. For Designs A and B, $Y_{i0}$ is generated from the
ordered logit model without explanatory variables. From a fixed effects
perspective, this makes Design B\ a little special, but it makes it fit the
assumptions for the correlated random effects approach. Design A\ is without
fixed effects, so it also satisfies the assumptions for the correlated
random effects approach, but in this case the true parameter value of one of
the parameters (the variance of the error in the specification of the fixed
effect) is at the boundary of the parameter space, which could render standard inference problematic.
We perform 400 Monte Carlo replications. The results are presented in Tables
\ref{Design A}, \ref{Design B}\ and \ref{Design C}. For comparison, we also
include the results for the correlated random effects estimator (cf.
\citealt{wooldridge2005}) that specifies the distribution of the unobserved
heterogeneity as $A=\sum_{t=0}^{4}X_{t}^{\prime }{\Greekmath 0112} \left( t\right)
+\sum_{q=1}^{4}1\left\{ Y_{0}=q\right\} {\Greekmath 0112} \left( q\right) +{\Greekmath 011B} Z$,
where $Z\sim N\left( 0,1\right) $. Design C violates the implicit assumption behind the correlated random effects
approach. Most importantly, the distribution of $A_{i}$ is discrete. On the
other hand, the relationship between the explanatory variables and $A_{i}$
is linear, so the violation is not extreme.
For each design and for each sample size, Tables \ref{Design A}, \ref{Design
B}\ and \ref{Design C} report the true values of the parameters, the median
bias of the method of moments estimator and the correlated random effects
estimator, the interquartile range of the estimators, and the median
absolute errors of the estimators. For each parameter, we also report the
ratio of the median absolute error of the correlated random effects
estimator relative to the method of moment estimator. Values of this ratio
greater than one suggest that the method of moments estimator is more
precise than the correlated random effects estimator. In Designs A and B,
the correlated random effects estimator is the correctly specified maximum
likelihood estimator. It is therefore not surprising that the median
absolute error ratio is less than one for all parameters and all sample
sized in this case. On the other hand, the ratio is above 0.65 in all cases,
suggesting that the loss of efficiency from the method of moments estimator
is not too large for this design. For these designs, both
estimators appear to be close to median unbiased, and the relative
performance of the estimators is driven by the difference in their
variability. For Design C with non-normal heterogeneity, the relative
performance of the two estimators is different for the different parameters.
The correlated random effects estimator is always less variable in terms of
interquartile range, but the biases in the estimates of the ${\Greekmath 010D} $'s and $
{\Greekmath 010E} $'s are large enough that the method of moments estimator tends to be
more precise when the sample size is large. For the coefficients on the
explanatory variables, ${\Greekmath 010C} $, the correlated random effects estimator is almost unbiased in Design C despite the
misspecification of the model for the fixed effect. This makes sense, because the specified model for the unobserved
heterogeneity will tend to control for any linear dependence between
explanatory variables and the level of the fixed effect. The specific results
for each of the ${\Greekmath 010D} $'s and for each of the ${\Greekmath 0115} $'s should be
considered with some care since the calculations are done under the specific
normalization that ${\Greekmath 010D} _{2}=0$ and ${\Greekmath 0115} _{2}=0$. With different
normalizations, the pattern of the results would have been different.
However, it is clear from Table \ref{Design C} that in the design with
heterogeneity, the misspecification embedded in the correlated random
effects approach generally speaking leads to biased estimates of the ${\Greekmath 010D}
$'s and the ${\Greekmath 0115} $, and that these biases will make the method of
moments estimator more precise for large sample sizes.
\begin{table}[tbp]
\caption{Design A}
\label{Design A}
\begin{center}
{\footnotesize \begin{tabular}{lrrrrrrrrrr}
\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.008 & 0.004 & 0.001 & 0.014 & 0.000 & -0.036 & -0.027 & -0.027 & 0.000 & 0.024 \\
IRQ & 0.096 & 0.057 & 0.055 & 0.181 & 0.000 & 0.174 & 0.201 & 0.097 & 0.000 & 0.108 \\
MAE & 0.046 & 0.028 & 0.028 & 0.097 & 0.000 & 0.092 & 0.104 & 0.051 & 0.000 & 0.060 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.001 & 0.004 & -0.002 & 0.019 & 0.000 & -0.018 & -0.010 & -0.003 & 0.000 & 0.009 \\
IQR & 0.112 & 0.068 & 0.066 & 0.249 & 0.000 & 0.232 & 0.252 & 0.135 & 0.000 & 0.144 \\
MAE & 0.056 & 0.034 & 0.033 & 0.124 & 0.000 & 0.117 & 0.123 & 0.067 & 0.000 & 0.072 \\
\\
MAE ratio & 0.832 & 0.833 & 0.830 & 0.780 & \multicolumn{1}{c}{---} & 0.787 & 0.845 & 0.760 & \multicolumn{1}{c}{---} & 0.835 \\
\\\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.006 & 0.000 & 0.002 & 0.010 & 0.000 & -0.014 & -0.023 & -0.013 & 0.000 & 0.013 \\
IRQ & 0.066 & 0.038 & 0.039 & 0.118 & 0.000 & 0.112 & 0.139 & 0.082 & 0.000 & 0.082 \\
MAE & 0.032 & 0.019 & 0.020 & 0.057 & 0.000 & 0.060 & 0.072 & 0.038 & 0.000 & 0.041 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.001 & -0.001 & 0.002 & 0.005 & 0.000 & -0.009 & -0.005 & 0.008 & 0.000 & 0.005 \\
IQR & 0.076 & 0.046 & 0.049 & 0.155 & 0.000 & 0.159 & 0.195 & 0.107 & 0.000 & 0.107 \\
MAE & 0.037 & 0.023 & 0.023 & 0.077 & 0.000 & 0.081 & 0.097 & 0.055 & 0.000 & 0.053 \\
\\
MAE ratio & 0.876 & 0.814 & 0.859 & 0.737 & \multicolumn{1}{c}{---} & 0.736 & 0.741 & 0.701 & \multicolumn{1}{c}{---} & 0.775 \\
\\\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.002 & -0.000 & -0.001 & 0.013 & 0.000 & -0.010 & -0.011 & -0.010 & 0.000 & 0.012 \\
IRQ & 0.046 & 0.032 & 0.025 & 0.100 & 0.000 & 0.084 & 0.098 & 0.049 & 0.000 & 0.059 \\
MAE & 0.022 & 0.016 & 0.013 & 0.052 & 0.000 & 0.041 & 0.051 & 0.025 & 0.000 & 0.032 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.001 & 0.001 & -0.002 & 0.005 & 0.000 & -0.001 & 0.008 & -0.004 & 0.000 & 0.001 \\
IQR & 0.046 & 0.033 & 0.030 & 0.123 & 0.000 & 0.101 & 0.129 & 0.067 & 0.000 & 0.072 \\
MAE & 0.023 & 0.016 & 0.015 & 0.061 & 0.000 & 0.050 & 0.066 & 0.033 & 0.000 & 0.036 \\
\\
MAE ratio & 0.955 & 0.979 & 0.842 & 0.839 & \multicolumn{1}{c}{---} & 0.810 & 0.775 & 0.768 & \multicolumn{1}{c}{---} & 0.898 \\
\\\end{tabular}
}
\par
{\footnotesize MAE is the median absolute error, IQR is the interquartile
range. ${\Greekmath 010D}_2$ and ${\Greekmath 0115}_2$ are normalized.}
\end{center}
\end{table}
\begin{table}[tbp]
\caption{Design B}
\label{Design B}
\begin{center}
{\footnotesize \begin{tabular}{lrrrrrrrrrr}
\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.003 & -0.003 & -0.003 & -0.007 & 0.000 & -0.010 & 0.007 & -0.007 & 0.000 & 0.009 \\
IRQ & 0.102 & 0.057 & 0.064 & 0.224 & 0.000 & 0.203 & 0.262 & 0.144 & 0.000 & 0.131 \\
MAE & 0.051 & 0.028 & 0.032 & 0.112 & 0.000 & 0.100 & 0.130 & 0.073 & 0.000 & 0.062 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.001 & -0.003 & -0.004 & 0.010 & 0.000 & -0.007 & -0.006 & -0.000 & 0.000 & 0.002 \\
IQR & 0.118 & 0.078 & 0.084 & 0.323 & 0.000 & 0.300 & 0.323 & 0.187 & 0.000 & 0.172 \\
MAE & 0.059 & 0.039 & 0.043 & 0.163 & 0.000 & 0.154 & 0.162 & 0.096 & 0.000 & 0.087 \\
\\
MAE ratio & 0.867 & 0.718 & 0.742 & 0.688 & \multicolumn{1}{c}{---} & 0.650 & 0.802 & 0.754 & \multicolumn{1}{c}{---} & 0.715 \\
\\\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.001 & -0.000 & -0.000 & -0.008 & 0.000 & 0.006 & 0.006 & -0.005 & 0.000 & 0.000 \\
IRQ & 0.073 & 0.047 & 0.048 & 0.173 & 0.000 & 0.141 & 0.183 & 0.106 & 0.000 & 0.098 \\
MAE & 0.037 & 0.023 & 0.023 & 0.086 & 0.000 & 0.070 & 0.089 & 0.054 & 0.000 & 0.049 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.001 & -0.002 & 0.001 & 0.004 & 0.000 & -0.007 & 0.006 & -0.002 & 0.000 & 0.004 \\
IQR & 0.086 & 0.060 & 0.055 & 0.205 & 0.000 & 0.196 & 0.231 & 0.126 & 0.000 & 0.128 \\
MAE & 0.043 & 0.030 & 0.028 & 0.104 & 0.000 & 0.098 & 0.119 & 0.062 & 0.000 & 0.063 \\
\\
MAE ratio & 0.867 & 0.782 & 0.852 & 0.829 & \multicolumn{1}{c}{---} & 0.720 & 0.751 & 0.883 & \multicolumn{1}{c}{---} & 0.775 \\
\\\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.004 & -0.000 & 0.000 & 0.010 & 0.000 & 0.011 & 0.005 & -0.004 & 0.000 & -0.001 \\
IRQ & 0.049 & 0.034 & 0.032 & 0.121 & 0.000 & 0.096 & 0.119 & 0.066 & 0.000 & 0.065 \\
MAE & 0.024 & 0.017 & 0.016 & 0.059 & 0.000 & 0.049 & 0.062 & 0.034 & 0.000 & 0.033 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.003 & -0.002 & 0.001 & 0.016 & 0.000 & 0.008 & 0.004 & -0.002 & 0.000 & 0.001 \\
IQR & 0.058 & 0.040 & 0.042 & 0.145 & 0.000 & 0.122 & 0.156 & 0.084 & 0.000 & 0.077 \\
MAE & 0.029 & 0.020 & 0.021 & 0.073 & 0.000 & 0.061 & 0.078 & 0.042 & 0.000 & 0.039 \\
\\
MAE ratio & 0.853 & 0.847 & 0.753 & 0.807 & \multicolumn{1}{c}{---} & 0.799 & 0.793 & 0.802 & \multicolumn{1}{c}{---} & 0.854 \\
\\\end{tabular}
}
\par
{\footnotesize MAE is the median absolute error, IQR is the interquartile
range. ${\Greekmath 010D}_2$ and ${\Greekmath 0115}_2$ are normalized.}
\end{center}
\end{table}
\begin{table}[tbp]
\caption{Design C}
\label{Design C}
\begin{center}
{\footnotesize \begin{tabular}{lrrrrrrrrrr}
\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.006 & 0.007 & 0.003 & 0.168 & 0.000 & 0.127 & 0.674 & -0.049 & 0.000 & -0.308 \\
IRQ & 0.108 & 0.073 & 0.070 & 0.216 & 0.000 & 0.260 & 0.365 & 0.156 & 0.000 & 0.173 \\
MAE & 0.054 & 0.037 & 0.037 & 0.170 & 0.000 & 0.156 & 0.674 & 0.081 & 0.000 & 0.308 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.018 & 0.005 & 0.009 & 0.074 & 0.000 & 0.007 & 0.303 & 0.026 & 0.000 & -0.146 \\
IQR & 0.185 & 0.112 & 0.106 & 0.327 & 0.000 & 0.479 & 0.702 & 0.271 & 0.000 & 0.376 \\
MAE & 0.097 & 0.058 & 0.054 & 0.173 & 0.000 & 0.232 & 0.445 & 0.139 & 0.000 & 0.227 \\
\\
MAE ratio & 0.556 & 0.638 & 0.677 & 0.983 & \multicolumn{1}{c}{---} & 0.671 & 1.515 & 0.586 & \multicolumn{1}{c}{---} & 1.355 \\
\\\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.004 & -0.004 & -0.000 & 0.189 & 0.000 & 0.137 & 0.691 & -0.045 & 0.000 & -0.308 \\
IRQ & 0.081 & 0.052 & 0.055 & 0.151 & 0.000 & 0.192 & 0.272 & 0.098 & 0.000 & 0.110 \\
MAE & 0.040 & 0.026 & 0.028 & 0.189 & 0.000 & 0.156 & 0.691 & 0.058 & 0.000 & 0.308 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.013 & -0.006 & -0.000 & 0.017 & 0.000 & 0.035 & 0.202 & 0.025 & 0.000 & -0.075 \\
IQR & 0.144 & 0.084 & 0.092 & 0.290 & 0.000 & 0.457 & 0.638 & 0.176 & 0.000 & 0.322 \\
MAE & 0.070 & 0.044 & 0.045 & 0.148 & 0.000 & 0.227 & 0.354 & 0.098 & 0.000 & 0.169 \\
\\
MAE ratio & 0.574 & 0.594 & 0.626 & 1.277 & \multicolumn{1}{c}{---} & 0.687 & 1.950 & 0.597 & \multicolumn{1}{c}{---} & 1.822 \\
\\\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & 0.007 & -0.003 & -0.002 & 0.179 & 0.000 & 0.133 & 0.712 & -0.043 & 0.000 & -0.315 \\
IRQ & 0.052 & 0.037 & 0.034 & 0.099 & 0.000 & 0.125 & 0.171 & 0.080 & 0.000 & 0.079 \\
MAE & 0.025 & 0.019 & 0.017 & 0.179 & 0.000 & 0.133 & 0.712 & 0.051 & 0.000 & 0.315 \\
\\\multicolumn{11}{c}{Method of Moments. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
True & 1.000 & 0.000 & 0.000 & -1.000 & 0.000 & 0.000 & 1.000 & -2.000 & 0.000 & 2.000 \\
Bias & -0.003 & -0.003 & -0.008 & 0.006 & 0.000 & 0.026 & 0.090 & 0.024 & 0.000 & -0.028 \\
IQR & 0.107 & 0.066 & 0.058 & 0.200 & 0.000 & 0.336 & 0.559 & 0.143 & 0.000 & 0.226 \\
MAE & 0.052 & 0.033 & 0.030 & 0.101 & 0.000 & 0.163 & 0.276 & 0.075 & 0.000 & 0.122 \\
\\
MAE ratio & 0.483 & 0.561 & 0.560 & 1.781 & \multicolumn{1}{c}{---} & 0.819 & 2.577 & 0.686 & \multicolumn{1}{c}{---} & 2.589 \\
\\\end{tabular}
}
\par
{\footnotesize MAE is the median absolute error, IQR is the interquartile
range. ${\Greekmath 010D}_2$ and ${\Greekmath 0115}_2$ are normalized.}
\end{center}
\end{table}
Tables \ref{Design A CI}-\ref{Design C CI} illustrate that the smaller bias of the method of
moments estimator can make it more reliable for inference than the
correlated random effects estimator when the assumptions underpinning the
correlated random effects estimator are violated. Specifically, the table
presents the fraction of times that 80, 90 and 95 percent confidence
intervals based on the correlated random effects estimator and on the method
of moments estimator cover the true unknown parameter. As one would expect
from the results in Table \ref{Design C}, the bias in the correlated random
effects estimator combined with its low variability can make it unlikely
that a confidence interval covers the true parameter value. Tables \ref
{Design A CI} and \ref{Design B CI} show the same results for Design A and
Design B. In these case, the confidence intervals based on the
correlated random effects estimator do very well. For Design B, this is not
surprising as this is a correctly specified maximum likelihood setting. It is
interesting that the correlated random effects estimator also does well in
Design A. Although it is the maximum likelihood estimator of a correctly
specified model, the estimation is made non-standard by the fact that the
true value of one of the parameters (the variance of the error in the
specification for the random effect) is on the boundary of the parameter
space and asymptotic normality would therefore not follow from textbook
asymptotic theory. As a general statement, Tables \ref{Design A CI}, \ref
{Design B CI} and \ref{Design C CI} also illustrate that the confidence
interval based on the method of moments estimator can be somewhat erratic even
with relatively large sample sizes.\footnote{
On the other hand, in simulations not reported here, we have found that the
natural estimator of the variance of the method of moments estimator can
perform poorly. The results reported here therefore use bootstrap
standard errors based on the interquartile range of 1000 bootstrap
replications. The standard errors reported for the correlated random effects
are based on the \textquotedblleft robust\textquotedblright\ expression for
the asymptotic variance of extremum estimators.}
\begin{table}[tbp]
\caption{Design A}
\label{Design A CI}
\begin{center}
{\footnotesize \begin{tabular}{lrrrrrrrrrr}
\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.797 & 0.812 & 0.807 & 0.790 & \multicolumn{1}{c}{---} & 0.770 & 0.780 & 0.807 & \multicolumn{1}{c}{---} & 0.787 \\
90\% CI & 0.912 & 0.897 & 0.902 & 0.892 & \multicolumn{1}{c}{---} & 0.892 & 0.875 & 0.912 & \multicolumn{1}{c}{---} & 0.900 \\
95\% CI & 0.948 & 0.953 & 0.950 & 0.938 & \multicolumn{1}{c}{---} & 0.940 & 0.938 & 0.960 & \multicolumn{1}{c}{---} & 0.968 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.870 & 0.850 & 0.850 & 0.902 & \multicolumn{1}{c}{---} & 0.870 & 0.875 & 0.930 & \multicolumn{1}{c}{---} & 0.953 \\
90\% CI & 0.935 & 0.932 & 0.950 & 0.950 & \multicolumn{1}{c}{---} & 0.945 & 0.930 & 0.973 & \multicolumn{1}{c}{---} & 0.988 \\
95\% CI & 0.965 & 0.975 & 0.983 & 0.970 & \multicolumn{1}{c}{---} & 0.978 & 0.968 & 0.995 & \multicolumn{1}{c}{---} & 0.995 \\
\\ \multicolumn{11}{c}{Correlated Random Effects. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.785 & 0.815 & 0.800 & 0.818 & \multicolumn{1}{c}{---} & 0.775 & 0.795 & 0.772 & \multicolumn{1}{c}{---} & 0.782 \\
90\% CI & 0.912 & 0.915 & 0.887 & 0.895 & \multicolumn{1}{c}{---} & 0.883 & 0.875 & 0.873 & \multicolumn{1}{c}{---} & 0.900 \\
95\% CI & 0.953 & 0.950 & 0.940 & 0.938 & \multicolumn{1}{c}{---} & 0.935 & 0.932 & 0.935 & \multicolumn{1}{c}{---} & 0.950 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.890 & 0.897 & 0.883 & 0.900 & \multicolumn{1}{c}{---} & 0.885 & 0.905 & 0.910 & \multicolumn{1}{c}{---} & 0.940 \\
90\% CI & 0.958 & 0.953 & 0.955 & 0.960 & \multicolumn{1}{c}{---} & 0.960 & 0.963 & 0.988 & \multicolumn{1}{c}{---} & 0.978 \\
95\% CI & 0.980 & 0.973 & 0.978 & 0.978 & \multicolumn{1}{c}{---} & 0.975 & 0.990 & 0.998 & \multicolumn{1}{c}{---} & 0.988 \\
\\ \multicolumn{11}{c}{Correlated Random Effects. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.765 & 0.772 & 0.850 & 0.777 & \multicolumn{1}{c}{---} & 0.815 & 0.805 & 0.782 & \multicolumn{1}{c}{---} & 0.807 \\
90\% CI & 0.863 & 0.892 & 0.917 & 0.890 & \multicolumn{1}{c}{---} & 0.927 & 0.897 & 0.883 & \multicolumn{1}{c}{---} & 0.895 \\
95\% CI & 0.927 & 0.940 & 0.945 & 0.940 & \multicolumn{1}{c}{---} & 0.963 & 0.927 & 0.932 & \multicolumn{1}{c}{---} & 0.953 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.863 & 0.865 & 0.892 & 0.892 & \multicolumn{1}{c}{---} & 0.920 & 0.885 & 0.905 & \multicolumn{1}{c}{---} & 0.890 \\
90\% CI & 0.915 & 0.935 & 0.958 & 0.950 & \multicolumn{1}{c}{---} & 0.973 & 0.955 & 0.965 & \multicolumn{1}{c}{---} & 0.958 \\
95\% CI & 0.958 & 0.968 & 0.985 & 0.980 & \multicolumn{1}{c}{---} & 0.990 & 0.985 & 0.975 & \multicolumn{1}{c}{---} & 0.988 \\
\\ \end{tabular}
}
\par
{\footnotesize Based on 1000 bootstrap replications. ${\Greekmath 010D}_2$ and $
{\Greekmath 0115}_2$ are normalized.}
\end{center}
\end{table}
\begin{table}[tbp]
\caption{Design B}
\label{Design B CI}
\begin{center}
{\footnotesize \begin{tabular}{lrrrrrrrrrr}
\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.828 & 0.843 & 0.815 & 0.777 & \multicolumn{1}{c}{---} & 0.792 & 0.795 & 0.785 & \multicolumn{1}{c}{---} & 0.820 \\
90\% CI & 0.900 & 0.925 & 0.915 & 0.900 & \multicolumn{1}{c}{---} & 0.892 & 0.895 & 0.880 & \multicolumn{1}{c}{---} & 0.925 \\
95\% CI & 0.943 & 0.955 & 0.955 & 0.945 & \multicolumn{1}{c}{---} & 0.938 & 0.945 & 0.935 & \multicolumn{1}{c}{---} & 0.968 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.887 & 0.880 & 0.860 & 0.875 & \multicolumn{1}{c}{---} & 0.870 & 0.873 & 0.883 & \multicolumn{1}{c}{---} & 0.943 \\
90\% CI & 0.958 & 0.945 & 0.943 & 0.935 & \multicolumn{1}{c}{---} & 0.938 & 0.943 & 0.940 & \multicolumn{1}{c}{---} & 0.983 \\
95\% CI & 0.985 & 0.965 & 0.983 & 0.965 & \multicolumn{1}{c}{---} & 0.965 & 0.970 & 0.973 & \multicolumn{1}{c}{---} & 0.995 \\
\\ \multicolumn{11}{c}{Correlated Random Effects. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.785 & 0.780 & 0.800 & 0.792 & \multicolumn{1}{c}{---} & 0.795 & 0.782 & 0.772 & \multicolumn{1}{c}{---} & 0.795 \\
90\% CI & 0.907 & 0.897 & 0.895 & 0.885 & \multicolumn{1}{c}{---} & 0.905 & 0.897 & 0.902 & \multicolumn{1}{c}{---} & 0.885 \\
95\% CI & 0.953 & 0.948 & 0.955 & 0.943 & \multicolumn{1}{c}{---} & 0.955 & 0.935 & 0.965 & \multicolumn{1}{c}{---} & 0.943 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.892 & 0.890 & 0.895 & 0.900 & \multicolumn{1}{c}{---} & 0.875 & 0.897 & 0.943 & \multicolumn{1}{c}{---} & 0.920 \\
90\% CI & 0.965 & 0.950 & 0.955 & 0.948 & \multicolumn{1}{c}{---} & 0.968 & 0.968 & 0.973 & \multicolumn{1}{c}{---} & 0.960 \\
95\% CI & 0.985 & 0.960 & 0.983 & 0.975 & \multicolumn{1}{c}{---} & 0.983 & 0.985 & 0.990 & \multicolumn{1}{c}{---} & 0.985 \\
\\ \multicolumn{11}{c}{Correlated Random Effects. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.800 & 0.810 & 0.802 & 0.805 & \multicolumn{1}{c}{---} & 0.820 & 0.785 & 0.787 & \multicolumn{1}{c}{---} & 0.802 \\
90\% CI & 0.880 & 0.897 & 0.905 & 0.920 & \multicolumn{1}{c}{---} & 0.922 & 0.887 & 0.887 & \multicolumn{1}{c}{---} & 0.907 \\
95\% CI & 0.945 & 0.948 & 0.950 & 0.973 & \multicolumn{1}{c}{---} & 0.960 & 0.935 & 0.932 & \multicolumn{1}{c}{---} & 0.943 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.860 & 0.902 & 0.880 & 0.895 & \multicolumn{1}{c}{---} & 0.883 & 0.883 & 0.910 & \multicolumn{1}{c}{---} & 0.932 \\
90\% CI & 0.948 & 0.965 & 0.960 & 0.950 & \multicolumn{1}{c}{---} & 0.953 & 0.948 & 0.963 & \multicolumn{1}{c}{---} & 0.983 \\
95\% CI & 0.968 & 0.988 & 0.978 & 0.978 & \multicolumn{1}{c}{---} & 0.983 & 0.985 & 0.988 & \multicolumn{1}{c}{---} & 0.995 \\
\\ \end{tabular}
}
\par
{\footnotesize Based on 1000 bootstrap replications. ${\Greekmath 010D}_2$ and $
{\Greekmath 0115}_2$ are normalized.}
\end{center}
\end{table}
\begin{table}[tbp]
\caption{Design C}
\label{Design C CI}
\begin{center}
{\footnotesize \begin{tabular}{lrrrrrrrrrr}
\multicolumn{11}{c}{Correlated Random Effects. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.780 & 0.825 & 0.787 & 0.593 & \multicolumn{1}{c}{---} & 0.728 & 0.103 & 0.742 & \multicolumn{1}{c}{---} & 0.113 \\
90\% CI & 0.892 & 0.922 & 0.897 & 0.695 & \multicolumn{1}{c}{---} & 0.835 & 0.200 & 0.850 & \multicolumn{1}{c}{---} & 0.218 \\
95\% CI & 0.960 & 0.963 & 0.953 & 0.787 & \multicolumn{1}{c}{---} & 0.900 & 0.328 & 0.940 & \multicolumn{1}{c}{---} & 0.295 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 500. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.792 & 0.820 & 0.792 & 0.797 & \multicolumn{1}{c}{---} & 0.762 & 0.682 & 0.828 & \multicolumn{1}{c}{---} & 0.730 \\
90\% CI & 0.915 & 0.935 & 0.915 & 0.895 & \multicolumn{1}{c}{---} & 0.867 & 0.792 & 0.922 & \multicolumn{1}{c}{---} & 0.830 \\
95\% CI & 0.955 & 0.958 & 0.958 & 0.940 & \multicolumn{1}{c}{---} & 0.915 & 0.883 & 0.963 & \multicolumn{1}{c}{---} & 0.877 \\
\\ \multicolumn{11}{c}{Correlated Random Effects. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.792 & 0.802 & 0.775 & 0.345 & \multicolumn{1}{c}{---} & 0.560 & 0.005 & 0.720 & \multicolumn{1}{c}{---} & 0.007 \\
90\% CI & 0.910 & 0.895 & 0.887 & 0.478 & \multicolumn{1}{c}{---} & 0.718 & 0.015 & 0.863 & \multicolumn{1}{c}{---} & 0.030 \\
95\% CI & 0.963 & 0.963 & 0.943 & 0.590 & \multicolumn{1}{c}{---} & 0.823 & 0.045 & 0.925 & \multicolumn{1}{c}{---} & 0.060 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 1000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.845 & 0.815 & 0.792 & 0.810 & \multicolumn{1}{c}{---} & 0.785 & 0.720 & 0.833 & \multicolumn{1}{c}{---} & 0.758 \\
90\% CI & 0.915 & 0.920 & 0.877 & 0.907 & \multicolumn{1}{c}{---} & 0.873 & 0.833 & 0.917 & \multicolumn{1}{c}{---} & 0.833 \\
95\% CI & 0.950 & 0.950 & 0.932 & 0.950 & \multicolumn{1}{c}{---} & 0.900 & 0.900 & 0.963 & \multicolumn{1}{c}{---} & 0.917 \\
\\ \multicolumn{11}{c}{Correlated Random Effects. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.780 & 0.802 & 0.795 & 0.135 & \multicolumn{1}{c}{---} & 0.480 & 0.000 & 0.645 & \multicolumn{1}{c}{---} & 0.000 \\
90\% CI & 0.883 & 0.885 & 0.900 & 0.230 & \multicolumn{1}{c}{---} & 0.637 & 0.000 & 0.765 & \multicolumn{1}{c}{---} & 0.000 \\
95\% CI & 0.940 & 0.948 & 0.945 & 0.353 & \multicolumn{1}{c}{---} & 0.735 & 0.000 & 0.875 & \multicolumn{1}{c}{---} & 0.002 \\
\\
\multicolumn{11}{c}{Method of Moments. Sample Size: 2000. (nrep = 400)} \\
& \multicolumn{1}{c}{${\Greekmath 010C} _{1}$} & \multicolumn{1}{c}{${\Greekmath 010C} _{2}$} &
\multicolumn{1}{c}{${\Greekmath 010C} _{3}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{2}$} & \multicolumn{1}{c}{${\Greekmath 010D} _{3}$} &
\multicolumn{1}{c}{${\Greekmath 010D} _{4}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{1}$} &
\multicolumn{1}{c}{${\Greekmath 0115} _{2}$} & \multicolumn{1}{c}{${\Greekmath 0115} _{3}$} \\
80\% CI & 0.805 & 0.807 & 0.843 & 0.843 & \multicolumn{1}{c}{---} & 0.815 & 0.745 & 0.860 & \multicolumn{1}{c}{---} & 0.762 \\
90\% CI & 0.915 & 0.927 & 0.945 & 0.930 & \multicolumn{1}{c}{---} & 0.920 & 0.830 & 0.932 & \multicolumn{1}{c}{---} & 0.885 \\
95\% CI & 0.965 & 0.973 & 0.965 & 0.958 & \multicolumn{1}{c}{---} & 0.960 & 0.877 & 0.960 & \multicolumn{1}{c}{---} & 0.922 \\
\\ \end{tabular}
}
\par
{\footnotesize Based on 1000 bootstrap replications. ${\Greekmath 010D}_2$ and $
{\Greekmath 0115}_2$ are normalized.}
\end{center}
\end{table}
\subsection{Empirical illustration\label{Section: Empirical Illustration}}
In this section, we illustrate the value of the moment conditions derived in
this paper in an empirical illustration inspired by \cite
{contoyannis_dynamics_2004}. The dependent variable is self-reported health
status. We use data from the first five waves of the British Household
Panel Survey, and we restrict the sample to individuals
who are between 26 and 70 years old in the first wave. This yields a data set with $5093$ individuals observed in 5
time periods, including the initial observation (so $T=4$). In the original
data set, the dependent variable can take five values. We aggregate these
into \textquotedblleft Poor or Very Poor\textquotedblright\ ($8.1\%$ of the
observations), \textquotedblleft Fair\textquotedblright\ ($18.6\%$),
\textquotedblleft Good\textquotedblright\ ($47.6\%$), and \textquotedblleft
Excellent\textquotedblright\ ($25.7\%$). We also consider specifications
where the first two are merged into one outcome.
We use two sets of explanatory variables. In the first, we use age and
age-squared (measured as $Age/10$ and $(Age-45)^{2}/{100}$, respectively,
where $Age$ is measured in years). In the second, we also include log-income.
The results are presented in Table \ref{Table: Empirical Results Version2}, which
also presents the estimates from a correlated random effects specification.
We have normalized the ${\Greekmath 010D}$-coefficient associated with
\textquotedblleft Good Health\textquotedblright\ and the threshold ($
{\Greekmath 0115} $) just below \textquotedblleft Good Health\textquotedblright\ to be zero.
The most consistent result presented in Table \ref{Table: Empirical Results Version2} is that the
coefficient on age is negative across all specifications and that the coefficient on age-squared is
insignificantly different from 0 in all specifications.
The point estimates for the effect of income on self-reported health are
positive and statistically significant for all the specifications. The most puzzling aspect of Table
\ref {Table: Empirical Results Version2} is that the standard error of the correlated random effects estimator of
the coefficient on age gets much more precise when log-income is included. The method of moments estimator does not display this pattern.
A comparison of the other standard errors reveals that the correlated random effects estimator is less variable
than the method of moments estimator. This is in line with the interquartile ranges reported in Tables \ref{Design A}, \ref{Design B}\ and \ref{Design C}.
\begin{table}[h!]
\caption{Empirical Results: Application to Self-Reported Health Status}
\label{Table: Empirical Results Version2}
\begin{center}
{\footnotesize
\begin{tabular}{lrrrrcrrrr}
& \multicolumn{4}{c}{Three Outcomes} & & \multicolumn{4}{c}{Four Outcomes}
\\
& \multicolumn{1}{c}{MoM} & \multicolumn{1}{c}{CRE} & \multicolumn{1}{c}{MoM}
& \multicolumn{1}{c}{CREs} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{MoM}
& \multicolumn{1}{c}{CRE} & \multicolumn{1}{c}{MoM} & \multicolumn{1}{c}{CRE}
\\
$Age/10$ & $ -1.070 $ & $ -1.110 $ & $ -1.271 $ & $ -0.179 $ & & $ -1.136 $ & $ -0.961 $ & $ -1.078 $ & $ -0.147 $ \\
& $( 0.296\rlap{)}$ & $( 0.145\rlap{)}$ & $( 0.470\rlap{)}$ & $( 0.035\rlap{)}$ & & $( 0.214\rlap{)}$ & $( 0.137\rlap{)}$ & $( 0.331\rlap{)}$ & $( 0.032\rlap{)}$ \vspace{5pt} \\
{\tiny ($Age-45)^2/100$} & $ 0.396 $ & $ -0.251 $ & $ 0.804 $ & $ 0.109 $ & & $ 0.278 $ & $ -0.226 $ & $ -0.293 $ & $ 0.025 $ \\
& $( 0.925\rlap{)}$ & $( 0.548\rlap{)}$ & $( 2.047\rlap{)}$ & $( 0.544\rlap{)}$ & & $( 0.749\rlap{)}$ & $( 0.503\rlap{)}$ & $( 1.037\rlap{)}$ & $( 0.499\rlap{)}$ \vspace{5pt} \\
log-income & & & $ 0.054 $ & $ 0.283 $ & & & & $ 0.130 $ & $ 0.217 $ \\
& & & $( 0.198\rlap{)}$ & $( 0.048\rlap{)}$ & & & & $( 0.089\rlap{)}$ & $( 0.045\rlap{)}$ \vspace{25pt} \\
${\Greekmath 010D}_1$ & & & & & & $ -1.477 $ & $ -1.594 $ & $ -1.427 $ & $ -1.674 $ \\
& & & & & & $( 0.179\rlap{)}$ & $( 0.109\rlap{)}$ & $( 0.247\rlap{)}$ & $( 0.111\rlap{)}$ \vspace{5pt} \\
${\Greekmath 010D}_2$ & $ 0.269 $ & $ -0.810 $ & $ -0.753 $ & $ -0.824 $ & & $ -0.759 $ & $ -0.618 $ & $ -0.691 $ & $ -0.657 $ \\
& $( 0.165\rlap{)}$ & $( 0.067\rlap{)}$ & $( 0.354\rlap{)}$ & $( 0.068\rlap{)}$ & & $( 0.100\rlap{)}$ & $( 0.063\rlap{)}$ & $( 0.116\rlap{)}$ & $( 0.064\rlap{)}$ \vspace{5pt} \\
${\Greekmath 010D}_3$ & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ & & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ \\
& $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ \vspace{5pt} \\
${\Greekmath 010D}_4$ & $ 0.147 $ & $ 0.484 $ & $ 0.559 $ & $ 0.590 $ & & $ 0.395 $ & $ 0.557 $ & $ 0.635 $ & $ 0.619 $ \\
& $( 0.133\rlap{)}$ & $( 0.063\rlap{)}$ & $( 0.619\rlap{)}$ & $( 0.064\rlap{)}$ & & $( 0.105\rlap{)}$ & $( 0.064\rlap{)}$ & $( 0.236\rlap{)}$ & $( 0.065\rlap{)}$ \vspace{25pt} \\
${\Greekmath 0115}_1$ & & & & & & $ -2.464 $ & $ -2.560 $ & $ -2.510 $ & $ -2.555 $ \\
& & & & & & $( 0.057\rlap{)}$ & $( 0.050\rlap{)}$ & $( 0.080\rlap{)}$ & $( 0.050\rlap{)}$ \vspace{5pt} \\
${\Greekmath 0115}_2$ & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ & & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ & $ 0.000 $ \\
& $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ & $( 0.000\rlap{)}$ \vspace{5pt} \\
${\Greekmath 0115}_3$ & $ 3.707 $ & $ 3.818 $ & $ 3.304 $ & $ 3.780 $ & & $ 3.374 $ & $ 3.758 $ & $ 3.340 $ & $ 3.728 $ \\
& $( 0.085\rlap{)}$ & $( 0.054\rlap{)}$ & $( 0.203\rlap{)}$ & $( 0.054\rlap{)}$ & & $( 0.060\rlap{)}$ & $( 0.053\rlap{)}$ & $( 0.082\rlap{)}$ & $( 0.054\rlap{)}$ \vspace{5pt} \\
\\ \hline
\end{tabular}
}
\end{center}
\par
{\footnotesize \singlespacing
The dependent variable is self-reported health status and the data is a
balanced panel from the first five waves of the British Household Panel
Survey. We restrict the sample to individuals who are between 26 and 70 years old in the first wave. In columns 1-4, the dependent variable takes the values ``Poor, Very Poor, or Fair'', ``Good'', and
``Excellent''.
The dependent variable in columns 5-8
takes values ``Poor or Very Poor'', ``Fair'',
``Good'' and ``Excellent''. MoM refers to the method of moments estimator and CRE is the
correlated random effects estimator.}
\end{table}
\section{Conclusions}
\label{Conclusions}
This paper has extended the analysis in \cite{honore2020dynamic} to provide conditional moment conditions for panel data fixed effects versions of the dynamic ordered logit models like the one considered in \cite{muris2020dynamic}. The moment conditions are interesting in their own right, and the paper also illustrates the potential for systematically deriving moment conditions for nonlinear panel models. The moment conditions presented here can be used for estimation as well as for testing more parametric specifications of the individual-specific effects in dynamic ordered logits. For point-identification, it is important to investigate whether the moment conditions are \textit{uniquely} satisfied at the true parameter values. The paper presents conditions under which this is the case.
The paper also proposes a practical strategy for turning the derived conditional moment conditions into unconditional moment conditions that can be used for GMM estimation, and it illustrates the use of the resulting estimator in a small Monte Carlo study as well as in an empirical application.
More broadly, this paper contributes to the literature on panel data estimation of nonlinear models with fixed effects. In this context, the main contribution is to illustrate the potential for applying the functional differencing insights of \cite{bonhomme2012functional} to logit-type models.
\ifx\undefined\leavevmode\rule[.5ex]{3em}{.5pt}\
\fi
\ifx\undefined\textsc
\let\tmpsmall\tmpsmall\sc
\fi
\begin{thebibliography}{}
\harvarditem[Aguirregabiria and Carro]{Aguirregabiria and
Carro}{2021}{aguirregabiriaIdentificationAverageMarginal}
{\sc Aguirregabiria, V., {\tmpsmall\sc and} J.~M. Carro} (2021):
``Identification of {{Average Marginal Effects}} in {{Fixed Effects Dynamic
Discrete Choice Models}},'' {\em unpublished\/}, pp. 1--31.
\harvarditem[Aguirregabiria, Gu, and Luo]{Aguirregabiria, Gu, and
Luo}{2021}{aguirregabiriaSufficientStatisticsUnobserved2021}
{\sc Aguirregabiria, V., J.~Gu, {\tmpsmall\sc and} Y.~Luo} (2021): ``Sufficient
Statistics for Unobserved Heterogeneity in Structural Dynamic Logit Models,''
{\em Journal of Econometrics\/}, 223(2), 280--311.
\harvarditem[Albarran, Carrasco, and Carro]{Albarran, Carrasco, and
Carro}{2019}{albarran_estimation_2019}
{\sc Albarran, P., R.~Carrasco, {\tmpsmall\sc and} J.~M. Carro} (2019):
``Estimation of {Dynamic} {Nonlinear} {Random} {Effects} {Models} with
{Unbalanced} {Panels},'' {\em Oxford Bulletin of Economics and Statistics\/},
81(6), 1424--1441.
\harvarditem[Arellano]{Arellano}{2003}{arellano2003discrete}
{\sc Arellano, M.} (2003): ``Discrete choices with panel data,''
{\em Investigaciones econ{\'o}micas\/}, 27(3), 423--458.
\harvarditem[Arellano and Bond]{Arellano and Bond}{1991}{arellano_tests_1991}
{\sc Arellano, M., {\tmpsmall\sc and} S.~Bond} (1991): ``Some {Tests} of
{Specification} for {Panel} {Data}: {Monte} {Carlo} {Evidence} and an
{Application} to {Employment} {Equations},'' {\em The Review of Economic
Studies\/}, 58(2), 277.
\harvarditem[Arellano and Bonhomme]{Arellano and
Bonhomme}{2011}{ArellanoBonhomme2011}
{\sc Arellano, M., {\tmpsmall\sc and} S.~Bonhomme} (2011): ``Nonlinear Panel
Data Analysis,'' {\em Annual Review of Economics\/}, 3(1), 395--424.
\harvarditem[Aristodemou]{Aristodemou}{2021}{aristodemou2018semiparametric}
{\sc Aristodemou, E.} (2021): ``Semiparametric Identification in Panel Data
Discrete Response Models,'' {\em Journal of Econometrics\/}, 220(2), 253--271.
\harvarditem[Baetschmann, Staub, and Winkelmann]{Baetschmann, Staub, and
Winkelmann}{2015}{Baetschmann2015}
{\sc Baetschmann, G., K.~E. Staub, {\tmpsmall\sc and} R.~Winkelmann} (2015):
``Consistent estimation of the fixed effects ordered logit model,''
{\em Journal of the Royal Statistical Society A\/}, 178(3), 685--703.
\harvarditem[Blundell and Bond]{Blundell and Bond}{1998}{blundell_initial_1998}
{\sc Blundell, R., {\tmpsmall\sc and} S.~Bond} (1998): ``Initial conditions and
moment restrictions in dynamic panel data models,'' {\em Journal of
Econometrics\/}, 87(1), 115--143.
\harvarditem[Bonhomme]{Bonhomme}{2012}{bonhomme2012functional}
{\sc Bonhomme, S.} (2012): ``Functional differencing,''
{\em Econometrica\/}, 80(4), 1337--1385.
\harvarditem[Botosaru, Muris, and Pendakur]{Botosaru, Muris, and
Pendakur}{2023}{bmp2021}
{\sc Botosaru, I., C.~Muris, {\tmpsmall\sc and} K.~Pendakur} (2023):
``Identification of Time-Varying Transformation Models with Fixed Effects,
with an Application to Unobserved Heterogeneity in Resource Shares,''
{\em Journal of Econometrics\/}, 232(2), 576--597.
\harvarditem[Carro and Traferri]{Carro and Traferri}{2014}{carro_state_2014}
{\sc Carro, J.~M., {\tmpsmall\sc and} A.~Traferri} (2014): ``State {Dependence}
and {Heterogeneity} in {Health} {Using} a {Bias}-{Corrected}
{Fixed}-{Effects} {Estimator},'' {\em Journal of Applied Econometrics\/},
29(2), 181--207.
\harvarditem[Chamberlain]{Chamberlain}{1980}{chamberlain_analysis_1980}
{\sc Chamberlain, G.} (1980): ``Analysis of {Covariance} with {Qualitative}
{Data},'' {\em The Review of Economic Studies\/}, 47(1), 225--238.
\harvarditem[Chamberlain]{Chamberlain}{1985}{Chamberlain1985}
{\sc \leavevmode\rule[.5ex]{3em}{.5pt}\ {}} (1985): ``Heterogeneity, Omitted Variable Bias, and
Duration Dependence,'' in {\em Longitudinal Analysis of Labor Market Data\/},
ed. by J.~J. Heckman, {\tmpsmall\sc and} B.~Singer, no.~10 in Econometric Society
Monographs series, pp. 3--38. Cambridge University Press, Cambridge, New York
and Sydney.
\harvarditem[Chen, Christensen, and Tamer]{Chen, Christensen, and
Tamer}{2018}{ChenChristensenTamer2018}
{\sc Chen, X., T.~Christensen, {\tmpsmall\sc and} E.~Tamer} (2018): ``Monte
Carlo Confidence Sets for Identified Sets,'' {\em Econometrica\/}, 86(6),
1965--2018.
\harvarditem[Contoyannis, Jones, and Rice]{Contoyannis, Jones, and
Rice}{2004}{contoyannis_dynamics_2004}
{\sc Contoyannis, P., A.~M. Jones, {\tmpsmall\sc and} N.~Rice} (2004): ``The
dynamics of health in the {British} {Household} {Panel} {Survey},''
{\em Journal of Applied Econometrics\/}, 19(4), 473--503.
\harvarditem[Das and van Soest]{Das and van Soest}{1999}{das_panel_1999}
{\sc Das, M., {\tmpsmall\sc and} A.~van Soest} (1999): ``A panel data model for
subjective information on household income growth,'' {\em Journal of
Economic Behavior \& Organization\/}, 40(4), 409--426.
\harvarditem[Davezies, D'Haultfoeuille, and Mugnier]{Davezies, D'Haultfoeuille,
and Mugnier}{2022}{davezies2022fixed}
{\sc Davezies, L., X.~D'Haultfoeuille, {\tmpsmall\sc and} M.~Mugnier} (2022):
``Fixed Effects Binary Choice Models with Three or More Periods,''
{\em Quantitative Economics (forthcoming)\/}.
\harvarditem[Dobronyi, Gu, and Kim]{Dobronyi, Gu, and
Kim}{2021}{dobronyi2021identification}
{\sc Dobronyi, C., J.~Gu, {\tmpsmall\sc and} K.~i. Kim} (2021): ``Identification
of Dynamic Panel Logit Models with Fixed Effects,'' {\em arXiv preprint
arXiv:2104.04590\/}.
\harvarditem[Fernández-Val, Savchenko, and Vella]{Fernández-Val, Savchenko,
and Vella}{2017}{fernandez-val_evaluating_2017}
{\sc Fernández-Val, I., Y.~Savchenko, {\tmpsmall\sc and} F.~Vella} (2017):
``Evaluating the role of income, state dependence and individual specific
heterogeneity in the determination of subjective health assessments,''
{\em Economics \& Human Biology\/}, 25, 85--98.
\harvarditem[Galama and Kapteyn]{Galama and Kapteyn}{2011}{galamakapteyn}
{\sc Galama, T., {\tmpsmall\sc and} A.~Kapteyn} (2011): ``Grossman’s missing
health threshold,'' {\em Journal of Health Economics\/}, 30, 1044--1056.
\harvarditem[Grossman]{Grossman}{1972}{grossmanJPE}
{\sc Grossman, M.} (1972): ``On the concept of health capital and the
demand for health,'' {\em Journal of Political Economy\/}, 80, 223--255.
\harvarditem[Hahn]{Hahn}{1997}{hahn_note_1997}
{\sc Hahn, J.} (1997): ``A {Note} on the {Efficient} {Semiparametric}
{Estimation} of {Some} {Exponential} {Panel} {Models},'' {\em Econometric
Theory\/}, 13(4), 583--588.
\harvarditem[Hansen]{Hansen}{1982}{Hansen1982}
{\sc Hansen, L.~P.} (1982): ``Large Sample Properties of Generalized Method
of Moments Estimators,'' {\em Econometrica\/}, 50(4), pp. 1029--1054.
\harvarditem[Honor{\'e}]{Honor{\'e}}{2002}{honore_nonlinear_2002}
{\sc Honor{\'e}, B.~E.} (2002): ``Nonlinear models with panel data,''
{\em Portuguese Economic Journal\/}, 1(2), 163.
\harvarditem[Honor{\'e} and Hu]{Honor{\'e} and Hu}{2004}{HonoreHu2004a}
{\sc Honor{\'e}, B.~E., {\tmpsmall\sc and} L.~Hu} (2004): ``Estimation of Cross
Sectional and Panel Data Censored Regression Models with Endogeneity,''
{\em Journal of Econometrics\/}, 122(2), 293--316.
\harvarditem[Honor{\'e} and Kyriazidou]{Honor{\'e} and
Kyriazidou}{2000}{honore2000panel}
{\sc Honor{\'e}, B.~E., {\tmpsmall\sc and} E.~Kyriazidou} (2000): ``Panel data
discrete choice models with lagged dependent variables,''
{\em Econometrica\/}, 68(4), 839--874.
\harvarditem[Honor{\'e} and Weidner]{Honor{\'e} and
Weidner}{2020}{honore2020dynamic}
{\sc Honor{\'e}, B.~E., {\tmpsmall\sc and} M.~Weidner} (2020): ``Moment
Conditions for Dynamic Panel Logit Models with Fixed Effects,'' {\em arXiv
preprint arXiv:2005.05942\/}.
\harvarditem[Johnson]{Johnson}{2004a}{johnson2004identification}
{\sc Johnson, E.~G.} (2004a): ``Identification in discrete choice models
with fixed effects,'' in {\em Working paper, Bureau of Labor Statistics\/}.
Citeseer.
\harvarditem[Johnson]{Johnson}{2004b}{JohnsonThesis2004}
{\sc Johnson, E.~G.} (2004b): ``Panel Data Models With Discrete Dependent
Variables,'' Ph.D. thesis, Stanford University.
\harvarditem[Khan, Ouyang, and Tamer]{Khan, Ouyang, and
Tamer}{2021}{khan_inference_2021}
{\sc Khan, S., F.~Ouyang, {\tmpsmall\sc and} E.~Tamer} (2021): ``Inference on
{Semiparametric} {Multinomial} {Response} {Models},'' {\em Quantitative
Economics\/}, 12, 743--777.
\harvarditem[Kitazawa]{Kitazawa}{2021}{kitazawa2021transformations}
{\sc Kitazawa, Y.} (2021): ``Transformations and moment conditions for
dynamic fixed effects logit models,'' {\em Journal of Econometrics\/}.
\harvarditem[Kruiniger]{Kruiniger}{2020}{kruiniger2020further}
{\sc Kruiniger, H.} (2020): ``Further results on the estimation of dynamic
panel logit models with fixed effects,'' {\em arXiv preprint
arXiv:2010.03382\/}.
\harvarditem[Magnac]{Magnac}{2000}{magnac2000subsidised}
{\sc Magnac, T.} (2000): ``Subsidised training and youth employment:
distinguishing unobserved heterogeneity from state dependence in labour
market histories,'' {\em The Economic Journal\/}, 110(466), 805--837.
\harvarditem[Muris]{Muris}{2017}{Muris2017}
{\sc Muris, C.} (2017): ``Estimation in the Fixed-Effects Ordered Logit
Model,'' {\em The Review of Economics and Statistics\/}, 99(3), 465--477.
\harvarditem[Muris]{Muris}{2020}{muris_incomplete}
{\sc \leavevmode\rule[.5ex]{3em}{.5pt}\ {}} (2020): ``{Efficient GMM Estimation with Incomplete
Data},'' {\em The Review of Economics and Statistics\/}, 102(3), 518--530.
\harvarditem[Muris, Raposo, and Vandoros]{Muris, Raposo, and
Vandoros}{2023}{muris2020dynamic}
{\sc Muris, C., P.~Raposo, {\tmpsmall\sc and} S.~Vandoros} (2023): ``A dynamic
ordered logit model with fixed effects,'' {\em Review of Economics and
Statistics\/}, forthcoming.
\harvarditem[Newey and McFadden]{Newey and McFadden}{1994}{NeweyMcFadden94:HoE}
{\sc Newey, W.~K., {\tmpsmall\sc and} D.~McFadden} (1994): ``Large Sample
Estimation and Hypothesis Testing,'' in {\em Handbook of Econometrics\/}, ed.
by R.~F. Engle, {\tmpsmall\sc and} D.~L. McFadden, no.~4 in Handbooks in
Economics,, pp. 2111--2245. Elsevier, {N}orth-{H}olland, Amsterdam, London
and New York.
\harvarditem[Neyman and Scott]{Neyman and Scott}{1948}{neyman1948consistent}
{\sc Neyman, J., {\tmpsmall\sc and} E.~L. Scott} (1948): ``Consistent estimates
based on partially consistent observations,'' {\em Econometrica\/}, 16, 1--32.
\harvarditem[Pakes, Porter, Shepard, and Calder-Wang]{Pakes, Porter, Shepard,
and Calder-Wang}{2022}{PakesPorterShepardCalderWang2022}
{\sc Pakes, A., J.~Porter, M.~Shepard, {\tmpsmall\sc and} S.~Calder-Wang}
(2022): ``Unobserved Heterogeneity, State Dependence, and Health Plan
Choices,'' {\em Working paper. Revised Sept. 2022\/}.
\harvarditem[Rigobon]{Rigobon}{2002}{rigobon2002}
{\sc Rigobon, R.} (2002): ``The curse of non-investment grade countries,''
{\em Journal of Development Economics\/}, 69, 423--449.
\harvarditem[Shi, Shum, and Song]{Shi, Shum, and
Song}{2018}{shi_estimating_2018}
{\sc Shi, X., M.~Shum, {\tmpsmall\sc and} W.~Song} (2018): ``Estimating
{Semi}-{Parametric} {Panel} {Multinomial} {Choice} {Models} {Using} {Cyclic}
{Monotonicity},'' {\em Econometrica\/}, 86(2), 737--761.
\harvarditem[Wooldridge]{Wooldridge}{2005}{wooldridge2005}
{\sc Wooldridge, J.~M.} (2005): ``Simple solutions to the initial
conditions problem in dynamic, nonlinear panel data models with unobserved
heterogeneity,'' {\em Journal of Applied Econometrics\/}, 20(1), 39--54.
\end{thebibliography}
\newpage