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Dynamic Ordered Panel Logit Models
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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 arellano_tests_1991 and 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 (kitazawa2021transformations and 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. Chamberlain1985 and 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 (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. das_panel_1999, JohnsonThesis2004, Baetschmann2015, Muris2017, and bmp2021). Unfortunately, the dynamic ordered logit model cannot be similarly reduced to a dynamic binary choice model (see 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 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 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 muris2020dynamic builds on methods for dynamic binary choice models in honore2000panel by restricting how past values of the dependent variable enter the model. In particular, in 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 (contoyannis_dynamics_2004, albarran_estimation_2019) or requires a large number of time periods for consistency (carro_state_2014, fernandez-val_evaluating_2017). An earlier version of 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 honore_nonlinear_2002, arellano2003discrete, and 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 das_panel_1999, JohnsonThesis2004, Baetschmann2015, and Muris2017; results for static and dynamic binomial and multinomial choice models are in chamberlain_analysis_1980, honore2000panel, magnac2000subsidised, shi_estimating_2018, aguirregabiriaSufficientStatisticsUnobserved2021, aguirregabiriaIdentificationAverageMarginal, PakesPorterShepardCalderWang2022 and 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) introduces the model and the moment conditions that are free of fixed effects. Section (ref) presents identification results for the common model parameters based on those moment conditions. Section (ref) discusses how to use the moment conditions for estimation and inference. Section (ref) 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) concludes the paper. The appendix provides proofs and further computational details.
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.
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), 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:
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
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 (ref) and (ref) 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), but otherwise impose (ref) 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 (ref) 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).}
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
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
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
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.
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
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 (ref). The model assumptions in the last subsection are therefore sufficient to evaluate the conditional expectation in (ref).
If we can establish the conditional moment condition (ref) then, by the law of iterated expectations, we also have the unconditional moment conditions
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 (neyman1948consistent), because the moment condition (ref) 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 (ref) 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 (ref) 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 johnson2004identification and davezies2022fixed, and also for a mixture of logistics (briefly discussed in 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 (ref). 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 honore2020dynamic. Indeed, for the binary choice case ($Q=2$), our moment functions below exactly coincide with those in honore2020dynamic, and we refer to that paper for more details on the derivation, which is closely related to the functional differencing method in 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 (ref) holds -- and on their implications for the identification and estimation of ${\Greekmath 0112}$.
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 (ref) 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 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
The moment functions presented below have the property that
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 Muris2017 and 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 honore2020dynamic, which builds on ideas in bonhomme2012functional. Once derived, we can prove by hand that the moment functions are valid (proof of Theorem (ref) 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 (ref) help to appreciate some aspects of the structure of the moment functions (see also Lemma (ref) 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).
For $q_1,q_3 \in \{1,\ldots,Q-1\}$ and $q_2 \in \{2,\ldots,Q-1\}$, we define
Any valid moment function satisfying (ref) can be multiplied by an arbitrary constant and remain a valid moment function. In (ref) 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
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 (ref).
Equation (ref) 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 (ref), 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
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
Notice that the moment functions for $q_2=1$ and $q_2=Q$ also satisfy (ref), 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 Muris2017 and muris2020dynamic, and they also incorporate the moment functions for the dynamic binary choice model in honore2020dynamic as special cases.
Together, the formulas (ref), (ref), and (ref) 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 (ref) 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 $A$.
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
for the model probabilities $p_{y_0}(y,x,{\Greekmath 0112}^0,{\Greekmath 010B})$ given by (ref), 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 honore2020dynamic and kitazawa2021transformations, and the completeness of those binary choice moment conditions is discussed in kruiniger2020further and dobronyi2021identification.
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 (ref), 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
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 (ref) equals the moment function in equation (ref). For larger values of $T$, the moment function in (ref) 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 (ref) 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
When $T$ equals 3 and $(t,s,r)=(1,2,3)$, these moment functions agree with the ones in equations (ref) and (ref), where all the arguments were made explicit. For $r=s+1$, analogous to (ref) and (ref) for $T=3$, the two moment conditions in (ref) for $T \geq 3$ can be obtained from (ref) 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 (ref) as our master equation, which summarizes all the moment conditions provided in this paper. In (ref) we can choose more general $r\geq s+1$, but otherwise the structure of (ref) can be derived from (ref).
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 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 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 (ref).
The following theorem establishes that the moment functions in (ref) and (ref) do indeed deliver valid moment conditions.
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). The moment functions in (ref) 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) 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 honore2020dynamic. By contrast, the moment functions in (ref) 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) 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 $s>r+1$.
Theorem (ref) states that the moment functions in (ref) and (ref) 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 (ref) 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 (ref) 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
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 (ref) and (ref). 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) 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 (ref), 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, (ref) suggests that the total number of available moment conditions for each value of the covariates $x$ and initial conditions $y_0$ is equal to
As explained in Section (ref), 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 (ref), 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 (ref). 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 (ref), and thereby check (ref) numerically. In Appendix (ref) we furthermore show that the total number of linearly independent conditional moment conditions for our model is at least the number obtained in (ref), 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 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 kruiniger2020further and 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).
The model probabilities in (ref) and the moment functions in (ref) and (ref) 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 (ref) 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 (ref) for this single index, but Theorem (ref) 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 (ref) as
The moment functions in (ref) and (ref) 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 (ref)) 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) remains valid under that replacement. } In other words, if we replace the latent variable specification in (ref) by
for an arbitrary function $z(\cdot,\cdot,\cdot)$, then the moment functions (ref), (ref), (ref), and Theorem (ref) remain fully valid.
We believe that the linear specification in (ref) 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:
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 (ref), we do not consider these more general models further.
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). All results in this section impose the following model assumption. \newtheorem{IDassumption}{Assumption}
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 (ref) 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). 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) 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.
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 (ref) reads
Theorem (ref) 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.
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) to the data shifted by one time period.
In addition to Assumption (ref), 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 (ref) 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 $Y_0$.
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 (ref) with $q_1=q_2=q_3=1$, but now for general regressor values
For $k\in \{1,\ldots ,K\}$ we define
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). The following proposition formalizes this.
The proof is given in the appendix. Again, in addition to Assumption (ref), the additional condition in Proposition (ref) simply guarantees that the conditional expectation in (ref) is well-defined.
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 (ref) with $q_2=q_3=1$ and $q_1 \in \{2,\ldots,Q-1\}$ can be written as
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.
The proof is given in the appendix.
Combining Proposition (ref), (ref), and (ref), 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) 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.
The moment conditions in Section\ (ref) 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) in order to turn these moment conditions into an estimator. The problem with such an approach is that the conditioning set in Proposition (ref) will often have probability 0. Alternatively, one can form a set of unconditional moment functions by constructing
where the vector-valued function, $m_{Y_{i0}}$, is composed of linear combinations of the moment functions in (ref), (ref), and (ref), 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), it is necessary to normalize one of the $Q$ elements of ${\Greekmath 010D} $ and one of the $Q-1$ elements of ${\Greekmath 0115} $.}
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 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) 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, 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, 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 (ref), (ref), and (ref). 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 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.
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), 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.
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 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), (ref)\ and (ref). For comparison, we also include the results for the correlated random effects estimator (cf. 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), (ref)\ and (ref) 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) 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.
Tables (ref)-(ref) 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), 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) and (ref) 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), (ref) and (ref) 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.}
In this section, we illustrate the value of the moment conditions derived in this paper in an empirical illustration inspired by 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), 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) 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) 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), (ref)\ and (ref).
This paper has extended the analysis in honore2020dynamic to provide conditional moment conditions for panel data fixed effects versions of the dynamic ordered logit models like the one considered in 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 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 bonhomme2012functional to logit-type models.
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