Identification of Dynamic Panel Logit Models with Fixed Effects

Abstract

We show that identification in a general class of dynamic panel logit models with fixed effects is related to the truncated moment problem from the mathematics literature. We use this connection to show that the identified set for structural parameters and functionals of the distribution of latent individual effects can be characterized by a finite set of conditional moment equalities subject to a certain set of shape constraints on the model parameters. In addition to providing a general approach to identification, the new characterization can deliver informative bounds in cases where competing methods deliver no identifying restrictions, and can deliver point identification in cases where competing methods deliver partial identification. We then present an estimation and inference procedure that uses semidefinite programming methods, is applicable with continuous or discrete covariates, and can be used for models that are either point- or partially-identified. Finally, we illustrate our identification result with a number of examples, and provide an empirical application to employment dynamics using data from the National Longitudinal Survey of Youth.

Figures (7)

• Figure 1: The orthogonal decomposition of the vector $\bm p(\bm w) - \bm G(\bm w,\theta)\bm r^*(\bm w,\theta)$ into the vectors $\bm p(\bm w) - \bm p_{\bm G}(\bm w)$ and $\bm p_{\bm G}(\bm w) - \bm G(\bm w,\theta)\bm r^*(\bm w,\theta)$. Functional differencing checks if $||\bm p(\bm w) - \bm p_{\bm G}(\bm w)||=0$, but this is not sufficient to verify whether $||\bm p(\bm w) - \bm G(\bm w,\theta)\bm r^*(\bm w,\theta)||=0$.
• Figure 2: The black curve is the set of $\theta$ that satisfies $\bm v_1(\theta)^\top\bm p(0, \bm x) = 0$ and the blue curve is the set of $\theta$ that satisfies $\bm v_2(\theta)^\top\bm p(0, \bm x) = 0$. There are two values of $\theta$ that satisfy both moment restrictions. The underlying data generating process imposes $P(Y_{i0} = 0) = 1$, that the fixed effect distribution $Q_{\alpha}$ is discrete with equal mass at $-2$ and $1$, and that $(\beta_0,\gamma_0)=(0.50,0.80)$.
• Figure 3: AR(1), $T=2$.
• Figure 4: AR(1), $T=3$.
• Figure 5: AR(1), $T=3$, time trend, $n=1,000$.

Theorems & Definitions (58)

• Example 1: $AR(1)$ dynamic logit binary choice
• Example 2: $AR(p)$ dynamic logit binary choice
• Example 3: Dynamic AR(1) ordered logit model
• Example 4: Dynamic AR(1) binary choice logit-type and mixed logit errors
• Example 1: $AR(1)$ dynamic logit binary choice, continued
• Example 2: $AR(p)$ dynamic logit binary choice, continued
• Example 3: Dynamic AR(1) ordered logit, continued
• Example 4: Dynamic AR(1) with logit-type or mixed logit errors, continued
• Definition 2.1: Identified Set
• Theorem 2.1