[Paper Review] Transition Probabilities and Moment Restrictions in Dynamic Fixed Effects Logit Models

Kevin Dano|arXiv (Cornell University)|Feb 28, 2023

Energy, Environment, Economic Growth6 citations

TL;DR

This paper introduces a systematic analytical method to derive moment restrictions in dynamic fixed effects logit models with fixed effects and strictly exogenous regressors, leveraging the rational function structure of logit transition probabilities. It enables consistent estimation at the √N rate without symbolic computation, extending to models of arbitrary lag order and multinomial or vector autoregressive forms.

ABSTRACT

Dynamic logit models are popular tools in economics to measure state dependence. This paper introduces a new method to derive moment restrictions in a large class of such models with strictly exogenous regressors and fixed effects. We exploit the common structure of logit-type transition probabilities and elementary properties of rational fractions, to formulate a systematic procedure that scales naturally with model complexity (e.g the lag order or the number of observed time periods). We detail the construction of moment restrictions in binary response models of arbitrary lag order as well as first-order panel vector autoregressions and dynamic multinomial logit models. Identification of common parameters and average marginal effects is also discussed for the binary response case. Finally, we illustrate our results by studying the dynamics of drug consumption amongst young people inspired by Deza (2015).

Motivation & Objective

• To address limitations in existing conditional likelihood and GMM approaches for dynamic fixed effects logit models, particularly in handling higher-order lags and continuous regressors.
• To develop a systematic, analytic procedure for deriving moment restrictions that scales naturally with model complexity, such as lag order or time periods.
• To enable efficient estimation of common parameters and average marginal effects in binary response models with fixed effects and strictly exogenous regressors.
• To extend the applicability of moment restriction methods beyond AR(1) models to general dynamic logit specifications, including panel VARs and multinomial logit models.
• To provide a framework that avoids reliance on numerical experimentation or symbolic computation, unlike prior approaches such as those using functional differencing or symbolic algebra tools.

Proposed method

• Exploits the common algebraic structure of logit-type transition probabilities as rational functions of the linear index, enabling systematic manipulation.
• Uses elementary properties of rational fractions to derive moment conditions that are analytically tractable and do not require numerical integration or symbolic computation.
• Derives moment restrictions by conditioning on observed covariates and leveraging the conditional independence structure induced by fixed effects.
• Applies the method to construct moment conditions for binary response models of arbitrary lag order, first-order panel VARs, and dynamic multinomial logit models.
• Derives identifying restrictions for common parameters and average marginal effects in binary response models via the derived moment conditions.
• Validates the method by showing that the resulting score function satisfies the efficiency conditions of Newey (1990), establishing semiparametric efficiency.

Experimental results

Research questions

• RQ1 How can moment restrictions be systematically derived in dynamic fixed effects logit models with arbitrary lag order and fixed effects, without relying on numerical or symbolic computation?
• RQ2 Can the rational function structure of logit transition probabilities be exploited to generate analytically tractable moment conditions for general dynamic panel models?
• RQ3 What are the implications of this method for estimation efficiency, particularly in models with continuous regressors or higher-order lags where existing methods fail?
• RQ4 To what extent can this approach be extended to multinomial and vector autoregressive dynamic logit models?
• RQ5 Does the derived moment condition structure lead to semiparametric efficiency in the AR(1) model, as confirmed by the efficiency bound calculation?

Key findings

• The method successfully derives moment restrictions for binary response models of arbitrary lag order by exploiting the rational function form of logit transition probabilities.
• The derived moment conditions are analytically closed-form and do not require symbolic computation or numerical integration, enabling scalable application to complex models.
• The method achieves √N estimation rates for parameters, even with continuous regressors, overcoming a key limitation of the conditional likelihood approach.
• For the AR(1) model, the derived score function is shown to be semiparametrically efficient, matching the theoretical bound from Newey (1990).
• The method extends naturally to panel vector autoregressions and dynamic multinomial logit models, providing a unified framework for moment restriction derivation.
• The paper demonstrates that the moment conditions are valid and consistent across different initial conditions (e.g., Y₀ = 0 or Y₀ = 1), ensuring robustness in practice.