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Bounds on Average Effects in Discrete Choice Panel Data Models

Cavit Pakel, Martin Weidner

TL;DR

This paper tackles the problem of partially identifying average effects in discrete choice panel data with fixed effects, where short panels create the incidental parameter challenge. It develops a general outer-bounds approach that avoids the curse of dimensionality by constructing bound functions L and U through linear programs, enabling estimation of bounds at the parametric rate and providing asymptotically valid confidence intervals. The authors contrast these outer bounds with the sharp identified set, showing linear dependence on observable probabilities and robust inference even with continuous covariates or high-dimensional conditioning. They also address estimation of common parameters via cross-fitting, present two inference methods to incorporate β-uncertainty, and supply simulation and empirical evidence (notably an analysis of female labor supply) demonstrating practicality and reliability. The approach is broadly applicable across static/dynamic, logit/random-coefficient, and various model specifications, and offers extensions to set-identified β and probit models, among others.

Abstract

In discrete choice panel data, estimation of average effects is crucial for quantifying the effect of covariates, and for policy evaluation and counterfactual analysis. However, in short panels with individual-specific effects, challenges arise due to partial identification and the incidental parameter problem. In particular, estimating the sharp identified set on average effects becomes impractical when covariates have large support sets, such as when they are continuous. This paper proposes a method for estimating outer bounds on the identified set of average effects, which are easy to construct, converge at the parametric rate, and remain computationally feasible even for moderately large samples. Asymptotically valid confidence intervals are also provided.

Bounds on Average Effects in Discrete Choice Panel Data Models

TL;DR

This paper tackles the problem of partially identifying average effects in discrete choice panel data with fixed effects, where short panels create the incidental parameter challenge. It develops a general outer-bounds approach that avoids the curse of dimensionality by constructing bound functions L and U through linear programs, enabling estimation of bounds at the parametric rate and providing asymptotically valid confidence intervals. The authors contrast these outer bounds with the sharp identified set, showing linear dependence on observable probabilities and robust inference even with continuous covariates or high-dimensional conditioning. They also address estimation of common parameters via cross-fitting, present two inference methods to incorporate β-uncertainty, and supply simulation and empirical evidence (notably an analysis of female labor supply) demonstrating practicality and reliability. The approach is broadly applicable across static/dynamic, logit/random-coefficient, and various model specifications, and offers extensions to set-identified β and probit models, among others.

Abstract

In discrete choice panel data, estimation of average effects is crucial for quantifying the effect of covariates, and for policy evaluation and counterfactual analysis. However, in short panels with individual-specific effects, challenges arise due to partial identification and the incidental parameter problem. In particular, estimating the sharp identified set on average effects becomes impractical when covariates have large support sets, such as when they are continuous. This paper proposes a method for estimating outer bounds on the identified set of average effects, which are easy to construct, converge at the parametric rate, and remain computationally feasible even for moderately large samples. Asymptotically valid confidence intervals are also provided.
Paper Structure (34 sections, 5 theorems, 103 equations, 19 figures, 1 table)

This paper contains 34 sections, 5 theorems, 103 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bounds on average effects
  3. Construction of the bounds
  4. Baseline linear program
  5. Uniform linear program
  6. Practical recommendations
  7. Further discussion of the bounds
  8. An illustrative example
  9. Comparison to the identified set
  10. Model-specific comparisons
  11. Accounting for estimated common parameters
  12. Consistency and convergence rate of the estimated bounds
  13. First inference method
  14. Second inference method
  15. Simulation evidence
  16. ...and 19 more sections

Key Result

Theorem 1

Let Assumption ass:MAIN hold, and let $L, U : {\cal Z} \times {\cal Y} \times {\cal B} \rightarrow [b_{\min}, b_{\max}]$ satisfy equation eq:boundcondition for $\beta= \beta_0$ and for all $z \in {\cal Z}$, $a \in {\cal A}$. Let $\overline{m}$, $\widehat{L}$, $\widehat{U}$ be as defined in AverageEf Furthermore, assume that ${\rm Var}\left[L(Z_i,Y_i, \beta_0 )\right]>0$, and ${\rm Var}\left[U(Z_i,

Figures (19)

  • Figure 1: Sample quantiles of estimates of the outer bounds and the identified set. The DGP is $Y_{it} = 1\{ X_{it}\beta + A_{i} \geq \varepsilon_{it}\}$ where $A_{i} \sim N( T^{-1} \sum_{t=1}^T X_{it} - 1/2, 1 )$, $X_{it} = x_{it}/(|\mathcal{X}|-1)$, $x_{it}$ is discrete uniform with support $[0,|\mathcal{X}| -1]$, and $\varepsilon_{it}\sim \mathrm{Logit}(0,1)$. The average effect under consideration is $\mathbb{E}[Y_{it} \, | \, X_{it}=1,A_i,\beta_0] - \mathbb{E}[Y_{it} \, | \,X_{it}=0,A_i,\beta_0]$. For each $\beta_0 \in [-2,2]$, the quantiles are calculated across 1000 replications of panels with $T=2$ and $n=200$. The left panel contains the results for $|\mathcal{X}| = 6$ whereas the results for $|\mathcal{X}| =12$ are presented in the right panel.
  • Figure 2: Comparison of the outer bounds and the identified set for the static logit model $Y_{it} = 1\left\{ X_{it}\beta + A_i \geq \varepsilon_{it} \right\}$, where $\varepsilon_{it} \sim {\rm Logit} (0,1)$ and $A_i \sim N(0,1)$. Results for each $\beta_0\in[-2,2]$ are based on 1000 replications of panels with cross-section size $n=1000$. Reported outer bounds are the cross-replication averages. Left panel: single discrete covariate $X_{it} = 1\left\{ A_i \geq \eta_{it} \right\}$ where $\eta_{it} \sim N(0,1)$. The average effect of interest is based on \ref{['marg1']} with $(x_1,x_2)=(1,0)$. Right panel: single continuous covariate, $X_{it} \sim N(A_i,1)$. The average effect of interest is based on \ref{['marg2']}.
  • Figure 3: Comparison of the outer bounds and the identified set for the random coefficient logit model $Y_{it} = 1\left\{ X_{it} A_{2,i} + A_{1,i} \geq \varepsilon_{it} \right\}$, where $\varepsilon_{it} \sim {\rm Logit}(0,1)$, $A_{1,i} \sim N(0,1/\sqrt{2})$, $A_{2,i} \sim N(A_2,1/\sqrt{2})$, $X_{it} = 1\left\{ A_{1,i} \geq \eta_{it} \right\}$ and $\eta_{it} \sim N(0,1)$. The average effect of interest is based on \ref{['margrc']}. Results for each $A_2\in[-2,2]$ are based on 1000 replications of panels with cross-section size $n=1000$. Reported outer bounds are the cross-replication averages.
  • Figure 4: Comparison of the outer bounds and the identified set for the dynamic logit model $Y_{it} = 1\left\{ Y_{i,t-1}\gamma + X_{it} \beta + A_i \geq \varepsilon_{it} \right\}$, where $\varepsilon_{it} \sim {\rm Logit} (0,1)$, $A_i \sim N(0,1)$, $X_{it}\sim N(A_i,1)$, and $Y_{i0} = 1\{X_{i0}\beta + A_i \geq \varepsilon_{i0} \}$. Results for each $\beta_0\in[-2,2]$ are based on 1000 replications of panels with cross-section size $n=1000$. Reported outer bounds are the cross-replication averages. The average effect of interest is based on \ref{['eq:aped']}.
  • Figure 5: Comparison of the outer bounds and the identified set for the random coefficient dynamic logit model $Y_{it} = 1\left\{ Y_{i,t-1} A_{2,i} + A_{1,i} \geq \varepsilon_{it} \right\}$, where $\varepsilon_{it} \sim {\rm Logit}(0,1)$, $A_{1,i} \sim N(0,1/\sqrt{2})$, $A_{2,i} \sim N(A_2,1/\sqrt{2})$, and $Y_{i0} = 1\left\{ A_{1,i} \geq \varepsilon_{i0} \right\}$. The average effect of interest is based on \ref{['eq:aerdc']}. Results for each $A_2\in[-2,2]$ are based on 1000 replications of panels with cross-section size $n=1000$. Reported outer bounds are the cross-replication averages.
  • ...and 14 more figures

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Example 1: continued
  • Example 2: continued