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Identification and Estimation in a Time-Varying Endogenous Random Coefficient Panel Data Model

Ming Li

Abstract

This paper proposes a correlated random coefficient linear panel data model, where regressors can be correlated with time-varying and individual-specific random coefficients through both a fixed effect and a time-varying random shock. I develop a new panel data-based method to identify the average partial effect and the local average response function. The identification strategy employs a sufficient statistic to control for the fixed effect and a control variable for the random shock. Conditional on these two controls, the residual variation in the regressors is driven solely by the exogenous instrumental variables, and thus can be exploited to identify the parameters of interest. The constructive identification analysis leads to three-step series estimators, for which I establish rates of convergence and asymptotic normality. To illustrate the method, I estimate a heterogeneous Cobb-Douglas production function for manufacturing firms in China, finding substantial variations in output elasticities across firms that can be related to various firm characteristics.

Identification and Estimation in a Time-Varying Endogenous Random Coefficient Panel Data Model

Abstract

This paper proposes a correlated random coefficient linear panel data model, where regressors can be correlated with time-varying and individual-specific random coefficients through both a fixed effect and a time-varying random shock. I develop a new panel data-based method to identify the average partial effect and the local average response function. The identification strategy employs a sufficient statistic to control for the fixed effect and a control variable for the random shock. Conditional on these two controls, the residual variation in the regressors is driven solely by the exogenous instrumental variables, and thus can be exploited to identify the parameters of interest. The constructive identification analysis leads to three-step series estimators, for which I establish rates of convergence and asymptotic normality. To illustrate the method, I estimate a heterogeneous Cobb-Douglas production function for manufacturing firms in China, finding substantial variations in output elasticities across firms that can be related to various firm characteristics.

Paper Structure

This paper contains 22 sections, 6 theorems, 145 equations, 2 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model
  4. Identification
  5. Identification Idea
  6. Assumptions
  7. Extensions
  8. Estimation and Large Sample Theory
  9. Estimation
  10. Convergence Rates and Asymptotic Normality
  11. Empirical Application
  12. Data and Methodology
  13. Results
  14. Conclusion
  15. Proofs and Supplementary Theoretical Results
  16. ...and 7 more sections

Key Result

Theorem 1

If Assumptions assu:mono--assu:control for eta and either Assumption assu:residual variation in X or assu:more_var_X are satisfied, then the APE $\overline{b}=T^{-1}\sum_{t=1}^{T}\mathbb{E}\beta_{it}$ and the LAR function $b_{t}(x)=\mathbb{E}\left[\left.\beta_{it}\right|X_{it}=x\right]$ are both ide

Figures (2)

  • Figure 1: Distribution of Conditional Means of Output Elasticities: Textile
  • Figure 2: Distribution of Conditional Means of Output Elasticities: Other Sectors

Theorems & Definitions (19)

  • Example 1: Production Function Estimation
  • Example 2: Labor Supply
  • Example 3: Almost Ideal Demand System (AIDS)
  • Remark 1
  • Theorem 1: Identification
  • Remark 2: Compare $\boldsymbol{\widetilde{b}_{1t}\left(v,w\right)}$ with $\boldsymbol{\widehat{b}_{1t}\left(v,w\right)}$
  • Remark 3: Computational Considerations
  • Lemma 1: Convergence Rates of $\boldsymbol{\widehat{V}}$ and $\boldsymbol{\widehat{G}}$
  • Theorem 2: Convergence Rates of $\boldsymbol{\widehat{\overline{b}}}$ and $\boldsymbol{\widehat{b}\left(x\right)}$
  • Lemma 2: Asymptotic Normality of $\boldsymbol{\widehat{b}_{1}\left(v,w\right)}$
  • ...and 9 more