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Testing for Endogeneity: A Moment-Based Bayesian Approach

Siddhartha Chib, Minchul Shin, Anna Simoni

Abstract

A standard assumption in the Bayesian estimation of linear regression models is that the regressors are exogenous in the sense that they are uncorrelated with the model error term. In practice, however, this assumption can be invalid. In this paper, using the exponentially tilted empirical likelihood framework, we develop a Bayes factor test for endogeneity that compares a base model that is correctly specified under exogeneity but misspecified under endogeneity against an extended model that is correctly specified in either case. We provide a comprehensive study of the log-marginal exponentially tilted empirical likelihood. We demonstrate that our testing procedure is consistent from a frequentist point of view: as the sample grows, it almost surely selects the base model if and only if the regressors are exogenous, and the extended model if and only if the regressors are endogenous. The methods are illustrated with simulated data, and problems concerning the causal effect of automobile prices on automobile demand and the causal effect of potentially endogenous airplane ticket prices on passenger volume.

Testing for Endogeneity: A Moment-Based Bayesian Approach

Abstract

A standard assumption in the Bayesian estimation of linear regression models is that the regressors are exogenous in the sense that they are uncorrelated with the model error term. In practice, however, this assumption can be invalid. In this paper, using the exponentially tilted empirical likelihood framework, we develop a Bayes factor test for endogeneity that compares a base model that is correctly specified under exogeneity but misspecified under endogeneity against an extended model that is correctly specified in either case. We provide a comprehensive study of the log-marginal exponentially tilted empirical likelihood. We demonstrate that our testing procedure is consistent from a frequentist point of view: as the sample grows, it almost surely selects the base model if and only if the regressors are exogenous, and the extended model if and only if the regressors are endogenous. The methods are illustrated with simulated data, and problems concerning the causal effect of automobile prices on automobile demand and the causal effect of potentially endogenous airplane ticket prices on passenger volume.
Paper Structure (51 sections, 35 theorems, 251 equations, 4 figures, 5 tables)

This paper contains 51 sections, 35 theorems, 251 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moment restrictions and feasible distributions
  4. Sample ETEL weights, tilted sample distribution, and likelihood
  5. Dual representation and log-ETEL identities
  6. Population KL projection, exponential tilting, and pseudo-true values
  7. Models
  8. Data, regression structure, and target parameter
  9. Base model $\mathcal{M}_b$ (possibly misspecified)
  10. Extended model $\mathcal{M}_e$ (correctly specified)
  11. Numerical illustration
  12. Testing procedure
  13. Bayes factor
  14. Rationale
  15. Consistency of the testing procedure
  16. ...and 36 more sections

Key Result

Theorem 4.1

Suppose that there is a ${ \if@compatibility \mathchar"0112 {} \mathchar"0112 }\in\Theta$ such that $\mathbf{E}[{ \if@compatibility \mathchar"0122 {} \mathchar"0122 }_{i}({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })(z_{1,i}'z_{2,i}')']= 0$ and that Assumption Ass_absolut Then, (i) is equivalent to (ii), and (iii) is equivalent to (iv).

Figures (4)

  • Figure 1: Base model under neglected endogeneity: Marginal posterior densities of ${ \if@compatibility \mathchar"010C {} \mathchar"010C }$ for different sample sizes. Posterior mean is indicated by dashed vertical line.
  • Figure 2: Extended model ($x_{i}$ moment is inactive): Marginal posterior densities of ${ \if@compatibility \mathchar"010C {} \mathchar"010C }$ for different sample sizes. Posterior mean is indicated by dashed vertical line.
  • Figure 3: Extended model under neglected endogeneity: Marginal posterior densities of $v = \text{cov}(x,{ \if@compatibility \mathchar"0122 {} \mathchar"0122 })$ for different sample sizes. Posterior mean is indicated by dashed vertical line.
  • Figure 4: BLP models: Marginal posterior distributions of the coefficient on the price variable, ${ \if@compatibility \mathchar"010C {} \mathchar"010C }$. Posterior mean and standard deviation of ${ \if@compatibility \mathchar"010C {} \mathchar"010C }$ are -0.089 and 0.004, respectively, for the base model with the original BLP (linear) specification, while they are -0.087 and 0.004, respectively, with the augmented BLP (nonlinear) specification. For the extended model, posterior mean and standard deviation of ${ \if@compatibility \mathchar"010C {} \mathchar"010C }$ are -0.183 and 0.015, respectively, for the linear specification and -0.143 and 0.009, respectively, for the nonlinear specification.

Theorems & Definitions (35)

  • Theorem 4.1
  • Theorem 4.2: Base model: stochastic expansion of log-ETEL
  • Theorem 4.3: Extended model.
  • Corollary 4.1: Base model.
  • Corollary 4.2: Extended model.
  • Theorem 4.4
  • Theorem 4.5
  • Theorem F.1: Stochastic LAN in the base model.
  • Theorem F.2: Stochastic LAN in the base model under exogeneity
  • Theorem F.3: Stochastic LAN in the extended model.
  • ...and 25 more