[Paper Review] Estimation in a Generalization of Bivariate Probit Models with Dummy Endogenous Regressors

Sukjin Han, Sungwon Lee|arXiv (Cornell University)|Aug 17, 2018

Statistical Methods and Inference30 references4 citations

TL;DR

This paper proposes a semiparametric estimation framework for bivariate threshold crossing models with dummy endogenous regressors, using parametric copulas and nonparametric marginal distributions to ensure robustness against mis-specification of the joint error distribution. The method achieves root-n asymptotic normality for sieve maximum likelihood estimators, enabling valid inference on structural parameters and the average treatment effect (ATE), with simulations showing that ATE estimates are highly sensitive to parametric assumptions but robust under the proposed semiparametric approach.

ABSTRACT

The purpose of this paper is to provide guidelines for empirical researchers who use a class of bivariate threshold crossing models with dummy endogenous variables. A common practice employed by the researchers is the specification of the joint distribution of the unobservables as a bivariate normal distribution, which results in a bivariate probit model. To address the problem of misspecification in this practice, we propose an easy-to-implement semiparametric estimation framework with parametric copula and nonparametric marginal distributions. We establish asymptotic theory, including root-n normality, for the sieve maximum likelihood estimators that can be used to conduct inference on the individual structural parameters and the average treatment effect (ATE). In order to show the practical relevance of the proposed framework, we conduct a sensitivity analysis via extensive Monte Carlo simulation exercises. The results suggest that the estimates of the parameters, especially the ATE, are sensitive to parametric specification, while semiparametric estimation exhibits robustness to underlying data generating processes. We then provide an empirical illustration where we estimate the effect of health insurance on doctor visits. In this paper, we also show that the absence of excluded instruments may result in identification failure, in contrast to what some practitioners believe.

Motivation & Objective

• Address the common empirical practice of assuming bivariate normality for unobservables in bivariate probit models, which may lead to severe misspecification.
• Provide a robust estimation framework that maintains point identification while allowing flexibility in the marginal distributions of unobservables.
• Investigate the sensitivity of average treatment effect (ATE) estimates to parametric assumptions on the joint distribution of errors.
• Demonstrate that the absence of excluded instruments can lead to identification failure, challenging a common practitioner belief.
• Establish asymptotic theory for sieve maximum likelihood estimators in a semiparametric setting with parametric copulas and nonparametric margins.

Proposed method

• Propose a semiparametric model with a parametric copula function for the dependence structure of unobservables (ε, ν), and nonparametric marginal distributions for ε and ν.
• Use sieve maximum likelihood (ML) estimation to handle infinite-dimensional parameters in the nonparametric margins, enabling consistent and asymptotically normal estimation.
• Apply copula-based dependence modeling (e.g., Gaussian, Clayton, Gumbel, Frank) to flexibly capture the dependence between the structural error and the endogenous regressor error.
• Establish root-n asymptotic normality for the sieve ML estimators of structural parameters and ATE under regularity conditions and an exclusion restriction.
• Conduct sensitivity analysis via Monte Carlo simulations under various data-generating processes, including correct and misspecified marginal distributions.
• Use bootstrap confidence intervals to assess coverage performance of inference procedures on ATE and structural parameters.

Experimental results

Research questions

• RQ1 How sensitive are average treatment effect (ATE) estimates to the parametric assumption on the joint distribution of unobservables in bivariate probit models?
• RQ2 Can a semiparametric framework with nonparametric margins and parametric copulas achieve robust estimation of ATE and structural parameters under model misspecification?
• RQ3 Under what conditions is identification of the ATE and structural parameters achieved in triangular models with dummy endogenous regressors?
• RQ4 Does the absence of excluded instruments lead to identification failure in such models, contrary to common practitioner assumptions?
• RQ5 How do bootstrap confidence intervals perform in terms of coverage probability for ATE and structural parameters in finite samples?

Key findings

• ATE estimates from parametric models (e.g., bivariate probit with normal errors) are highly sensitive to incorrect parametric assumptions on the joint distribution of unobservables, with RMSE values for ATE reaching up to 0.0902 under misspecification.
• The semiparametric estimator with nonparametric margins and parametric copulas exhibits strong robustness, with RMSE for ATE as low as 0.0040 under misspecified marginals (mixture of normals) and correct copula.
• Bootstrap confidence intervals for ATE achieve nominal coverage (95%) with 90.5% probability under normal approximation and 93.0% under percentile bootstrap, indicating reliable inference.
• When the true marginal distribution is a mixture of normals but assumed to be normal, parametric estimation yields large biases (e.g., -0.7069 for δ₁) and high RMSE (0.1699), while semiparametric estimation reduces bias and RMSE significantly.
• The semiparametric framework maintains good performance across different copula families (Gaussian, Clayton, Gumbel, Frank), with RMSE for ATE consistently below 0.01 in correctly specified cases.
• Identification failure occurs when excluded instruments are absent, contradicting the belief among some practitioners that identification is guaranteed under triangular structure alone.