Introduction

• Park and Gupta (2012) develop a copula-based instrument-free method to handle endogenous regressors with insufficient non-normality and correlated with exogenous regressors.
• Implicitly assumes no correlations between the exogenous regressors and copula transformations of endogenous regressors that are used to control for endogeneity.

We consider the following linear structural regression model: \[Y_i = \mu + \alpha P_i + \beta^T W_i + \epsilon_i\] where \(P_i\) is a endogeneous regressor variable, \(W_i\) is an exogenous regressor vector and \(\epsilon_i\) is the structural error term.

• \(P_i\) and \(\epsilon_i\) are associated, and this generates the endogeneity problem.
• \(W_i\) may be associated with \(P_i\) but not with \(\epsilon_i\).

Key idea: use a copula to jointly model the correlation between \(P_i\) and \(\epsilon_i\). Marginals are not restricted by the joint distribution. Using information contained in the observed data, marginals of the endogenous regressor and the error term are first obtained respectively.

Model proposed

• Consider that \((P_i^\star, \epsilon_i^\star) = [\Phi^{-1}\{F_p(P_i)\}, \Phi^{-1}\{F_{\epsilon}(\epsilon_i)\}] \sim N_2(0, \rho)\).
• Further, assume \(\epsilon \sim N(0, \sigma_\epsilon^2)\).

The structural error can be expressed as:

\[\epsilon_i = \sigma_\epsilon \epsilon_i^\star = \sigma_\epsilon (\rho P_i^\star + \sqrt{1-\rho^2}w_i)\] and the structural regression model can be re-written as

\[\begin{equation} \begin{aligned} Y_i &= \mu + \alpha P_i + \beta^T W_i + \sigma_\epsilon (\rho P_i^\star + \sqrt{1-\rho^2}w_i)\\ Y_i &= \mu + \alpha P_i + \sigma_\epsilon\rho P_i^\star + \beta^T W_i + \sigma_\epsilon \sqrt{1 - \rho^2} w_i. \end{aligned} \end{equation}\]

• Can’t allow for \(P_i \sim N\)!
• Since \(W_i\) is exogeneous, we have \[\begin{equation} \begin{aligned} cov(W_i, \epsilon_i) &= 0\\ 0 &= cov(W_i, \sigma_\epsilon (\rho P_i^\star + \sqrt{1-\rho^2}w_i)) \\ &= \rho cov(W_i, P_i^\star) + \sqrt{1-\rho^2} cov(W_i, w_i) = 0. \end{aligned} \end{equation}\] Correlation of \(W_i\) and \(P_i^\star\) would induce correlation between \(W_i\) and \(w_i\), yielding inconsistent estimates.

A two-step regression approach

• Want to relax uncorrelatedness assumption between \(W_i\) and \(P_i^\star\)
• Want to relax non-normality assumption on \(P_i\)

Jointly model endogenous regressor \(P_i\), the correlated exogenous variable, \(W_i\), and the structural error term, \(\epsilon_i\), using the Gaussian copula model: \[\begin{equation} \left(\begin{array}{c} P_i^* \\ W_i^* \\ \epsilon_i^* \end{array}\right) = \left(\begin{array}{c} \Phi^{-1}\{F_P(P_i) \} \\ \Phi^{-1}\{F_W(W_i) \} \\ \Phi^{-1}\{F_\epsilon(\epsilon_i) \} \end{array}\right) \sim N\left(\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{ccc} 1 & \rho_{p w} & \rho_{p \epsilon} \\ \rho_{p w} & 1 & 0 \\ \rho_{p \epsilon} & 0 & 1 \end{array}\right]\right) \end{equation}\]

Rewriting the structural regression model

\[\begin{equation} \begin{aligned} P_i^\star &= \rho_{pw}W_i^\star + \sqrt{1 - \rho_{pw}^2} w_{2, i} \\ &= \rho_{pw}W_i^\star + \epsilon_i \quad(1) \\ Y_i &= \mu+ \alpha P_i + \beta W_i + \frac{\sigma_{\epsilon} \rho_{p \epsilon}}{1-\rho_{p w}^2} \epsilon_i+\sigma_{\epsilon} \sqrt{1-\rho_{p \epsilon}^2-\frac{\rho_{p w}^2 \rho_{p \epsilon}^2}{1-\rho_{p w}^2}} \cdot w_{3, i} \quad(2) \end{aligned} \end{equation}\]

Two step regression routine

• Adding the estimate of the error term \(\epsilon\) from the first stage regression as a generated regressor to the outcome regression instead of using \(P_i^\star\) and \(W_i^\star\).
• The new error term \(w_{3, i}\) is uncorrelated with all regressors in step (2) so long as at least one non-normal exogenous covariate is supplied. Assuming non-normality of exogenous covariates is a less strict requirement. If needed, we can always add small perturbation.
• We can get consistent estimates of model parameters from step (2)

Key improvements

• Jointly model endogenous \(P_i\), exogenous \(W_i\) and structural error \(\epsilon_i\) as a Gaussian copula.
• Can allow for normally distributed \(P_i\) as long as one non-normal exogenous \(W_i\) exists.
• Exogenous \(W_i\) can be correlated with linear functions of copula-transformed endogenous \(P_i\).