Informational Content of Special Regressors in Heteroskedastic Binary Response Models

We quantify the identifying power of special regressors in heteroskedastic binary regressions with median-independent or conditionally symmetric errors. We measure the identifying power using two criteria: the set of regressor values that help point identify coefficients in latent payoffs as in (Manski 1988); and the Fisher information of coefficients as in (Chamberlain 1986). We find for median-independent errors, requiring one of the regressors to be “special" (in a sense similar to (Lewbel 2000)) does not add to the identifying power or the information for coefficients. Nonetheless it does help identify the error distribution and the average structural function. For conditionally symmetric errors, the presence of a special regressor improves the identifying power by the criterion in (Manski 1988), and the Fisher information for coefficients is strictly positive under mild conditions. We propose a new estimator for coefficients that converges at the parametric rate under symmetric errors and a special regressor, and report its decent performance in small samples through simulations.