Informational Content of Special Regressors in Heteroskedastic Binary Response Models1

We quantify the informational content of special regressors in heteroskedastic binary regressions with median-independent or conditionally symmetric errors. We measure informational content by two criteria: the set of regressor values that help point identify coe¢ cients in latent payo¤s as in (Manski 1988); and the Fisher information of coe¢ cients as in (Chamberlain 1986). We …nd for median-independent errors, requiring one of the regressors to be “special" in a sense similar to (Lewbel 2000) does not add the identifying power or the information for coe¢ cients. Nonetheless it does help identify the error distribution and the average structural function. For conditionally symmetric errors (which were shown to add no informational content by (Manski 1988) and (Zheng 1995) without special regressors), the presence of a special regressor improves the identifying power by the criterion of (Manski 1988), and the Fisher information for coe¢ cients is strictly positive under mild conditions. We propose a new estimator for coe¢ cients that converges at the parametric rate under symmetric errors and a special regressor, and report its decent performance in small samples through simulations.