Essays in Econometrics and Finance

Abstract

Binary choice models can be easily estimated (using, e.g. maximum likelihood estimation) when the distribution of the latent error is known, as in Logit or Probit. In contrast, most estimators with unknown error distribution (e.g., maximum score, maximum rank correlation, or Klein-Spady) are computationally difficult or numerically unstable, making estimation impractical with more than a few regressors. The first chapter proposes an estimator that is convex at each iteration, and so is numerically well behaved even with many regressors and large sample sizes. The proposed estimator, which is root-n consistent and asymptotically normal, is based on batch gradient descent, while using a sieve to estimate the unknown error distribution function. Simulations show that the estimator has lower mean bias and root mean squared error than Klein-Spady estimator. It also requires less time to compute. The second chapter discusses the same estimator in high dimensional setting. The estimator is consistent with rate lower than root-n when the number of regressors grows slower than the number of observations and asymptotic normal when the square of the number of regressors grows slower than the number of observations. Both theory and simulation show that higher learning rate is needed with higher number of regressors. The third chapter provides an application of the proposed estimator to bankruptcy prediction. With more than 20 regressors, the proposed estimator performs better than logistic regression in terms of Area Under the Receiver Operating Characteristics using firm data one year or two years prior to bankruptcy, but worse than logistic regression using firm data three years prior to bankruptcy.