Asymmetric AdaBoost for Maximum Score Estimation of High-dimensional Binary Choice Regression Models

Carter Hill's numerous contributions (books and articles) in econometrics stand out especially in pedagogy. An important aspect of his pedagogy is to integrate "theory and practice'' of econometrics, as coined into the titles of his popular books. The new methodology we propose in this paper is consistent with these contributions of Carter Hill. In particular, we bring the maximum score regression of \citet{Manski1975,Manski1985} to high dimension in theory and show that the "Asymmetric AdaBoost'' provides the algorithmic implementation of the high dimensional maximum score regression in practice. Recent advances in machine learning research have not only expanded the horizon of econometrics by providing new methods but also provided the algorithmic aspects of many of traditional econometrics methods. For example, Adaptive Boosting (AdaBoost) introduced by \citet{Freund1996} has gained enormous success in binary/discrete classification/prediction. In this paper, we introduce the ``Asymmetric AdaBoost'' and relate it to the maximum score regression in the algorithmic perspective. The Asymmetric AdaBoost solves high-dimensional binary classification/prediction problem with state-dependent loss functions. Asymmetric AdaBoost produces a nonparametric classifier via minimizing the "asymmetric exponential risk'' which is a convex surrogate of the non-convex 0-1 risk. The convex risk function gives a huge computational advantage over non-convex risk functions of \citet{Manski1975,Manski1985} especially when the data is high-dimensional. The resulting nonparametric classifier is more robust than the parametric classifiers whose performance depends on the correct specification of the model. We show that the risk of the classifier that Asymmetric AdaBoost produces approaches the Bayes risk which is the infimum of risk that can be achieved by all classifiers. Monte Carlo experiments show that the Asymmetric AdaBoost performs better than the commonly used LASSO-regularized logistic regression when parametric assumption is violated and sample size is large. We apply the Asymmetric AdaBoost to predict business cycle turning points as in \citet{Ng2014a}.