This dissertation focuses on the research on the semiparametric and nonparametric identification and estimation of the binary choice model and the general heteroskedastic transformation regression model. The proposed method is unique and thus contributes to the literature against the existing methods. The dissertation contains two chapters.

The first chapter provides a novel framework to systematically address the drawbacks associated with the smoothed maximum score estimator Horowitz (1992) for the binary choice model. The smoothed maximum score estimator typically has large finite sample bias and is quite sensitive to the choice of smoothing parameter, which makes it difficult to implement in practice. Till now, these issues have largely remained unresolved. It is useful to...[ Read more ]

This dissertation focuses on the research on the semiparametric and nonparametric identification and estimation of the binary choice model and the general heteroskedastic transformation regression model. The proposed method is unique and thus contributes to the literature against the existing methods. The dissertation contains two chapters.

The first chapter provides a novel framework to systematically address the drawbacks associated with the smoothed maximum score estimator Horowitz (1992) for the binary choice model. The smoothed maximum score estimator typically has large finite sample bias and is quite sensitive to the choice of smoothing parameter, which makes it difficult to implement in practice. Till now, these issues have largely remained unresolved. It is useful to recognize that these problems are commonly associated with nonparametric kernel density and regression estimators. Indeed, as pointed out by Horowitz (1992), there is an intimate link between the smoothed maximum score estimator and the kernel nonparametric density and regression estimation. In the nonparametric estimation literature local linear and local polynomial regression estimators (e.g., Fan and Gijbels (1996)) are known to have favorable finite sample and asymptotic properties over the kernel regression estimator. In this chapter we propose a novel framework which leads naturally to a local polynomial smoothing based estimator. As expected, our estimator is shown to possess the favorable properties typically associated with the local polynomial regression estimator.

Along with the development of maximum score estimation for the cross-sectional binary choice model, there has been parallel development of distribution free estimation of the binary choice panel data model with fixed effects. Drawbacks associated with smoothed maximum score estimators, such as large finite sample biases and sensitivity to the smoothing parameter, still plague the panel data version of the smoothed maximum score estimator, and even seem to be more serious than its cross-sectional counterpart. It shows that our local polynomial based estimator can be extended to the panel data model, and the new estimator is shown to have similar favorable properties over the panel version of the smoothed maximum score estimator.

The second chapter proposes a √n − semiparametric estimation of a general heteroskedastic transformation regression model. Generalized transformation regression models have received a great deal of attentions in both theoretical and applied econometrics as well as biostatistics. Chen (2002), Han (1987) and Horowitz (1996), among others, require independent error terms, thus rule out possible heteroskedasticity. Chen (2010a,b) and Khan (2001) consider quantile regression for the transformation model that permits general heteroskedasticity, but the resulting quantile predictor converges at rates slower than the parametric rate √n . In combining the existing literature to overcome the shortcomings above, in this chapter, we consider the estimation of a general transformation model (with censoring) with median location restriction subject to a nonlinear multiplicative form of heteroskedasticity and the resulting quantile predictor converges at the parametric rate. The basic idea of our estimation method is based on the discretizing approach in transforming this general heteroskedastic transformation model into a maximum rank based model and a sequence of binary choice model with a family of quantiles of the error term. We integrate and sum over the objective functions based on the constructed binary choice model over y and different quantiles of the error term. The Monte Carlo simulation experiments show that our estimator works substantially well in finite samples.