Nonparametric expectile regression and testin

The expectile regression is defined as the best predictor under the asymmetric square loss function. This thesis proposes a kernel type nonparametric estimator of the expectile regression, termed nonparametric expectile regression estimator (NPERE). The NPERE has properties similar to the nonparametric quantile regression estimator (NPQRE), but it has computational advantages over the NPQRE just as the parametric expectile regression does over the parametric quantile regression, and it can be easily applied to testing problems.;The asymptotic properties of the NPERE are derived by following the standard consistency proof of M-estimators and the asymptotic normality proof of Bickel (1967). It is shown that the NPERE is consistent, and its asymptotic distribution is normal with the usual rate of convergence of nonparametric kernel estimators. The asymptotic variance depends on the conditional distribution function of the error terms given the explanatory variables.;The NPERE can be applied to two important testing problems: heteroscedasticity and symmetry of error terms. The proposed test statistic for heteroscedasticity is constructed by comparing the variances of different expectile regressions. The proposed test for the symmetry of the error terms is constructed by comparing the mean of the $tau$-th and (1-$tau$)-th expectile regression with the 0.5-th expectile regression. While the majority of the test statistics for heteroscedasticity and symmetry are based on the specification of the mean regression and/or the variance up to a finite dimensional unknown parameter vector, the proposed test statistics are robust to the both types of possible misspecifications. By following the linearization approach and using the properties of U-statistics, it is shown that the asymptotic distribution of both test statistics is the chi-square distribution.;Some finite sample behavior of the proposed test statistics is presented.