| Quantile regression concerns the conditional quantile functions of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of the conditional distributions, but also in the upper and lower tails. It offers a more complete picture than does the least squares regression between the response variable and the covariates.;In this thesis, we consider a family of parametric power transformations for the dependent variable such that a linear or partially linear quantile regression model holds after transformation. The two models being considered are the power-transformed linear quantile regression model and power-transformed partially linear quantile regression model, respectively. We use a cusum process of residuals to measure lack of fit for a given quantile function. A power transformation is chosen to minimize the lack of fit. For the power-transformed linear quantile regression model, we show that the proposed estimator is consistent and asymptotically normal under some mild conditions. We demonstrate that the proposed approach works better than competing methods in the presence of heteroscedasticity and heavy-tails. Inferences about the transformation parameter and about the covariate effects are considered mathematically as well as empirically. A test for the adequacy of the power-transformation models is also proposed. For the power-transformed partially linear quantile regression model, we establish the consistency property for the proposed estimator. |
