Generalized quasi-likelihood ratio tests for varying coefficient quantile regression models

Quantile regression models which can track the relationship of predictive variables and the response variable in specific quantiles are especially useful in applications when extreme quantiles instead of the center of the distribution are interesting. Compared to classical conditional mean regressions, quantile regression models can provide a more comprehensive structure of the conditional distribution of the response variable. Also, they are more robust to skewed distributions and outliers. Therefore, quantile regression models have been applied extensively in many applied areas. Due to its greater flexibility, a varying coefficient regression technique has been extended to the quantile regression models recently. In this dissertation, my aim is to propose a new test procedure, termed as generalized quasi-likelihood (GQLR) test, to test whether all or partial coefficients are indeed constant or of some specific functions for the varying coefficient quantile regression models. The test statistics are constructed based on the comparison of the quasi-likelihood functions under null and alternative hypotheses. The asymptotic distributions of the proposed test statistics are also derived.;First, the functional coefficients in a varying coefficient quantile regression model are estimated by applying local linear fitting technique with jackknife method. Then, I construct the generalized quasi-likelihood ratio test statistics to test whether the varying coefficients are of some specific functional forms, including two special cases: testing whether the varying coefficients are known or unknown constants. The asymptotic normality of the proposed test statistic is derived upon the Bahadur representation of the estimators. I also discuss how to estimate the asymptotic variance-covariance matrix and investigate the power of the proposed test procedures in Chapter 2.;Secondly, I consider the similar testing procedure to test if partial coefficients in a varying coefficient quantile regression model are constant or of some specific form with other coefficients completely unspecified in Chapter 3. The corresponding generalized quasi-likelihood ratio test statistic is constructed based on comparing the quasi-likelihood functions under the null and alternative hypotheses. The asymptotic distributions of the proposed test statistics for both constancy and specific functional form are derived respectively and the power of the proposed test procedures is also investigated.;Finally, to exam the finite sample performance of all test statistics proposed. In Chapters 2 and 3, Monte Carlo simulation studies are conducted respectively at the end of each chapter. I also apply the proposed test methodologies to test if the existing models in the literature used to analyze the Boston house price data are appropriate or not. The simulation results and the real example illustrate the effectiveness and practical usefulness of the proposed test statistics. Chapter 4 concludes the dissertation. I also discuss some future research topics related to this dissertation.