On quantile inference for conditional heteroscedastic models

Abstract

In this thesis, three conditional heteroscedatic models are investigated under a quantile regression framework. Among these three models, two are first proposed as new conditional heteroscedatic models while the remaining one is considered from a new perspective of quantile estimation.

First, this thesis proposes a new linear double autoregression model which is compatible with the quantile regression. The existence of strictly stationary solutions is discussed, and a necessary and sufficient condition is established. A doubly weighted conditional quantile estimation is introduced, where the first set of weights ensures the asymptotic normality of the estimators, and the second improves its efficiency through balancing individual conditional quantile estimators across multiple quantile levels. Moreover, goodness-of-fit tests based on the quantile autocorrelation function are suggested for checking the adequacy of the fitted models. Simulation studies indicate that the proposed inference tools perform well in finite samples, and an empirical example illustrates the usefulness of the new model.

Secondly, this thesis introduces a novel quantile double autoregression model which is driven by functionally dependent autoregressive coefficients. A simple but nontrivial transformation is employed to the scale part, which removes positive restrictions on the scale functional coefficients. The strict stationarity is discussed under mild conditions, and a global self-weighted conditional quantile estimator is considered. The strong consistency and asymptotic normality of the proposed estimator are established with only fractional moments on the process, which allows the model to handle heavy-tailed data. Simulation results demonstrate the finite sample performance of the proposed methodology, and an empirical example is presented to illustrate the usefulness of the new model.

Finally, this thesis considers a regression model with ARCH or GARCH errors. Traditionally the conditional quantiles are estimated using a two-step procedure: first, the conditional mean is estimated, and in the second step, the conditional quantile is estimated by a quantile regression on the estimated regression residuals from the first step. The efficiency and limiting distributions of these two-step quantile regression estimates are affected by the first-step preliminary estimation. Moreover, the two-step quantile regression may suffer from the efficiency loss due to the conditional heteroscedasticity. In this thesis, a joint estimation approach, where both the conditional mean and the ARCH or GARCH error structure are incorporated in the conditional quantile estimation, is proposed. The joint estimation procedure is expected to be more efficient than the conventional two-step approach based on the estimated residuals. Meanwhile, additional efficiency gain can be achieved from the weighted quantile regression. Asymptotic properties are developed for the joint weighted estimator and its resulting conditional quantile predictor. Simulation studies indicate that the proposed weighted joint procedure outperforms the two-stage approach in estimating conditional quantiles. An empirical application illustrates the usefulness of the joint weighted approach in modeling and predicting the conditional quantiles.