Robust methods and quantile inference for econometric models

Abstract

This thesis studies the robust diagnostic checking, quantile inference, and the least absolute deviations (LAD) estimation for some time series models. Some new inference tools are developed which might shed new light on our understanding of financial and economic time series.

While the estimation for time series models with heavy-tailed innovations has been widely discussed in the literature, the corresponding robust goodness-of-fit tests are still lacking. A major problem is that the commonly used autocorrelation function in constructing goodness-of-fit tests necessarily imposes certain moment conditions on the innovations. Using the fact that a bounded random variable has finite moments of all orders, this thesis addresses this problem by first transforming the (absolute) residuals with a bounded function. Specifically, this thesis considers the widely used generalized autoregressive conditional heteroscedastic (GARCH) model. By transforming the absolute residuals with their corresponding empirical distribution function, robust goodness-of-fit test is constructed based on the sample autocorrelation function of the transformed residuals sequence.

Apart from the diagnostic checking problem, this thesis also studies the quantile inference for the GARCH model. Quantile-based risk measures, such as the Value-at-Risk, are widely used by financial firms and regulators nowadays. The GARCH model is arguably the most widely used conditional heteroscedastic time series models in financial applications. It has proved highly successful in capturing the volatility dynamics of asset returns, and thus has great potential in the quantile inference for financial time series. However, so far feasible quantile regression methods for conditional heteroscedastic time series models have been confined to a variant of the GARCH model, the linear GARCH model, owing to its tractable conditional quantile structure. This thesis develops an easy-to-implement conditional quantile estimation procedure for the GARCH model, based on a simple yet nontrivial transformation. A new bootstrapping procedure is proposed to approximate the asymptotic distribution of the quantile regression estimator, and diagnostic tools based on the residual quantile autocorrelation function are constructed to check the adequacy of the fitted conditional quantiles.

For nonstationary vector autoregressive (AR) time series models with pure unit roots, this thesis considers the panel unit root tests based on the LAD estimation. The motivation is twofold. First, the vector AR modeling is a way to allow cross-sectional dependence in the panel data. Second, the LAD estimation is a robust complement to the ordinary least squares method for heavy-tailed time series. To approximate the complicated asymptotic distribution of the LAD estimator, a novel bootstrap method is proposed, which is a hybrid of the wild bootstrap and the randomly weighted bootstrap method. Based on the proposed LAD estimation and bootstrap method, three bootstrapping panel unit root tests are constructed.