Quantile inference and bootstrapping approximation for some time series models

Abstract

In this thesis, we discuss some robust estimation methods for nonlinear time series from both frequency and time domain perspectives. A weighted bootstrap method is also introduced to facilitate model inference.

Firstly, we propose a conditional quantile estimation method for the hysteretic autoregressive model, which has been proven to be valuable for modeling nonlinear time series with hysteresis when switching between regimes; see \cite{Li_et_al2015a}. The motivation comes from the presence of phenomenon known as hysteresis, which prevails in many economic series, such as the US unemployment rate, the industrial output series, etc. By applying the conditional quantile estimation, we manage to extend the statistical inference on the mean process to that on the conditional quantiles across the whole distribution of the underlying process. We have established the asymptotic theories for the conditional quantile estimators, and introduced a weighted bootstrap method to circumvent any unknown quantities involved in the asymptotic covariance matrix. We have conducted extensive simulation studies to evaluate the finite sample performance of our method. The efficiency of the quantile approach is fully illustrated by a real data example on the post-WW II US monthly unemployment return series.

Secondly, the semi-quantile spectral analysis has been introduced. It is based on quantile autocorrelation introduced by \cite{Li_Li_Tsai2015}. The proposed semi-quantile spectral densities enjoy the same strengths as the classical $L^2$-spectrum and the quantile spectra, recently proposed by \cite{Hagemann2011}. Specifically, the semi-quantile spectra are robust against outliers, and are able to detect hidden periodicity across the whole distribution of the underlying process. Moreover, they can preserve crucial information contained in the original time series that could be otherwise ignored by the quantile spectral densities. Apart from simulation studies, we also include a real data example on the US monthly total number of building permits issued by the permit-issuing authorities for illustration.

Thirdly, we have proposed a weighted bootstrap method for approximating the asymptotic covariance matrices of the least-absolute-deviation estimator and the portmanteau test statistic for the generalized autoregressive conditional heteroscedasticity model. The newly introduced bootstrap approach is relatively easy to implement, and can avoid tedious estimation for unknown quantities that are present in the asymptotic distributions derived by the analytical approach of \cite{Peng_Yao2003, Li_Li2005}. The asymptotic properties of the proposed bootstrap method has been fully addressed. We have also conducted extensive simulation studies together with a real data example using the HSI index to demonstrate the usefulness of the proposed bootstrap method.