Inferences for Heteroscedastic Location-Scale Time Series Models with Application to VaR and ES Estimation

Abstract

This thesis first considers nonlinear expectile regression models to estimate conditional expected shortfall (ES) and value-at-risk (VaR). In the literature, the asymmetric least squares (ALS) regression method has been widely used to estimate expectile regression models. However, no literatures rigorously investigated the asymptotic properties of the ALS estimates in nonlinear models with heteroscadasticity. Motivated by this aspect, this thesis studies the consistency and asymptotic normality of the ALS estimates and conditional VaR and ES in those models. To illustrate, a simulation study and real data analysis are conducted. Secondly, this thesis studies the asymptotic properties of a class of conditionally heteroscedastic location-scale time series models with innovations following a generalized asymmetric Student-t distribution (ASTD) or asymmetric exponential power distribution (AEPD). We show the consistency and asymptotic normality of the conditional maximum likelihood estimator (MLE) of model parameters under certain regularity conditions, and then, based on the MLE, we estimate the conditional VaR and ES using their closed forms induced from the model. Their performance is compared with that of conditional autoregressive value-at-risk (CAViaR) and expectile (CARE) methods. Meanwhile, one should be convinced of the model adequacy in advance of the VaR and ES calculation. For this task, we develop an entropy-based goodness of fit test based on residuals and a residual-based cumulative sum (CUSUM) test to conduct a parameter change test. To handle the former, we also investigate the asymptotic behavior of the residual empirical process. For illustration, a simulation study and real data analysis are provided.