This paper extends the classical two-regime threshold autoregressive model by introducing hysteresis to its regime-switching structure, which leads to a new model: the hysteretic autoregressive model. The proposed model enjoys the piecewise linear structure of a threshold model but has a more flexible regime switching mechanism. A sufficient condition is given for geometric ergodicity. Conditional least squares estimation is discussed, and the asymptotic distributions of its estimators and information criteria for model selection are derived. Simulation results and an example support the model.

The recent paper by Peng & Yao (2003) gave an interesting extension of least absolute deviation estimation to generalised autoregressive conditional heteroscedasticity, GARCH, time series models. The asymptotic distributions of absolute residual autocorrelations and squared residual autocorrelations from the GARCH model estimated by the least absolute deviation method are derived in this paper. These results lead to two useful diagnostic tools which can be used to check whether or not a GARCH model fitted by using the least absolute deviation method is adequate. Some simulation experiments give further support to the asymptotic theory and a real data example is also reported.

We consider a unified least absolute deviation estimator for stationary and nonstationary fractionally integrated autoregressive moving average models with conditional heteroscedasticity. Its asymptotic normality is established when the second moments of errors and innovations are finite. Several other alternative estimators are also discussed and are shown to be less efficient and less robust than the proposed approach. A diagnostic tool, consisting of two portmanteau tests, is designed to check whether or not the estimated models are adequate. The simulation experiments give further support to our model and the results for the absolute returns of the Dow Jones Industrial Average Index daily closing price demonstrate their usefulness in modelling time series exhibiting the features of long memory, conditional heteroscedasticity and heavy tails.

This article proposes a mixture double autoregressive model by introducing the flexibility of mixture models to the double autoregressive model, a novel conditional heteroscedastic model recently proposed in the literature. To make it more flexible, the mixing proportions are further assumed to be time varying, and probabilistic properties including strict stationarity and higher order moments are derived. Inference tools including the maximum likelihood estimation, an expectation–maximization (EM) algorithm for searching the estimator and an information criterion for model selection are carefully studied for the logistic mixture double autoregressive model, which has two components and is encountered more frequently in practice. Monte Carlo experiments give further support to the new models, and the analysis of an empirical example is also reported.

The idea of statistical learning can be applied in financial risk management. In recent years, value-at-risk (VaR) has become the standard tool for market risk measurement and management. For better VaR estimation, Engle and Manganelli ( 2004 Engle , R. , Manganelli , S. ( 2004 ). CAViaR: Conditional value at risk by regression quantiles . Journal of Business & Economic Statistics 22 : 367 – 381 . [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) introduced the conditional autoregressive value-at-risk (CAViaR) model to estimate the VaR directly by quantile regression. To entertain the nonlinearity and structural change in the VaR, we extend the CAViaR idea using two approaches: the threshold GARCH (TGARCH) and the mixture-GARCH models. The estimation method of these models are proposed. Our models should possess all the advantages of the CAViaR model and enhance the nonlinear structure. The methods are applied to the S&P500, Hang Seng, Nikkei and Nasdaq indices to illustrate our models.