Wai Keung Li

Recent Theoretical Results for Time Series Models with GARCH Errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA-GARCH are summarized. Various new ARCH-type models, including double threshold ARCH and GARCH, ARFIMA-GARCH, CHARMA and vector ARMA-GARCH, are also reviewed.

On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity

Abstract This article considers fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity, which combines the popular generalized autoregressive conditional heteroscedastic (GARCH) and the fractional (ARMA) models. The fractional differencing parameter d can be greater than 1/2, thus incorporating the important unit root case. Some sufficient conditions for stationarity, ergodicity, and existence of higher-order moments are derived. An algorithm for approximate maximum likelihood (ML) estimation is presented. The asymptotic properties of ML estimators, which include consistency and asymptotic normality, are discussed. The large-sample distributions of the residual autocorrelations and the square-residual autocorrelations are obtained, and two portmanteau test statistics are established for checking model adequacy. In particular, non-stationary FARIMA(p, d, q)-GARCH(r, s) models are also considered. Some simulation results are reported. As an illustration,...

On a Double-Threshold Autoregressive Heteroscedastic Time Series Model

Tongs threshold models have been found useful in modelling nonlinearities in the conditional mean of a time series. The threshold model is extended to the so-called double-threshold ARCH(DTARCH) model, which can handle the situation where both the conditional mean and the conditional variance specifications are piecewise linear given previous information. Potential applications of such models include financial data with different (asymmetric) behaviour in a visiting versus a falling market and business cycle modelling. Model identifications estimation and diagnostic checking techniques are developed Maximum likelihood estimation can be achieved via an easy-to-use iteratively weighted least squares algorithm. Portmanteau-type statistics are also derived for checking model adequacy. An illustrative example demonstrates that asymmetric behaviour in the mean and the variance could be present in financial series and that the DTARCH model is capable of capturing these phenomena. Copyright 1996 by John Wiley & Sons, Ltd.

A stochastic volatility model with Markov switching

This article presents a new way of modeling time-varying volatility. We generalize the usual stochastic volatility models to encompass regime-switching properties. The unobserved state variables are governed by a first-order Markov process. Bayesian estimators are constructed by Gibbs sampling. High-, medium- and low-volatility states are identified for the Standard and Poors 500 weekly return data. Persistence in volatility is explained by the persistence in the low- and the medium-volatility states. The high-volatility regime is able to capture the 1987 crash and overlap considerably with four U.S. economic recession periods.

Diagnostic Checks in Time Series

INTRODUCTION DIAGNOSTIC CHECKS FOR UNIVARIATE LINEAR MODELS Introduction The Asymptotic Distribution of the Residual Autocorrelation Distribution Modifications of the Portmanteau Statistic Extension to Multiplicative Seasonal ARMA Models Relation with the Lagrange Multiplier Test A Test Based on the Residual Partial Autocorrelation test A Test Based on the Residual Correlation Matrix test Extension to Periodic Autoregressions THE MULTIVARIATE LINEAR CASE The Vector ARMA model Granger Causality Tests Transfer Function Noise (TFN) Modeling ROBUST MODELING AND ROBUST DIAGNOSTIC CHECKING A Robust Portmanteau Test A Robust Residual Cross-Correlation Test A Robust Estimation Method for Vector Time Series The Trimmed Portmanteau Statistic NONLINEAR MODELS Introduction Tests for General Nonlinear Structure Tests for Linear vs. Specific Nonlinear Models Goodness-of-Fit Tests for Nonlinear Time Series Choosing Two Different Families of Nonlinear Models CONDITIONAL HETEROSCEDASTICITY MODELS The Autoregressive Conditional Heteroscedastic Model Checks for the Presence of ARCH Diagnostic Checking for ARCH Models Diagnostics for Multivariate ARCH models Testing for Causality in the Variance FRACTIONALLY DIFFERENCED PROCESS Introduction Methods of Estimation A Model Diagnostic Statistic Diagnostics for Fractional Differencing MISCELLANEOUS MODELS AND TOPICS ARMA Models with Non-Gaussian Errors Other Non-Gaussian time Series The Autoregressive Conditional Duration Model A Power Transformation to Induce Normality Epilogue

Diagnostic checking of nonlinear multivariate time series with multivariate arch errors

Multivariate time series with multivariate ARCH errors have been found useful in many applications. In order to check the adequacy of these models, we define the sum of squared (standardized) residual autocorrelations and derive their asymptotic distribution. The results are used to derive several new multivariate portmanteau tests. Simulation results show that the asymptotic standard errors are quite satisfactory compared with empirical standard errors and that the tests have reasonable empirical size and power. The distribution of the standardized residual autocorrelations is also derived.

Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence

Abstract Least squares (LS) and maximum likelihood (ML) estimation are considered for unit root processes with GARCH (1, 1) errors. The asymptotic distributions of LS and ML estimators are derived under the condition α + β < 1. The former has the usual unit root distribution and the latter is a functional of a bivariate Brownian motion, as in Ling and Li [Ling, S., Li, W. K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving‐average time series with GARCH errors. Ann. Statist.26:84–125]. Several unit root tests based on LS estimators, ML estimators, and mixing LS and ML estimators, are constructed. Simulation results show that tests based on mixing LS and ML estimators perform better than Dickey–Fuller tests which are based on LS estimators, and that tests based on the ML estimators perform better than the mixed estimators.

On a threshold autoregression with conditional heteroscedastic variances

Abstract This paper considers a time series model with a piecewise linear conditional mean and a piecewise linear conditional variance which is a natural extension of Tongs threshold autoregressive model. The model has potential applications in modelling asymmetric behaviour in volatility in the financial market. Conditions for stationarity and ergodicity are derived. Asymptotic properties of the maximum likelihood estimator and two model diagnostic checking statistics are also presented. An illustrative example based on the Hong Kong Hang Seng index is also reported.

Asymptotic Inference for Unit Root Processes with GARCH(1,1) Errors

This paper investigates the so-called one-step local quasi‐maximum likelihood estimator for the unit root process with GARCH~1,1! errors+ When the scaled conditional errors ~the ratio of the disturbance to the conditional standard deviation! follow a symmetric distribution, the asymptotic distribution of the estimated unit root is derived only under the second-order moment condition+ It is shown that this distribution is a functional of a bivariate Brownian motion as in Ling and Li ~1998, Annals of Statistics 26, 84‐125! and can be used to construct the unit root test+

A note on the corrected Akaike information criterion for threshold autoregressive models

A bias-corrected Akaike information criterion AICC is derived for self-exciting threshold autoregressive (SETAR) models. The small sample properties of the Akaike information criteria (AIC, AICC) and the Bayesian information criterion (BIC) are studied using simulation experiments. It is suggested that AICC performs much better than AIC and BIC in small samples and should be put in routine usage.