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

Shiqing Ling · W. K. Li

243 citations0/0 files storedJournal of the American Statistical Association

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Financial Risk and Volatility Modeling Complex Systems and Time Series Analysis Time Series Analysis and Forecasting

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, the proposed model is also applied to the daily returns of the Hong Kong Hang Seng index (1983–1984).

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References

1986Generalized autoregressive conditional heteroskedasticity 1982Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation 1977Time Series Analysis: Forecasting and Control 1977Time Series Analysis: Forecasting and Control 1992ARCH modeling in finance 1980AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING 1988A Capital Asset Pricing Model with Time-Varying Covariances 1983THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS

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