Modeling Sequences of Long Memory Non-negative Covariance Stationary Random Variables

Abstract

This paper extends the class of covariance stationary GARCH processes of Engle (1982) and Bollerslev (1986) to the case of non-summable autocovariances. We improve on the results of two previous studies in this field: FIGARCH model of Baillie, Bollerslev and Mikkelsen (1996), which generates sequences of non-negative random variables with infinite first and higher-order moments, and hyperbolic decay rate of the impulse response function, and linear ARCH model of Giraitis, Robinson and Surgailis (2000), which does not nest the class of short-memory GARCH processes. We use infinite series representation of GARCH models in terms of martingale differences innovations referred to as MD-ARCH(infinity) representation. This allows for the case of hyperbolically decaying square-summable weighting coefficients. Conditions for non-negativity and covariance stationarity of MD-ARCH(infinity) sequences are derived, and functional limit of the normalized partial sums of the process is studied. Applications of long-memory MD-ARCH(infinity) processes include volatility modeling and high-frequency financial data econometrics.