On the distributional properties of GARCH processes (Q2740039)

Many economic time series exhibit periods of unusually large volatility followed by periods of relative tranquility. The problem of modeling stochastic volatility and conditional heteroskedasticity in the financial context has been an active area of research over the past decade. A number of models for the conditional distribution and the conditional variance have been proposed with the most celebrated case being the autoregressive conditional heteroskedasticity (ARCH) model introduced by \textit{R.F. Engle} [Econometrica 50, 987-1007 (1982; Zbl 0491.62099)]. The model is often referred to as the linear ARCH(p) process \(Y(t).\) It is interesting that the process \(Y^{2}(t)\) has the structure of a covariance stationary AR(p) process. This yields volatility clustering, a phenomenon observed in a number of high frequency financial data records. The GARCH(p,q) specification of \textit{T. Bollerslev} [J. Econ. 31, 307-327 (1986; Zbl 0616.62119)] provides a more flexible structure by adding lags in the conditional variance equation.NEWLINENEWLINENEWLINEIn this paper, nonasymptotic characterizations of the tail behavior of the unconditional distribution function of GARCH processes are given. A large class of non-normal noise processes is considered including the important cases of the Student and normal inverse Gaussian distributions. Two main results are obtained.NEWLINENEWLINENEWLINEFirst, a nontrivial lower bound for the probability \(P\{Y^{2}(t)\leq y\}\) is established. The bound holds for \(0\leq y\leq y_{\max},\) where \(y_{\max}\) depends on the distribution of a non-normal noise process and the variance of \(Y(t).\) Furthermore, a lower bound for the joint probability \(P\{Y^{2}_{1}\leq y_{1},...,Y^{2}_{n}\leq y_{n}\}\) is also given. In fact, a result is derived regarding the run length of a classical Shewhart control chart for correlated ARCH/GARCH type data. The theory is illustrated by numerical examples.