ARCH models

Abstract

The key properties of financial time series appear to be that: (a) marginal distributions have heavy tails and thin centres (leptokurtosis); (b) the scale appears to change over time; (c) return series appear to be almost uncorrelated over time but to be dependent through higher moments (see Mandelbrot, 1963; Fama, 1965). Linear models like the autoregressive moving average (ARMA) class cannot capture well all these phenomena, since they only really address the conditional mean µ _t = E(y _t |y_{t − 1},…) and in a rather limited way. This motivates the consideration of nonlinear models. For a discrete time stochastic process y _t , the conditional variance σ ² _t = var (y _t )|y_{t − 1},… of the process is a natural measure of risk for an investor at time t…1. Empirically it appears to change over time and so it is important to have a model for it. Engle (1982) introduced the autoregressive conditional heteroskedasticity (ARCH) model where for simplicity we rewrite y _t → y _t − µ _t and suppose that the process started in the infinite past. This model makes σ ² _t vary over time depending on the realization of past squared returns. Forσ ² _t to be a valid conditional variance it is necessary that ω>0 and γ ≥ 0, in which case σ ² _t for all t. Suppose also that y _t = ε _t σ _t with ε _t i.i.d. mean zero and variance one. Provided ω < 1, the process y_t is weakly (covariance) stationary and has finite unconditional variance σ _t = E(σ ² _t ) = E(y ² _t ) = ω/(1 − γ). This can be proven rigorously under a variety of assumptions on the initialization of the process (see Nelson, 1990).