Autoregressive processes with exponentially decaying probability distribution functions: Applications to daily variations of a stock market index

We consider autoregressive conditional heteroskedasticity (ARCH) processes in which the variance σy2 depends linearly on the absolute value of the random variable y as σy2 = a + b|y|. While for the standard model, where σy2 = a + by2, the corresponding probability distribution function (PDF) P(y) decays as a power law for |y|→∞, in the linear case it decays exponentially as P(y)∼exp(-α|y|), with α = 2/b. We extend these results to the more general case σy2 = a + b|y|q, with 0<2. We find stretched exponential decay for 1<2 and stretched Gaussian behavior for 0<1. As an application, we consider the case q= 1 as our starting scheme for modeling the PDF of daily (logarithmic) variations in the Dow Jones stock market index. When the history of the ARCH process is taken into account, the resulting PDF becomes a stretched exponential even for q = 1, with a stretched exponent β = 2/3, in a much better agreement with the empirical data. © 2002 The American Physical Society.