LARCH, Leverage, and Long Memory

We consider the long-memory and leverage properties of a model for the conditional variance V-sub-t-super-2 of an observable stationary sequence X-sub-t, where V-sub-t-super-2 is the square of an inhomogeneous linear combination of X-sub-s, s < t, with square summable weights b-sub-j. This model, which we call linear autoregressive conditionally heteroskedastic (LARCH), specializes, when V-sub-t-super-2 depends only on X-sub-t - 1, to the asymmetric ARCH model of Engle (1990, Review of Financial Studies 3, 103--106), and, when V-sub-t-super-2 depends only on finitely many X-sub-s, to a version of the quadratic ARCH model of Sentana (1995, Review of Economic Studies 62, 639--661), these authors having discussed leverage potential in such models. The model that we consider was suggested by Robinson (1991, Journal of Econometrics 47, 67--84), for use as a possibly long-memory conditionally heteroskedastic alternative to i.i.d. behavior, and further studied by Giraitis, Robinson and Surgailis (2000, Annals of Applied Probability 10, 1002--1004), who showed that integer powers X-sub-t-super-ℓ, ℓ ≥ 2 can have long-memory autocorrelations. We establish conditions under which the cross-autocovariance function between volatility and levels, h-sub-t = covV-sub-t-super-2,X-sub-0, decays in the manner of moving average weights of long-memory processes on suitable choice of the b-sub-j. We also establish the leverage property that h-sub-t < 0 for 0 < t ≤ k, where the value of k (which may be infinite) again depends on the b-sub-j. Conditions for finiteness of third and higher moments of X-sub-t are also established. Copyright 2004, Oxford University Press.