Larch, Leverage, and Long Memory

We consider the long-memory and leverage properties of a model for the conditional variance V of an observable stationary sequence X, where V is the square of an inhomogeneous linear combination of X, s lt; t, with square summable weights b. This model, which we call linear autoregressive conditionally heteroskedastic (LARCH), specializes, when V depends only on X, to the asymmetric ARCH model of Engle (1990, Review of Financial Studies 3, 103-106), and, when V depends only on finitely many X, 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, [ell ] e 2 can have long-memory autocorrelations. We establish conditions under which the cross-autocovariance function between volatility and levels, h = covlt;fengt;lt;cp type=quot;lparquot;gt;V,Xlt;cp type=quot;rparquot;gt;lt;/fengt;, decays in the manner of moving average weights of long-memory processes on suitable choice of the b. We also establish the leverage property that h lt; 0 for 0 lt; t d k, where the value of k (which may be infinite) again depends on the b. Conditions for finiteness of third and higher moments of X are also established.t2tt2sjt2t-1t2st[ell ]tt20jtjt