LARCH, Leverage, and Long Memory

Abstract

We consider the long-memory and leverage properties of a model for the conditional variance \(V_{t}^{2}\) of an observable stationary sequence Xt, where \(V_{t}^{2}\) is the square of an inhomogeneous linear combination of Xs, s < t, with square summable weights bj. This model, which we call linear autoregressive conditionally heteroskedastic (LARCH), specializes, when \(V_{t}^{2}\) depends only on Xt−1, to the asymmetric ARCH model of Engle (1990, Review of Financial Studies 3, 103–106), and, when \(V_{t}^{2}\) depends only on finitely many Xs, 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 \(X_{t}^{{\ell}}\) ⁠, ℓ ≥ 2 can have long-memory autocorrelations. We establish conditions under which the cross-autocovariance function between volatility and levels, \(h_{t}\ =\ \mathrm{cov}\left(V_{t}^{2},X_{0}\right)\) ⁠, decays in the manner of moving average weights of long-memory processes on suitable choice of the bj. We also establish the leverage property that ht < 0 for 0 < t ≤ k, where the value of k (which may be infinite) again depends on the bj. Conditions for finiteness of third and higher moments of Xt are also established.