LARCH, Leverage and Long Memory

We consider the long memory and leverage properties of a model for the conditional variance V 2 t of an observable stationary sequence Xt, where V 2 t is the square of an inhomogeneous linear combination of Xs, s < t, with square summable weights bj . This model, which we call linear ARCH (LARCH), specializes, when V 2 t depends only on Xt−1, to the asymmetric ARCH model of Engle (1990), and, when V 2 t depends only on finitely many Xs, to a version of the quadratic ARCH model of Sentana (1995), these authors having discussed leverage potential in such models. The model which we consider was suggested by Robinson (1991), for use as a possibly long memory conditionally heteroscedastic alternative to i.i.d. behaviour, and further studied by Giraitis, Robinson and Surgailis (2000), who showed that integer powers X t , ` ≥ 2, can have long memory autocorrelations. We establish conditions under which the crossautocovariance function between volatility and levels, ht = Cov(V 2 t , X0), 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. JEL Classification: C22 Corresponding Author: Donatas Surgailis We thank two referees and the Associate Editor for helpful comments. ∗Research supported by ESRC Grant R000238212. †Research supported by Lithuanian State Science and Studies Foundation Grant K-014. ‡Research supported by a Leverhulme Trust Personal Research Professorship and ESRC Grant R000238212.