Sampling, Embedding and Inference for CARMA Processes

Abstract

A CARMA(p,q) process Y is a strictly stationary solution Y of the pth‐order formal stochastic differential equation a(D)Yt = b(D)DLt, where L is a two‐sided Lévy process, a(z) and b(z) are polynomials of degrees p and q respectively, with p > q, and D denotes differentiation with respect to t. Since estimation of the coefficients of a(z) and b(z) is frequently based on observations of the Δ‐sampled sequence , for some Δ > 0, it is crucial to understand the relation between Y and YΔ. If then YΔ is an ARMA sequence with coefficients depending on those of Y and the crucial problems for estimation are the determination of the coefficients of YΔ from those of Y (the sampling problem) and the determination of the coefficients of Y from those of YΔ (the embedding problem). In this article we consider both questions and use the results to determine the asymptotic distribution, as n→∞, with Δ fixed, of , where is the quasi‐maximum‐likelihood estimator of the vector of coefficients of a(z) and b(z), based on n consecutive observations of YΔ.