Sampling, Embedding and Inference for CARMA Processes

A CARMA( p , q ) process Y is a strictly stationary solution Y of the pth ‐order formal stochastic differential equation a ( D ) Y t  =  b ( D ) DL t , 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 Y Δ : = ( Y n Δ ) n ∈ Z , for some Δ > 0, it is crucial to understand the relation between Y and Y Δ . If E L 1 2 < ∞ then Y Δ is an ARMA sequence with coefficients depending on those of Y and the crucial problems for estimation are the determination of the coefficients of Y Δ from those of Y ( the sampling problem ) and the determination of the coefficients of Y from those of Y Δ ( the embedding problem ). In this article we consider both questions and use the results to determine the asymptotic distribution, as n → ∞ , with Δ fixed, ofn Δ ( β ^ − β ) , where β ^ is the quasi‐maximum‐likelihood estimator of the vector of coefficients of a ( z ) and b ( z ), based on n consecutive observations of Y Δ .