Sampling, Embedding and Inference for CARMA Processes. (19th November 2018)

• Record Type: Journal Article
• Authors: Brockwell, Peter J.; Lindner, Alexander
• Is Part Of: Journal of time series analysis. Volume 40:Number 2(2019)
• Page Start: 163
• Page End: 181
• Publication Date: 2018-11-19
• Subjects: CARMA process -- sampling -- embedding -- complex‐valued information matrix -- quasi‐maximum‐likelihood estimation; Time-series analysis -- Periodicals; 519.232
• DOI: 10.1111/jtsa.12433
• Languages: English
• ISSNs: 0143-9782

Abstract

Abstract : 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, of n 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 .