Abstract
This chapter discusses the formulation and analysis of unobserved-components models. It discusses how unobserved-components models, which capture much of the flavor of those used by economic statisticians of the 19th and early 20th centuries, may be formulated by superimposing simple mixed moving-average autoregressive models with independent white noise inputs. If a process is invertible, that is, if the process has a purely autoregressive representation, the autocovariance generating function has no zeros on the unit circle. If the generating function of a sequence is an analytic function in some region, then the infinite series form represents the Laurent expansion of that function in the region. Thus, the autocovariances of a zero-mean stationary purely linearly nondeterministic time series are the coefficients in a Laurent series expansion of the autocovariance generating function in the annulus about the unit circle in which the series converges. Therefore, Cauchy's integral formula and the residue theorem may be used to evaluate these autocovariances in a simple fashion for time series having a rational spectral density.
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