Analysis of Economic Time Series. A Synthesis by Marc Nerlove

By Marc Nerlove

During this version Nerlove and his co-authors illustrate ideas of spectral research and strategies according to parametric types within the research of monetary time sequence. The publication offers a way and a style for incorporating financial instinct and idea within the formula of time-series types

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It follows from Wold's theorem that any purely nondeterministic process can be written as a one-sided moving average with a white noise input. This representation is extremely important in spectral theory and in the theory of optimal prediction and extraction. ;,}, which is determined, as indicated, only by the remote past of { x j , can be shown to be linearly deterministic (Anderson, 1971, p. 421). It can be shown that the deterministic component ξί in the Wold decomposition (15) xt = it + nt is essentially of the f o r m 12 m Σ É.

It follows that So the integral of the spectral density function over ( — π, π) is the variance of the process. The spectral density, in effect, gives the decomposition of the variance of { x j by frequency bands, and \if(X)dX can be thought of as the part of the variance "due t o " or associated with frequencies in the interval (α, β). This interpretation will become clearer following the discussion of the theory of the spectral representation. Suppose that the process { x j is white noise so that y 0 ) = 4 io, ;*a See Appendix C, Section 5, " F o u r i e r Transforms and ' W i n d o w s .

W e are assuming y(0) = 1. 48 III. The Spectral Representation and Its Estimation If { x j is a real-valued process, xt = x, for all t e T. It follows from (9) that ίλί xt =ρ_πβ άζ(λ) = Ρ_ β- άΟλ) = p_ e dÖ^I\ ίλί at π n (17) so that ζ(λ) = ζ( — λ) if ζ(λ) is the spectral process associated with a real time series. Write (18) ζ(λ) = ϊ[υ(λ)-ίν(λ)1 where U(X) and V(X) are real functions. , ϋ(-λ)=υ(λ\ ν(-λ)=-Υ(λ). (19) Since the product of two even functions or two odd functions is even a n d the product of an even and an odd function is odd, since cos At is even and sin Xt is odd, and since the integral of an o d d function over an interval symmetrical about the origin is zero, (9) may be rewritten x t = J* cos At dV(k) + J* sin ÀtdV(À), (20) where ϋ(λ) and V{X) are mutually orthogonal real processes on the interval [ — π, π ] with orthogonal increments, such that dF(X) = i{E[dU(À)] 2 + Ε[άΥ(λ)Υ}, (21) Clearly, dF(k) = dF{ — A), except at λ = —π where a discontinuous j u m p may occur by convention, so dF(X) is symmetric about λ = 0 and certainly real ; hence, γ(τ) = 2 cos ÀzdFttl (22) which also follows directly from (16) when {xt} is a real-valued process.

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