Analysis of Economic Time Series. A Synthesis by Marc Nerlove

By Marc Nerlove

During this version Nerlove and his co-authors illustrate suggestions of spectral research and techniques in keeping with parametric types within the research of monetary time sequence. The ebook presents a way and a mode for incorporating financial instinct and idea within the formula of time-series versions

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304). N o w U t- l = X t - i — a AX t - 2 — ' ' ' , U t - 2 = X t - 2 — aX l t - 3 — ' ' ' » -> e t so that Eutut-k 9 = Eut{xt-k — axxt-k-1 — · · · } = 0, k Φ 0. (10) Strictly speaking, Σ, is linearly deterministic. An example is the time series generated by ξί = + e "\ where u is uniformly distributed over the interval ( — π, π) a n d ν is independently distributed with arbitrary distribution function. Indeed, under the assumptions made, x f is itself white noise, in the sense that the best linear predictor of the current value is 0 (see Cox a n d Miller, 1965, p.

A process {x,} may be thought of as an infinite dimensional r a n d o m variable. Suppose that we observe one member of this population. Such an observation is called a realization of the process {x f }. Let us denote the value of such a single observation by {£,}. The infinite sample mean is N 1 m 3 ofi. = ml i 9v / , ι Σ ξ» 6 () A process is said to be s t a t i o n a r y to the o r d e r ρ if m o m e n t s u p to the p t h o r d e r are i n d e p e n d e n t 26 II. Introduction to the Theory of Stationary Time Series if the limit on the right exists.

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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