By Andrew C. Harvey, Tommaso Proietti

ISBN-10: 0199278695

ISBN-13: 9780199278695

This quantity offers a suite of readings which offer the reader an idea of the character and scope of unobserved parts (UC) types and the tools used to house them. The e-book is meant to provide a self-contained presentation of the tools and applicative concerns. Harvey has made significant contributions to this box and offers sizeable introductions during the publication to shape a unified view of the literature.

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**Extra resources for Readings in Unobserved Components Models**

**Example text**

Forecast of the next value s1 is now simply sˆ1,0 = φ sˆ0,0 , and since the error in this forecast is s1 − sˆ1,0 = φ(s0 − sˆ0,0 ) + ǫ1 its mean square error is given as 2 p1,0 = E{(s1 − sˆ1,0 )2 } = φ 2 (p0,−1 − p0,−1 −1 2 0 ) + σǫ . This procedure may be repeated as the observations y1 , y2 , . . 18b) 2 (pt,t−1 − pt,t−1 2 −1 t ) + σǫ . 18a) gives a recursion for the current estimate of the signal, with coefficients that vary over time, but again independently of the observations. 18a) approaches a limit.

It is assumed that φ = θ , but these parameters are otherwise unrestricted. s. forecast of yt+1 , denoted yˆt+1,t , together with quantities such as the mean square error or innovation variance E{(yt+1 − yˆt+1,t )2 } = t+1 . It is convenient to express y as the sum of two random variables, one of which, x, is forecastable while the other, ǫ, is not (other than at a value of zero). 1) as xt = φyt−1 − θǫt−1 yt = xt + ǫt . 3) and forecasts of x and y coincide. The recursive procedure is started off at time 0.

2) to the first gives r r x(1)t = j =1 φj yt−j − θk ǫt−k k=1 as required. A simple relation between the transition matrices and the ARMA coefficient polynomials is |I − F z| = φ(z), |I − Fθ z| = θ(z). 2), with wt = vt = ǫt and Q = S = R = σǫ2 (all scalars). 7) the matrix F ∗ = F −GSR−1 H ′ is equal to Fθ , defined above, in this case, and cov(wt∗ ) = G(Q − SR−1 S ′ )G′ = 0. 3) in the present case. First, the covariance recursion can be greatly simplified. 7), as specialized in the preceding paragraph, we obtain Pt+1,t = Fθ {Pt,t−1 − Pt,t−1 H −1 ′ ′ t H Pt,t−1 }Fθ .

### Readings in Unobserved Components Models by Andrew C. Harvey, Tommaso Proietti

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