By Andrew C. Harvey, Tommaso Proietti

This quantity offers a suite of readings which offer the reader an idea of the character and scope of unobserved parts (UC) versions and the tools used to accommodate them. The publication is meant to provide a self-contained presentation of the tools and applicative matters. Harvey has made significant contributions to this box and gives mammoth introductions through the ebook to shape a unified view of the literature.

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

**Example text**

7) that as the forecast horizon, j , increases so the observed history ceases to be informative, and the forecast yˆt+j,t tends to the unconditional mean, zero. Similarly, Vj tends to the variance of y. Conversely, if φ(L) is not invertible, neither the forecast nor its error variance in general approach limits as j increases. 4) we have Kvt = Gwt since K = G and vt = wt = ǫt . s. 7) and obtain a generating function for the weight on yt−i in the forecast of yt+j as fj (z) = H ′ F j −1 (I − zFθ )−1 G, j = 1, 2, .

12) can be rearranged to give the familiar “adaptive expectations” formula yˆt+1,t − yˆt,t−1 = (1 − θ )(yt − yˆt,t−1 ). Forecasting j steps ahead (j > 1) is straightforward in the present framework, since yˆt+j,t is simply xˆt+j,t . 3) as xˆt+j,t = φ j −1 xˆt+1,t , j = 2, 3, . . xt+j − xˆt+j,t = (φ − θ )(ǫt+j −1 + φǫt+j −2 + · · · + φ j −2 ǫt+1 ) + φ j −1 (xt+1 − xˆt+1,t ). Thus the j -step-ahead forecast yˆt+j,t = φ j −1 yˆt+1,t has mean square error E{(yt+j − yˆt+j,t )2 } = (φ − θ)2 j −2 i=0 φ 2i σǫ2 + φ 2j −2 pt+1,t + σǫ2 .

23) and is the moving average coefficient in the ARMA(1,1) representation of y. 23), whence σǫ2 + ση2 (1 − φβ)(1 − φ/β) = 0. 22) coincide. s. recursion, which is valid more generally. The principal case of this model discussed in the literature occurs when φ = 1, so that the signal follows a random walk. For example, st might then represent the “persisting” or “permanent” component of income, and observed income yt comprises this together with a purely random “transitory” component, ηt . 22), which in this case gives the familiar exponentially weighted moving average (Muth, 1960).