By Andrew C. Harvey, Tommaso Proietti

This quantity offers a set of readings which provide the reader an concept of the character and scope of unobserved elements (UC) types and the equipment used to accommodate them. The booklet is meant to provide a self-contained presentation of the tools and applicative matters. Harvey has made significant contributions to this box and offers mammoth introductions in the course of the ebook to shape a unified view of the literature.

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1) where φ(L) and θ(L) are polynomials in the lag operator L of degrees p and q respectively, and ǫt is white noise. A state-space representation in which the first element of the state vector is the forecastable part of yt , that is x(1)t = yt − ǫt , is obtained from the following equivalences. Define r = max(p, q), and 30 Readings in Unobserved Components Models φp+1 = · · · = φr = 0 if p < q, or θq+1 = · · · = θr = 0 if p    θ1 φ1 1 1 0 ··· 0 0  φ2  θ2  0 0 1 0 0     ..  .. .. ..

4) indicates that, unlike the onestep-ahead forecast errors for the observed series, yt , which form an innovation sequence, the errors in the one-step-ahead estimates of the unobserved state vector follow a first-order autoregression. The difference is essentially that observing yt allows the previous estimate to be fully corrected before the forecast of yt+1 is made, but only partial correction to the state forecast is possible, resulting in the persistence of errors. Forecasting the state vector more than one period ahead is very simple in the present framework.

3c) ′ = H Pt,t−1 H + R. 3a) expresses the fact that the innovation in xt , which is not observed, is estimated by its orthogonal projection on that in yt . Similarly, the second term on the right-hand side of equation (b) is the projection of Gwt on y˜t ; equations (c) and (d) are definitional. The covariance update is the Riccati difference equation Pt+1,t = FPt,t−1 F ′ + GQG′ − Kt Kt = FC t,t + GS −1 t . 3f) The final term of equation (e) is the reduction in the conditional variance of xt+1 attributable to the information in y˜t , and Kt is the Kalman gain, so called because it gives the amount by which the new information affects the one-step-ahead forecast of the state: combining (a) and (b) we have xˆt+1,t = F xˆt,t−1 + Kt y˜t .

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