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By Falk M.

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6. Suppose that (Zt )t∈Z is a stationary process with mean µZ := E(Z0 ) and autocovariance function γZ and let (at ) be an absolutely summable filter. Then Yt = u au Zt−u , t ∈ Z, is also stationary with au µZ µY = E(Y0 ) = u and autocovariance function au a ¯w γZ (t + w − u). γY (t) = u w Proof. 1 Linear Filters and Stochastic Processes 47 and, thus, sup E(|Zt |2 ) < ∞. 5. 5 immediately implies E(Yt ) = ( u au )µZ and part (iii) implies for t, s ∈ Z n n E((Yt − µY )(Ys − µY )) = lim Cov n→∞ n au Zt−u , u=−n n = lim n→∞ au a ¯w Cov(Zt−u , Zs−w ) u=−n w=−n n n au a ¯w γZ (t − s + w − u) = lim n→∞ aw Zs−w w=−n u=−n w=−n au a ¯w γZ (t − s + w − u).

K. 15). Since this solution is unique, we obtain k 1 = β0 = cu u=−k and thus, (cu ) is a moving average. As can be seen in Exercise 13 it actually has symmetric weights. We summarize our considerations in the following result. 24 Chapter 1. 2. Fitting locally by least squares a polynomial of degree p to 2k + 1 > p consecutive data points yt−k , . . , yt+k and predicting yt by the resulting intercept β0 , leads to a moving average (cu ) of order 2k + 1, given by the first row of the matrix (X T X)−1 X T .

Satisfy E(Yt ) = µ for 0 ≤ t ≤ N − 1, and E(Yt ) = λ for t ≥ N . Then we have for t ≥ N t−N E(Yt∗ ) = α t−1 (1 − α)j λ + α j=0 j=t−N +1 t−N +1 = λ(1 − (1 − α) −→t→∞ λ. , the smoothness of the filtered series Yt∗ , where we assume for the sake of a simple computation of the variance that the Yt are uncorrelated. If the variables Yt have common expectation µ, then this expectation carries over to Yt∗ . After a change point N , where the expectation of Yt changes for t ≥ N from µ to λ = µ, the filtered variables Yt∗ are, however, biased.

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A First Course on Time Series Analysis Examples with SAS by Falk M.


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