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The forecast error Yt+1 − Yt∗ =: et+1 then satisfies the ∗ equation Yt+1 = αet+1 + Yt∗ . 3 Chapter 1. Elements of Exploratory Time Series Analysis Autocovariances and Autocorrelations Autocovariances and autocorrelations are measures of dependence between variables in a time series. Suppose that Y1 , . . , Yn are square integrable random variables with the property that the covariance Cov(Yt+k , Yt ) = E((Yt+k −E(Yt+k ))(Yt − E(Yt ))) of observations with lag k does not depend on t. Then γ(k) := Cov(Yk+1 , Y1 ) = Cov(Yk+2 , Y2 ) = .
2) with given constants a1 , . . , ap and white noise (εt )t∈Z has a stationary solution (Yt )t∈Z if all p roots of the equation 1 − a1 z − a2 z 2 − . . − ap z p = 0 are outside of the unit circle. In this case, the stationary solution is almost surely uniquely determined by Yt := bu εt−u , t ∈ Z, u≥0 where (bu )u≥0 is the absolutely summable inverse causal filter of c0 = 1, cu = −au , u = 1, . . , p and cu = 0 elsewhere. Proof. 11. 6, and its uniqueness follows from εt = Yt − a1 Yt−1 − . .
Ap ∈ R with ap = 0, and a white noise (εt ) such that Yt = a1 Yt−1 + . . + ap Yt−p + εt , t ∈ Z. 2) The value of an AR(p)-process at time t is, therefore, regressed on its own past p values plus a random shock. 6 MA(q)-processes are automatically stationary, this is not true for AR(p)-processes (see Exercise 26). The following result provides a sufficient condition on the constants a1 , . . 2). 3. 2) with given constants a1 , . . , ap and white noise (εt )t∈Z has a stationary solution (Yt )t∈Z if all p roots of the equation 1 − a1 z − a2 z 2 − .