By Falk M.

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**Example text**

The recursively defined filter ∆p Yt = ∆(∆p−1 Yt ), t = p, . . , n, is the difference filter of order p. The difference filter of second order has, for example, weights a0 = 1, a1 = −2, a2 = 1 ∆2 Yt = ∆Yt − ∆Yt−1 = Yt − Yt−1 − Yt−1 + Yt−2 = Yt − 2Yt−1 + Yt−2 . p If a time series Yt has a polynomial trend Tt = k=0 ck tk for some constants ck , then the difference filter ∆p Yt of order p removes this trend up to a constant. Time series in economics often have a trend function that can be removed by a first or second order difference filter.

Stationary Processes A stochastic process (Yt )t∈Z of square integrable complex valued random variables is said to be (weakly) stationary if for any t1 , t2 , k ∈ Z E(Yt1 ) = E(Yt1 +k ) and E(Yt1 Y t2 ) = E(Yt1 +k Y t2 +k ). 1 Linear Filters and Stochastic Processes 43 The random variables of a stationary process (Yt )t∈Z have identical means and variances. The autocovariance function satisfies moreover for s, t ∈ Z γ(t, s) := Cov(Yt , Ys ) = Cov(Yt−s , Y0 ) =: γ(t − s) = Cov(Y0 , Yt−s ) = Cov(Ys−t , Y0 ) = γ(s − t), and thus, the autocovariance function of a stationary process can be viewed as a function of a single argument satisfying γ(t) = γ(−t), t ∈ Z.

Then γ(k) := Cov(Yk+1 , Y1 ) = Cov(Yk+2 , Y2 ) = . . is called autocovariance function and ρ(k) := γ(k) , γ(0) k = 0, 1, . . is called autocorrelation function. Let y1 , . . , yn be realizations of a time series Y1 , . . , Yn . The empirical counterpart of the autocovariance function is c(k) := 1 n n−k (yt+k − y¯)(yt − y¯) with bary = t=1 1 n n yt t=1 and the empirical autocorrelation is defined by r(k) := c(k) = c(0) n−k ¯)(yt t=1 (yt+k − y n (y − y¯)2 t=1 t − y¯) . See Exercise 8 (ii) in Chapter 2 for the particular role of the factor 1/n in place of 1/(n−k) in the definition of c(k).

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

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