By Michael Falk

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**Example text**

Yn can be viewed as a clipping from a sequence of random variables . . , Y−2 , Y−1 , Y0 , Y1 , Y2 , . . In the following we will introduce several models for such a stochastic process Yt with index set Z. 1 Linear Filters and Stochastic Processes For mathematical convenience we will consider complex valued random variables Y , whose range is √ the set of complex numbers C = {u + iv : u, v ∈ R}, where i = −1. Therefore, we can decompose Y as Y = Y(1) + iY(2) , where Y(1) = Re(Y ) is the real part of Y and Y(2) = Im(Y ) is its imaginary part.

Thus, the matrix X T X is invertible iff the columns of X are linearly independent. 12). 16) have, therefore, the unique solution β = (X T X)−1 X T y. 18) The linear prediction of yt+u , based on u, u2 , . . , up , is p β j uj . p yˆt+u = (1, u, . . , u )β = j=0 Choosing u = 0 we obtain in particular that β0 = yˆt is a predictor of the central observation yt among yt−k , . . , yt+k . The local polynomial approach consists now in replacing yt by the intercept β0 . 2 Linear Filtering of Time Series 27 it is actually a moving average.

Yt+k )T . 11). Thus, the matrix X T X is invertible iff the columns of X are linearly independent. 12). 16) have, therefore, the unique solution β = (X T X)−1 X T y. 18) The linear prediction of yt+u , based on u, u2 , . . , up , is p β j uj . p yˆt+u = (1, u, . . , u )β = j=0 Choosing u = 0 we obtain in particular that β0 = yˆt is a predictor of the central observation yt among yt−k , . . , yt+k . The local polynomial approach consists now in replacing yt by the intercept β0 . 2 Linear Filtering of Time Series 27 it is actually a moving average.