By Michael Falk

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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. 18) k cu yt+u β0 = u=−k with some cu ∈ R which do not depend on the values yu of the time series and hence, (cu ) is a linear filter. Next we show that the cu sum up to 1. Choose to this end yt+u = 1 for u = −k, . . , k. 15). Since this solution is unique, we obtain k 1 = β0 = cu u=−k and thus, (cu ) is a moving average. 13 it actually has symmetric weights.

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. The random variable Y is called integrable if the real valued random variables Y(1) , Y(2) both have finite expectations, and in this case we define the expectation of Y by E(Y ) := E(Y(1) ) + i E(Y(2) ) ∈ C.

S. Bureau of Census has recently released an extended version of the X-11 Program called Census X-12-ARIMA. 1 and higher as PROC X12; we refer to the SAS online documentation for details. 4 that linear filters may cause unexpected effects and so, it is not clear a priori how the seasonal adjustment filter described above behaves. Moreover, end-corrections are necessary, which cover the important problem of adjusting current observations. This can be done by some extrapolation. 2a: Plot of the Unemployed1 Data yt and of yt , seasonally adjusted by the X-11 procedure.

### A First Course on Time Series Analysis : Examples with SAS by Michael Falk

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