By Michael Falk

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Two AXIS and two SYMBOL statements are used to customize the graphic containing two plots, the original data and the by X11 seasonally adjusted data. A LEGEND statement defines the text that explains the symbols. Best Local Polynomial Fit A simple moving average works well for a locally almost linear time series, but it may have problems to reflect a more twisted shape. This suggests fitting higher order local polynomials. Consider 2k + 1 consecutive data yt−k , . . , yt , . . , yt+k from a time series.

1) ¯ t are adjusted to approximately sum up to 0 over any (4) The D 12-months period by putting (1) ¯ t(1) − 1 1 D ¯ (1) + D ¯ (1) + · · · + D ¯ (1) + 1 D ¯ (1) . Sˆt := D t−6 t−5 t+5 12 2 2 t+6 (5) The differences (1) Yt (1) := Yt − Sˆt ∼ Tt + Rt then are the preliminary seasonally adjusted series, quite in the manner as before. 2 Linear Filtering of Time Series 23 (1) (6) The adjusted data Yt are further smoothed by a Henderson moving average Yt∗∗ of order 9, 13, or 23. (7) The differences (2) Dt := Yt − Yt∗∗ ∼ St + Rt then leave a second estimate of the sum of the seasonal and irregular components.

9. 3 lists West Germany’s public expenditures (in billion D-Marks) between 1961 and 1990. Compute simple moving averages of order 3 and 5 to estimate a possible trend. Plot the original data as well as the filtered ones and compare the curves. 10. (Unemployed Females Data) Use PROC X11 to analyze the monthly unemployed females between ages 16 and 19 in the United States from January 1961 to December 1985 (in thousands). 11. Show that the rank of a matrix A equals the rank of AT A. 2: Unemployed2 Data.