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C−u = cu . 14. (Unemployed1 Data) Compute a seasonal and trend adjusted time series for the Unemployed1 Data in the building trade. To this end compute seasonal differences and first order differences. Compare the results with those of PROC X11. 15. Use the SAS function RANNOR to generate a time series Yt = b0 +b1 t+εt , t = 1, . . , 100, where b0 , b1 = 0 and the εt are independent normal random variables with mean µ and variance σ12 if t ≤ 69 but variance σ22 = σ12 if t ≥ 70. Plot the exponentially filtered variables Yt∗ for different values of the smoothing parameter α ∈ (0, 1) and compare the results.

The following plots show the total annual output of electricity production in Germany between 1955 and 1979 in millions of kilowatt-hours as well as their first and second order differences. While the original data show an increasing trend, the second order differences fluctuate around zero having no more trend, but there is now an increasing variability visible in the data. 3a: Annual electricity output, first and second order differences. 1 2 3 /* e l e c t r i c i t y _ d i f f e r e n c e s .

12. Note that the roots z1 , . . , zp of A(z) = 1 + a1 z + · · ·+ap z p are complex valued and thus, the coefficients bu of the inverse causal filter will, in general, be complex valued as well. The preceding proof shows, however, that if ap and each zi are real numbers, then the coefficients bu , u ≥ 0, are real as well. The preceding proof shows, moreover, that a filter (au ) with complex coefficients a0 , a1 , . . , |z| = 1 for each root. The additional condition |z| > 1 for each root then implies that the inverse filter is a causal one.

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

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