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Problems 43
1.13. Find a filter of the form 1 + αB + βB 2 + γB 3 (i.e., find α, β, and γ ) that
passes linear trends without distortion and that eliminates arbitrary seasonal
components of period 2.
1.14. Show that the filter with coefficients [a−2,a−1,a0,a1,a2] 1[−1,
4, 3, 4, −1]
9
passes third-degree polynomials and eliminates seasonal components with period
3.
1.15. Let {Yt} be a stationary process with mean zero and let a and b be constants.
a. If Xt a + bt + st + Yt, where st is a seasonal component with period
12, show that ∇∇12Xt (1 − B)(1 − B12 )Xt is stationary and express its
autocovariance function in terms of that of {Yt}.
b. If Xt (a + bt)st + Yt, where st is a seasonal component with period 12,
show that ∇2 12Xt (1 − B12 ) 2Xt is stationary and express its autocovariance
function in terms of that of {Yt}.
1.16. (Using ITSM to smooth the strikes data.) Double-click on the ITSM icon, select
File>Project>Open>Univariate, click OK, and open the file STRIKES.
TSM. The graph of the data will then appear on your screen. To smooth the
data select Smooth>Moving Ave, Smooth>Exponential,orSmooth>FFT.Try
using each of these to reproduce the results shown in Figures 1.18, 1.21, and
1.22.
1.17. (Using ITSM to plot the deaths data.) In ITSM select File>Project>Open>
Univariate, click OK, and open the project DEATHS.TSM. The graph of
the data will then appear on your screen. To see a histogram of the data, click
on the sixth yellow button at the top of the ITSM window. To see the sample
autocorrelation function, click on the second yellow button. The presence of a
strong seasonal component with period 12 is evident in the graph of the data
and in the sample autocorrelation function.
1.18. (Using ITSM to analyze the deaths data.) Open the file DEATHS.TSM, select
Transform>Classical, check the box marked Seasonal Fit, and enter 12
for the period. Make sure that the box labeled Polynomial Fit is not checked,
and click, OK. You will then see the graph (Figure 1.24) of the deseasonalized
data. This graph suggests the presence of an additional quadratic trend function.
To fit such a trend to the deseasonalized data, select Transform>Undo Classical
to retrieve the original data. Then select Transform>Classical and
check the boxes marked Seasonal Fit and Polynomial Trend, entering 12
for the period and Quadratic for the trend. Click OK and you will obtain the
trend function
ˆmt 9952 − 71.82t + 0.8260t 2 , 1 ≤ t ≤ 72.