Forecasting Exercises
Exercise 11.1
Sections 11.1 and 11.3 in the textbook have the appropriate definitions.
Exercise 11.2
See Section 11.2 of the textbook.
Exercise 11.3
The first step is to regress (using sample period data) actual Yt against a
constant, ft1,
ft2,
and ft3,
and obtain the weights 
0, 1, 2, and 3. Next obtain h-step ahead forecasts ft+h,1, ft+h,2, and ft+h,3. The combined h-step ahead forecast is ft+h = 1 ft+h,1 + 2 ft+h,2 + 3ft+h,3.
Exercise 11.4
Deseasonalizing refers to the
process of removing the seasonal effects from a series. Define seasonal
dummy variables D1, D2, and D3, which take the
value 1 during the first, second and third quarters, respectively, and
0 in other quarters. Next regress sales St against a constant, D1, D2, and D3. The deseasonalized
series is given by St*
= St - 0 - 1D1 - 2 D2- 3D3, where the 's are the estimated regression
coefficients.
Exercise 11.5
Detrending stands for the
process of removing a time trend from a series. First graph Yt against time and
identify the shape of the relationship (see Section 11.4 of the
textbook for possible functional form). Let f(t)
be the fitted function. Then the detrended series is Yt* = Yt - f(t).
For example if a quadratic relation was fit, the detrended series will
be Yt*
= Yt - 0 - 1 t-
2 t2, where the 's are the estimated regression
coefficients.
Exercise 11.7
Var (ut) = Var (
t - 
t-1)
= 
2(1+2).
Cov (ut, ut-1) = Cov (t - t-1,
t-1
- t-2) = - 2.
Finally, Cov (ut, ut-s) = 0 for all s > 1 because t is a white noise
series. Therefore the only non-zero correlation is with t-1 for which the correlation
coefficient is - /(1+2).
It is easy to extend this analysis to prove that if the moving average
is of order p, the first p autocorrelation values will be
non-zero but the rest will be zero.
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Updated March 14, 2006
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