Statistics 451
Spring 2015
Examination 2
1. The Box-Cox transformation is given by
( (Z ∗ +m)γ −1
t
Zt =
γ
log(Zt∗
Name
You must show all of your work
γ 6= 0
+ m) γ = 0
Usually we take m = 0 but there are some circumstances where we would use another value
of m. Give an example if a situation where we would want to use m > 0.
2. If you had a time series process consisting of counting the number of events that are relatively
rare (e.g., the number of serious automobile accidents in Ames in a month), what transformation might you expect to provide more constant variance than the original data. Explain
why.
3. Consider the ARMA(1,1) model
(1 − φ1 B)Zt = (1 − θ1 B)at
As we know, this model can be expressed as
Zt = ψ1 at−1 + ψ2 at−2 + ψ3 at−3 + · · · + at .
(a) Derive expressions for ψ1 and ψ2 for this model.
(b) Derive Cov(Zt , at−2 ) for this model.
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4. A time series Zt can be described by a random walk Zt = Zt−1 + at . Assume that a realization
Z1 , Z2 , . . . Z100 is available to compute forecasts and that a forecast is needed for Z103 , using
Z100 as the forecast origin.
(a) Give an expression for Z103 , based on this random walk model.
(b) Give an expression that can be used to compute Zb100 (3), the forecast for Z103 .
(c) Derive e100 (3), the forecast error in Zb100 (3).
(d) Give an expression for the variance of e100 (3).
(e) Give an expression for an approximate 95% prediction interval for Z103 .
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5. Which diagnostic is most useful for detecting a violation of the assumption that the at values
are independent?
6. The AR(1) model can be expressed by
Zt = θ0 + φ1 Zt−1 + at ,
at ∼ nid(0, σa2 ).
(a) Derive and expression for the variance σZ2 = Var(Z) for this model.
(b) Derive expressions for the autocorrelation function values ρ1 and ρ2 for this model.
(c) Explain how to compute (i.e., give a formula for) the first two true PACF values φ11 and
φ22 for this model.
7. The AIC has been suggested as a useful tool for choosing among time series models.
(a) Why is AIC a better criterion than choosing the model with the largest likelihood?
(b) Explain why AIC should not be used exclusively in making such a choice. Be as specific
as possible in your answer.
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8. Suppose that the model for Zt is (1 − B)Zt = θ0 + at and that the working series is defined
by Wt = (1 − B)2 Zt .
(a) Derive an expression for the model for Wt (note that Zt (for any lags) should not appear
in this expression).
(b) Derive the mean of Wt .
(c) Derive the variance of Wt .
(d) Is Wt stationary? Why or why not?
(e) Is Wt invertible? Why or why not?
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