Statistics 451
Spring 2014
Examination 1
Name
You must show all of your work
1. Consider the normal distribution, time series model Zt = µ + at , at ∼ nid(0, σa2 ). An exact
95% prediction interval for a future observation from based on a past realization z1 , . . . , zn is
r
1
Z̄ ± t(.975,n−1) Sz 1 +
n
while
Z̄ ± t(.975,n−1) Sz
provides a simple approximation. Here Z̄ is the sample mean and Sz denotes the sample
standard deviation of the realization z1 , . . . , zn .
(a) Briefly explain the conceptual difference between these two formulas. Under what circumstances do they give similar answers?
(b) The quantity n − 1 is known as “degrees of freedom” is used as an index for the tdistribution quantile in the above formulas. What is the practical interpretation of this
quantity?
(c) Briefly explain why it is that we prefer to report the sample standard deviation Sz rather
than the sample variance Sz2 .
(d) Briefly explain the important differences between a confidence interval and a prediction
interval.
2. Briefly explain why variability that is a percentage of the level of a time series will tend to
cause a time series regression model to exhibit non-constant variance.
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3. The following figure shows quarterly data on the amount of exports from the US to the EEC
from the first quarter of 1958 to the second quarter of 1968 (42 observations).
0.6
0.8
1.0
exports
1.2
1.4
Exports To EEC
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
Year
The following model was fit to the data
Yt = β0 + β1 Time + β2 Q2 + β3 Q3 + β4 Q4 + at ,
at ∼ nid(0, σa2 )
where Yt is exports in billions of dollars, Time is year (1958.00, 1958.25, . . . , 1968.25) and
Q2 is 1 in quarter 2 and 0 otherwise, Q3 is 1 in quarter 3 and 0 otherwise, and Q4 is 1 in
quarter 4 and 0 otherwise. The results of the regression fit using ordinary least squares showed
βb0 = −145, βb1 = .0749, βb2 = −8.40, βb3 = −.44, βb4 = −23.24, σ
ba = .09.
(a) What is the practical interpretation of the parameter β1 in this model? (That is, how
would you explain the meaning to someone who did not know much statistics?)
(b) What is the practical interpretation of the parameter β3 in this model?
(c) What is the practical interpretation of the parameter σa in this model?
(d) Based on the above model, give an expression for a simple approximate prediction interval for the amount of exports in the third quarter of 1968 (set it up, actual calculations
not needed).
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(e) You do not have enough information to actually do the test, but outline the steps that
you would follow (include an equation) to test if there is evidence of seasonality in these
data. Be as complete as possible.
4. Consider the following MA(2) time series model
Zt = (1 − θ2 B2 )at ,
at ∼ nid(0, σa2 ).
(a) Derive an expression for the variance of Zt .
(b) Derive an expression for ρ3 , the autocorrelation between observations separated by three
time periods.
5. The normal probability plot (also known as a Q-Q plot) is an extremely useful tool in data
analysis. What is the primary purpose of this tool?
6. Dervive an expression for the variance of the AR(2) model Żt = φ1 Żt−1 + φ2 Żt−2 + at
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7. The following sample autocorrelations were computed from a realization of length n =100.
k:
0
ρbk : 1.000
1
0.450
2
0.012
3
4
5
6
7
8
0.022 −0.013 −0.011 0.020 −0.011 −0.014
(a) Compute the sample partial autocorrelation values φb11 and φb22
(b) Compute standard errors for φb11 and φb22 .
(c) Use your work in parts 7a and 7b to check to see if there is evidence that φ11 and φ22
are different from 0.
(d) What conclusions can you draw from the sample ACF and sample PACF in this problem?
8. For each of the following differencing schemes, write down Wt as a function of past and present
values of Zt .
(a) Wt = (1 − B)2 Zt
(b) Wt = (1 − B)(1 − B12 )1 Zt
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