UNIVERSITY OF SASKATCHEWAN
Stats 349.3(02) – Sample Final Examination
Open Book Examination
Instructor: W. H. Laverty
Each question is worth 20 marks
I
Suppose that {xt: t  T} is an AR(1) time series with  = 0, 1 = 0.7 and u = 1.4.
Thus {xt: t  T} satisfies the equation (1) with {ut: t  T} denoting a white noise
time series with standard deviation u = 1.4. Suppose the {zt: t  T} is an MA(1)
time series with mean  = 0, 1 = -0.4 and v = 2.0 and is independent of {xt: t  T}.
Thus {zt: t  T} satisfies the equation (2) with {vt: t  T} denoting a white noise time
series with standard deviation v = 2.0.
xt  0.7 xt 1  ut
1
zt  vt  0.4vt 1
2
Suppose that yt = xt + zt for t  T.
a) Determine the spectral density of {xt: t  T}
b) Determine the spectral density of {zt: t  T}
c) Determine the spectral density of {yt: t  T}
d) Determine the squared coherency function of{yt: t  T}with {xt: t  T}
II Suppose that {ut : t T}is a time series of independent observations with mean
zero and common variance 2= 4. Suppose further that {xt : t T} is a time series
satisfying:
xt = 0.6 xt-1 + ut + 0.1 ut-1 - 0.2 ut-2 + 5 .
a)
b)
c)
d)
Find the mean value, = E(xt), of the time series.
Compute (h), the autocovariance function and (h), the autocorrelation
function of the process. Plot their graphs.
If x100 = 10.0 would you expect x102 to be above or below the mean of the
process?
Find the values of 11, 22 and 33 - the values of the partial autocorrelation
function (PAFC) at lags 1, 2 and 3 respectively.
III Suppose that the time series {xt : t T} has been identified as an MA(1) time series
and satisfies the equation:
xt =  + ut + 1 ut-1 .
Let r1 the value of the sample autocorrelation function at lag 1. Then the methods of
moments estimator of the parameter 1 satisfies the equation:
^2 - 
^ + r = 0 (1)
r
1 1
1
1
Show that one of the roots of equation (1) is less than or equal to 1 in absolute value
while the other root is greater than or equal to 1.
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UNIVERSITY OF SASKATCHEWAN
Stats 349.3(02) – Final Examination
April 2008
IV Suppose that {ut: t  T} is an independent time series with mean zero, variance u2= 9.0.
Suppose that the time series {xt: t  T} satisfies the equation.:
xt = 1.5 xt - 1 – 0.5 xt - 2 + 3.0 + ut - 2/3 ut-1.
a) Identify the time series.
b) Determine the autocovariance function and autocorrelation function of the time
series xt - xt-1.
c) Find the random shock form of the time series.
d) Suppose that the first five observations of the time series are x1 = 2.5, x2 = 4.0,
x3 = 3.7, x4 =1.0 and x5 = 1.5.
Use these observations to compute prediction intervals for the next 5
observations. (Compute 95% prediction limits)
V.
Suppose that {ut: t  T} is an independent time series with mean 0,
variance u2 = 4.0. Suppose that the time series {xt: t  T} satisfies
the equation.:
xt = .5 xt-1 + 1.0 + ut - .4 ut-1.
a) Determine the autocovariance function and autocorrelation function of the time
series xt.
b) Find the random shock form of the time series.
c) Suppose that the first four observations of the time series are x1 = 2.75,
x2 = 1.10, x3 = 1.38 and x4 =0.94
i) Use these observations to compute prediction intervals for the next 4
observations. (Compute both 95% and 66.7% prediction limits)
ii) If the fifth observation turns out to be x5 = 4.22 use this information to
re-compute prediction intervals for the next 3 observations. (Compute both
95% and 66.7% prediction limits)
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