Stats 349.3(02) Review Problems
1. Suppose that {xt: t  T} is a stationary time series with autocovariance function
x(h) and spectral density fx(). Suppose that we apply two filtering operations to
obtain the time series{yt: t  T} and {wt: t  T}:
yt 
wt 

ax
r 
t t r

by
s 
t
t s
a. Show that the spectral density of {wt: t  T is given by
f w     A    B    f x    where
2
A  

2
 at eit and B    
t 

 be
t 
t
 i t
.
b. Suppose that {xt: t  T} satisfies the equation
xt  ut  0.5ut 1
where{ut: t  T} is a white noise time series with 2 = 2.0.
1
1
1
Suppose that yt  xt 1  xt  xt 1 and wt  yt  .8 yt 1 . Find and plot
4
2
4
the spectral density, fw()., of {wt: t  T}
2. Suppose that {xt: t  T} is a stationary time series with autocovariance function
x(h). Suppose that we apply a filtering operation to obtain the time series{yt: t  T}.
yt  xt  xt 1  xt
Find the autocovariance function, y(h)., of {yt: t  T} in terms of x(h).
h
Find y(h) if  x  h    .
3. Consider the following model with ARIMA identification (p, d, q) = (1, 0, 0) and
Seasonal ARIMA identification (ps , ds, qs) = (0, 1, 1) and period k = 12.
a. Write out the model using the backshift operator, B, notation.
b. Expand the equation and write out the model in terms of {xt: t  T} and
the white noise time series {ut: t  T}.
4. Suppose that {ut: t = ...,-2,-1,0,1,2,...} is an independent time series with mean
zero, variance u2= 4.0. Suppose that the time series {xt: t = ...,-2,-1,0,1,2,...}
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.
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Stats 349.3(02) Review Problems
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 2observations. (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
observation.
(Compute both 95% and 66.7% prediction limits)
5. Suppose that for N = 80 observations on the time series {xt : t T} the
following statistics were calculated:
_
x = 10.54
C(0) = the sample variance = 4.99
In addition the sample autocorrelation function, rh, was computed for h = 1,
2, 3, ..., 10 and is tabulated below:
h
rh
1
0.60
2
0.37
3
4
5
6
0.13 -0.04 -0.08 0.02
7
0.09
8
0.07
9
0.15
10
0.15
ˆ , for k = 1, 2, 3.
Compute the sample partial autocorrelation function 
kk
i) Suppose that this time series is identified as an ARIMA(1,0,1) time
series.
Use the method of moments to estimate all the parameters of the model.
ii) Suppose that this time series is identified as an AR(3) time series.
Use the method of moments to estimate all the parameters of the model.
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