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Unformatted text preview: ST525 TIME SERIES I
First Mid—term, October 2nd, 2002, 12 noon
Time allowed : 2 hrs Open Book and Notes. 60 points=100% 1. (20 points) If {Zt,t = O,i1, . . } is IID(O,1), Z is independent of {Zt} with
mean 0 and variamce 1 and w is a ﬁxed number between 0 and 71 , ﬁnd EXt
and E(Xt+hXt) for each of the processes {Xht = 0,:121, . ..} deﬁned below. In
each case state whether or not the process is stationary. (a) X: = Z + Z0 cos(wt) + Z1 sin(wt),
(b) X221: Zt—1(Zt  Zr—z), () Xt*Xt1=Zt, Xo=0 (CUX =(—1)tz 2. (20 points) (a) Let Y: = a + bi? + 5,: + Zt, t = O,:l:1,..., where a and b are constants, {Zt} «9 WN(O, 02 )and {st} has period 4 Find a ﬁlter of the 'form 1 —
Bd to eliminate the periodic component of {11;} and ﬁnd the mfgnand
autocovariance function of the ﬁltered series, Xt— — (1 — Bd)Y; "’ r1”: .1~ «1‘13" "'31:; Suppose that {Xi} is a stationary time series satisfying
Xt = .5X1;_1 "l“ Wt, where Wt = l + Z + Z21,
{Zt} ~ WN( 0, a 2) and Z is uncorrelated with {Z} with mean zero and variance 02 .Express Xuntgrnrrs of W22 3 < t, and hence ﬁnd the mean ' MW and autocovariance function of {Xt}. 3. (20 points) For the MA(1) process, ‘ Xi =2 Z7; — 0.5Zt_1, {Z75} N WN(O,1),
(a) ﬁnd the best linear predictor of X3 in terms of 1 ,X2 and X1 and its mean squared error (to three decimal places)7 (b) ﬁnd the best linear predictor of X3 in terms of 1 and Xk, —00 < k g 2,
and its mean squared error (to three decimal places), Phi/ll? (0) ﬁnd the mean squared error (to three decimal places) of the predictor obtained by truncating the predictor in (b) to exclude the terms involving
X15, k S 0. 4. (20 points) The sample mean and variance of observations {$1, . . . $5100} of
a stationary time series are I E: 10, 7(0) = 0.25,
and the sample autocorrelations at lags 1, . . . , 10 are ﬁ(1) M2) M3) 0(4) 5(5). 13(6) W) M8) M9) x3(10)
0.800 0.500 0.481 0.253 0.038 0.015 —0.052 0.026 —0.031 —0.012 (a) Are these results consistent with the hypothesis that {2:1, . . . , {1:100} are
observed values of an HD sequence {X13}? Explain your answer. (b) Write down EXt and Cov(Xt_ph,Xt) for the AR(1) process deﬁned by (Xt _ a) ~ ¢(Xt—1  M) = Zn {Zia} ~ WMOJZ), ¢ < 1. (c) By equating the sample mean and covariances "f, '"y(0) and '?(l) to the
corresponding. quantities for the model in (b), ﬁnd estimates of the pa— rameters d), 02 and ,u in a model of the form (b) to represent the observed '
V data. ' 7 (d) Assuming your ﬁtted model is correct, use it to ﬁnd the best linear pre—
dictor of X101 in terms of 1 and Xk, k g 100. (e) What is the mean quarecl error of the predictor found in (d) (assuming,
as in (d), that your ﬁtted model is correct)? ...
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 Spring '09
 BROCKWELL
 Forecasting

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