STK 320 / WST 321 time series analysis Exercise 3 memo

1


EXERCISE 3 SUGGESTED SOLUTION
1.(a)
t −1
Z t = ∑ Wt − j
j =0
t −1
(
= ∑ at − j + 12 at − j −1 )
j =0

= (at + 12 at −1 ) + (at −1 + 12 at − 2 ) + (at − 2 + 12 at − 3 ) + ... + (a1 + 12 a0 )
= at + 32 at −1 + 32 at − 2 + ... + 32 a1 + 12 a0

1.(b) (i)
E (Z t ) = E (at + 32 at −1 + 32 at − 2 + ... + 32 a1 + 12 a0 )
=0

1.(b) (ii)
Var (Z t ) = Var (at + 32 at −1 + 32 at − 2 + ... + 32 a1 + 12 a0 )
= [1 + 2.25(t − 1) + 0.25]σ a2
= (2.25t − 1)σ a2

1.(b) (iii)
Var (∆Z t ) = Var (Z t − Z t −1 )
[
= Var (at + 32 at −1 + 32 at − 2 + ... + 32 a1 + 12 a0 )
− (at −1 + 32 at − 2 + 32 at −3 + ... + 32 a1 + 12 a0 ) ]
= Var (at + 0.5at −1 )
= (1 + 0.25)σ a2
= 1.25σ a2

1.(b) (iv)
[
Var (Z1 + Z 2 + Z 3 ) = Var (a1 + 12 a0 ) + (a2 + 32 a1 + 12 a0 )
+ (a3 + 32 a2 + 32 a1 + 12 a0 ) ]
= Var (a3 + 2.5a2 + 4a1 + 1.5a0 )
= (1 + 6.25 + 16 + 2.25)σ a2
= 25.5σ a2




WST321

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1.(b) (v)
Cov(Z t − 2 Z t +1 , Z t +1 + Z t + 2 ) = Cov(Z t , Z t +1 ) + Cov(Z t , Z t + 2 )
− 2Var (Z t +1 ) − 2Cov(Z t +1, Z t + 2 )
= Cov(at + 32 at −1 + 32 at − 2 + ... + 32 a1 + 12 a0 ,
at +1 + 32 at + 32 at −1 + ... + 32 a1 + 12 a0 )
+ Cov(at + 32 at −1 + 32 at − 2 + ... + 32 a1 + 12 a0 ,
at + 2 + 32 at +1 + 32 at + ... + 32 a1 + 12 a0 )
− 2Var (at +1 + 32 at + 32 at −1 + ... + 32 a1 + 12 a0 )
− 2Cov(at +1 + 32 at + 32 at −1 + ... + 32 a1 + 12 a0 ,
at + 2 + 32 at +1 + 32 at + ... + 32 a1 + 12 a0 )
= [1.5 + 2.25(t − 1) + 0.25]σ a2
+ [1.5 + 2.25(t − 1) + 0.25]σ a2
− 2(1 + 2.25t + 0.25)σ a2
− 2(1.5 + 2.25t + 0.25)σ a2
= (− 4.5t − 7 )σ a2

1.(c) SAS Program
data ima;
n=200;
seed=0;
theta=-0.5;
var_at=2.5;
zt_1=0;
at_1=sqrt(var_at)*rannor(seed);
do t = 1 to n;
at=sqrt(var_at)*rannor(seed);
zt=zt_1+at-theta*at_1;
wt=zt-zt_1;
output;
zt_1=zt;
at_1=at;
end;
run;


1.(d) SAS Program
goptions reset=all i=join;
symbol1 color=blue;
title1 'Simulated IMA(1,1) series';
proc gplot data=ima;
plot zt*t;
run;
proc arima data=ima plots(only)=series(acf);
identify var=zt nlag=12;
run;




WST321