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Sta457 Time Series Analysis

Homework 1

Jan. 18, 2013

Due Jan. 29, 2012 in class

You should work out this Homework individually. Group works or discussions are not

acceptable.

No late Homework will be accepted.

(1) Problem 1.4 on Page 41 of the Textbook.

(2) Problem 1.5 on Page 41 of the Textbook.

(3) Problem 1.11 on Page 42 of the Textbook.

(4) Problem 1.15 on Page 43 of the Textbook.

(5) Suppose that we have the following times series model with a time-varying trend:

t

Z = 2 +W ; for t = 1;2; ;n;

t t

n

where W is an AR(1) process W =:5W +a and a ’s are i.i.d. (independent and

t t t 1 t t

identically distributed) standard normal random variables.

a . Calculate Corr(Z ;Z ) and Corr(Z ;Z ). Are they equal?

1 2

n=2 n=2+1

b . Calculate the ?rst order sample ACF ˆ(1) for (Z ) by assuming that n goes

t

to in?nity. Is it the same as the the ?rst order ACF of (W )?

t

c . Let Y = Z Z . Prove that Y , t = 2;3; ;n is a weakly stationary time

t t t 1 t

series. Therefore we can transform Z into a stationary time series by taking the

t

?rst order di?erence of it.

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