STAT 679: Time Series Analysis II Assignment 1

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STAT 679 - Assignment 1 - Due date is on the course outline

1. Let γˆXY (m) be the usual sample cross-covariance function for jointly stationary series
{Xt, Yt}, with data {xt, yt}n
t=1. Show that
E [ˆγXY (m)] = µn − m
n
¶
{γXY (m) − cov [¯y, x¯∗
] − cov [¯x, y¯∗] + cov [¯x, y¯]} ,
where x¯∗ is the average of {xt}n
t=m+1 and y¯∗ is the average of {yt}n−m
t=1 .
2. Assume μX = 0; consider the problem of minimizing the function
fm(α1,m, ..., αm,m) = E £
{Xt − α1,mXt−1 − ... − αm,mXt−m}2¤
,
which is the MSE when Xt is forecast by α1,mXt−1+...+αm,mXt−m. Let the minimizers
be α∗
1,m, ..., α∗
m,m. The lag-m PACF value, written φmm, is defined to be α∗
m,m. Show
that
φmm = corr h
Xt − Xˆt, Xt−m − Xˆt−m
i
,
where each Xˆ denotes the best (i.e. minimum MSE) predictor which is a linear function
of Xt−1, ..., Xt−m+1.
3. The obvious estimate of
f(νk) = X∞
m=−∞
e−2πiνkmγ(m),
using data {xt}n
t=1 is
ˆf(νk) = Xn−1
m=−(n−1)
e−2πiνkmγˆ(m).
Show that this reduces to the periodogram:
ˆf(νk) = I(νk) = |X(k)|2.
4. Suppose that {Xt} is AR(2), mean zero and stationary.

(a) In the representationXt = wt + ψ1wt−1 + ψ2wt−2 + ...show that ψk = φ1ψk−1 + φ2ψk−2, where ψ1 = φ1, ψ0 = 1.

(b) Obtain a closed form expression for ψk. (Hint: write ψk−bψk−1 = a

¡ ψk−1 − bψk−2 ¢ for suitable constants a and b; then iterate.)

5. Suppose that filter coefficients are approximated by

aM
s = 1
M
M
X−1
k=0
A(ωk)e2πiωks
and then Y M
t = P
|s|<M/2 aM
s Xt−s for t = M/2 − 1, ..., n − M/2.
(a) Show that these coefficients are real and symmetric (aM s = aM −s), if A is real and symmetric: A (ω) = A (−ω).

(b) Show that aM s = X∞ t=−∞ at [I(t − s is a multiple of M)] .

(c) Establish the bound in Problem 4.32:

E h¡ Y M t − Yt ¢2 i ≤ 4γX(0) 
  X
|k|≥M/2
|ak|
2
.