Time Series Exam, 2010: Solutions - STAT

(c) We have
ρ(t−1,t) =
=
γ(t−1,t)
√
γ(t−1,t−1)γ(t,t)
(t−1)σ 2
√
(t−1)σ2 tσ 2
=
…
t−1
−→ 1, as t → ∞.
t
This means two observations far apart are strongly correlated.
(d) We showed in (b) that µ t depends on t and γ(s,t), on min(s,t).
(e) Let us consider the differenced series Ỹt = Y t −Y t−1 = ε t +δ. We have then µ t = δ, which
does not depend on t and cov(Y s ,Y t ) = cov(ε s +δ,ε t +δ) = σ 2 δ(s−t), where δ(s−t) = 1 if
s = t and 0 otherwise.
2. (a) See Lemma 18.
(b) Conditional on A = {Y t−r ,...,Y t−1 }, we find that E(Y t | A) = φy t−1 and E(Y t−r−1 | A) =
φ −1 y t−r , the latter because we can write Y t−r−1 = φ −1 (Y t−r −ε t−r ). Moreover,
E(Y t Y t−r−1 | A) = E{(φY t−1 +ε t )φ −1 (Y t−r −ε t−r ) | A} = y t−1 y t−r , r ≥ 1,
so the conditional covariance cov(Y t ,Y t−r−1 | A) = E(Y t Y t−r−1 | A)−E(Y t | A)E(Y t−r−1 | A) = 0.
If we see a series in which the partial autocorrelation function has zeros after a certain point, then
we use this property to diagnose an AR model, whereas the same property for the ACF suggests
an MA model. See slide 154 and previous material.
(c) The plots show the correlogram (empirical ACF) and partial correlogram empirical PACF)
for the data. The estimates on the left are moment-based, and those on the right are obtained
from them using the Yule–Walker equations. The horizontal dashed lines show significance limits
for the correlogram and partial correlogram elements, based on assumptions of white noise with
finite fourth moments; the limits are at ±2/ √ n.
The correlogramshows geometric decline with alternating sign, and suggests that this is an AR(1)
model with φ ≈ −0.9. This is confirmed by the PACF, which has just one significant value, at
h = 1, with value around −0.9.
3. If {γ h } h∈Z is the ACF of a stationary random sequence, then there exists a unique function F
defined on [−1/2,1/2] such that F(−1/2) = 0, F is right-continuous and non-decreasing, with
symmetric increments about zero, and
∫
γ h = e 2πihu dF(u), h ∈ Z.
(−1/2,1/2]
The function F is called the spectral distribution function of γ h , and its derivative f, if it exists,
is called the spectral density function. If ∑ h|γ h | < ∞, then f exists. A function f(ω) defined on
[−1/2,1/2] is the spectrum of a stationary process if and only if f(ω) = f(−ω), f(ω) ≥ 0, and
∫
f(ω)dω < ∞.
We have
γ(h) = E[(Y t+h −E[Y t+h ])(Y t −E[Y t ])]
[ ]
∑
= E ψ j ε t+h−j ·∑
ψ l ε t−l
j
= ∑ j
ψ j ψ j−h σ 2 ,
l
2