Time Series Exam, 2010: Solutions - STAT

(b) Since the roots of 1 −x+0−5x 2 are (1±i)/2, this is an ARMA(2,1) process. Moreover,
it is neither causal nor invertible since the two roots of φ(z) lie inside the unit disk and the
root of θ(z) is 1.
(c) Here, we have
γ(h) = 5σ 2 δ(h)−2σ 2 (δ(h+1)+δ(h−1))
and so we have the same ACF as we had in (a). The two processes are the same (provided
the variance of the white noise is such that the two autocovariance functions are identical),
but the model in (a) is invertible, whereas the model in (c) is not.
5. See the notes.
6. All linear state space models involve two equations, the state equation, which determines the
evolution of an underlying unobserved state, and the observation equation, which determines how
the observed data are related to the state. The local trend model (a simple special case) has
State equation:
Observation equation:
µ t+1 = µ t +η t , η t
iid
∼ N(0,σ
2
η ),
y t = µ t +ε t , ε t
iid
∼ N(0,σ 2 ),
where the η t and ε t are mutually independent. We suppose that data y 1 ,...,y n are available.
Let H t denote the information available at time t. Filtering is the estimation of µ t using H t ,
smoothing is the estimation of µ t using H n and prediction is the forecasting µ t+h fot h > 0 using
H t .
(a) We have
State equation:
Observation equation:
X t = −0.9X t−2 +ε t , ε t
iid
∼ N(0,σ
2
ε ),
Y t = X t +η t , η t
iid
∼ N(0,σ
2
η ).
(b) Since η t is an independent white noise, Y t is stationary if and only if X t is stationary.
Moreover, X t is an AR(2) model, so provided the variances σ0 2 and σ2 1 are such that var(X t)
does not depend on t, Y t is stationary. Since
®
(−0.9) t/2 X
X t = 0 + ∑ t/2
k=1 ε 2k(−0.9) t/2−k , t even,
(−0.9) (t+1)/2 X −1 + ∑ (t−1)/2
k=0
ε 2k+1 (−0.9) (t−1)/2−k , t odd,
the variance of X t is given by
{
var(X t ) =
σ0 2(−0.9)t +σε
2 1−(0.81) t/2
1−0.81
, t even,
σ1 2(−0.9)t+1 +σε
2 1−(0.81) (t+1)/2
1−0.81
, t odd,
and so X t and Y t are stationary if and only if
σ 2 0 = σ2 1 =
σ 2 ε
1−0.81 .
(c) The left time plot (X t ) shows clearly the AR(2) structure, whereas on the right time plot, it
is more difficult to see, because of the noise η t . The range is also more important (from -10
to 5 instead of -8 to 2). The left ACF is typical from an AR(2) model with such parameters.
On the right one, the added noise reduces the proportion of information in the observation
and the correlation, so the values are diminished on the plot. Finally, on the left PACF, we
clearly find the model structure, whereas on the right one, we also have the consequences of
the added noise.
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