Time Series (MATH5/30085) 2004
Exercises 1
1. Properties of covariance. Using the definition
Cov
(
X, Y
) =
E
[(
X

μ
X
)(
Y

μ
Y
)]
Prove the following:
(a)
Cov
(
X, Y
) =
Cov
(
Y, X
)
(b)
Cov
(
a
+
bX, c
+
dY
) =
bdCov
(
X, Y
)
(c)
Cov
(
X, Y
) =
E
(
XY
)

μ
X
μ
Y
2. Find the autocorrelation sequences for the following processes
(a) A white noise process with
E
(
X
t
) =
μ
,
V ar
(
X
t
) =
σ
2
∀
t
(b)
X
t
=
epsilon1
t

epsilon1
t

1
(c) For an MA(1) process,
X
t
=
epsilon1
t

θ
1
epsilon1
t

1
, show that you cannot identify
an MA(1) process uniquely from the autocorrelation by comparing the
results using
θ
1
with those if you replaced
θ
1
by
θ

1
1
3. Considering a first order AR process
X
t
=
φ
1
X
t

1
+
epsilon1
t
AR(1) – Markov process
(a) Find the mean and an expression for the variance
(b) Show that for the variance to be finite,

θ
1

must be less than one
(c) Find the autocorrelation sequence (you may assume stationarity)
4.
(a) What is meant by saying that a stochastic process is secondorder sta
tionary?
(b) Determine whether the following stochastic process is secondorder sta
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 Spring '09
 jsdkasj
 Covariance, Variance, Autocorrelation, 1 K, Stationary process, τ, Xt

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