SOLUTIONS for Examples IV for Time Series (S8)
NB: Stationarity by itself always means second-order stationarity
[1] (a) Now
N 1
X
S (p) (f ) =
s(p) e
i2f
.
= (N 1)
So
Z
1/2
S (p) (f ) df =
1/2
N 1
X
s(p)
= (N 1)
Z
1/2
e
i2f
df.
1/2
When = 0, the inte

Chapter 3
Spectral Representation theorem for discrete
time stationary processes
Spectral analysis is a study of the frequency domain characteristics of a process, and
describes the contribution of each frequency to the variance of the process.
Let us den

SOLUTIONS for Examples I for Time Series (S8)
[1](a)(i)
E cfw_Xt = E cfw_Y1 cos(ct) + E cfw_Y2 sin(ct) = 0.
Also for the covariance (which for = 0 gives the variance),
E cfw_Xt Xt+ = E cfw_[Y1 cos(ct) + Y2 sin(ct)][Y1 cos(c[t + ]) + Y2 sin(c[t + ])]
=

SOLUTIONS for Examples II for Time Series (S8)
NB: Stationarity by itself always means second-order stationarity
[1]
(a) We cannot assume the means are zero here. But, since cfw_Xt and cfw_Yt are stationary,
Ecfw_Zt = Ecfw_Xt + Yt = X + Y = Z
Ecfw_Zt+

Chapter 6
Parametric model tting: autoregressive processes
Here we concentrate on models of the form
Xt 1,p Xt1 . . . p,p Xtp = t .
As we have seen the corresponding sdf is
S (f ) =
|1 1,p ei2f
2
.
. . . p,p ei2f p |2
This class of models is appealing to

Chapter 5
A na non-parametric spectral estimator the periodogram
ve
Suppose the zero mean discrete stationary process cfw_Xt has a purely continuous
spectrum with sdf S (f ). We have,
S (f ) =
1
|f | .
2
s ei2f
=
With = 0, we can use the biased estimat

SOLUTIONS for Examples III for Time Series (S8)
NB: Stationarity by itself always means second-order stationarity
1
1
[1] (a) The corresponding characteristic polynomial is (z) = (1 + 12 z 24 z 2 ) which can
1
1
be factorized as (1 6 z)(1 + 4 z) so that t

Time Series (S8) 2013/2014
Coursework II: SOLUTIONS
[1] (a) note that, for any complex number z ,
(1
z)
N1
X
z=
t
t=0
N1
X
(1
z )z =
t
t=0
N1
X
z
N1
X
t
t=0
z
t+1
N1
X
=
t=0
z
N
X
t
t=0
zt = 1
zN .
t=1
PN 1
If z 6= 1, we can divide through by 1 z to get t

SOLUTIONS for Examples V for Time Series (S8)
NB: Stationarity by itself always means second-order stationarity
[1] (a) Let
J(f )
N
X
ht (Xt
i2f t
)e
.
t=1
By the spectral representation theorem
Z
Xt =
1/2
0
ei2f t dZ(f 0 ),
1/2
where cfw_Z() is a proces

Time Series (S8) 2013/2014
Coursework I: SOLUTIONS
[1]
(a)(i) Since Xt = + 0 for all t, X =
1
N
PN
1
t=0
Xt = + 0 , and hence
var X = var cfw_ + 0 = var cfw_0 =
2
.
[2 marks]
2
2
(ii) Then, cov cfw_Xt+ , Xt = cov cfw_ + 0 , + 0 = var cfw_0 = , so s =

Chapter 2
Real-Valued discrete time stationary processes
Denote the process by cfw_Xt . For xed t, Xt is a random variable (r.v.), and hence
there is an associated cumulative probability distribution function (cdf ):
Ft (a) = P(Xt a),
and
Ecfw_Xt =
varcf

Chapter 8
Forecasting
Suppose we wish to predict the value of Xt+l of a process, given Xt , Xt1 , Xt2 , . . .
Let the appropriate model for cfw_Xt be an ARMA(p, q ) process:
(B )Xt = (B ) t .
Consider a forecast Xt (l) of Xt+1 (an l-step ahead forecast)

Chapter 4
Estimation of mean and autocovariance function
Ergodic property
Methods we shall look at for estimating quantities such as the autocovariance function will use observations from a single realization. Such methods are based on the
strategy of rep

Chapter 7
Bivariate Time Series
The two real-valued discrete time stochastic processes cfw_X1,t and cfw_X2,t are said to
be jointly stationary stochastic processes if cfw_X1,t and cfw_X2,t are each, separately,
second-order stationary processes, and c