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Chapter

STA457H1 Chapter Notes -If And Only If


Department
Statistical Sciences
Course Code
STA457H1
Professor
Zhou Zhou

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Stationary Models
Let
Xt
be a time series with
E
[
Xt
2
]
<
The mean function of
Xt
is
μX
(
t
)
=E[Xt]
The covariance function of
{
Xt
}
is
γX
(
r , s
)
=Cov
[
Xr, Xs
]
=E
[
(
XrμX
(
r
)
) (
XsμX
(
s
)
)
]
r , s
Xt
is weakly stationary if
1.
μX
(
t
)
is independent of
t
; and
2.
is independent of
t
for each
h
Note:
γX
(
h
)
=γX
(
h ,0
)
=γX
(
t ,t +h
)
ACVF and ACF
Let
{
Xt
}
be a stationary time series
The autocovariance function (ACVF) of
{
Xt
}
at lag
h
is
γX
(
h
)
=Cov
[
Xt, Xt+h
]
The autocorrelation function (ACF) of
{
Xt
}
at lag
h
is
ρX
(
h
)
=γX
(
h
)
γX
(
0
)
=Corr
[
Xt, Xt+h
]
Properties of ACVF:
1.
γX
(
h
)
≤ γX
(
0
)
h
2.
γX
(
h
)
=γX
(
h
)
h

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Sample ACVF and ACF
Let
x1, x2, …, xn
be observations of a time series
The sample autocovariance function is
̂
γX
(
h
)
=1
n
t=1
n
h
(
xt+
h
́x
)
(
xt́x
)
,n<h<n
The sample autocorrelation function is
̂
ρX
(
h
)
=
̂
γX
(
h
)
̂
γX
(
0
)
,n<h<n

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Some Stationary Models
iid noise
{
Xt
}
IID
(
0, σ2
)
- observations are independent and identically distributed with zero mean
μX
(
t
)
=E
[
Xt
]
=0
γX
(
t , t+h
)
=
{
σ2,h=0
0,h ≠ 0
- iid noise with finite second moment is stationary
White noise
{
Xt
}
WN (0, σ2)
- sequence of uncorrelated random variables each with zero mean and variance
σ2
- has same autocovariance function as iid noise, therefore stationary
- every
IID
(
0, σ2
)
sequence is
WN
(
0, σ2
)
but not conversely
First-order moving average
{
Xt
}
MA
(
1
)
Xt=Zt+a Zt1, t=0, ±1,
where
{
Zt
}
WN
(
0, σ2
)
and
aR
μX
(
t
)
=E
[
Xt
]
=0
γX
(
h
)
=
{
σ2(1+a2),h=0
σ2a ,h=±1
0,
h
>1
ρX
(
h
)
=
{
1,h=0
a
1+a2,h=±1
0,
h
>1
First-order autoregression
{
Xt
}
AR(1)
Xt=a X t1+Zt, t =0, ±1,
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