# STA457H1 Chapter Notes -If And Only If

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

[

(

Xr−μX

(

r

)

) (

Xs−μX

(

s

)

)

]

∀r , s

Xt

is weakly stationary if

1.

μX

(

t

)

is independent of

t

; and

2.

γX

(

t , t+h

)

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

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

)

(

xt−́x

)

,−n<h<n

The sample autocorrelation function is

̂

ρX

(

h

)

=

̂

γX

(

h

)

̂

γX

(

0

)

,−n<h<n

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 Zt−1, t=0, ±1, …

where

{

Zt

}

WN

(

0, σ2

)

and

a∈R

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 t−1+Zt, t =0, ±1, …

## Document Summary

X (r ,s)=cov[ x r , xs]=e[( x r x (r))(x s x (s))] r , s is. X (t ,t+h) is independent of t for each h. Note: x (h)= x (h ,0)= x (t ,t +h) X (h)=cov [ xt , x t +h] X (h)= 1 (xt+ h x)(xt x), n