Sample mean

n

1

Xt

x = n

xt

=1

Sample variance

n

jnj

x(h)

V [x] = n h= n 1

1

X

h

Sample autocovariance function

b

1 n h

Xt

(xt+h x)(xt x)

(h) =

n

=1

Sample autocorrelation function

(h) =

(h)

b

b

(0)

Sample cross-variance function

b

b

1 n h

Xt

xy(h) =

n

=1 (xt+h x)(yt

y

)

Sample cross-correlation function

xy(h)

xy(h) =

b

x(0) y(0)

b

Properties

pb

b

1

x(h) =

p

if xt is white noise

n

b

1

xy(h) =

p

if xt or yt is white noise

n

b

1

k

vt =

xt 1

2k + 1

i= k

X

1

k

If

i= k wt j 0, a linear trend function t = 0

+ 1t passes

2k+1

Autoregressive polynomial

(z) = 1 1z pzp

z 2 C ^ p 6= 0

Autoregressive operator

(B) = 1 1B pBp

Autoregressive model order p, AR (p)

xt = 1xt 1 + + pxt p + wt () (B)xt = wt

AR (1)

k 1

k!1;j j<1

1

k

Xj

j

j

xt = (xt k) +

(wt j)

=

X

(wt j)

=0

j=0

E [xt] =

j=0 (E [wt j]) = 0

|

{z

}

1

linear process

(h) =

j

P

t+h t

1 2

Cov [x

; x ] =

w2 h