Moving average polynomial

(z) = 1 + 1z + + qzq

z 2 C ^ q 6= 0

Moving average operator

(B) = 1 + 1B + + pBp

MA (q) (moving average model order q)

xt = wt + 1wt 1 + + qwt q () xt = (B)wt

q

Xj

E [xt] = jE [wt j] = 0

=0

P

t+h

t

(0

q h

h > q

2

j j+h 0 h q

(h) = Cov [x

; x ] =

w

j=0

MA (1)

xt = wt + wt 1

h = 1

(h) = 8 w2

>

(1 + 2) 2

h = 0

w

<0

h > 1

>

:

h = 1

(1+ 2)

(h) = (0

h > 1

ARMA (p; q)

xt = 1xt 1 + + pxt p + wt + 1wt 1 + + qwt q

(B)xt = (B)wt

Partial autocorrelation function (PACF)

xih 1 , regression of xi on fxh 1; xh 2; : : : ; x1g

hh = corr(xh xhh 1; x0 x0h 1)

h 2

E.g., 11 = corr(x1; x0) = (1)

ARMA (p; q) is causal (future-independent) () 9f

1

jg : Pj=0 j < 1 such that

1

Xj

wt j = (B)wt

xt =

=0

() 9f jg

1

ARMA (p; q) is invertible

: Pj=0 j < 1 such that

1

Xj

(B)xt = Xt j = wt

=0

Properties

ARMA (p; q) causal

() roots of (z) lie outside the unit circle

1

(z)

Xj

jzj 1

(z) =

jzj = (z)

=0

ARMA (p; q) invertible () roots of (z) lie outside the unit circle

1

(z)

Xj

jzj 1

(z) =

jzj = (z)

=0

Behavior of the ACF and PACF for causal and invertible ARMA models

AR (p)

MA (q)

ARMA (p; q)

ACF

tails o

cuts o after lag q

tails o

PACF

cuts o after lag p

tails o q

tails o