Time Series - STAT - EPFL

General linear process
Theorem 37 The spectrum of the general linear process
Y t =
∞∑
a j ε t−j , (8)
j=0
where {ε t } is white noise, may be written as
f(ω) = b 0 +
∞∑
b m cos(2πmω), −1/2 ≤ ω ≤ 1/2. (9)
m=1
□ Any real-valued even continuous function that satisfies f(ω) = f(ω + k) for integer k can be
expressed as the (implicit) Fourier series (9), so any stationary random sequence with a
continuous spectrum can be represented as a general linear process—at least so far as
second-order properties are concerned.
□ The general linear representation (8) is only useful if it involves only a few parameters—hence the
use of ARMA models, which are quite flexible linear models with finite parameters.
□ The computations leading to (9) implicitly presuppose that ∑ a 2 j < ∞, which then implies that
{Y t } is stationary with finite variance and covariance function.
Time Series Spring 2010 – slide 136
ARMA models slide 137
Autoregressive process
Definition 38 An autoregressive process of order p, AR(p), model, is of the form
Y t = φ 1 Y t−1 + φ 2 Y t−2 + · · · + φ p Y t−p + ε t , (10)
where {Y t } is stationary and φ 1 ,... ,φ p are constants and φ p ≠ 0. Unless otherwise mentioned, we
iid
assume here that ε t ∼ N(0,σ 2 ). A process with non-zero mean is obtained by replacing Y t in (10) by
Y t − µ, etc.
The backshift operator B can be used to write (10) in the form
(1 − φ 1 B − · · · − φ p B p )Y t = φ(B)Y t = ε t ,
where φ(B) is the autoregressive operator, and this suggests writing
Y t = φ(B) −1 ε t
to get the causal representation Y t = ∑ ∞
j=0 ψ jε t−j = ψ(B)ε t , if it exists. To find the coefficients of
ψ(B), we suppose that such a representation exists, and then match terms on the left and right of the
equation φ(B)Y t = φ(B)ψ(B)ε t = ε t .
Time Series Spring 2010 – slide 138
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