The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 303 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 303

∞ ∞

adj[A ()L ]

--------------------------B ()Lεεεεt = ∑HiLiεt, with ∑Hi absolutely summable

det[A ()L] i = 1 i = 1

which demonstrates that the process xt is stationary.^1 As in the univari-

ate case, if one would consider the equations in 1/z, the same reasoning

applies but with roots strictly inside the unit circle.

A stationary ARMA(p,q) process is an autocorrelated process. Its

time-independent autocorrelation function satisfies a set of linear differ-

ence equations. Consider an ARMA(p,q) process which satisfies the fol-

lowing equation:

A 0 xt + A 1 xt – 1 + ...+ APxtP– = B 0 εεεεt+ B 1 εεεεt – 1 + ...+ Bqεεεεtq–

where A 0 = I. By expanding the expression for the autocovariance func-

tion, it can be demonstrated that the autocovariance function satisfies

the following set of linear difference equations:

0if hq>

qh

A 0 ΓΓΓΓh + A 1 ΓΓΓΓh – 1 + ...+ APΓΓΓΓhp=





∑BjhΩΩΩΩH′ j

• j = 0

where ΩΩΩΩand Hi are, respectively, the covariance matrix and the Markov

coefficients of the process in its infinite moving-average representation:

∞

xt = ∑Hiεεεεti–

i = 0

From the above representation, it is clear that if the process is purely MA,

that is, if p = 0, then the autocovariance function vanishes for lag h > q.

It is also possible to demonstrate the converse of this theorem. If a

linear stationary process admits an autocovariance function that satis-

fies the following equations,

A 0 ΓΓΓΓh + A 1 ΓΓΓΓh – 1 + ...+ APΓΓΓΓhp– = 0 if h > q

(^1) Christian Gourieroux and Alain Monfort, Time Series and Dynamic Models (Cam-
bridge: Cambridge University Press, 1997).