The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 304 Wednesday, February 4, 2004 12:58 PM

304 The Mathematics of Financial Modeling and Investment Management

then the process admits an ARMA(p,q) representation. In particular, a sta-

tionary process is a purely finite moving-average process MA(q), if and

only if its autocovariance functions vanish for h > q, where q is an integer.

Nonstationary Multivariate ARMA Models

Let’s now consider nonstationary series. Consider a series defined for t ≥

0 that satisfies the following set of difference equations:

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

where, as in the stationary case, εεεεt–i is an n-dimensional zero-mean,

white noise process with nonsingular variance-covariance matrix ΩΩΩΩ, A 0

= I, B 0 = I, Ap ≠0, Bq ≠0. Suppose, in addition, that initial conditions

(x–1,...,x–p,εεεεt,...,εεεε–q) are given. Under these conditions, we say that the pro-

cess xt, which is well defined, admits an ARMA representation.

A process xt is said to admit an ARIMA representation if, in addi-

tion to the above, it satisfies the following two conditions: (1) det[B(z)]

has all its roots strictly outside of the unit circle, and (2) det[A(z)] has

all its roots outside the unit circle but with at least one root equal to 1.

In other words, an ARIMA process is an ARMA process that satisfies

some additional conditions. Later in this chapter we will clarify the

meaning of integrated processes.

Markov Coefficients and ARMA Models

For the theoretical analysis of ARMA processes, it is useful to state

what conditions on the Markov coefficients ensure that the process

admits an ARMA representation. Consider a process xt, stationary or

not, which admits a moving-average representation either as

∞

xt = ∑Hiεεεεti–

i = 0

or as a linear model:

t

xt = ∑Hiεεεεti– + h ()tz

i = 0

The process xi admits an ARMA representation if and only if there

is an integer q and a set of p matrices Ai, i = 0, ..., p such that the