The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 301 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 301

A process xt defined for t ≥0 is called an Autoregressive Integrated

Moving Average process—ARIMA(p,d,q)—if it satisfies a relationship

of the type

AL()( I – λL)dxt = BL()εt

where:

■ The polynomials A(L) and B(L) have roots strictly greater than 1.

■ εt is a white noise process defined for t ≥0.

■ A set of initial conditions (x–1, ..., x–p–d, εt, ..., ε–q) independent from

the white noise is given.

Later in this chapter we discuss the interpretation and further properties

of the ARIMA condition.

Stationary Multivariate ARMA Models

Let’s now move on to consider stationary multivariate processes. A sta-

tionary process which admits an infinite moving-average representation

of the type

∞

xt = ∑Hiεεεεti–

i = 0

where εt–i is an n-dimensional, zero-mean, white-noise process with

nonsingular variance-covariance matrix ΩΩΩΩ is called an autoregressive

moving average—ARMA(p,q)—model, if it satisfies a difference equa-

tion of the type

A ()Lxt = B ()εL t

where A and B are matrix polynomials in the lag operator L of order p

and q respectively:

p

A ()L = (^) ∑AiL
i
, A 0 = I, Ap ≠ 0
i = 1