The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 302 Wednesday, February 4, 2004 12:58 PM

302 The Mathematics of Financial Modeling and Investment Management

p

B ()L = (^) ∑BjLj , B 0 = I, Bq ≠ 0
j = 1
If q = 0, the process is purely autoregressive of order p; if q = 0, the pro-
cess is purely a moving average of order q. Rearranging the terms of the
difference equation, it is clear that an ARMA process is a process where
the i-th component of the process at time t, xi,t, is a linear function of all
the components at different lags plus a finite moving average of white
noise terms.
It can be demonstrated that the ARMA representation is not unique.
The nonuniqueness of the ARMA representation is due to different rea-
sons, such as the existence of a common polynomial factor in the
autoregressive and the moving-average part. It entails that the same pro-
cess can be represented by models with different pairs p,q. For this rea-
son, one would need to determine at least a minimal representation—
that is, an ARMA(p,q) representation such that any other ARMA(p′,q′)
representation would have p′> p, q′> q. With the exception of the
univariate case, these problems are very difficult from a mathematical
point of view and we will not examine them in detail.
Let’s now explore what restrictions on the polynomials A(L) and
B(L) ensure that the relative ARMA process is stationary. Generalizing
the univariate case, the mathematical analysis of stationarity is based on
the analysis of the polynomial det[A(z)] obtained by formally replacing
the lag operator L with a complex variable z in the matrix A(L) whose
entries are finite polynomials in L.
It can be demonstrated that if the complex roots of the polynomial
det[A(z)], that is, the solutions of the algebraic equation det[A(z)] = 0,
which are in general complex numbers, all lie outside the unit circle,
that is, their modulus is strictly greater than one, then the process that
satisfies the ARMA conditions,
A ()Lxt = B ()Lεεεεt
is stationary. The demonstration is based on formally solving the ARMA
equation, writing (see Chapter 5 on matrix algebra)
()]
xt = A ()εεεε

• 1
  ()B L
  adj[A L
  L ()εεεεt = --------------------------B L t
  det[A ()L]

If the roots of the polynomial det[A(z)] lie outside the unit circle,

then it can be shown that