The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 312 Wednesday, February 4, 2004 12:58 PM

312 The Mathematics of Financial Modeling and Investment Management

Note also that ARIMA processes are not invertible as infinite mov-

ing averages, but as discussed, they can be inverted in terms of a generic

linear moving average model with stochastic initial conditions. In addi-

tion, the process in the d-differences is asymptotically stationary.

In both trend stationary and difference stationary processes, innova-

tions can be serially autocorrelated. In the ARMA representations dis-

cussed in the previous section, innovations are serially uncorrelated white

noise as all the autocorrelations are assumed to be modeled in the ARMA

model. If there is residual autocorrelation, the ARMA or ARIMA model

is somehow misspecified.

The notion of an integrated process is essentially linear. A process is

integrated if stationary innovations keep on adding indefinitely. Note

that innovations could, however, cumulate in ways other than addition,

producing essentially nonlinear processes. In ARCH and GARCH pro-

cesses for instance, innovations do not simply add to past innovations.

The behavior of integrated and nonintegrated time series is quite dif-

ferent and the estimation procedures are different as well. It is therefore

important to ascertain if a series is integrated or not. Often a prelimi-

nary analysis to ascertain integratedness suggests what type of model

should be used.

A number of statistical tests to ascertain if a univariate series is inte-

grated are available. Perhaps the most widely used and known are the

Dickey-Fuller (DF) and the Augmented Dickey-Fuller (ADF) tests. The

DF test assumes as a null hypothesis that the series is integrated of order

1 with uncorrelated innovations. Under this assumption, the series can

be written as a random walk in the following form:

Xt + 1 =ρXt ++b εt

ρ = 1

εt IID

where IID is an independent and identical sequence (see Chapter 6).

In a sample generated by a model of this type, the value of ρ esti-

mated on the sample is stochastic. Estimation can be performed with the

ordinary least square (OLS) method. Dickey and Fuller^4 determined the

theoretical distribution of ρ and computed the critical values of ρ that

(^4) See William H. Greene, Econometric Analysis: Fifth Edition (Upper Sadle River,
NJ: Prentice-Hall, 2003).