The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 541 Wednesday, February 4, 2004 1:10 PM

Multifactor Models and Common Trends for Common Stocks 541

dynamic cointegration introduces a small number of lags in the cointe-

grating relationship. In other words, cointegration reduces the order of

integration by applying linear regressions between variables; dynamic

cointegration reduces the order of integration by applying autoregres-

sive modeling. A VAR model with n lags

xt =A 1 xt – 1 + A 2 xt – 2 + ... + Anxtn– + εεεεt

exhibits dynamic cointegration if there exists a stationary autoregressive

combination of the variables of the type

α′xt + β′∆xt

Cointegration and dynamic cointegration can coexist in the same

model in the sense that variables can be cointegrated and dynamically

cointegrated. Note that, if the log price process is integrated of order 1,

then the return process is stationary so that factor models for returns and

cointegrated models for prices can coexist. In addition, linear combina-

tions of prices and returns can also be stationary.

Cointegration is equivalent to the existence of common stochastic

trends. This property is also expressed by the equivalence between an

ECM and a state-space model. Recall that a state-space model is written

as

xt = aAz+ t + Bμμμμt

zt + 1 = Czt + Dεεεεt

where state-space variables are either stationary or integrated variables.

Although a cointegrated price system of price processes can always be

expressed as a state-space model, the variables in the state-space repre-

sentation might include lagged prices. This fact was shown in Chapter

12 when addressing the question of the equivalence between ARMA

models and state-space models. In general, prices might be expressed in

the following factor form:

p q

pt = st + ∑ Aipti– + ∑ Bjftj– + ut

i = 1 j = 0