The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 342 Wednesday, February 4, 2004 12:59 PM

342 The Mathematics of Financial Modeling and Investment Management

n –^1

∆xt = ∑ AL

i





+ 1 ∆xt + ααααββββ′xt + ηηηηt

i = 1 

where ααααis a p×r matrix, ββββis a a p×r matrix with ααααββββ′ = ΠΠΠΠand ηηηηt is a vec-

tor of stationary disturbances.

Within the basic framework of ECM, different cointegration models

have been proposed. Two major models need mention:

■ The Autoregressive Distributed Lag (ARDL) model which explicitly

takes into account exogenous variables that are not cointegrated

among themselves.^17

■ The Dynamic Cointegration Approach which models the long-run

cointegration relationships not as a static regression but as a dynamic

model with a small number of lags.

Cointegration of log price processes makes sense from an economic

point of view. Prices must somehow follow a common trend otherwise

they will, in the long run, diverge indefinitely. This is not a real eco-

nomic justification of cointegration. Even if in the long run all processes

end up as fluctuations around some common trend, it does not mean

that they are cointegrated. Many other possible mechanisms might be at

work, such as discrete adjustment.

State-Space Modeling and Cointegration

The notion of state-space modeling is that empirically measurable eco-

nomic variables are a linear regression over a set of hidden variables

modeled as an autoregressive process. State-space models represent

dynamical factor models as the states are the hidden factors of the

model. The state-space representation introduced above can be general-

ized in many different ways, in particular by letting the noise terms be

different in the state equations and in the regressions.

As we have seen earlier in this chapter, there is equivalence between

state-space models and ARMA models. In particular, there is equiva-

lence between cointegrated models represented by ECM models, and

state-space models. The factors are the common trends.

(^17) See M.H. Pesaran and Y. Shin, “An Autoregressive Distributed Lag Modeling Ap-
proach to Cointegration Analysis,” Chapter 11 in S. Strom (ed.), Econometrics and
Economic Theory in the 20th Century: The Ragnar Fresh Centennial Symposium
(Cambridge: Cambridge University Press, 1999).