The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 341 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 341

The concept of cointegration, introduced by Granger in 1981,^16 can

be expressed in the following way. Suppose that a set of n time series,

integrated of order 1, is given. If there is a linear combination of the

series

n

δt = ∑βixit,

i = 1

which is stationary, then the series xi,t are said to be cointegrated. Any

linear combination as the one above is called a cointegrating relation-

ship. Given n time series, there can be from none to at most n – 1 coin-

tegrating relationships.

Though a definition of cointegration of this type is often given in the

literature, it should be clear that it is strictly applicable only to pro-

cesses that extend in time from –∞ to +∞. Series that start from some ini-

tial instant cannot be stationary but can be, at most, asymptotically

stationary. To make the definition of cointegration more general, one

should allow asymptotic stationarity instead of strict stationarity.

Cointegrating relationships express long-run equilibrium between

time series. As noted above, in financial terms, cointegrating relation-

ships represent stationary portfolios. Suppose there are n time series xi,t,

i = 1,...,n and k < n cointegrating relationships. It can be demonstrated

that there are n – k integrated time series uj,t, j = 1,...,n – k, called com-

mon trends, such that every time series xi,t can expressed as a linear

combination of the common trends plus a stationary disturbance:

nk–

xit = ∑γ jujt, + ηit,

j = 1

This is clearly a multifactor representation of integrated processes.

Is there a general representation of cointegrated processes? The

answer is affirmative. Granger was able to demonstrate the fundamental

theorem according to which a multivariate integrated process is cointe-

grated if and only if it can be represented in the Error Correction Model

(ECM) form. The ECM representation is a representation of a multi-

variate process in first differences with corrections in levels as follows:

(^16) C.W.J. Granger, “Some Properties of Time Series Data and Their Use in Econo-
metric Model Specification,” Journal of Econometrics 16 (1981), pp. 121–130.