The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 540 Wednesday, February 4, 2004 1:10 PM

540 The Mathematics of Financial Modeling and Investment Management

∆xt = (ΦΦΦΦ 1 L + ΦΦΦΦ 2 L^2 + ... + ΦΦΦΦn – 1 Ln –^1 )∆xt + ΠΠΠΠxt + m + εεεεt

The error correction restrictions apply to the matrix ΠΠΠΠ. An ECM is a

VAR model with ΠΠΠΠ = αβ′ where α, β are n×r matrices. The term in level

provides the error correction. The Granger Representation Theorem

demonstrated by Granger in 1987^10 states that if a process is cointe-

grated with r cointegrating relationships then the above ECM holds.

James Stock and Mark Watson^11 first observed in 1988 that a coin-

tegrated model with r cointegrating relationships admits n–r common

trends. The implication is that all time series can be written in the form

xt = aAz+ t + ηηηηt

where the zt are the common stochastic trends, which are I(1) integrated

processes, and the ηt are stationary processes.

Models for cointegration can be extended in various ways. In the

context of cointegration, Hashem Pesaran and Yongcheol Shin^12 intro-

duced the Autoregressive Distributed Lag (ARDL) models. An ARDL

model contains exogenous variables that are not cointegrated among

themselves. It has the following form:

xt = α 0 + α 1 t + (ΦΦΦΦ 1 L + ΦΦΦΦ 2 L^2 + ... + ΦΦΦΦpLp )xt + βzt

+ (β 1 L + β 2 L +

2

+ ...βqL

q

)∆zt + ut

0 = (P 1 LP+ 2 L

2

+ ... + P L

s

s )∆zt + εεεεt

where the z are I(1) noncointegrated variables and the y exhibit r cointe-

grating relationships. Pesaran and Shin demonstrated that the classical

approach to ARDL systems that are valid for stationary processes can

be extended to integrated processes.

Cointegration models can also be extended in the sense of dynamic

cointegration (or polynomial cointegration). Cointegrating relationships

are static relationships between variables taken at the same time;

(^10) R.F. Engle and C.W.J. Granger, “Cointegration and Error Correction: Represen-
tations, Estimation and Testing,” Econometrica 55 (1987), pp. 252–276.
(^11) James Stock and Mark Watson, “Testing for Common Trends,” Journal of the
American Statistical Association 83 (December 1988), pp. 1097–1107.
(^12) Hashem M. Pesaran and Yongcheol Shin, “An Autoregressive Distributed Lag
Modelling Approach to Cointegration Analysis,” Chapter 11 in S. Strom (ed.),
Econometrics and Economic Theory in the 20th Century (Cambridge, U.K.: Cam-
bridge University Press, 1999).