The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 531 Wednesday, February 4, 2004 1:10 PM

Multifactor Models and Common Trends for Common Stocks 531

constants. The first formulation expresses a linear relationship between

the unconditional means of the returns and of the factors. The second

formulation is the linear regression function which expresses a linear

relationship between the mean of returns at time t conditional on the

realization of the factors at the same time; the third is the standard for-

mulation of the linear regression of returns on factors.

Obviously returns and factors are all defined on the same probabil-

ity space and have a joint pdf.^1 Recall from Chapter 6 on probability

theory that joint multivariate normal distributions factorize in a linear

regression. In the case of joint normality, returns, factors and noise all

have normal joint distributions. However other distributions, for

instance the Student-t, factorize in a linear regression while distribu-

tions such as lognormal and Pareto distributions do not factorize in a

linear regression.

Factors range from innovations to exogenous variables, such as

macroeconomic variables, to abstract factors formed as linear combina-

tions of the processes. The multifactor model is a regression between

variables at the same time and does not specify a dynamics for these

variables. In other words, a multifactor model is not, per se, a predictive

model. To perform forecasts and parameter estimates, a process dynam-

ics of factors must be specified. The simplest dynamic assumption is that

factors are independent and identically distributed (IID) variables. In

this case, the noise is a white noise. Other specifications of factors

dynamics have been proposed; these will be discussed later.

Factor market models can be generalized to include linear condi-

tional factor models where factors and returns are conditional on some

information set I known at time t – 1. The information set will contain

the history of returns and factors up to time t – 1 and, possibly, other

variables. Linear conditional factor models are written as follows:

E[rt It – 1 ]= αααα+ ββββE[ftIt – 1 ]

where the constants are now time-dependent and conditional on the

information set:

cov(ritfst It – 1 )

βis = ---------------------------------------

var(fst It – 1 )

(^1) For a discussion of what families of joint pdfs admit a linear regression function,
see, amongst other, A. Spanos, Statistical Foundations of Econometric Modeling
(Cambridge, U.K.: Cambridge University Press, 1986).