The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 338 Wednesday, February 4, 2004 12:59 PM

338 The Mathematics of Financial Modeling and Investment Management

Factors can also be obtained through another statistical procedure

called factor analysis. Factor analysis estimates factors using a maxi-

mum likelihood procedure. Suppose that factors are not portfolios but

exogenous variables, such as macroeconomic variables. In this case, the

factor structure is given and the estimation problem becomes one of esti-

mating a regression relationship. This problem can be solved through

maximum likelihood estimates.

Let’s now summarize the previous discussion on multifactor models.

From the point of view of econometrics, the key justification of factor

models is dimensionality reduction. It can be empirically ascertained

that the empirical variance-covariance matrices computed over reason-

able time windows are unstable and noisy. This might be due to various

reasons, in particular to the fact that functional dependence between

variables is more complex than a simple structure of linear correlation.

The key problem is to extract maximum information from noise. Multi-

factor models attempt to provide a solution to this problem within the

domain of simple regressive models. There are different families of mul-

tifactor models: regression over given exogenous variables, factor analy-

sis under the assumption of multivariate random walks, state-space

models. In addition, multifactor models might be applied to both

returns and prices.

VECTOR AUTOREGRESSIVE MODELS

The next step is to model factors. This requires introducing a broad

family of ARMA models called Vector Autoregressive (VAR) Models. A

VAR model is a multivariate AR(n) model. In a VAR model the current

value of each variable is a linear function of the past values of all vari-

ables plus random disturbances. In full generality, a VAR model can be

written as follows:

xt = A 1 xt – 1 + A 2 xt – 2 + ...+ Apxtp– + Dst + εεεεt

where xt = (x 1 ,t, ...,xnt, ) is a multivariate stochastic time series in vec-

tor notation, Ai, i = 1,2,...,p, and D are deterministic n×n matrices,

εεεεt = ε 1 ,t, ...ε, nt, is a multivariate white noise with variance-covariance

matrix ΩΩΩΩ= { σij } and st = s 1 ,t, ...,snt, is a vector of deterministic

terms. Using the lag-operator L notation, a VAR model can be written

in the following form:

xt = (A 1 L + A 2 L

2

+ ...+ AL

N

n )xt + Dst + εεεεt