The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 542 Wednesday, February 4, 2004 1:10 PM

542 The Mathematics of Financial Modeling and Investment Management

s

ft = ∑ Ckftk– + ηηηηt

k= 1

where the price processes pt have an autoregressive distributed-lag

dynamics, the factors ftfollow a VAR or VARMA model, the terms ut

are idiosyncratic (i.e., they are mutually uncorrelated), and stare deter-

ministic terms. The terms utand ηηηηtmight be white noise or might be

autocorrelated (i.e., they are a stationary process that obeys ARMA

equations).

Factor models can be cast in the state-space representation. Con-

sider, for example, the following model:

pt = Bft+ ut

s

ft = ∑ Ckftk– + ηηηηt

k= 1

and

q

ut = ∑ Hkutk– + εεεεt

k= 1

An equivalent state-space model can be obtained by defining the follow-

ing state vector:

zt ′ = [ft...ftp– ut...utq– ]

and the following transition matrix:

C 1 ... Cs– 1 Cs.0 0

ηηηηt

I 0 0.

.. .. .. 0

.. .. ..
0
.
0 ... I 0. 0
z^0
t =^......... zt– 1 +
0 0. H 1 ... Hq– 1 Hq
εεεεt
0
. I 0 0 ..
.. .. ...
....
0
0 0. 0 ... I 0