The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 537 Wednesday, February 4, 2004 1:10 PM

Multifactor Models and Common Trends for Common Stocks 537

EXHIBIT 18.2 PCA Performed on Correlations Matrix of S&P 500 Stocks,

January 2, 2001–September 19, 2003

DYNAMIC MARKET MODELS OF RETURNS

Now let’s consider stationary return processes with a dynamics more

complex than that of an IID sequence of variables. A reasonable gener-

alization of factor market models are state-space models of the form

that was described in Chapter 12:

rt = αααα+ Azt + Bεεεεt

zt + 1 = Czt + Dεεεεt

Note that z is non-observable. Therefore, the noise term can be

placed either at t or t – 1. The first equation is the usual regression of a

factor market model while the second equation is a one-lag stationary

Vector Auto Regressive—denoted by VAR(1)—model that describes the

autoregressive dynamics of the factors. Note that we assume that the

above equations describe the dynamics of returns; the following section

discusses how similar equations might describe prices.