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.