12-FinEcon-Model Sel Page 335 Wednesday, February 4, 2004 12:59 PM
Financial Econometrics: Model Selection, Estimation, and Testing 335Asset Pricing Theory (APT) Models
Asset pricing models based on a single factor have been criticized as
unduly restrictive and truly multifactor models have been proposed. In a
multifactor model of asset prices, the restriction of absence of arbitrage
must be imposed. The Arbitrage Pricing Theory (APT) of Roll and Ross
allows multiple factors and fixes all other price processes on the basis of
absence of arbitrage (see Chapter 14).
APT models can be divided into two different categories in function
of how factors are treated. In the one, factors are portfolios or exoge-
nous variables such as macroeconomic factors; in the other, factors are
either modeled or not.
First consider the case of given exogenous factors. In this case, the APT
model must be estimated as a constrained regressive model. Constraints
typically forbid the possibility of using simple ordinary least square (OLS)
estimates. Thus the estimation procedures are generally based on the direct
application of Maximum Likelihood principles.PCA and Factor Models
If factors are not given, they must be determined with statistical learning
techniques. Given the variance-covariance matrix, if factors are portfo-
lios, one can determine factors using the technique of Principal Compo-
nents Analysis (PCA).
Principal Components Analysis (PCA) implements a dimensionality
reduction of a set of observations. The concept of PCA is the following.
Consider a set of n time series Xi, for example the 500 series of returns
of the S&P 500. Consider next a linear combination of these series, that
is, a portfolio of securities. Each portfolio P is identified by an n-vector
of weights ωωωω^2
2 P and is characterized by a variance σP. In general, the
variance σP will depend on the portfolio’s weights ωωωωP. Lastly consider a
normalized portfolio which has the largest possible variance. In this
context, a normalized portfolio is a portfolio such that the squares of
the weights sum to one.
If we assume that returns are IID sequences, jointly normally dis-
tributed with variance-covariance matrix ΩΩΩΩ, a lengthy direct calculation
demonstrates that each portfolio’s return will be normally distributed
with varianceσ^2
P = ωωωωPTΩΩΩΩωωωω
PTherefore the normalized portfolio of maximum variance can be deter-
mined in the following way: