Factor Analysis and Principal Components Analysis 245
Obviously this is a mathematical abstraction because no real market is
infinite. However, it is a useful abstraction because it simplifies the statement
of many properties. In practice, we use the concept of both infinite mar-
kets and infinite factor models to approximate the behavior of very large
markets. If a market is very large, say, thousands of stocks, we assume that
its properties can be well approximated by the properties of the abstract
infinite market. We will see in the following sections a number of properties
that apply to infinite markets.
estimating the Number of Factors
There are several criteria for determining the number of factors but there
is no rigorous method that allows one to identify the number of factors
of classical factor models. Rigorous criteria exist in the limit of an infinite
number of series formed by an infinite number of time points.
For finite models, a widely used criterion is the Cattell scree plot, which
can be described as follows. In general, there are as many eigenvalues as
stocks. Therefore, we can make a plot of these eigenvalues in descending
order. An example is in Table 12.1. In general, we empirically observe that
the plot of eigenvalues exhibits an elbow, that is, it goes down rapidly, but
it slows down at a certain point. Heuristically, we can assume there are as
many factors as eigenvalues to the right of the elbow, assuming eigenvalues
grow from left to right. However, the scree plot is a heuristic criterion, not
a formal criterion. The Akaike information criterion and the Bayesian infor-
mation criterion are also used, especially to determine the number of factors
for large models. We describe these two information criteria in Appendix E.
estimating the Model’s parameters
Let’s start by estimating the model’s parameters. The usual estimation
method for factor models is maximum likelihood estimation (MLE) which
we explain in Chapter 13. However, MLE requires that we know, or that we
make an assumption on the probability distribution of the observed vari-
ables. We will therefore make the additional assumption that variables are
normally distributed. Other distributional assumptions can be used but the
assumption of normal distribution simplifies calculations.
As explained in Chapter 13, the likelihood is defined as the probability
distribution computed on the data. MLE seeks those values of the param-
eters that maximize the likelihood. As we assume that data are normally
distributed, the likelihood depends on the covariance matrix. From equa-
tion (12.9) we know that ΣΨ=+BB' and we can therefore determine the
parameters B,Ψ maximizing the likelihood with respect to these parameters.