Factor Analysis and Principal Components Analysis 251
If we compute the covariance matrix of F, we obtain:
ΣF=
−
0.9975
0.0000
0.0000
0.0000
0. 0
00000
09737
0000
0.0000
− 0.0000
−
−
00000
00000
09843
00000
000000
08548
0.0000
To summarize, our illustration demonstrates what is done in practice.
We started with a matrix of eight series of 20 standardized daily return data
series and tried to find a smaller set of four series of factors that explain the
data as a multiple regression. To do so, we first determine with MLE, B, and
Ψ, and then estimate factors.
Factors are not observed; they are reconstructed from data and are
abstract. For example, in the factor model that we have estimated, it would
be difficult to interpret factors. We can say that the original returns are
exposed to four risk factors but we cannot easily interpret these factors. As
we have remarked, we cannot say that our risk factors are unique. We will
see later how we can partially modify this proposition.
Other types of Factor Models
In financial modeling, there are factor models that are not obtained through
factor analysis. These models are multiple regressions on specific families of
regressors. Widely used regressors include macroeconomic variables, funda-
mental information on companies such as market capitalization and book-
to-value ratio, as well as countries and sectors. In factor models based on
countries and sectors, stock return time series are partitioned into countries
and/or industrial sectors. The factor loadings are equal to unity if a stock
belongs to a country or a sector, zero in all other cases (just as when using
categorical variables in regression analysis as explained in Chapter 6) and
factor scores are obtained through regressions.
Principal Components Analysis
Let’s now look at principal components analysis (PCA). PCA is used to par-
simoniously represent data. There are similarities and differences between
PCA and factor analysis. The main difference is that PCA is a data-reduction
technique: PCA can be applied to any set of data without assuming any
statistical model while factor analysis assumes a statistical model for the
data. In addition, principal components are linear combinations of the data