Factor Analysis and Principal Components Analysis 247
We can thus estimate the factor loadings and the residual variances under
the assumptions that factors are uncorrelated with unitary variances.
While we can estimate the factor loadings and the residuals’ variances
with MLE methods, we cannot estimate factors with maximum likelihood.
There is a fundamental factor indeterminacy so that, in finite models, factors
cannot be uniquely determined. The usual solution consists in estimating a
set of factor scores. This is done by interpreting equation (12.2) as a regres-
sion equation, which allows one to determine factor scores f.
Before proceeding to illustrate other factor models it is useful to illus-
trate the above with an example. Table 12.1 lists eight series of 20 daily
stock returns Rtti,,== 12 ...,, 01 i , ..., 8. The first row reports the stock
symbols, the rows that follow give stock daily returns from December 2,
2011, to December 30, 2011.
Let’s begin by standardizing the data. The matrix X of standardized
data is obtained by subtracting the means and dividing by the standard
deviations as follows:
{
[]
()
()
}
μ=
σ= −μ
=
−μ
σ
=
= =
ER
ER
X
R
XXXX
X XX X
'
iti
iti
ti
ti i
i
iiti Ti
N ti
2
1
1
(12.10)
The empirical covariance matrix ΣX of the standardized data, which
is the same as the correlation matrix of the original data, is the following
8 × 8 matrix:
Σ=
1.0000 0.6162 0.3418 0.3991 0.8284 0.2040 0.4461 0.1273
0.6162 1.0000 0.5800 0.7651 0.6932 0.4072 0.7145 0.5209
0.3418 0.5800 1.0000 0.7925 0.6807 0.7864 0.8715 0.8312
0.3991 0.7651 0.7925 1.0000 0.7410 0.7449 0.8656 0.8771
0.8284 0.6932 0.6807 0.7410 1.0000 0.5630 0.7577 0.5496
0.2040 0.4072 0.7864 0.7449 0.5630 1.0000 0.7480 0.8420
0.4461 0.7145 0.8715 0.8656 0.7577 0.7480 1.0000 0.7952
0.1273 0.5209 0.8312 0.8771 0.5496 0.8420 0.7952 1.0000
X
