248 The Basics of financial economeTrics
Assume that the standardized data X can be represented with four fac-
tors. Hence, we can use equation (12.9) to represent the above covariance
matrix as ΣΨX=+BB' where B is a 4 × 8 matrix of factor loadings and
Ψ is an 8 × 8 diagonal matrix of variances. Using the function factoran of
MATLAB, we estimate B and Ψ:
=
−−
−−−
−
−
−
−
B
0.7411 0.5245 0.1871 0.1707
0.9035 0.0440 0.4202 0.0107
0.7614 0.5634 0.2033 0.2069
0.8630 0.3504 0.0092 0.2900
0.9215 0.1575 0.3477 0.0137
0.6028 0.5476 0.2658 0.1600
0.8503 0.3637 0.0898 0.0025
0.6616 0.6449 0.1086 0.2814
Ψ=
0.1115 0000000
00.0050 000000
00 0.0188 00000
000 0.0482 0000
0000 0.0050 000
00000 0.2405 00
000000 0.1367 0
0000000 0.0553
We can see that
+Ψ=
BB'
1.0000 0.6159 0.3421 0.4046 0.8283 0.1819 0.4566 0.1244
0.6159 1.0000 0.5800 0.7651 0.6932 0.4072 0.7145 0.5208
0.3421 0.5800 1.0000 0.7926 0.6806 0.7884 0.8710 0.8309
0.4046 0.7651 0.7926 1.0000 0.7408 0.7561 0.8597 0.8776
0.8283 0.6932 0.6806 0.7408 1.0000 0.5638 0.7574 0.5497
0.1819 0.4072 0.7884 0.7561 0.5638 1.0000 0.7352 0.8259
0.4566 0.7145 0.8710 0.8597 0.7574 0.7352 1.0000 0.8061
0.1244 0.5208 0.8309 0.8776 0.5497 0.8259 0.8061 1.0000
is a good estimate of the covariance matrix of the standardized data. To see
this point, in order to make a quantitative evaluation of how well the matrix
BB' +Ψ approximates the covariance matrix of the standardized data, we
