250 The Basics of financial economeTrics
Because xBtt=+f εt, the vector []xftt has a normal distribution with
mean zero and the following covariance matrix:
= +Ψ
x
f
BB B
BI
cov t ''
t
From this it can be demonstrated that an estimate of factor scores is
given by
fBˆtt=+()BB' Ψx (12.11)
Scores are only an estimate of factors and, in general, do not have the prop-
erties of factors; that is, scores do not have unit variance and are not orthogo-
nal. For example, using the MATLAB function factoran, in our illustration, we
obtain the following 20 × 4 matrix estimate of the matrix of scores F:
F=
−
−
−
0 1307
1 3760
0 0302
0 0744
1 1057
1 1724
1
......
.11110
0 2808
1 0935
0 2160
0 8722
1 3746
2 065
−
−
−
.
.
.
.
.
.55
1 6935
0 4430
0 7366
0 2023
0 8073
0 7959
0
−
−
−
−
......
..
......
1836
0 1885
0 8360
0 5621
0 6716
1 5757
00
−
−
−
− 4423
0 7784
0 8173
0 1184
0 1450
0 4340
0 1600
−
−
−
......
11 5356
3 0525
0 2481
0 0515
0 3304
0 8878
038
.......
−
−
330
0 4548
1 1355
0 9245
0 6448
1 0122
0 1156
0
−
−
−
.
.
.
.
.
.
..
......
1176
0 3086
2 3988
0 5359
0 9409
1 6825
14
−
−
−
−
6673
0 1549
1 3063
0 7989
0 0451
0 2513
0 6037
......
−
−
−
00 1956
0 4972
1 1246
0 2546
0 0978
2 4148
0
.
.
.
.
.
.
.
−
−
−
−
99868
0 5556
0 5283
0 3218
0 1613
2 5238
050
......
−
−
−
− 003
0 2178
0 1682
0 5341
0 5219
0 0660
0 2079
0
−
−
......
.11950
0 0616
0 1904
.
.
