258 The Basics of financial economeTrics
From equation (12.12):
XP 11 =+CV 11 PC 21 VP 28 ++ CV 18 and XP 81 =+CV 81 PC 28 VP 28 ++ CV 88
Therefore,
eP 11 =+CV 11 PC 21 VP 24 ++ CV 14 and eP 81 =+CV 81 PC 28 VP 24 ++ CV 84
Consider, for example, the scalar product
()eV 11 '' 51 =+PCVV11 15 +PC 41 VV' 415
Each of the products VV',11 15...,'VV14 15 is equal to zero because the eigen-
vectors are orthogonal. The same reasoning can be repeated for any prod-
uct ()eVij', 1 ij==1 234,,,, 5 ,,, 678 , which shows that the error terms are
orthogonal to the last four principal components.
Consider now the scalar product ()ee 18 '. This scalar product is a
weighted sum of products VV', 11 ijij==1 234,,,, 1 ,,, 234. These products
are zero if ij≠ but they are different from zero when ij=. Hence the cova-
riance and the correlations between residuals are not zero. For example, in
our illustration, the covariance of the residuals is:
Σ=
0.0205 0.0038 0.0069 –0. 0062–0. 0250 0.0044–0. 0064 0.0108
0.0038 0.0362 0.0113 –0. 0322–0. 0196 0.0421–0. 0138–0. 0173
0.0069 0.0113 0.0334 0.0005 –0. 0053–0. 0000–0. 0470 0.0066
–0. 0062–0. 0322 0.0005 0.0425 0.0172–0. 0335 0.0010–0. 0002
–0. 0250–0. 0196–0. 0053 0.0172 0.0436–0. 0330–0. 0009 0.0096
0.0044 0.0421 –0. 0000–0. 0335–0. 0330 0.0646 0.0080–0. 0417
–0. 0064–0. 0138–0. 0470 0.0010–0. 0009 0.0080 0.0692–0. 0178
0.0108 –0. 0173 0.0066–0. 0002 0.0096–0. 0417–0. 0178 0.0515
e
which is not a diagonal matrix. Hence the fundamental equation of factor
models ΣΨ=+BB' (equation (12.7)) does not, in general, hold for princi-
pal components.
It can be demonstrated that the weights of the last four principal compo-
nents in equation (12.14) minimize the sum of squared residuals. Therefore,
the weights in equation (12.14) are the same as those one would obtain esti-
mating a linear regression of the data on the last four principal components.
