254 The Basics of financial economeTrics
where each column is the product of the data X and the corresponding
eigenvector. The columns of the matrix PC are what we referred to earlier as
the principal components. Table 12.2 shows the eight principal components.
Principal components are linear combinations of the original data.
Because the original data are a series of stock returns, we can think of prin-
cipal components as portfolios formed with those same stocks. The values
of principal components are therefore the returns of these portfolios.
Step 3: Getting the data Back exactly from principal Components Thus far we have
transformed the original time series into a new set of mutually uncorrelated
time series are called principal components. We can now ask if and how we
can reconstruct data from principal components. Since VV' = I, we can write
the following:
XX==VV''()XVVP= CV' (12.14)
taBLe 12.2 The Set of All Principal Components of the Sample Standardized Data
PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 PC 7 PC 8
0.03 –0.61 –0.25 –0.13 0.48 –0.60 0.09 0.19
0.23 –0.19 0.24 0.67 –0.11 0.39 –1.11 2.28
–0.04 –0.30 0.13 –0.18 0.07 –0.28 –0.51 –0.43
0.31 0.06 –0.49 0.00 1.18 –0.78 0.08 0.27
0.19 –0.20 0.07 –0.05 –0.71 0.04 –0.72 –3.60
–0.27 –0.11 –0.50 0.28 –0.86 –0.33 –0.42 2.82
–0.17 0.32 0.23 –0.17 0.42 0.15 –0.91 –3.10
0.12 0.17 –0.44 –0.04 0.11 1.60 –1.14 –2.23
–0.13 0.14 –0.14 0.21 –0.07 0.08 0.34 –2.41
0.22 0.00 0.10 –1.14 –0.73 0.20 0.86 0.90
–0.28 –0.23 0.17 –0.07 0.46 1.38 –1.21 0.96
–0.36 0.12 –0.10 –0.38 0.11 –1.01 0.30 –2.50
–0.12 0.24 0.28 –0.04 0.30 0.12 0.76 6.03
0.00 –0.11 0.04 0.34 0.11 0.83 4.19 –1.77
0.05 –0.01 0.76 –0.36 0.25 –0.14 –0.16 1.20
0.21 0.40 –0.20 0.11 –0.15 –0.12 –0.24 1.77
–0.06 0.06 0.07 0.54 –0.34 –0.21 0.59 0.06
–0.04 0.06 0.48 0.50 –0.12 –0.83 –0.52 –1.90
–0.13 0.04 –0.50 –0.24 –0.16 –0.03 –0.11 1.98
0.24 0.14 0.07 0.14 –0.22 –0.50 –0.15 –0.48