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## Principal components

Stata’s pca allows you to estimate parameters of principal-component models.

. webuse auto
(1978 Automobile Data)

. pca price mpg rep78 headroom weight length displacement foreign

Principal components/correlation            Number of obs    =        69
Number of comp.  =         8
Trace            =         8
Rotation: (unrotated = principal)       Rho              =    1.0000

Component     Eigenvalue   Difference         Proportion   Cumulative

Comp1         4.7823      3.51481             0.5978       0.5978

Comp2         1.2675      .429638             0.1584       0.7562

Comp3        .837857      .398188             0.1047       0.8610

Comp4        .439668     .0670301             0.0550       0.9159

Comp5        .372638      .210794             0.0466       0.9625

Comp6        .161844     .0521133             0.0202       0.9827

Comp7        .109731      .081265             0.0137       0.9964

Comp8       .0284659            .             0.0036       1.0000

Principal components (eigenvectors)

Variable      Comp1     Comp2     Comp3     Comp4     Comp5     Comp6

price     0.2324    0.6397   -0.3334   -0.2099    0.4974   -0.2815

mpg    -0.3897   -0.1065    0.0824    0.2568    0.6975    0.5011

rep78    -0.2368    0.5697    0.3960    0.6256   -0.1650   -0.1928

headroom     0.2560   -0.0315    0.8439   -0.3750    0.2560   -0.1184

weight     0.4435    0.0979   -0.0325    0.1792   -0.0296    0.2657

length     0.4298    0.0687    0.0864    0.1845   -0.2438    0.4144

displacement     0.4304    0.0851   -0.0445    0.1524    0.1782    0.2907

foreign    -0.3254    0.4820    0.0498   -0.5183   -0.2850    0.5401

Variable      Comp7     Comp8   Unexplained

price     0.2165   -0.0891             0

mpg     0.1625    0.0115             0

rep78    -0.0813    0.0065             0

weight     0.1104    0.8228             0

length     0.5437   -0.4921             0

displacement    -0.7733   -0.2608             0

foreign    -0.1173    0.0639             0



We typed pca price mpg ... foreign. All Stata commands share the same syntax: the names of the variables (dependent first and then independent) follow the command's name, and they are, optionally, followed by a comma and any options. In this case, we did not specify any options.

Having estimated the principal components, we can at any time type pca by itself to redisplay the principal-component output. We can also type screeplot to obtain a scree plot of the eigenvalues, and we can use the predict command to obtain the components themselves.

screeplot, typed by itself, graphs the proportion of variance explained by each component:

. screeplot

Typing screeplot, yline(1) ci(het) adds a line across the y-axis at 1 and adds heteroskedastic bootstrap confidence intervals.

. screeplot, yline(1) ci(het)

We can obtain the first two components by typing

. predict pc1 pc2, score
(6 components skipped)

Scoring coefficients

Variable      Comp1     Comp2     Comp3     Comp4     Comp5     Comp6

price     0.2324    0.6397   -0.3334   -0.2099    0.4974   -0.2815

mpg    -0.3897   -0.1065    0.0824    0.2568    0.6975    0.5011

rep78    -0.2368    0.5697    0.3960    0.6256   -0.1650   -0.1928

headroom     0.2560   -0.0315    0.8439   -0.3750    0.2560   -0.1184

weight     0.4435    0.0979   -0.0325    0.1792   -0.0296    0.2657

length     0.4298    0.0687    0.0864    0.1845   -0.2438    0.4144

displacement     0.4304    0.0851   -0.0445    0.1524    0.1782    0.2907

foreign    -0.3254    0.4820    0.0498   -0.5183   -0.2850    0.5401

Variable      Comp7     Comp8

price     0.2165   -0.0891

mpg     0.1625    0.0115

rep78    -0.0813    0.0065

weight     0.1104    0.8228

length     0.5437   -0.4921

displacement    -0.7733   -0.2608

foreign    -0.1173    0.0639



The score option tells Stata's predict command to compute the scores of the components, and pc1 and pc2 are the names we have chosen for the two new variables. We could have obtained the first three factors by typing, for example, predict pc1 pc2 pc3, score.

An important feature of Stata is that it does not have modes or modules. We typed pca to estimate the principal components. We then typed screeplot to see a graph of the eigenvalues — we did not have to save the data and change modules. Similarly, we typed predict pc1 pc2, score to obtain the first two components. The new variables, pc1 and pc2, are now part of our data and are ready for use; we could now use regress to fit a regression model.

The two components should have correlation 0, and we can use the correlate command, which like every other Stata command, is always available for use. To verify that the correlation between pc1 and pc2 is zero, we type

. correlate pc1 pc2
(obs=69)

pc1      pc2

pc1     1.0000

pc2     0.0000   1.0000