Can I add several additional variables to my principal component regression? I did a principal component analysis, which resulted in 7 components that I am now using for a principal component regression on my independent variable.
However, I want to add 2 control variables, but these are not component scores, but "normal" variables (on a Likert scale, the same sort of variables my independent variables were).
Is that okay to do or do I have to make these into components as well? I have done the regression both with those two as variables and combined in a component and the results are practically the same.
 A: It depends.
As you may know PCR starts with PCA on the independent variables.
$$X = TP' + E$$
After obtaining the scores ($T$), one carries out regression between $T$ and $Y$ so that:
$$Y = TB + F$$
The first PCA step assures the scores (columns of $T$) are uncorrelated so that you can find "healthy" regression coefficients rather than dealing with the originally problematic matrix (rank deficiency, multicollinearity, $p \gg n$ etc..) which can, for example, yield very large regression coefficients and cause overfitting.
Thus, if you add some variables, depending on the nature of those variables, you may end up with a similar problem that caused you to use PCR rather than OLS in the first place. On the other hand, it may be just OK. I suggest to confirm your each model's success via testing it on an independent validation set or at least by using CV.
Personally, I would add those variables prior to PCR. If interpretability by looking at the regression coefficients is your concern, then
$$Y = XPB + F$$ thus $\hat{B} = PB$ which you can directly use on your (probably at least mean-centered) data and can be interpreted just as easly.
