I am comparing two regressions:
$y= \beta_0+ \beta_1 x_1 + \beta_2 x_2 + \epsilon $
$ y = \gamma_0 + \gamma_1 PC_1 + \gamma_2 PC_2 +\eta $.
The regressors in the second regression, $PC_1, PC_2$, are principal components generated by a PCA of $x_1,x_2$.
Presumably if the PCA just linearly transforms the original regressors, the intercepts should be identical. However, I find them very different. What is the reason behind this difference?