Suppose we have the following regression model:
y = b0 + b1*X1 + b2*X2 + error1 (1)
The partial regression plot or added variable plot of x1 is (i.m.h.o.) constructed as follows:
regress y on x2 and save the residuals: residy
regress x1 on x2 and save the residuals: residx1
Finally, plot residy (vertical axis) against residx1 (horizontal axis). In this plot, the slope associated with residx1 is equal to b1 in equation (1). This can be confirmed by estimating:
residy = a0 + a1*residx1 + error2 (2)
For this last equation, the estimates will be: a0=0, which makes sense as both residy and residx1 are estimated residuals which have zero means. Also, a1=b1 i.e. the slope a1 of residx1 equals the slope in the " full" equation (1).
However, error2 does NOT equal error1. In a number of online documents discussing "partial regression plots" or "added variable plots" the procedure outlined above is explained, but in addition one often points to the similarity of the two errors, 1 and 2 above. See e.g.
Some papers, however, seem to convey that these errors 1 and 2 are NOT equal. They show that, in general,
error2 = (I - Hj)*error1
where Hj is the hat-matrix, equal to Hj = Xj*(Xj'Xj)*Xj' with Xj denoting the X matrix of the full model (1) without the variable for which the partial plot is created, being X1 in the above example.
Which should be the right conclusion here? My guess would be that the residuals a the partial regression plot, also often called "added variable plot" (such plots can e.g. be shown by Spss with the regression command) are NOT equal to those of the orginal " full" model.
Thanks for any help or explanation on this issue!!