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Additional resources for A Guide to Modern Econometrics (2nd Edition)
They want economic relationships that are more generally valid than the sample they happen to have. They want to draw conclusions about what happens if one of the variables actually changes. That is: they want to say something about things that are not observed (yet). In this case, we want the relationship that is found to be more than just a historical coincidence; it should reﬂect a fundamental relationship. To do this it is assumed that there is a general relationship that is valid for all possible observations from a well-deﬁned population (for example all US households, or all ﬁrms in a certain industry).
It projects the vector y upon the columns of X (the column space of X ). This is just the geometric translation of ﬁnding the best linear approximation of y from the columns (regressors) in X. The residual vector of the projection e = y − Xb = (I − PX )y = MX y is the orthogonal complement. It is a projection of y upon the space orthogonal to the one spanned by the columns of X. This interpretation is sometimes useful. For example, projecting twice on the same space should leave the result unaffected, so that it holds that PX PX = PX and MX MX = MX .
Similarly, the coefﬁcient β3 for school i gives the expected wage difference between two individuals with the same experience and gender where one has one additional year of schooling. In general, the coefﬁcients in a multiple regression model can only be interpreted under a ceteris paribus condition, which says that the other variables that are included in the model are constant. 2. 17 before. 11, this difference is still statistically highly signiﬁcant. 48. Clearly the null hypothesis has to be rejected.