*Prove that the least-squares estimates of the coefficient 2 for Xij is the same in the following...

*Prove that the least-squares estimates of the coefficient β2 for Xij is the same in the following two fixed-effects models (numbered as in Section 23.7.1):

Recall the context: The data are divided into groups i ¼ 1; ... ; m, with individuals j ¼ 1; ... ; ni in the ith group. The first model (Model 1) fits a different intercept βð1Þ
1i in each group, along with the common slope βð1Þ
2 . The second model (Model 5) fits a common intercept βð5Þ
1 and common slope βð5Þ
2 but controls for the compositional variable Xi#. (Hint:
Consider the added-variable plot that determines the coefficient βb2.
57 In Model 1, this addedvariable plot is the scatterplot for residuals from the regressions of Yij and Xij & Xi# on a set of m dummy regressors for groups. In Model 5, the added-variable plot is the scatterplot for residuals from the regressions of Yij and Xij & Xi# on the compositional variable Xi# and the intercept, but the compositional variable is itself the projection of Xij onto the space spanned by the group dummy regressors—that is, the group means Xi# are of course perfectly determined by the groups.)