Consider the regression model Y = Xβ + ε. Assume X is stochastic and ε is such that E(X′ε) ̸= 0. However, there is a matrix of variables Z such that E(Z′ε) = 0 and E(Z′X) ̸= 0. The dimension of the matrix X is T ×k (T is the number of observations and k is the number of regressors) whereas the dimension of the matrix Z is T × q with q > k.
1.Is β ^ from a regression of Y on X consistent for β?
2.Regress the matrix X on the matrix Z (i.e., you want to regress each column of X on the matrix Z). Express the fitted values compactly as a function of X and Z.
3.Regress the observations Y on the fitted values from the previous regression (a T × k matrix). Express compactly the new estimator as a function of X, Z, and Y. (Note: you could use a very specific idempotent matrix here).
4.Is the new estimator consistent for β?
5.Assume k = q. Does the form of the estimator simplify?
6.(Interpret all of your previous results from an applied stand- point. Why are they useful?