ECON 2020. SAMPLE QUESTION # 9 1. Consider the following linear regression: y = X1 β1 + X2 β2 + u, y = n × 1, X1 = n × k1 , X2 = n × k2 (1) For a a × b matrix Z of rank b, let’s use the notations PZ = Z(Z 0 Z)−1 Z 0 and QZ = Ia − PZ Let Ky = KX1 β1 + KX2 β2 + u (2) Ky = X1 β1 + KX2 β2 + u (3) be two regression equations. Replace K successively by PX1 , QX1 , PX , QX , and tell in each case if the OLS estimators of the coefficients β1 and β2 in (1), (2), and (3) are the same or are different. Explain your answers. 2. 1- a) For the regression model yi = µ + i , i ∼ IID N [0, σ 2 ], ˆ2 = b) Consider the two estimators of µ, µ ˆ1 = y, µ P Pi iyi ,. ii Which of the 2 estimators is consistent? Study the asymptotic normality of µ ˆ1 . Which of the 2 estimators is unbiased? Which of the 2 estimators is more efficient, i.e., has a smaller variance? 1 2) Consider the following 2 models: M ODEL 1 : Y1 = X1 β1 + 1 (4) Y2 = X2 β2 + 2 (5)  Y = Xβ + , =  1 2  ⇔  , Y =  Y1 Y2   , X =  X1 0 0 X2   , β =  with the dimensions and the specification Yi = (ni × 1); Xi = (ni × k), ∼ N [0, σ 2 I(n1 +n2 ) ].  M ODEL2 :   where X0 =  X1 Y1 Y2    =  X1 X2   β0 + ⇔ Y = X0 β0 + , (7)   and Y1 , Y2 , X1 , X2 have the same dimensions as in X2 MODEL 1, and β0 is (k × 1). Using the notations Q = In1 +n2 − X(X 0 X)−1 X 0 and Q0 = In1 +n2 − X0 (X00 X0 )−1 X00 , show that the statistic F = (0 Q0 −0 Q)/[tr(Q0 )−tr(Q)] 0 Q)/tr(Q) has a F (tr(Q0 ) − tr(Q), tr(Q)) distribution, and compute the values of tr(Q0 ) − tr(Q), tr(Q) as an expression of n and k. 3. Exercices 5.1- 5.5, page 112. 2 β1 β2   (6). 4. Exercices 6.1- 6.3 and 6.6., page141. 5. Exercice 7.3 page 156. 3