1. OLS: Theoretical Review Assumption 1: Linearity –y = x′  +  or y = X  +  ) (where x,  : Kx1) Assumption 2: Exogeneity –E(  |X) = 0 –E(x  ) = lim n   i x i  i /n = lim n  X′  /n = 0 –Var(x  ) = E(xx′  2 ) = lim n   i x i x i ′  i 2 /n

2. OLS: Theoretical Review Assumption 3: No Multicolinearity –rank(X) = K –E(xx') = lim n   i x i x i ′/n = lim n  X′X/n –plim X’X/n exists and nonsingular

3. OLS: Theoretical Review Non-Spherical Disturbances: General Heteroschedasticity –Var(  –Var(x  ) = E(xx′  2 ) = lim n  X′  X/n –X′  /  n  d N(0,  X′  X/n)

4. OLS: Theoretical Review Assumption 4: Homoschedasticity and No Autocorrelation –Var(  2 I –Var(x  ) = E(xx′  2 ) = lim n   2 X′X/n –X′  /  n  d N(0,  2 X′X/n)

5. OLS: Theoretical Review Least Squares Estimator –b = (X'X) -1 X'y Variance-Covariance Matrix of b –General Heteroscedasticity: Var(b) = (X'X) -1 X′  X(X'X) -1 –Homoschedasticity: Var(b) =  2 (X'X) -1

6. OLS: Theoretical Review Least Squares Estimator –b = (X'X/n) -1 (X'y/n) –b =  + (X'X/n) -1 (X'  /n), b  p  –  n(b -  ) = (X'X/n) -1 (X'  /  n) –  n(b -  )  d N(0,A -1 BA -1 ) A = E(xx′) = lim n  X'X/n B = E(xx′  2 ) = lim n  X'  X/n –  n(b -  )  d N(0,  2 A -1 ) under homoschedasticity

7. OLS: Theoretical Review Asymptotic Normality –b ~ a N( , (X'X) -1 X′  X(X'X) -1 ) –b ~ a N( ,  2 (X'X) -1 ) under homoschedasticity –The unknown  or  2 needs to be consistently estimated.

8. OLS: Theoretical Review Estimate of Asymptotic Var(b) –Under Homoschedasticity

9. OLS: Theoretical Review Estimate of Asymptotic Var(b) –White Estimator (Heteroschedasticity-Consistent Estmate of Asymptotic Covariance Matrix)

10. OLS: Theoretical Review GLS (Generalized Least Squares) –If  is known and nonsingular, then  -1/2 y =  -1/2 X  +  -1/2  or y * = X *  +  * –E(  * |X * ) = 0, E(  *  * '|X * ) = I –b GLS = (X'  -1 X) -1 X'  -1 y –Var(b GLS ) = (X'  -1 X) -1 –b GLS ~ a N( , (X'  -1 X) -1 )

11. Example U. S. Gasoline Market, 1953-2004 –EXPG = Total U.S. gasoline expenditure –PG = Price index for gasoline –Y = Per capita disposable income –Pnc = Price index for new cars –Puc = Price index for used cars –Ppt = Price index for public transportation –Pd = Aggregate price index for consumer durables –Pn = Aggregate price index for consumer nondurables –Ps = Aggregate price index for consumer services –Pop = U.S. total population in thousands

12. Example y = X  +  y = G; X = [1 PG Y] where G = (EXPG/PG)/POP y = ln(G); X = [1 ln(PG) ln(Y)] Elasticity Interpretation