6-16 5 Tests for Exogeneity How do we tell whether variables really need to be treated as endogenous or not? Consider again equations(14)-(16). Equation(14)contains Y and Y3-but do we really need equations for them? We can formally test this using a hausman test as follows: 1. Obtain the reduced form equations corresponding to(14)- (16). The reduced forms turn out to be: H=丌10+丌1X1+丌1X2+v 2=20+m21X1 (17)-(19) 3=m30+31X1 Estimate the reduced form equations(17)-(19)using OLs, and obtain the fitted values. YY.Y C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-16 • How do we tell whether variables really need to be treated as endogenous or not? • Consider again equations (14)-(16). Equation (14) contains Y2 and Y3 - but do we really need equations for them? • We can formally test this using a Hausman test as follows: 1. Obtain the reduced form equations corresponding to (14)- (16). The reduced forms turn out to be: (17)-(19) Estimate the reduced form equations (17)-(19) using OLS, and obtain the fitted values, 5 Tests for Exogeneity Y X X v Y X v Y X v 1 10 11 1 12 2 1 2 20 21 1 2 3 30 31 1 3 = + + + = + + = + + , , Y Y Y 1 2 3
6-17 Tests for Exogeneity 2. Run the regression corresponding to equation(14) 3. Run the regression (14) again, but now also including the fitted values YY as additional regressors: Y,=Cn+c1Y,+c2Y,+c,X1+c:X,+Y,+,Y2+l1 20) 4. Use an F-test to test the joint restriction that n2=0, and n3=0. If the null hypothesis is rejected, Y, and Y3 should be treated as endogenous C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-17 2. Run the regression corresponding to equation (14). 3. Run the regression (14) again, but now also including the fitted values as additional regressors: (20) 4. Use an F-test to test the joint restriction that 2 = 0, and 3 = 0. If the null hypothesis is rejected, Y2 and Y3 should be treated as endogenous. Tests for Exogeneity , Y Y 2 3 1 1 3 3 1 1 0 1 2 3 3 4 1 5 2 2 2 Y = + Y + Y + X + X + Y ˆ + Y ˆ + u
6-18 6 Recursive systems Consider the following system of equations Y=Bo +y1 Y2=B20+B21 +y21X+y2X2+l2 (21-23) 1=B30+B31+B2Y2+y31Xx1+y32X2+l2 Assume that the error terms are not correlated with each other Can we estimate the equations individually using ols Equation 21: Contains no endogenous variables, so X, and X are not correlated with u. So we can use Ols on(21) Equation 22: Contains endogenous Y together with exogenous X and X,. We can use ols on(22) if all the rhs variables in 22)are uncorrelated with that equations error term. In fact, Y1 is not correlated with u, because there is no Y, term in equation (21). So we can use OLs on(22). C Chris brooks2002,陈磊204
© Chris Brooks 2002, 陈磊 2004 6-18 • Consider the following system of equations: (21-23) • Assume that the error terms are not correlated with each other. Can we estimate the equations individually using OLS? • Equation 21: Contains no endogenous variables, so X1 and X2 are not correlated with u1 . So we can use OLS on (21). • Equation 22: Contains endogenous Y1 together with exogenous X1 and X2 . We can use OLS on (22) if all the RHS variables in (22) are uncorrelated with that equation’s error term. In fact, Y1 is not correlated with u2 because there is no Y2 term in equation (21). So we can use OLS on (22). 6 Recursive Systems Y X X u Y Y X X u Y Y Y X X u 1 10 11 1 12 2 1 2 20 21 1 21 1 22 2 2 3 30 31 1 32 2 31 1 32 2 3 = + + + = + + + + = + + + + +