Consider the following structural equation: y1 = 12y2 + 13y3 + 11X1 + 12X2 + u1 where y2 and y3 are endogenous and X1 and

Consider the following structural equation: y1 = α12y2 + α13y3 + β11X1 + β12X2 + u1 where y2 and y3 are endogenous and X1 and X2 are exogenous. Also, suppose that the excluded exogenous variables include X3 and X4.

(a) Test the null hypothesis that y2 and y3 are exogenous against the alternative that they are not. Show that Hausman’s test statistic can be obtained from the augmented regression:

y1 = α12y2 + α13y3 + β11X1 + β12X2 + γ2ˆy2 + γ3ˆy3 + 1 where ˆy2 and ˆy3 are predicted values from regressing y2 and y3 on X = [X1,X2,X3,X4].

Hausman’s test is equivalent to testing Ho; γ2 = γ3 = 0. See equation (11.54).

(b) Show that the same results in part

(a) hold if we had used the following augmented regression:
y1 = α12y2 + α13y3 + β11X1 + β12X2 + γ2ν2 + γ3ν3 + η1 where ν2 and ν3 are the residuals from running y2 and y3 on X = [X1,X2,X3,X4]. See equation (11.55). Hint: Show that the regressions in

(a) and

(b) have the same residual sum of squares.