Q1) When testing for overidentifying restrictions, the following procedure was suggested:
1. Estimate the structural equation by Two Stage Least Squares and obtain the Two Stage Least Squares residuals, (uˆ)1 (U hat subscript 1).
2. Regress (uˆ)1 on all exogenous variables. Obtain the R-squared, say, (R^2) subscript(1).
3. Under the null hypothesis that all Instrumental variable's are uncorrelated with u1, n(R^2) is approximately distributed by a chi-squared distribution with q degrees of freedom, where q is the number of instrumental variables from outside the model minus the total number of endogenous explanatory variables. If n(R^2)1 exceeds the critical value for the desired significance level in the chi-squared distribution, we reject the null hypothesis and conclude that at least some of the Instrumental Variables are not exogenous.
Here, the (R^2) obtained in step 2 would be equal to zero always. Obtain this result in a simple regression model with an endogenous regressor X,
Y = β0 + β1X + U1
and a single instrumental variable Z for X.
Q2)Suppose that annual earnings and alcohol consumption are determined by the Simultaneous Equation Model:
log(earnings) = β0 + β1alcohol + β2educ + u1
alcohol = γ0 + γ1 log(earnings) + γ2educ + γ3 log(price) + u2
where price is a local price index for alcohol, which includes state and local taxes. Assume that educ and price are exogenous. If β1, β2, γ1, γ2 and γ3 are all different from zero, which equation is identified? How would you estimate that equation?