Q1. The following procedure was suggested when testing for over identifying restrictions:
1. Calculate the structural equation by Two Stage Least Squares and get the Two Stage Least Squares residuals, (uˆ)1 (U hat subscript 1).
2. Regress (uˆ)1 on all exogenous variables. Get the R-squared, say, (R^2) subscript (1).
3. All the 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. We reject the null hypothesis and conclude that at least some of the Instrumental Variables are not exogenous if n(R^2)1 exceeds the critical value for the desired significance level in the chi-squared distribution.
Find this result in a simple regression model with an endogenous repressor X,
Y = β0 + β1X + U1 and a single instrumental variable Z for X.
Q2. Assume that annual earnings and alcohol consumption are measured by the Simultaneous Equation Model:
alcohol = γ0 + γ1 log(earnings) + γ2educ + γ3 log(price) + u2
log(earnings) = β0 + β1alcohol + β2educ + u1
Assume that educe and price are exogenous. If β1, β2, γ1, γ2 and γ3 are all different from zero, which equation is identified? Where price is a local price index for alcohol, which includes state and local taxes, how would you estimate that equation?