Consider once again the model analyzed in Question 2 of assignment 1.The two structural equations of interest are
It is assumed that the variables educ, age , age²,lcigpric and restaurn are exogenous variables in the sense that they are uncorrelated with the error terms in both (1) and (2).
(a) Do you have any reason to believe that any of the regressors in (1) and (2) are endogenous? Briefly explain.
(b) Using the notation employed in your lecture notes, determine whether or not each of the structural equations above satisfies the order condition for identification. Carefully define any notation that you use.
(c) Formulate equations (1) and (2) above as a system of the form
Carefully define each of the vectors and matrices in (3) and specify their dimension.
(d) i) Estimate (3) as a system of SUR and report the estimated equation.
ii) Ignoring for the moment any endogeneity issues, is there any evidence to justify estimating (3) by SUR rather than OLS? Briefly explain.
iii) Do you have any reservations about estimating (3) by SUR? Briefly explain.
(e) Estimate (3) by 3SLS and report the estimated equations
(f) Consider the null hypothesis that the marginal effect of age on the dependent variable is the same in each of the structural equations and the marginal effect of cigs on income is equal but opposite in sign to the marginal effect of lincome on cigs. Formulate this hypothesis using the notation
Carefully define each of the vectors and matrices in (4) and specify their dimension.
(g) Test the null hypothesis specified in (g) for an individual of (sample) median age. State the null and alternative hypotheses in terms of restrictions on the parameters of the model, the sample and critical values of the test statistic and your test conclusion.
In Topic 5 of the lecture notes it is stated that the estimator
may be rewritten as
Use the fact that
together with the properties of the Kronecker product to verify this statement.