Question 4
i) Yes, it has an approximate interpretation as an elasticity since it is a log-log regression
ii) See do file
iii) The results are quite different. This could be attributed to (1) Correlation of regressors
with individual effects, so the estimators for the RE model (1,2,5) would be inconsistent,
whereas (3,4) would not. Moreover, the FD estimator uses a different sample.
iv) The clustered standard errors are systematically larger. This is because they account for
temporal correlation of the errors, which implies the data contains less information on
the parameters than if we had a cross-section of size NT, rather than a panel.
v) See do file. The Hausman test rejects equality of the two estimators at the 0.05 level,
and hence the RE model. We favour the FE model (within estimator) since it is
consistent.
vi) There is no single correct answer. Here are two possibilities
a. If we assume x_it is strictly exogenous, we can perform the within transformation and use instrumental variables (e.g. 2SLS or GMM) with instruments (x_it- xbar_i),(x_it-1-xbar_i). See my do file for STATA implementation. It seems that these instruments are possibly weak, since the first stage F-statistic is 1.77<10. It is important to use clustered standard errors.
b. If we assume lnwg_it is weakly exogenous, we can perform the first differences transformation and use instrumental variables estimation as in a with instruments (x_it-x_it-1), lnwg_it-2. These instruments seem even weaker than in a.