Prove the result claimed in Example 4.7. Greene (1980a) considers estimation in a regression model with an asymmetrically distributed disturbance, where has the gamma

Prove the result claimed in Example 4.7.

Greene (1980a) considers estimation in a regression model with an asymmetrically distributed disturbance,

where ε has the gamma distribution in Section B.4.5 [see (B-39)] and σ = ??P/λ is the standard deviation of the disturbance. In this model, the covariance matrix of the least squares estimator of the slope coefficients (not including the constant term) is

whereas for the maximum likelihood estimator (which is not the least squares estimator),

But for the asymmetry parameter, this result would be the same as for the least squares estimator. We conclude that the estimator that accounts for the asymmetric disturbance distribution is more efficient asymptotically.

Another example that is somewhat similar to the model in Example 4.7 is the stochastic frontier model developed in Chapter 18. In these two cases in particular, the distribution of the disturbance is asymmetric. The maximum likelihood estimators are computed in a way that specifically accounts for this while the least squares estimator treats observations above and below the regression line symmetrically. That difference is the source of the asymptotic advantage of the MLE for these two models.

y = (a +ovP) +x' +( oVP) = a* +x' +e*,