Regression Analysis When the Variance of the Dependent Variable is Proportional to the Square of its Expectation

In this article we consider a regression model where the variance of the dependent variable is proportional to the square of its expectation. First, we obtain the asymptotic efficiency of the weighted least squares estimator as compared to the maximum likelihood estimator for the cases when the dependent variable follows a normal, lognormal, or Gamma distribution. Second, we derive a test of whether the dependent variable follows a lognormal or a Gamma distribution. An example is worked out for the purpose of illustration.