The first two problems in this problem set were repeated pretty much unchanged from Fall 1997. See the corresponding questions in Problem Set #6 and its Solution Key.

Problem 3 is more of an exercise, where you were to observe what happened when you followed instructions and reported on the results.

You should find that explicit transformation of all the variables in the model, followed by OLS on the transformed data, will yield the identical parameter estimates and inferences as the corresponding OLS command applied to the raw data, but using weights. This just illustrates the tasks that are going on in the background when you ask SHAZAM to do a weighted least squares regression.

This exercise is also intended to reinforce the insight about how we deal with violations of the maintained hypotheses for OLS concerning the nature of the error term. We can still use OLS, we just have to apply it to a transformed version of the model, where the transformation is selected to "undo" whatever mischief exists in the error term. This same philosophy is used to correct for both heteroscedasticity and (as we shall see later in the course) for serially correlated errors. We transform all the data in order to effect a transformation of the associated residuals so that the transformed errors are "fine." We can then use OLS on the transformed data with impunity.

For this simulation exercise, we know the "true" form of the heteroscedasticity that created the data (a luxury virtually unheard of in practice). In some cases, residuals analysis on the errors from a preliminary naive OLS model using a sample of data from a population will reveal the nature of the heteroscedasticity relationship in that population; other times, we will get a peculiar sample and might think that the relationship is something quite different from the "true" one. This is one of the best reasons for not going overboard in trying to tailor your heteroscedasticity correction procedure to exactly capture the problems with the individual squared errors in your sample. You might be "barking up the wrong tree."