NETWORK FILES NEEDED:  n:het1.dat, n:het2.dat, n:het1.sha, n:het2.sha, n:wls_sim.sha

1. Weighted least squares (WLS) is the most common technique to use for (mostly) cross-sectional data where heteroskedasticity is a problem. We will look at a pair of contrived data sets in the files n:het1.dat and n:het2.dat. In n:het1.dat, there are 50 observations for recently hired production line workers on two variables: weeks of experience (weeks_i), and percent perfect gizmos produced (perfect_i). To save you some programming anxiety, I have begun the program to deal with these data in a file called n:het1.sha. NOTE: the type of heteroskedasticity explored in these data differs from the usual "fanning out" form, where the magnitude of the conditional error variance in the regression increases as some explanatory variable increases.

2. At other times, we will not have the luxury of repeated observations at each value of the explanatory variable(s) that let us calculate estimates of the conditional error variance for each group of observations to use in our weighting computations. The data set in n:het2.dat contains another sample of 50 new workers for the same firm, but this time, experience is reported in days (days_i). Percent perfect gizmos (perfect_i) is reported as before. In this type of situation, we need to figure out a way to construct weights using numbers that are at least proportional to the conditional error variances, s_i². These methods are always somewhat ad hoc, since it is impossible to know the true exact relationship between s_i² and any of the observable data, but it is important to make a concerted effort to correct for heteroskedasticity. WHY? The program in n:het2.sha reads the second data set.

3. Verifying what goes on in the background when you specify the weight= on an OLS command:  Run the simulation program contained in n:wls_sim.sha and review the output. This program creates a single sample of size 300 for a dependent variable d and an explanatory variable r with a known form of heteroscedasticity in the population regression function from which the data were drawn. Each time the program is run, an entirely different set of data will be created, so your results will differ from those of other people in your study group.

Review the commentary and questions contained in the program, then run it and see if your results are typical. Does the ols d r / weight=... method of producing weighted least squares regression estimates produce the same results as used to be obtained by making explicit transformations of all the variables in the model and running the regression on the transformed data? Note also that the transformed data, when plotted, probably look much more like they satisfy the maintained hypothesis of ordinary least squares regressions. A scatterplot for the transformed data certainly looks different than a scatterplot of the raw data.

Mission: insert lines to perform an initial "naive" OLS regression without any weights. Save the fitted residuals and square them. Then regress these residuals alternately on r (the explanatory variable in the main model), on the square of r, and on both at the same time. Do you recover the type of heteroscedasticity that was used in the production of the data? The lesson here is that a sample will not necessarily always reveal exactly the type of heteroscedasticity that afflicts the population from which the sample is drawn. BOTTOM LINE: We usually just do the best we can to characterize the approximate nature of our heteroscedasticity problem. We then use this information to effect the most appropriate WLS solution that we can come up with.