* SHAZAM code downloaded from UCLA Econ 143 (CAMERON) WebSite: 
* HTML file called e143sh26.htm, and should have 
*  been downloaded as wls_sim.sha

set nocolor
* We will each create our own sample from a known "population"
* with a known "population regression function," (PRF).  By construction, 
* these data are heteroscedastic.  The true population error variance 
* is approximately proportional to the squared value of the 
* explanatory variable.  

popsize:300
sample 1 [popsize]

* CREATE SOME "POPULATION" DATA:

* Let non-profit organization revenues (r) be distributed uniformly between
* $500,000 and $1,500,000.

genr r=500+uni(1000)

* Let fund-raising expenditures or "development" costs (d) be given by
* a known "data-generating process" plus an error u that depends on revenue.
* The standard deviation of the PRF regression error is "sig" times r.
* Thus, E[u-squared] is proportional to r-squared.

sig:.05
genr u=[sig]*r*nor(1)

genr d=5 + 0.2*r + u 

plot d r

ols d r / resid=e

genr e2=e*e
plot e2 r
genr r2=r*r

*
* For your particular data set (everyone's will be different), is
*  the squared regression error more closely associated with r
*  or with r2?  Which would you expect, given the way the data
*  were constructed?
ols e2 r
ols e2 r2

*
* Assume that the regression  ols e2 r2  has the highest R-squared value.
* Note that we expect it to, since that was the way the data set was
* generated.  It is possible that, for your particular sample, this true
* underlying relationship will not be clear from the sample you have drawn.
*
* If your data do not reveal the true underlying form of
* the heteroscedasticity in the PRF, you have learned that even our best
* efforts with a sample of data will sometimes not reveal the TRUE form
* of heteroscedasticity afflicting our data.  Here, we know the true form,
* but in any real-life application, we will not.
*
* Now, multiply both sides of the population regression function by 1/r.
* This means that d becomes (d/r), the constant term becomes (1/r), and
* the r term becomes (r/r)=1 (the new constant).  
* The next question is:  If we do a regression of the transformed 
* dependent variable on the transformed explanatory
* variables (where the "variable" is now 1/r), what do we get?

* First, generate the transformed variables:
*    td= transformed d
*    tconst= transformed constant term
*    tr= transformed r, now just a constant equal to 1
* We will use OLS td tconst tr / noconstant to prevent an automatic constant


genr td=d/r
genr tconst=1/r
genr tr=1

*
* First plot transformed (td) against transformed constant (tconst)
* Does this more closely satisfy the maintained hypotheses for OLS?

plot td tconst

* Next, run the regression through these points and NOTE the point estimates

ols td tr tconst / noconstant

*
* Now create the appropriate weighting variable and see if you get the same 
* results as by explicit transformation of all variables and weighted 
* least squares

genr wt=1/(r*r)

ols d r / weight=wt

stop