Subhash -

My apologies for not noticing, but it does look like the second example you gave does appear to show a case where one might use a zero coefficient of heteroscedasticity,  and if those data are real observations, as opposed to that author's artificial data illustration, then this exactly answers my question regarding real data where OLS might be appropriate. 

It looks like a real-life, real data example. But if anyone knows whether or not that is the case, I'd like to hear about it. If so, the data could be used to estimate the coefficient of heteroscedasticity, at least in ranges of x, to see if close to zero. It appears that it is, but that is not always correctly assessed by looking at the graph alone. 

In that example, there is a single regressor, x, where x is a quarterly percent change in unemployment rate, while y is the quarterly percent change in GDP.  

This looks like a potentially good example, not only to answer this question, but also an earlier question I have on ResearchGate regarding when negative x or y values might be appropriate. If you, Subhash,  would care to show that example as an answer to apply to that question also, I would appreciate it. 

Again, my apologies for not seeing this earlier. If there is heteroscedasticity,  it is not apparent from the graph, though it might be when actually estimated from the data using the iterated reweighted least squares (IRLS) method, or some other method, such as one I've published. 

Cheers - Jim 

PS - My guess is a slight heteroscedasticity.   This generally impacts variance more than the slope itself, but looking more closely at the graph, I'm guessing it will make the magnitude of the negative slope slightly greater.  However,  it is a nice example. 

James,

I'm not sure I understand the heteroscedasticity point. Standard errors scale with variances, not means or predicted values. So if the three examples you give have the same variance of Y, homoscedasticity is plausible (though not guaranteed).

Pat

Sorry.  I was trying to get at the idea of factoring the estimated residuals, as is shown at the bottom of page 397 in the article at the attached link, rather than the usual ei. 

So, to rephrase the question, When would that weighting factor ever be a constant in reality?  

Article Ken Brewer and the coefficient of heteroscedasticity as used...

Ordinary least-squares (OLS) regression is a generalized linear modelling technique that may be used to model a single response variable which has been recorded on at least an interval scale. The technique may be applied to single or multiple explanatory variables and also categorical explanatory variables that have been appropriately coded.

At a very basic level, the relationship between a continuous response variable (Y) and a continuous explanatory variable (X) may be represented using a line of best-fit, where Y is predicted, at least to some extent, by X. If this relationship is linear, it may be appropriately represented mathematically using the straight line equation 'Y = α + βx'.

To assess the effect that a single explanatory variable has on the prediction of Y, one simply compares the deviance statistics before and after the variable has been added to the model.

For a simple OLS regression model, the effect of the explanatory variable can be assessed by comparing the RSS statistic for the full regression model (Y = α + βx) with that for the null model (Y = α). The difference in deviance between the nested models can then be tested for significance using an F-test.

OLS is used in economics (econometrics), political science and electrical engineering (control theory and signal processing), among many areas of application. The Multi-fractional order estimator is an expanded version of OLS.

https://datajobs.com/data-science-repo/OLS-Regression-[GD-Hutcheson].pdf

https://en.wikipedia.org/wiki/Ordinary_least_squares

Subhash -

Thank you, but have you ever actually seen a case in reality that was homoscedastic, as opposed to heteroscedastic?  OLS is a special case of WLS, but have you ever seen it to be true?  That is, the coefficient of heteroscedasticity is certainly not going to be exactly zero.  I think it is generally substantial, yet ignored.  Can you think of any situation you have ever seen where, in reality, the coefficient of heteroscedasticity was virtually zero, and could be safely ignored, and thus OLS regression used instead of WLS regression?  Or perhaps the heteroscedasticity changed in various regions, so that OLS was an OK approach overall? 

Thank you -

Jim

Subhash -

My apologies for not noticing, but it does look like the second example you gave does appear to show a case where one might use a zero coefficient of heteroscedasticity,  and if those data are real observations, as opposed to that author's artificial data illustration, then this exactly answers my question regarding real data where OLS might be appropriate. 

It looks like a real-life, real data example. But if anyone knows whether or not that is the case, I'd like to hear about it. If so, the data could be used to estimate the coefficient of heteroscedasticity, at least in ranges of x, to see if close to zero. It appears that it is, but that is not always correctly assessed by looking at the graph alone. 

In that example, there is a single regressor, x, where x is a quarterly percent change in unemployment rate, while y is the quarterly percent change in GDP.  

This looks like a potentially good example, not only to answer this question, but also an earlier question I have on ResearchGate regarding when negative x or y values might be appropriate. If you, Subhash,  would care to show that example as an answer to apply to that question also, I would appreciate it. 

Again, my apologies for not seeing this earlier. If there is heteroscedasticity,  it is not apparent from the graph, though it might be when actually estimated from the data using the iterated reweighted least squares (IRLS) method, or some other method, such as one I've published. 

Cheers - Jim 

PS - My guess is a slight heteroscedasticity.   This generally impacts variance more than the slope itself, but looking more closely at the graph, I'm guessing it will make the magnitude of the negative slope slightly greater.  However,  it is a nice example. 

James, I agree that media may occur at left or right side of median. I have a short paper called "Parametric induction applied to small samples -method and model with inverse function-" It is written in spanish and waiting for publication. I send here a short version in case you may read it and share your comments.. Best wishes, emilio