What do you mean by "natural heteroscedasticity"?

There are models that will require heteroscedasticity. Actually almost all models require this, except the models based on the assumption of normal distributed errors.

Generally, I think, the "lack of fit" of the observed variance structure to the variance structure assumed by the model (be it homogeneous or some other relationship [see for instance Poisson models, where s² should equal µ]), always indicates that the model is in some way mis-specified. This does not need to be wrong or bad. It just tells us that the model does in some respect (an important one or an unimportan one) not use all the information from the data adequately  - or at least that we have data that shows that this is the case (because models are never correct).

You have a variance, especially the error-variance always and only w.r.t. some model. There is nothing like "variance as such", also "errors" are defined only w.r.t. some model. So I don't really see what a "natural heteroscedasticity" would be, since I don't even see what a "natural variance" should be.

So my point is that the whole fuzz about variance, heteroscedasticity and what else is more related to our models than to the data.If the assumption of homoscedastic errors should be reasonable but the observed errors (w.r.t. to the fitted model!) do show something else, it can only mean that our model is mis-specified (what can include, but is not limited to, the lack of relevant predictors).

By "natural" heteroscedasticity, I mean what the first page at the first link above called "pure" heteroscedasticity (rather than anything due to an omitted variable). There it included this statement: " 2) Heteroskedasticity is more likely to take place on cross-sectional models. Cross-sectional models often have observtions of widely different sizes in a sample."  -   I have long experience with such data, and if you use one of the techniques available to estimate the coefficient of heteroscedasticity, one of which I devised, you will virtually always find it measures to be within general bounds noted theoretically on page 111 of Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press.  

It is natural that the variance of the prediction error be greater when the size measure is 1000 times greater, than in the smaller case. 

Thanks for the comments, Jochen.

Dear James, thanks for your help, but I am afraid I still don't get it (yet).

I will use a physical example, not because it woul be in principle different to social, economical, biomedical data - only bvecause it is simpler to understand and it is easier to see and to exclude confounders.

If I measure the radioactivity (as counts on a Geiger counter in fixed time intervals) depending on the time, I get some declining trend of the counts. I could fit a straight line (although we know that this won't be appropriate) and find a strange non-linear dependency of the variance (of the errors or residuals) over time. One could chose a very small time window, so the variance may look homokedastic, but the relation between variance and time is still there (just not recognizealbe, given a sufficiently short time interval). Is this something you would call "natural heteroscedasticity"?

Now conside we recognize that the trend is an exponentially decreasing function, so we fit an exponential curve to the time course. This is appropriate. And yet we find heteroscedasticity: the variance of measurements at higher means will be higher. In fact we may see that the value of the variance is always a fixed multiple of the mean, independent of the time. Is this something you would call "natural heteroscedasticity"?

But who says that we have to measure counts? We could also measure log(counts) instead (this is in fact possible when we use an instrument that measures some relative activity; in other contexts like the quentification of nucleic acids with real-time PCR I have better examples how "log(concentrations)" are measured directly - not as concentration that is log-transformed). Plotting the log(counts) against the time shows a straight-line relationship, and fitting a straight-line model results in homoscedastic errors.

Now consider we had not measured the time and just recorded the (subsequent) readings of the counter. We get a sample of values and no predicor. All we can say is that the distribution is right-skewed. The distribution of the logarithmic values is uniform...

At the end, the structure (and heteroscedasticity) of the errors depends on how we measure (operationalize), what we measure, and what model we apply.

Actually, I was not referring to a time series.  

Sorry that I was not clear.  

Seasons Greetings to all!  

The estimated residuals will capture some heteroscedastic variance of  an omitted variable, therefore the natural heteroscedasticity won't be correctly estimated. Even if correctly estimated, the subsequent GLS estimation won't solve the estimation bias problem stemming form the omission. Naturally, I believe, the omitted variable problem should be taken care of first. Otherwise, it would be a waste of time, getting nowhere.

James, I don't think that it makes any difference if "time" or something else is considered as a predictor (unless I am completely missing something here!). You can take my example and just consider two radioactive samples of different mass, or different constitution, so you have some simple categorical variable as possible predictor for the activity. The problem remains the same.

First, Jochen, let's just say that we have heteroscedasticity that is actually caused by omitted variable bias, and that which is not.  I do agree, Eric, that it could be better to take care of omitted variables first, and then consider the remaining heteroscedasticity (which is what I also basically suggested to someone recently, but forgot). 

....................

A slight (but long) digression: 

I'll note here that the heteroscedasticity each of us may have seen, that is not caused by omitted variables, may depend on various fields of application, but i have rarely seen any place in establishment  survey statistics where homoscedasticity was anywhere near correct, and in that case there were severe data quality problems with smaller size reporting units.  You should not generally see homoscedasticity in survey statistics, and that has been indicated by various people, theoretically and in practice. You could do a transformation, but that would tend to complicate the issue. 

There is a large body of work which uses models to assist survey estimation.  See

Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang,     

and

Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press, 

for example. 

Survey weights can be "calibrated" to include heteroscedastic models, though I think it is true that many may not feel heteroscedasticity to be crucial for that method. 

There are reliable ways to estimate the coefficient of heteroscedasticity, the most well-known likely being iterated reweighted least squares. 

(One reference for that:

Carroll and Ruppert(1988), Transformation and Weighting in Regression, Chapman & Hall, Ltd. London, UK.)

There are other methods, including one I developed that has some useful additional properties. 

Different measures of this coefficient only disagree somewhat because models are never perfect, but I found two different methods yielding exactly the same results when artificial data, exactly following a model, were used.

For inference from skewed data establishment surveys (finite populations), I have this, for example: 

 https://www.researchgate.net/publication/261596397_Ken_Brewer_and_the_coefficient_of_heteroscedasticity_as_used_in_sample_survey_inference

So in my view, it seems odd that much of other literature seems to treat heteroscedasticity as a problem to be solved, when in applications I saw for many years, its absence would indicate a problem, and seem very odd indeed.  I don't think it worth transforming so that hypothesis tests may be used.  In general, hypothesis tests may be problematic:

 https://www.researchgate.net/publication/262971440_Practical_Interpretation_of_Hypothesis_Tests_-_letter_to_the_editor_-_TAS

So I have found it hard, after decades of working with heteroscedastic data, to communicate well with those who routinely have used OLS in their fields, and apparently found it useful.   

I could have digressed a great deal further, but only wish to say that it is hard to adopt a different mindset from your own experience, but I think that is a valuable service/use of ResearchGate, and I think I have learned a few things from the vastly different experiences of others.  However, there is always a good deal more to learn, and I know I won't always understand everything that others are saying.  

 - Thank you. 

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

Article Practical Interpretation of Hypothesis Tests - letter to the...

Regarding heteroscedasticity in establishment survey, finite population sampling applications: 

Consider page 111 of Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press.  There Ken Brewer states that if reporting units, in the case of finite population sampling, were such that some could be considered "...loose aggregations of what were originally small units...," then the sigma_i-squared would be proportional to x_i (the size measure, with one regressor).  That would be a lower bound.  Otherwise, he argues that the other extreme could be the sigma_i-squared proportional to x_i-squared (or, as I recall he told me, not much larger).   I have done empirical studies measuring this heteroscedasticity, and found this to generally be correct, case-after-case.  Others have worked in this area as well.  Homoscedasticity should not be expected.

So if you are interested in a survey statistics point-of-view of heteroscedasticity, you might want to get an interlibrary loan of Ken's book.  - Of course, even if you narrow the focus to survey statistics, there are other considerations/other approaches. 

Thank you James

You are always helping me in knowing more . 

Have a nice Cristmas

Helena