If the residual variance is not homoscedastic, the assumption of normal distributed errors is usually inappropriate, and one should better go to find a more appropriate error-model.

And even if the normal error model seems appropriate, it is not clear (or not eásy to see) what weights should be used. The "inverse variance weights" can be a useful standard, but the variances would need to be estimated (or guessed), and that is very difficult/imprecise (estimating a variance is much more difficult than estimating a mean!). If it is not clear if the weights make sense, it's not clear if the WLS gives any advantage. However, this point might be not that severe. I think the first point is more important.

The sigma for the random factors of the estimated residuals is generally appropriate to use with a normal distribution in a prediction interval.  No problem. And relative standard errors are always good indicators of accuracy. 

As to guessing a good weight, there are methods to estimate the coefficient of heteroscedasticity from the data, which would vritually always be better than setting it equal to zero.

Note: Looking back through my previous questions on ResearchGate, I found the following, from three years ago, which are on this topic:

https://www.researchgate.net/post/When_would_you_naturally_expect_OLS_regression_to_be_completely_appropriate

https://www.researchgate.net/post/What_might_be_anticipated_in_linear_or_other_regression_with_regards_to_heteroscedasticity_when_negative_values_of_x_and_y_are_well_within_range

All: Please provide your perspective on this issue, if you care to consider it. 

Thank you. 

These might be helpful:

 https://www.researchgate.net/publication/263809034_Alternative_to_the_Iterated_Reweighted_Least_Squares_Method_-_Apparent_Heteroscedasticity_and_Linear_Regression_Model_Sampling

 https://www.researchgate.net/publication/263032446_Weighting_in_Regression_for_Use_in_Survey_Methodology (Note that the use of "w" in the notation on page 2 was not a good choice when avoiding confusion.  Sorry.)

 https://www.researchgate.net/publication/263265199_CRE_Prediction_'Bounds'_and_Graphs_Example_for_Section_4_of_Properties_of_WLS_article

I can imagine that heteroscedasticity might be different for time series data, but I was not really thinking about that.

See https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression. 

The following is for finding a good value for the coefficient of heteroscedasticity:

 https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

A spreadsheet tool (with references) is found here:  

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

The estimated coefficient of heteroscedasticity, as explained in

 https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity

can be easily used in a regression weight expression.  The regression weight, for example, can be entered as "w" into SAS PROC REG.   This, I assume, may be similar for other statistical software. 

OLS regression is a special case of WLS (weighted least squares) regression.  For OLS, the coefficient of heteroscedasticity is zero and weights are all equal.  According to Brewer, below, and my paper, "Essential Heteroscedasticity," above, OLS is not what you should naturally expect.  And really, why wouldn't a larger predicted value in finite population sampling have a larger estimated variance of the prediction error?  Often one might think they have OLS, but as I showed in "Estimating the Coefficient of Heteroscedasticity" above, this may not be true.  In another example for a planned forthcoming paper, I can show how the choice of independent variables can lose information, and near homoscedasticity results.  But if the best set of predictors is available, and there are no data quality issues, and the data are real, not an artificial example, one can generally expect the range of coefficient of heteroscedasticity Brewer indicated, and 0 is not included. 

See Brewer, K.R.W.(2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press, especially pages 111, and 87, 130, 137, 142, and 203.

This could matter not just to the  estimated variance of the prediction error, but also to predicted totals in finite population sampling.  Yes the OLS estimator is unbiased, but in practical applications, the predictions obtained can matter.

For example:  

Suppose you have many small populations of establishments for which official statistics are to be collected on a frequent basis.  (Small area estimation may only go so far.  Consider establishment surveys whose appropriate modeling generally varies substantially from one group or type of respondent to another.  For energy establishments there can be various peculiarities.)  There could be monthly or even weekly samples, and much less frequently collected censuses which could be used for regressor data.  With many small sample sizes, the estimated coefficient of heteroscedasticity will vary, but you may find that for these kinds of data, or at least for various groups of them, the estimated coefficient of heteroscedasticity is generally within a certain range, and a somewhat different value is robust against the kind of data quality issues which tend to present in the production of these frequently published official statistics.  For any one published predicted total, the sample size may be small enough and the data skewed enough that having a reasonable weight for the largest observation(s), which has the largest estimated residual(s) can make a difference to that official statistic. 

(One check on the results, for example, is to add, say, monthly predicted totals and compare them, say, to an annual census for that period, made available later.) 

The above methodology is what you will see in the following:

"Quasi-Cutoff Sampling and the Classical Ratio Estimator - Application to Establishment Surveys for Official Statistics at the US Energy Information Administration - Historical Development," Knaub, 2017, invited presentation at US EIA, https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development

In Brewer(2002), noted previously, he mentions some default values for the coefficient of heteroscedasticity for different kinds of application, and a range in which you would expect them to fall. 

See Brewer, K.R.W.(2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press, especially pages 111, and 87, 130, 137, 142, and 203.