I might consider some linear transformations first. If that doesn't work, then I might consider performing an EFA to assess whether I have a dimensionality problem with my scales.

Cory

Anmol -

Heteroscedasticity is a natural feature of regression, which is explained by differences in predicted y values, which is to be expected, just as y-values are not all the same in a population either. (See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity.) This can be modified by model and data issues. (See https://www.researchgate.net/publication/324706010_Nonessential_Heteroscedasticity.)

For any regression of form

y = y* + e,

where y* is predicted y, considering that G.S. Maddala associated an asterisk with weighted least square (WLS) regression, we have

y = y* + (e_0)(z^gamma),

where this is illustrated in examples in

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

and you can use the following tool to do this with your data:

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

That tool will help you see the degree of heteroscedasticity that you have (the coefficient of heteroscedasticity, gamma), which is used in the regression weight, w, where

w = z^(-2gamma),

and z is the preliminary, say OLS, prediction of y.

You use your OLS results in that tool to decide on gamma, to get the regression weight formula, w, to enter into your software, such as SAS PROC REG, for example, which would then use that to modify your regression coefficients and produce WLS predictions, and WLS estimated variances of prediction errors, which can be used in prediction intervals.

For more information, please see https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and updates there.

Cheers - Jim

Regarding transformations: A transformation may not account for the 'right' amount of heteroscedasticity, and you may not be able to tell that by looking at a graphical residual analysis. Sometimes you really have to know what you are looking for. (See my last response, July 2, 2020, to https://www.researchgate.net/post/Hello_Do_you_think_that_this_is_pattern_of_heteroscedasticity_How_can_i_statistically_test_for_homoscedasticity.) It is best to measure heteroscedasticity by means of the coefficient of heteroscedasticity, gamma.

Further, once you have used a transformation, there might be problems of interpretation.

Aesthetically, transformations are somewhat artificial, and seem to me to be a lot like duct tape people use to 'fix' things. Heteroscedasticity in regression is a feature, not a bug, so there is nothing that needs to be 'fixed.' Homoscedasticity is a convenient shortcut when you can get away with it, with all regression weights equal, but it isn't the way things should be. Homoscedasticity is an artificial construct, the ubiquitous use of which, like the p-value, is an historical artifact that was not a really great idea in the first place. It has been abused and overused. Transformations, in this context, are also overused.

If you consider

y = y* + (e_0)(z^gamma)

and the information I linked in my previous response to this question, you will see that if gamma is zero, you have homoscedasticity. But gamma should be between 0.5 and 1.0 for finite population applications. (See "Essential Heteroscedasticity " a paper noted in my earlier response, with reference to Ken Brewer. - Note that this impacts regression coefficients.) In the simple model, for highly skewed energy establishment surveys with which I worked a great deal, gamma = 0.8 (or 0.7 to 0.9) was common when data were good, but I found that 0.5 helped guard against data quality problems with the smallest responders in an official statistics production mode, with many categories being published. But gamma=0 should not happen, though data might be that bad, or there could be some other issue. Issues, especially with more complex models, might effectively reduce gamma, but gamma can even be effectively increased, say if data from two different subpopulations are mixed, or there is an omitted variable. (Please remember that "gamma" is the "coefficient of heteroscedasticity.")

The Excel tool I provided earlier can help find a good gamma for your situation. If you have too little data for a good estimate of gamma, you should still estimate better than using zero, and you might pick a good gamma by considering the application. Further information is in the tool spreadsheet and papers linked in my first response posted above. Note reference to Ken Brewer's book.

Regarding the hypothesis test mentioned for this question, there is an update in the project I linked previously, https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, regarding hypothesis tests for heteroscedasticity. Note that a lone p-value tells you almost nothing. It changes with change in sample size, as does a standard error for a parameter, though a lone p-value is less practically interpretable. That is generally true, but in this case it is also difficult to make the hypothesis test useful, because finding an effect size that is helpful here is problematic. If you decide there is substantial heteroscedasticity, how do you judge that, and further, what do you do about it? But if you ignore hypothesis testing, and just determine a meaningful coefficient of heteroscedasticity, you can use it in the regression weight, as I previously described, and see how much difference it makes to results. It should usually not change predictions very much, usually, but will often change the estimated variance of the prediction error substantially. So why not just use a good coefficient of heteroscedasticity, and go ahead with that?

Finally, a side comment: In general, you should not only look at a "graphical residual analysis" regarding model fit, including heteroscedasticity, but also a "cross-validation" could help avoid overfitting your model to your particular sample.

Note that in the simplest case, for a ratio estimator, we just have y*=bx, and since b will be some constant, we can use z=x, since relative size is all that matters in regression weights. Using the above information, for a model-based classical ratio estimator (CRE), where the coefficient of heteroscedasticity, gamma, is 0.5, we have

w=1/x.

One of the updates to the above mentioned project is on regression weights, w, in statistical software, just as another update is with regard to there being no need for an hypothesis test for heteroscedasticity.

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

Here, for this question, note that we are not actually talking about "removing" heteroscedasticity.

Instead we are including heteroscedasticity in the model, where it belongs, through the use of regression weights, w. These weights are the inverse square values of the nonrandom factors of the estimated residuals. Thus, heteroscedasticity is part of the error structure. The regression weights are used in the estimation of regression coefficients. The regression coefficients are estimated by simultaneously minimizing the sums of weighted squared estimated residuals with respect to each regression coefficient (setting partial derivatives equal to zero). For homoscedasticity, that process occurs with equal weights, likely unity in practice, but any constant would do.

If your sample is large enough, use robust standard errors (Huber-White estimator).

You could use robust standard errors if all of your crucial OLS predictions are close enough, and you only want to try to improve on your (often extremely erroneous) standard errors. However, it may be hard to know when you have assumed too much, and done well enough.

Weighted least squares (WLS) regression is not messy that way. Rather than work with something that you know is faulty to some such unknown degree, it seems better to just model better. The excel tool I provided will help you convert from OLS to WLS results easily, once you see what you are doing, from here forward.

Again -

Here are the examples:

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

Here is the tool:

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

Cheers.