By definition, if the estimation is performed using robust standard errors, there is no heteroskedasticity. It is just that your standard errors are usually larger (less efficiency).

Best,

José-Ignacio

may be you can covert the given data into the natural log form

José-Ignacio Antón

I would assume that after using White, the heteroskedasticity tests would show no heteroskedasticity. But, you are right: If I report that my estimations are done with White robust errors, I guess 'I am covered.

Thank you.

Amritkant Mishra

The variables are already in natural logs.

Pandelis -

I have worked with finite population sampling, and am not very familiar with panels, but if you say fixed effects are appropriate here, and looking on Stackexchange it seems that could mean fixed coefficients, then maybe what I have on heteroscedasticity in regression for finite population sampling may directly apply here, or perhaps someone could explain how that should work.

At any rate, for finite population sampling, one can expect heteroscedasticity, though that may be impacted by model and data issues. This is explained here:

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

and here:

https://www.researchgate.net/publication/324706010_Nonessential_Heteroscedasticity

for regressions of form

y = y* + e

where y* is a weighted least squares (WLS) predicted y,

and e is the estimated residual for each i, and it is factored into a random factor, e_0, and a nonrandom factor, z^gamma, where z is a size measure (ideally the best y*, but in practice, while estimating regression weights, a preliminary value can be used, perhaps y_hat, the OLS predicted y), and gamma is the coefficient of heteroscedasticity. (I use y* and y_hat for WLS and OLS predicted y, employing notation used by G.S. Maddala.)

Regression weights are then

w = z^(-2gamma). This is used in the estimates of regression coefficients.

Examples of finding a coefficient of heteroscedasticity, gamma, are found in here:

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

and here is a tool for applying this for other cases:

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

For more information, please see the following:

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

The accuracy of the estimates for the coefficient of heteroscedasticity, gamma, are discussed in https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity. But even for a small sample, I would expect that it is generally better to do this than to always assume homoscedasticity, where gamma = 0. Default values for gamma are also discussed in https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx.

Thus, you handle heteroscedasticity in the error structure, where it belongs, and it then impacts the estimates of regression coefficients. You need software that will let you enter the regression weights, w. (Here is a simple example: For a ratio estimate,

w = 1/x.) The regression coefficients are found by minimizing (setting derivatives or partial derivatives equal to zero) Q, the sum of weighted square estimated residuals, with respect to each of the coefficients, including an intercept term if one is justified, and solving these equations simultaneously.

In most cases predictions may not change very much, but the impact will generally be substantial on the estimated variances of the prediction errors, if gamma is substantially different from zero. For smaller samples, the chance of larger differences in predictions is larger. The symmetry of points about the regression line (or curved line) matters.

I hope some of the above may be helpful here.

Cheers - Jim

PS - If you put all your data into log form only to try to reduce heteroscedasticity, but it wasn't enough, then you might want to go back and undo that transformation.

PSS - If you can do a "graphical residual analysis" to check model fit, and a "cross-validation" to check for overfitting to your particular sample, that might be very helpful.

As suggested above, try taking logs as the first step, but this may be not enough. For some examples and visual illustrations, let me recommend this poster presenting a framework for dealing with panel data:

Poster A Visual Framework for Longitudinal and Panel Studies (with ...

Thank you James R Knaub and Andrey Davydenko ,

the reason the variables are in logs is not to circumvent heterogeneity, but because of different measurement units.

I decided to overcome the issue by checking whether my estimations are robust to alternative specifications of the error term - if they do not change much, that should do.

Thanks, again.

Using GLS (than OLS) is the solution for your heteroscedasticity. Also, Gujarati and Porter suggested this option in their book of econometrics.

Fyi, if you are using STATA, the syntax of "xtgls depvar indepvars, igls" will be useful. For further, you may visit this link

https://www.stata.com/support/faqs/statistics/panel-level-heteroskedasticity-and-autocorrelation/