I have come across few Research Papers, where authors have mentioned, " We report heteroskedasticity corrected standard error or Standard errors are corrected for heteroskedasticity. I know what is heteroskedasticity, I know what is standard error as well. But, I don't know how SE is corrected? What is the method or trick ? Please explain me and if you can provide source/book which have explaination.
Ajay -
Perhaps what you have in mind is the sigma upon which the variance of a prediction error is based. This sigma is in turn based upon residuals. When you estimate a residual, it may be factored into a random factor and a nonrandom factor, the latter being the basis for the regression weight.
I once noticed that for a SAS output of estimated residuals, from SAS PROC REG, I think, it was really providing the estimated random factors of those residuals.
In econometrics they sometimes transform a regression to provide a new regression, with constant sigmas. Perhaps that is what you mean. I think I have seen that in more than one econometrics book. I would not think of that as a "correction," as heteroscedasticity is a natural part of the error structure, but many likely do consider it as such, so that they may apply hypothesis tests. However, given the way hypothesis tests are generally so misunderstood, misapplied, and overused, that is still not generally a good idea.
You may find something in the paper at the attached link that might help, or in the long reference list and bibliography.
https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys
Cheers - Jim
Article Properties of Weighted Least Squares Regression for Cutoff S...
The basic least squares model is characterized by the following coefficient variance-covariance matrix:
S = E(^B – B )(^B – B)' = ..... = ( X ' X )–1 T Q ( X ' X )–1 = σ2 ( X ' X )–1
where ^ indicates estimates. The assumption of homoskedasticity implies that:
Q = σ2 ( X ' X / T )
This specification does not holds with heteroskedasticity. As an example, the White heteroskedasticity-corrected standard error is based on the following estimate of Q:
^Q = T/(T-K) SUMt [^et2 ( XtXt' / T)] ; with SUMt as sum of [...] for t=1...T
where T is the number of observations, K is the number of regressors, ^e are the estimated residuals and T/T-K is the degree-of-freedom correction [White H. (1980) - A heteroskedasticity-consistent covariance matrix and a direct test for heteroskedasticity, Econometrica, 48, 817–838]
Ajay it is simple.You should see Econometric Models and Economic Forecasts by Pindyck and Rubinfeld,4ed,pages 148-159
If you apply transformation to the variables in the presence of heteroscedasity ,the disturbance terms become homoscedastic but R square does not have the same interpretation.Further,heteroscedasiticity does not affect unbiased property of the estimates of coefficient but it affects only its standard error.If corrections suggested by White or Newey West are applied then R square has the same interpretation and usual t-test can be applied to regression coefficient.See Pindyck and Rubinfeld.
In simple terms, when a regression suffers from heteroscedasticity, the OLS coefficients are still unbiased and consistent, but their standard errors are biased and inconsistent, so that hypothesis testing and confidence intervals are NOT valid. This is why it has become customary to use the so called "robust" estimation methods, which still give you the same (OLS) coefficient estimates, but the standard errors are "corrected" usually in accordance with a method proposed by H. White (Econometrica, 1980, v. 48, pp. 817-838). A good reference is J. Wooldridge, Introductory Econometrics, 3rd Ed., 2006, p. 274.
standard errors of which the effects of heteroscedasticity have been elimated. it is quite different from usual OLS standard errors containing heteroscedasticity
As shown at the bottom of page 2 and on page 5 in the first link, OLS is just a special case of WLS, often inappropriately used as a default. (There it is shown for a model through the origin, with only one regressor, but it is true more generally.)
https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys
https://www.researchgate.net/publication/262972023_HETEROSCEDASTICITY_AND_HOMOSCEDASTICITY
Also, here is how this can relate to survey statistics:
https://www.researchgate.net/publication/261596397_KEN_BREWER_AND_THE_COEFFICIENT_OF_HETEROSCEDASTICITY_AS_USED_IN_SAMPLE_SURVEY_INFERENCE
Article Properties of Weighted Least Squares Regression for Cutoff S...
Article HETEROSCEDASTICITY AND HOMOSCEDASTICITY
Article Ken Brewer and the coefficient of heteroscedasticity as used...
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