Sheetal  -

I don't follow all of your question or your subject matter, but I can tell you that heteroscedasticity is often a naturally occurring part of the error structure for the 'prediction' of your dependent variable, y.  I encountered it routinely with continuous data for establishment cross-sectional surveys.  

The estimated residuals should be factored into random and nonrandom parts, where the nonrandom factor is based on a regression weight.   Often that weight is well written in a form using a size measure, and an exponent making use of a "gamma" value called the coefficient of heteroscedasticity.  (For OLS, the nonrandom factor is any constant.) 

The size measure for a one-regressor model would be that regressor, x.   For more than one regressor, any combination of regressors might do.  A good example might be a preliminary prediction of y, say perhaps the OLS one, while you come up with your better prediction from weighted least squares  (WLS) regression.  

However, if only one regressor is causing heteroscedasticity, you could simply use that one regressor as a size measure. 

Your situation might be more complex, but the main concept here is that heteroscedasticity is used in reference to y and predicted y-values.  The regressors are used to constitute a size measure in the error structure.  I thought perhaps that might have been a source of confusion.   Perhaps not??  

But yes, heteroscedasticity in y can be due to just one regressor, though I'd expect more to be involved. And heteroscedasticity is not generally a matter of having it or not having it,  yes or no, but rather how much is there, and estimates of the coefficient of heteroscedasticity are possible. There are multiple methods, and actually I developed one. 

Cheers - Jim

Sheetal -

A caution about p-values: You cannot make an informed decision based on an isolated p-value. 

Please see the following:

Press release for the American Statistical Association:

http://www.amstat.org/newsroom/pressreleases/P-ValueStatement.pdf

and 

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

Best wishes - Jim

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

Dear James

First of all thanks for your prompt and valuable reply. I am a novice in research area, however, I m familiar with OLS regression but WLS regression is difficult to comply with as weights are needed to be assigned very cautiously. You have understood my problem quite well. Can you put some more light on concept of “one regressor as a size measure”. Is it about selecting the variable, defining the case labels and assigning WLS weights to the respective variable (which is causing heteroscedasticity)?

I am attaching one word file showing plots of standardized residuals. Guide me about the trend -are they okay? It seems that errors are dispersed in a random way.   

Thanks and Regards

Sheetal

Here are a few resources which may be of some help:

At a glance, this looks like it might be OK:

 https://onlinecourses.science.psu.edu/stat501/node/352

and

 https://onlinecourses.science.psu.edu/stat501/node/36

For an introduction to WLS, you could see

Bottom of page 1 and bottom half of page 2:

 https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys

And for size measure when more complex:

Appendix IV, page 36:

 https://www.researchgate.net/publication/242677296_Quasi-Cutoff_Sampling_and_Inference

That is just shown for multiple linear regression, but is not restricted to that. 

Intro to heteroscedasticity for cross-sectional surveys:

 https://www.researchgate.net/publication/262972023_HETEROSCEDASTICITY_AND_HOMOSCEDASTICITY

To estimate coefficient of heteroscedasticity:

 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

The model-based classical ratio estimator is often robust. 

Article Properties of Weighted Least Squares Regression for Cutoff S...

Article Quasi-Cutoff Sampling and Inference

Article HETEROSCEDASTICITY AND HOMOSCEDASTICITY

Conference Paper Alternative to the Iterated Reweighted Least Squares Method ...

Article Weighting in Regression for Use in Survey Methodology

 First resource is well informative about WLS weights and all others are enlightening the theoretical as well as practical aspects of WLS. U have done great job in your research field (mathematics).

Thanks again sir

Sheetal -

Thank you, but your application seems somewhat different from what is in my experience, so there may be other aspects to it of which I am unfamiliar.  For example, I don't really know why your second graph looks as it does.  So, I advise that, as always, you just keep an open mind to find out new/other ways to approach topics, or build on old ideas, as I can see you already are doing, by having asked this question.  Also please note that any kind of testing you can do may be enlightening, especially using graphics.  But note that subject matter theory is important.  Exploratory Data Analysis can be helpful, but spurious results will occasionally occur. 

I hope you enjoy your project.  Perhaps you might report back what you eventually find, to help enlighten us all. 

Best wishes - Jim

Dear Jim

I have applied WLS and found robust results. R square has increased significantly and some of the hypothesized variables got statistical support. I ll concentrate more on theory matter as suggested by you. Thanks for guidance.

Kind Regards

Sheetal

sheetal-

is it your problem resolved?

Yes Mr. Sabtu, i used it two years ago in my thesis.

Kind regards

Sheetal -

You might find this project to be of interest in your future work. I have only looked at what it means to use WLS rather than the special case, OLS, so there could be a bit more to consider when applying it to more complex situations. 

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

The basis for this is found here:

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

This helps with application:

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

and this

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

From there we get the regression weight expression which can be entered as "w" into SAS PROC REG.  I assume that other statistical software can be used in a similar fashion.  (The regression coefficients are made to minimize the sum of the weighted squared estimated residuals with respect to each regression coefficient, simultaneously.)

Perhaps you might find this useful sometime. 

Cheers - Jim

PS -  A very late congratulations on your thesis! 

Thank you so much Sir for heartfelt wishes. Obviously, this information, i shall incorporate in my coming projects.

Great day!!

Kind regards

Sheetal

Thank you. I hope this is of some use to you. Best wishes.