Akanda - the right question would, I think, be how to deal with heteroscedasticity.

One is to apply an appropriate transformation - derived, for example, from the family of Box-Cox transformations. This will help the 'transformed' data to have equal variance, and, as usually happens, will also make the transformed data to follow a normal distribution. 

The other is to account for heterscedasticity (and any non-normality) as it exists. This avoids the need to back-transform your results to original scale. For example, if your data follow a binomial distribution, you could, rather than transforming data to an angular scale, fit a Generalized Linear Model with binomial error on the original data.  

Akanda - the right question would, I think, be how to deal with heteroscedasticity.

One is to apply an appropriate transformation - derived, for example, from the family of Box-Cox transformations. This will help the 'transformed' data to have equal variance, and, as usually happens, will also make the transformed data to follow a normal distribution. 

The other is to account for heterscedasticity (and any non-normality) as it exists. This avoids the need to back-transform your results to original scale. For example, if your data follow a binomial distribution, you could, rather than transforming data to an angular scale, fit a Generalized Linear Model with binomial error on the original data.  

As stated by Subhash, your question has an assumption that heteroscedasticity needs to be removed. By definition, heteroscedasticity means the variance in the dependent variable depends on the value of the independent variable so you would need to characterize this dependence in order to transform to remove it.

Hello, 

I give you an example of  how to remove  heterocedasticity (

Please see the exemple in the file attached

No need to : see [1-2] and the IPAG paper linked below]. Googling 'granger causality test nonparametric" will also give you other possible interesting links...

See also the "causality" function of the R package "vars", whose doc says "For the Granger causality test, a robust covariance-matrix estimator can be used in case of heteroskedasticity through argument vcov. It can be either a pre-computed matrix or a function for extracting the covariance matrix. See vcovHC from package sandwich for further detail". This robustness might be sufficient in your case (possibly bootstrapping the analysis).

You might also try modeling this directly in terms of your (supposed) distributions, and either get an MLE of your parameters and their varoance-covarance matrix or a Bayesian estimate.

HTH,

--

Emmanuel Charpentier

1. Hacker RS, Hatemi-J A. Tests for causality between integrated variables using asymptotic and bootstrap distributions: theory and application. Applied Economics. 2006 Jul 20;38(13):1489–500.

2. De Gooijer JG, Sivarajasingham S. Parametric and nonparametric Granger causality testing: Linkages between international stock markets. Physica A: Statistical Mechanics and its Applications. 2008 Apr 15;387(11):2547–60.

http://www.tandfonline.com/doi/abs/10.1080/00036840500405763#.VRu67sKn2Bs

http://www.ipag.fr/wp-content/uploads/recherche/WP/IPAG_WP_2014_162.pdf

http://www.sciencedirect.com/science/article/pii/S0378437108000265

You might consider the alternative that the unequal variances is possibly a message and not just a problem.  It looks like you have tried valiantly to remove skewness.  Perhaps the distributions are reasonably symmetric  but with unequal variances.  If that is the case you ask yourself why.  Sometimes it is due to there being a mixture of distributions with similar means but differing variance.  Take a look at group membership to see if subgroups are present and are the cause.  The presence and nature of the subgroups might (or might not) be of great interest.  

A passing comment.  I am suspicious of any test for causality.  To me causality comes from the exchangability of observations and is a characteristic of the design.  I haven't read the H & H paper so I may be wrong.

When using kernel estimates of the distribution functions instead of unknown type of some standard (a priori chosen) ones, you can treat some heteroscedastic data by using kernels provided with individual (estimated) scale parameters to get probability distributions of such data.

http://www.math.gnostics.com

there is an alternative way to deal with heteroskedasticity you use (quqntile regression) or also called median regression instead of mean regression (OLS) median regression is robust  against heteroskedasticity you have no need any transformation 

Log transformation method, White's Heteroscedasticity consistency standard errors, weighted least sqaures etc

do LBNN or MPV methods

Some ideas may be useful:

1. Transforming the data into logs, that has the effect of reducing the effect of large errors relative to small ones and reducing the extreme value.

2. Consider the estimated/Feasible GLS/WLS methods

3. Using heteroscedasticity robust standard errors in which correct for the problem by enlarging the standard errors relative to what they would have been for the situation where the error variance is positively related to one of the explanatory variables, for example, you can consider the White correction form Huber-White or HAC consistent variance from Newey-West. Good luck.