Simultaneous heteroscedasticity and heavy tails in a regression model

I'm trying to create a prediction model using regression. This is the diagnostic plot for the model that I get from using lm() in R:

What I read from the Q-Q plot is that the residuals have a heavy-tailed distribution, and the Residuals vs Fitted plot seems to suggest that the variance of the residuals is not constant. I can tame the heavy tails of the residuals by using a robust model:

fitRobust = rlm(formula, method = "MM", data = myData)

But that's where things come to a stop. The robust model weighs several points 0. After I remove those points, this is how the residuals and the fitted values of the robust model look like:

The heteroscedasticity seems to be still there. Using

logtrans(model, alpha)

from the MASS package, I tried to find an $\alpha$ such that

rlm(formula, method = "MM")

with formula being $\log(Y + \alpha) \sim X_1+\cdots+X_n$ has residuals with constant variance. Once I find the $\alpha$, the resulting robust model obtained for the above formula has the following Residuals vs Fitted plot:

It looks to me as if the residuals still do not have constant variance. I've tried other transformations of response (including Box-Cox), but they don't seem like an improvement either. I am not even sure that the second stage of what I'm doing (i.e. finding a transformation of the response in a robust model) is supported by any theory. I'd very much appreciate any comments, thoughts, or suggestions.

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I think you're being a bit picky about the non-constant variance. It appears ok to me. What is the purpose of the regression? Explanation/hypothesis testing or prediction? –  probabilityislogic Sep 25 '12 at 7:38
@probabilityislogic, thank you for your comment. I very much appreciate it. My goal is prediction. You're right. I'm probably being too picky. Is there a measure for heteroscedasticity that I can look at? I thought of plotting variance vs fitted values but there aren't many points for each predicted value to calculate variance. I'm also curious to understand what is the solution to this problem in general. Are Box-Cox and log transforms applicable to robust models as well? –  user765195 Sep 25 '12 at 12:32
You can do pairwise test for equality of variances using the F test for a model with Gaussian error terms or if they have a non-Gaussian distribution there are robust tests for dispersion such as Levene's test. –  Michael Chernick Sep 25 '12 at 15:30
Thank you @MichaelChernick. I very much appreciate your comment. I finally used Koenker's generalization of Breusch-Pagan's test for heteroscedasticity as implemented in lmtest package in R (hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/lmtest/html/…). –  user765195 Sep 26 '12 at 3:15