Is it possible to calculate variable confidence intervals, conditional on $\hat{Y}$ to address heteroscedasticity? Estimating confidence intervals for non-normally distributed residuals can be accomplished using bootstrapping procedures, sandwich estimators or quantile regression.
But is there a way to calculate $\beta$ confidence intervals given the estimated value? That is, for each $\hat{Y}$ we know the residuals variance for this predicted value region, so rather than applying the same (robust) intervals to all predicted values, isn't it possible to adjust the intervals conditional to $\hat{Y}$?
In other words, the residuals could be split in different bins with their own distribution as the value of $\hat{Y}$ increases, then specific confidence intervals could be applied to each bin.
UPDATE
I'm going to elaborate a bit more because I'm surprised there isn't a simple answer to this question (or an obvious flaw). Let's take the residuals analysis from this thread:

Here we have a loss of predictive power at the sides of the chart, that is for $\hat{Y} ~ (0.694, 2.23], \hat{Y} ~ (3.51, 9.53]$. Clearly the confidence intervals are not uniform across all values of $\hat{Y}$.
Wouldn't make sense to adjust the $\beta$ confidence intervals given $\hat{Y}$?
 A: Perhaps you could take a look at the GAMLSS model of Rigby and Stasinopoulos. As in regression, this model allows to model how E[Y] varies as a function of predictor values. In addition, GAMLSS can model the spread of the residuals by a second, independent function of predictors. If needed, you could even model third and fourth moments of a distribution to account for skewness and kurtosis in the same way. The GAMLSS package in R has around 70 built-in distributions, so the model is extremely flexible, but still easy to use.
A: Have you looked at the Delta Method?  It relates change in variance of one variable to another through the derivative (or partial derivative for multivariate methods) using the Taylor series.
http://www.stanford.edu/class/cme308/notes/TaylorAppDeltaMethod.pdf
The truth is that you can relate any scaled value of the standard deviation of one variable to another.  The constant transports through the integral sign.
In light of the "pseudo-sigma" this suggests that quantiles can also be transported, as long as the Taylor Series approximation is valid.
A: I don't think you need different confidence intervals for the model parameters at different points. Your estimates for the model parameters will depend on the whole set of data and on the assumptions made for the distribution of the errors, but they will be one set of estimates for the entire model (independent of location in parameter space). Clearly, in this case you should not use the identically-independently-distributed assumption (you do not have 'identically').
Your tolerance intervals for the predictions would be different at different points. This is not simple to do and I am not sure this is covered in standard software packages. You may want to look at: Tolerance Intervals in a Heteroscedastic Linear Regression Context with Applications to Aerospace Equipment Surveillance, International Journal of Quality, Statistics, and Reliability
Volume 2009 (2009), Article ID 126283, 8 pages
http://dx.doi.org/10.1155/2009/126283 
