Model with complications A regular linear regression model is $y = c'x + \varepsilon$, where $c$ are unknown coefficients and $\varepsilon$ is Gaussian noise with zero mean and constant variance. I am building a model where the error term, $\varepsilon$, has two complications:


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*Its distribution is not normal.

*The error variance is not constant.


I know that the first issue can be addressed by some linear regression models, while the second issue can be addressed by linear regression (e.g., Tofallis, C (2008), "Least Squares Percentage Regression"). But I have never seen a model that would address both issues at the same time.  
 A: Sandwich based robust error estimation handles both heteroscedasticity and non-normal error distribution asymptotically. That also happens to mean that you get approximately valid inference in relatively samples.
One criticism might be that a method which is so robust must be of low power. Generally, not as true as one might think. But...could you make weaker or different assumptions about the distribution of the errors? For instance, instead of being normal, perhaps they could come from a general family of errors inclusive of the normal distribution like a t-distribution family or a 3 parameter normal family. This blurs the lines between classical inference, which in small samples relies upon strong distributional assumptions, and robust error estimation which is pretty much bullet proof in relatively large samples.
An example of blurring these lines for a hybrid approach, is maximizing a conditional likelihood that allows for platykurtic error distributions like a $t$-distribution with relatively low degrees of freedom. For the case of heteroscedasticity, you can inspect variograms to model the errors as a function of the mean, such as with a mean-variance relationship that is linear (alternately consider a Poisson GLM with an identity link). 
A: Both heteroscedasticity and heavy-tailedness can be considered violations of the distributional assumptions of a standard linear model.  If the distribution is nonetheless symmetrical, and the relationship is between $x$ and $y$ is rectilinear, your model should not be biased.  Instead, interval estimates and inferences would be incorrect.  With enough data, they may be approximately right anyway.  Unfortunately, it is difficult to know how much data would be 'enough', and the amount may be prohibitively large without your awareness one way or the other.  Thus, you need methods that do not rely on the standard distributional assumptions.  @AdamO's suggestions are viable.  Two additional approaches jump to mind:  


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*You could bootstrap your model to get better confidence intervals and p-values.  The advantage here is that your model is otherwise similar (particularly with respect to interpretability).  The disadvantages are that you need enough data to adequately represent the population, and that this probably requires you to write original code (i.e., there may not be convenient routines already).  

*The ultimate distribution-free regression method is to use ordinal logistic regression.  Ordinal models do not make any assumptions about the conditional distribution, they only require that you can claim, say, that a $7$ is $>$ a $6$.  That is not very restrictive.  The upside is considerable robustness, and there will be convenient functions for this in your software of choice.  The downside is that OLR models tend to be hard to interpret.  

