Heteroscedastic data: make error term proportional to x or y? Suppose I have heteroscedastic data in which error terms increase for larger data points.
Assuming that either of these appear to fit the data well, which is the correct model to use, and why?
$Y = \beta X + \epsilon X$
or 
$Y = \beta X + \epsilon Y$
where in both cases $\epsilon$ comes from $\mathcal{N}(\mu=1,\sigma)$
For prediction the second model is clearly less useful as the value of $Y$ is unknown, but if we are doing inference is there any advantage to it - such as, for example, the ability to use the fitted value of $\sigma$ as a measure of model performance?
 A: Once you fit the model which assumes homoscedastic errors to your data, you can check the plot of residuals versus fitted values. If that plot exhibits the "funnel" effect (i.e., increased variability as you move along the horizontal axis corresponding to the fitted values), then you need to investigate what kind of "fix" will be required to address the presence of the "funnel" effect (which signals heteroscedastic model errors).
If your model only includes one predictor, X, which is continuous, you can update the model to allow the error variability to increase with X. If X is categorical, you can allow different variabilities across categories.
If your model includes multiple predictors, it might be easier to have the error variability depend on the fitted values. 
I have not seen any situations where the variability would depend on the values of Y and that's because usually we're aiming to model the conditional variability of Y given the predictor variables X, etc.  For this reason, it makes more sense have the variability depend on known things like X or the fitted values.
