Linear model Heteroscedasticity I have the following linear model:


To address the residuals heteroscedasticity I have tried to apply a log transformation on the dependent variable as $\log(Y + 1)$ but I still see the same fan out effect on the residuals. The DV values are relatively small so the +1 constant addition before taking the log is probably not appropriate in this case.
> summary(Y)
Min.   :-0.0005647  
1st Qu.: 0.0001066  
Median : 0.0003060  
Mean   : 0.0004617  
3rd Qu.: 0.0006333  
Max.   : 0.0105730  
NA's   :30.0000000

How can I transform the variables to improve the prediction error and variance, particularly for the far right fitted values?
 A: You would want to try Box-Cox transformation. It is a version of a power transformation:
$$ y \mapsto \left\{ \begin{eqnarray} \frac{y^\lambda-1}{\lambda (\dot y)^{\lambda-1}}, & \lambda \neq 0 \\ \dot y \ln y, & \lambda = 0 \end{eqnarray} \right.
$$
where $\dot y$ is the geometric mean of the data. When used as a transformation of the response variable, its nominal role is to make the data closer to the normal distribution, and skewness is the leading reason why the data may look non-normal. My gut feeling with your scatterplot is that it needs to be applied to (some of) the explanatory and the response variables.
Some earlier discussions include What other normalizing transformations are commonly used beyond the common ones like square root, log, etc.? and How should I transform non-negative data including zeros?. You can find R code following How to search for a statistical procedure in R?
Econometricians stopped bothering about heteroskedasticity after seminal work of Halbert White (1980) on setting up inferential procedures robust to heteroskedasticity (which in fact just retold the earlier story by a statistician F. Eicker (1967)). See Wikipedia page that I just rewrote.
A: What is your goal? We know that heteroskedasticity does not bias our coefficient estimates; it only makes our standard errors incorrect. Hence, if you only care about the fit of the model, then heteroskedasticity doesn't matter.
You can get a more efficient model (i.e., one with smaller standard errors) if you use weighted least squares. In this case, you need to estimate the variance for each observation and weight each observation by the inverse of that observation-specific variance (in the case of the weights argument to lm). This estimation procedure changes your estimates.
Alternatively, to correct the standard errors for heteroskedasticity without changing your estimates, you can use robust standard errors. For an R application, see the package sandwich.
Using the log transformation can be a good approach to correct for heteroskedasticity, but only if all your values are positive and the new model provides a reasonable interpretation relative to the question that you are asking.
A: There is a very simple solution to heteroskedasticity issue associated with dependent variables within time series data.  I don't know if this is applicable to your dependent variable.  Assuming it is, instead of using nominal Y change it to % change in Y from the current period over the prior period.  For instance, let's say your nominal Y is GDP of $14 trillion in the most current period.  Instead, compute the change in GDP over the most recent period (let's say 2.5%).  
A nominal time series always grows and is always heteroskedastic (the variance of the error grows over time because the values grow). 
A % change series is typically homoskedastic because the dependent variable is pretty much stationary.   
