Trouble fitting a simple linear regression I have been trying to do a simple linear regression of x3=Weeks Claimed against x4=Weeks compensated.  I am having problems because my residual standard error is very high and also my residuals are not homoscedastic.  If anyone could provide some guidance on how to fix these problems I would really appreciate it.
Here is the output from R.



Thanks!!
 A: There's a clear division of your data into two groups - the small group at the low end of weeks claimed for whom weeks compensated exceeds weeks claimed, and everyone else, for which weeks compensated is less than weeks claimed (to see this neat separation, draw in the y=x line on your first plot; the line goes neatly between the two point clouds).
If you have some indicator variable which explains/describes/predicts the situation in which people can be compensated for more weeks than they claim (i.e. that identifies which people it will be without looking at their weeks compensated), it should neatly serve as a predictor in your regression to pull those people's residuals nicely in alignment. Failing that, some good proxy for whatever is causing that should help.

Edit:
I see from comments that these are observations from early in the study. If you can identify by time when people stopped being in the situation of being paid more weeks than claimed (more specifically identify when the cause of that ceased to be present), that would serve as the basis for a suitable dummy.
It may be an artifact of misalignment of what's included in the two counts (if you claim before you are paid, and no claims or payments before a particular date are considered), or it may be a change in policy or procedures (in which case the date should be known).
A: Your standard errors are not large relative to the size of the variables (in the 10,000 - 50,000 range). The main issue is the heteroscedasticity. This is being driven by the small set of points at the low end of Weeks Claimed. This does not seem like a statistical problem, but a need to further understand the problem. I would identify these observations and see whether they are different and whether that difference can be captured by some other variables in your data.
On a different note, are your variables well coded? 50,000 weeks seem like a long time (like  millennium). 
A: Eventually, you could also run a quantile regression in order to understand what happens at some specific quantiles. This should help to better understand the phenomenon you are studying.
