What to do with p-values when standard errors are obviously biased What am I supposed to do when I want to interpret significances, although I know that standard errors are biased because of wrong error term assumptions? I know that there is the possibility to use White estimators, weighted OLS. But my prof told me not to do so.
Maybe some extra information:
1.) I am analyzing an OLS with a whole bunch of dummy variables.
3.) the assumption of normal distributed error terms is wrong (they are t-distributed) and heteroskedasticity occurs.
4.) I am doing cross sectional analysis
5.) I have a lot of Interaction in the model. And most of the variables are non significant (p-values around 0.8, so that the coefficients are close to zero). My prof doesn't want me to get rid of these variables, although they are not significant (He said that this is not a good way, because stepwise elimination is trouble because of deleting wrong variables and choosing the right criteria).
On the one hand I understand why there is no way to interpret the significances. But on the other hand it makes interpreting not easier. Sure I can change the model, but I have to do OLS first, before I am allowed to switch!
 A: You can't interpret the $p$-values. The long-tailed errors you're describing often act to underestimate the standard errors, making your $p$-values too small (not to mention that $\hat{\beta}$ isn't normally distributed in finite samples). I suggest a non-parametric bootstrap so you can characterize the sampling distribution of your coefficient estimates without making an unwarranted assumptions about the error distribution. 
A: Regarding point 5.):
I don't know exactly what you're working on, but generally, keeping nonsignificant terms (when they're not of genuine prior interest) is also not good practise. It leads to overfitting and meaningless results. 
You should consider potentially important interactions, but if they aren't significant, they need to go. Although you should keep nonsignificant main effects if they're involved in significant interactions.
There are less subjective alternatives to stepwise deletion in any case, like searches for the most parsimonious model given a set of variables. In R, packages like AICcmodavg, MuMIn, or glmulti might be able to help you with that.
There are some good papers out there on model selection. If you're a biologist, try this one: http://faculty.washington.edu/skalski/classes/QERM597/papers/Johnson%20and%20Omland.pdf
