How to present a empirical study when using econometric models?

Question

I've got a (probably easy) question in how to handle empirical studies, when there are a lot of effects involved. I have a whole bunch of variables and I'd like to analyze just a few of them. But the problem is, that the model is wrong... So standard errors and coeficients itself are probably biased and so t-statistics are as well. So general: Everything is wrong. Pretty frustrating task to find out how to handle this problem, when it is not possible to say what coeficeints have a clear influence on $y$. What would you do in this case? It's possible to compact the coeficients, but the problem is still present. Do you have some experiences how to handle this problem? Or is there anyone who knows a good paper where it's been discussed? Fyi: I'm going to do Cross Validation afterwards, to compare models... But it's still required to make an analysis of the estimation before looking which model is good. And I'm bounded to do OLS, before looking for better models. The general question is: How are other studies dealing with biased std. errors or coefficients? Please help :(

Edit: I'm analyzing a whole bunch of effects on wage. Thererefor I've a lot of effects. I know that heteroskedasticity occurs, wage is skewed and the sample size is relatively small. I'm not interested in changing the model, since I've to do OLS without transforming variables or s.th. Just the regular OLS. Unfortunately I don't really know how to interpret all the effects, when I can't get rid of the non significant ones, because significance isn't good defined (because of bias). Is there a theory that says in general:"Although bias occurs you can assume that the effects with high significances are more clear different from 0, than other effects that are not significant?"

Answer 1

You can always interpret OLS as estimating the best linear approximation to the conditional expectation of the outcome given your explanatory variables. With this interpretation OLS is never biased. The downside is that the OLS estimates only describe the relationship among variables in your data. In particular, the OLS estimates don't necessarily tell you anything about causality. In your example, you could say something like, "the coefficient on education is positive and significant, suggesting that education might increase wages. However, this positive relationship could also be due to an omitted variable, such as intelligence or work ethic, which could be increase both education and wages. To account for this possibility, I <estimate some other model>."

Even while interpreting OLS as only approximating a conditional expectation, you do need to worry about estimating the standard errors correctly. With heteroskedastic and independent observations, the variance of OLS coefficients is consistently estimated by:

$$ (X'X)^{-1} (\sum x_i'x_i \hat{\epsilon}_i^2 ) (X'X)^{-1} $$

If you have dependent data, there are other ways to estimate the standard errors.