Zero conditional mean assumption How can I check the zero conditional mean assumption for a multiple linear regression? I read that when the assumption is violated, your model is misspecified. And your estimators are biased. So we need to check for it, but I'm completely oblivious as to how this is done.
 A: Edit: I assume you mean the conditional mean of the errors is zero. 
Imagine if the errors had a common nonzero mean - $\mu_\varepsilon$, say, and you fitted a least square model. What would happen? It would be absorbed by the constant, and the residuals would on average be zero. So you can't test whether the residuals have a common mean that's not zero.
What you can check is whether the residuals (and by implication the errors that they estimate) have constant mean; on average they're still zero, but conditionally they may have means some distance from zero.
The usual way to check that is a plot of residuals against the predictor(s), or if there are more than a couple of predictors, at least against fitted values. Here's the first diagnostic plot of the ones R will give by default when you plot the result of a regression:
plot(lm(dist~speed,data=cars))


It has a loess smooth superimposed, but without the curve you can still discern that the points tend to sit above the zero line at each end and perhaps below it in the middle. 
You can immediately see that the linearity assumption is somewhat suspect, and that perhaps some curved relationship is present.
A: In addition to what @glen_b said, there is a test for missspecifications (in fact there are many) one well known is the Ramsey RESET test. 
Run the original model, and obtain the fitted values. Add them along with fitted values squared and cubed to original model and test for joint significance (f-test). If you reject it implies you have neglected no linearity (specification problem).
No test can tell you what to do, and even if you do not reject the null there are no guarantees. 
Also another important problem is omitted variables, in fact this is even more of a problem since there is no test (you cannot test what you do not observe). Basiaclly it comes down intuition. Unless you have data from an experiment, it will almost always be the case that you have omitted variable bias, and that you model is biased since the mean independence assumption hinges of this. 
A: Test the model without constant. If the resulted equation has beta, r-squared and adjusted r-squared similar to those from the initially chosen model (i.e. model with constant) this means that the model has 0 mean error. If not, means not. (Brooks, Introductory Econometrics, page 131)
