My professor said that a regression is biased only if there is a correlation between the independent variable and the omitted variable (cov(x,u)=0). But wouldn't the omitted variable also have to be correlated with the dependent variable for the regression to be biased?
Thank you in advance!
Omitted-variable bias in standard linear regression occurs, as you recognize, if the omitted predictor is both correlated with the included predictors and is related to outcome. Your professor's statement that "a regression is biased only if there is a correlation between the independent variable and the omitted variable" is technically correct. The claim wasn't "if and only if"; "only if" represents a necessary but not necessarily sufficient requirement. For linear regression the correlation with included predictors is necessary but not sufficient for omitted-variable bias in the model.
Your focus on relation with outcome, however, is well placed in general. Other types of regression like logistic regression or Cox regression in survival analysis can have omitted-variable bias if any predictor related to outcome is left out of the model. In such regressions correlation with included predictors is no longer necessary for omitted-variable bias.
