Bias of OLS estimators due to exclusion of relevant variables and inclusion of irrelevant variables

In this paper I discuss three issues related to bias of OLS estimators in a general multivariate setting. First, I discuss the bias that arises from omitting relevant variables. I offer a geometric interpretation of such bias and derive sufficient conditions in terms of sign restrictions that allows us to determine the direction of bias. Second, I show that inclusion of some omitted variables will not necessarily reduce the magnitude of OVB as long as some others remain omitted. Third, I show that inclusion of irrelevant variables in a model with omitted variables can also have an impact on the bias of OLS estimators. I use the running example of a simple wage regression to illustrate my arguments.