Over the years I've posted a number of times about various aspects of using dummy variables in regression models. You can use the "Search" window in the right sidebar of this page if want to take a look at those posts.
One of my earlier working papers on this topic has now been accepted for publication.
The paper is titled, "On the Inconsistency of Instrumental Variables Estimators for the Coefficients of Certain Dummy Variables". Here's the abstract:
"In this paper we consider the asymptotic properties of the Instrumental Variables (IV) estimator of the parameters in a linear regression model with some random regressors, and other regressors that are dummy variables. The latter have the special property that the number of non-zero values is fixed, and does not increase with the sample size. We prove that the IV estimator of the coefficient vector for the dummy variables is inconsistent, while that for the other regressors is weakly consistent under standard assumptions. However, the usual estimator for the asymptotic covariance matrix of the I.V. estimator for all of the coefficients retains its usual consistency. The t-test statistics for the dummy variable coefficients are still asymptotically standard normal, despite the inconsistency of the associated IV coefficient estimator. These results extend the earlier results of Hendry and Santos (2005), which relate to a fixed-regressor model, in which the dummy variables are non-zero for just a single observation, and OLS estimation is used".
You can download the final working paper version of the paper from here.
The paper will be appearing in an upcoming issue of Journal of Quantitative Economics.