# Relationship between Conditional Mean and Dummy Variables in the presence of Additional Regressors

Consider the following regression model$$y_{it}=\beta_{1}M_{i}+\beta_{2}F_{i}+x_{it}'\gamma+\epsilon_{it}$$ where the LHS is some individual specific, time varying regressand the RHS variables consist of a vector of covariates $x_{it}'$ and 2 dummy variables for the sex of the individual. Notice that I have not included a constant to prevent multicollinearity. It is clear that if $x_{it}$ was not present, an OLS regression of the LHS on the RHS would result in the estiamtes representing conditional means by groups. For instance:$$\hat{\beta_{1}}=E[y_{it}|M=1]$$

This can be readily done by hand. My question is how does the inclusion of $x_{it}$ affect the procedure/interpretation? What would be the interpretation of $\beta_{1}$ in this case? Thanks!

• Are you allowing for the possibility that $M$ and $F$ are time-varying (they carry a $t$ index) :-)? – Christoph Hanck Jan 8 '16 at 16:02
• @ChinG I am not sure why you decide to include M and not to include B0. "F" is a dummy and equals 1 if "i" is female, 0 otherwise (i.e., a man). I do not know what you are studying, but usually you should include the constant and omit M (which I guess is the dummy for being male), or F. If you did so, the costant represented the average value of Y through time and across male "i"'s. Finally, when you exclude the constant, you assume that the value of Y is 0 when everything else is 0...on average this is true if the error term is normally distributed, but still, is this what you want to do? – Fuca26 Jan 8 '16 at 17:20
• @ChinG More importantly. You include in your regression M and F, so I ask you this: do you have in your dataset one or more "i"'s that are both female and male? that's a real question, I cannot exclude this possibility apriori, since I do not know what you are studying – Fuca26 Jan 8 '16 at 17:30