# Fixed effects in treatment model

Hello in a simple treatment model

$$y=\beta_0+\beta_1w+e$$

where $w$ is unity for treatment by group. Should you add dummies for the groups to control for differences. I read somewhere that you should never but I cannot remember where or why.

• Some clarifications please: Are there two groups, one treated, one non-treated? Also, by "add dummies for the groups to control for differences", you mean various control variables that reflect the existence or not of binary characteristics in the two groups? For example,if the treatment is "vaccination", you ask whether you should include also a dummy to indicate male/female subject? – Alecos Papadopoulos Apr 2 '15 at 3:36
• I mean several groups some treated some not, however the treatment is not random and there are some group specific characteristics that may be correlated with the treatment and affect the potential outcomes. To control for these characteristics can you add binary dummies by group. – user4326 Apr 2 '15 at 3:42
• You should update the question, not just elaborate in the comments. – BKay Apr 2 '15 at 11:05

This sounds like the standard "drop one dummy" requirement when binary characteristics are included in linear regression - because otherwise, we will get perfect mutlicollinearity and no-solution.

Assume you have three subgroups, separated by age: Y(oung), A(dult), O(ld). You have reasons to believe that the effect of the treatment correlates with age-group, and you want to control for this association. If you include three dummies in the regression to that effect, you will create perfect mutlicollinearity - because if you sum these three columns of the regresor matrix row-by-row, you will get a series of Ones, which is already present in the matrix, since there exists already a constant term and so a regressor that consists of a series of Ones.

In such cases, we exclude from the regressor matrix any one of the three control dummies (for many reasons it is not a good idea to drop the constant term instead of one of the dummies). This has the effect that the interpretation of the results becomes conditional on the group whose dummy we excluded: if say we specify

$$y=\beta_0+\beta_1w + \gamma_1A + \gamma_2O+e$$

then $\gamma_1$ measures how much more (or less, if negative) being Adult affects the outcome, compared to being Young, and $\gamma_2$ analogously for Old.

Situations of perfect (or near) multicollinearity may nevertheless occur even in this approach, depending also on the nature and distribution of values of the dependent variable. If the dependent variable is also binary, such possibilities increase, since we may have phenomena of "complete separation/perfect prediction" and the like.

In general, including controls in treatment regressions is a standard approach, and very reasonable. Including dummies as controls, may have the above mentioned complications.

• So lets say everyone in a group has the same treatment then do you still include the dummies? – user4326 Apr 2 '15 at 4:27
• Once you go into specific cases, there is usually not a definite "yes" or "no", due to algebraic reasons and the way the estimators are computed. Sometimes, you have to run the model to check if such problems arise. I added a little something to the answer. – Alecos Papadopoulos Apr 2 '15 at 4:45