Avoiding multicollinearity in a multi group diff-in-diff model Suppose I have an experiment where I have three groups $A, B , C$, such that groups $A$ and $B$ have specific (but different) characteristics and group $C$ is just a control group, with no special characteristics. We also have that $A, B$ and $C$ are disjoint groups.
Now, I want to find the effect of a treatment on group $A$ and on group $B$ (specifically, I want to find how the treatment affects a person, given that they have characteristic $A$ vs characteristic $B$).
If I specify the model as follows, I have multicollinearity:
$$Y_{it} = \alpha_0 + \beta_0 T_t + \beta_1 A_i + \beta_2 B_i + \beta_3 C_i + \gamma_1 (A_i\cdot T_t)+ \gamma_2 (B_i\cdot T_t)+ \gamma_3 (C_i\cdot T_t) + \epsilon_{it},$$
If I omit the $C_i$ dummy, I still have a multi collinearity if I include $C_i\cdot T_t$. How can I avoid such multicollinearity, and still specify a model that I can interpret meaningfully?
 A: Since each case belongs to exactly one of group A, B, or C, if the case doesn't belong to either A or B then it must belong to group C. So there are only 2 independent predictors among the three of A, B, and C. Choose 2 for your dummy variables; then your intercept will represent the value for the third group. Similar considerations for the interaction terms.
In response to comment: Note that if the values of $T$ start at 0 then all of the interaction terms have values of 0 at time $T = 0$, as they are products of group-membership dummies with time. Say that you choose to set A and B as your dummies. With the way you formulate the model, the intercept will be the value for Group C at time 0; it includes no interactions. Coefficients for A and B will represent differences from the value of Group C at time 0. The coefficient for $T$ will be the slope in time for Group C. The interaction terms with $T$ for Groups A and B will represent the differences in slope from that of Group C.
As $T$ represents time, make sure that you are handling the time series and potential repeated measures on the same individuals appropriately. Those are not always trivial tasks.
