Consider the case of time fixed effects only, i.e., Yit = ?0 + ?1Xit + ?3St + uit,

First replace ?0 + ?3St with ?t. Next show the relationship between the ?t and ?t in the following equation

Yit = ?0 + ?1Xit + ?2B2t + ... + ?TBTt + uit,

where each of the binary variables B2, …, BT indicates a different time period. Explain in words why the two equations are the same. Finally show why there is perfect multicollinearity if you add another binary variable B1. What is the intuition behind the fact that the OLS estimator does not exist in this case? Would that also be the case if you dropped the intercept?
What will be an ideal response?

Answer: Yit = β1Xit + φt + uit. The relationship is φ1 = β0, and φt = β0 + δt for t ≥ 2. Consider time period t, then the population regression line for that period is φt + β1Xit, with β1 being the same for all time periods, but the intercept varying from time period to time period. The variation of the intercept comes from factors which are common to all entities in a given time period, i.e., the St. The same role is played by δ2, … δT, since B2t …BTt are only different from zero during one period. There is perfect multicollinearity if one of the regressors can be expressed as a linear combination of the other regressors. Define B0t as a variable that equals one for all period. In that case, the previous regression can be rewritten as

Yit = β0 B0t + β1Xit + δ2B2t + ... + δTBTt + uit.

Adding B1 with a coefficient here results in

Yit = β0 B0t + β1Xit + δ1B1t + δ2B2t ... + δTBTt + uit.

But B0t = B1t + B2t + ... + BTt, and hence there is perfect multicollinearity. Intuitively, whenever any one of the binary variable equals one in a given period, so does the constant. Hence the coefficient of that variable cannot pick up a separate effect from the data. Dropping the intercept from the regression eliminates the problem.