Consider the case of time fixed effects only, i.e.,
Y_it = β₀ + β₁X_it + β₃S_t + u_it,
First replace β₀ + β₃S_t with φ_t. Next show the relationship between the φ_t and δ_t in the following equation
Y_it = β₀ + β₁X_it + δ₂B2_t + ... + δ_TBT_t + u_it,
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?

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The Answer of Consider the case of time fixed effects only, i.e., Y_it = β₀ + β₁X_it + β₃S_t + u_it, First replace β₀ + β₃S_t with φ_t. Next show the relationship between the φ_t and δ_t in the following equation Y_it = β₀ + β₁X_it + δ₂B2_t + ... + δ_TBT_t + u_it, 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?

Consider the Case of Time Fixed Effects Only, I