Consider the Case of Time Fixed Effects Only, I $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \beta _ { 3 } S _ { t } + u _ { i t } ,$$

Essay

Consider the case of time fixed effects only, i.e., $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \beta _ { 3 } S _ { t } + u _ { i t } ,$$
First replace $\beta _ { 0 } + \beta _ { 3 } S _ { t }$ with $\phi _ { t }$ . Next show the relationship between the $\phi _ { t }$ and $\delta _ { i }$ in the following equation $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \delta _ { 2 } B 2 _ { t } + \ldots + \delta _ { T } B T _ { t } + u _ { i t }$$ 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?