(Requires Matrix Algebra)Consider the Time and Entity Fixed Effect Model

Essay

(Requires Matrix Algebra)Consider the time and entity fixed effect model with a single
explanatory variable $$Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \gamma _ { 2 } D 2 _ { i } + \ldots + \gamma _ { n } D n _ { i } + \delta _ { 2 } B 2 _ { t } + \ldots + \delta _ { T } B T _ { t } + u _ { i t }$$ For the case of $n = 4$ and $T = 3$ , write this model in the form $\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$ , where, in general,
$$\boldsymbol { Y } = \left( \begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\\vdots \\Y _ { n }\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l } u _ { 1 } \\u _ { 2 } \\\vdots \\u _ { n }\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c c c c } 1 & X _ { 11 } & \cdots & X _ { k 1 } \\1 & X _ { 12 } & \cdots & X _ { k 2 } \\\vdots & \vdots & \ddots & \vdots \\1 & X _ { 1 n } & \cdots & X _ { k n }\end{array} \right) = \left( \begin{array} { l } \boldsymbol { X } _ { 1 } ^ { \prime } \\\boldsymbol { X } _ { 2 } ^ { \prime } \\\vdots \\\boldsymbol { X } _ { n } ^ { \prime }\end{array} \right) \text {, and } \boldsymbol { \beta } = \left( \begin{array} { l } \beta _ { 0 } \\\beta _ { 1 } \\\vdots \\\beta _ { k }\end{array} \right)$$ How would the $\boldsymbol { X }$ matrix change if you added two binary variables, $D 1$ and $B 1$ ? Demonstrate that in this case the columns of the $\boldsymbol { X }$ matrix are not independent. Finally show that elimination of one of the two variables is not sufficient to get rid of the multicollinearity problem. In terms of the OLS estimator, $\hat { \boldsymbol { \beta } } = \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { Y }$ , why does perfect multicollinearity create a problem?