(Requires Matrix Algebra)Consider the Time and Entity Fixed Effect Model $\gamma _ { 2 }$

Essay

(Requires Matrix Algebra)Consider the time and entity fixed effect model with a single explanatory variable
Y_it = ?₀ + ?₁X_it + $\gamma _ { 2 }$ D2_i + ... + $\gamma _ { n }$ Dn_i + ?₂B2_t + ... + ?_TBT_t + u_it,
For the case of n = 4 and T = 3, write this model in the form Y = X? + U, where, in general,
Y = $\left( \begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\Y _ { n }\end{array} \right)$ , U = $\left( \begin{array} { l } u _ { 1 } \\u _ { 2 } \\u _ { n }\end{array} \right)$ , X = $\begin{array} { l l l l } 1 & X _ { 11 } \ldots & X _ { k 1 } \\1 & X _ { 12 } \ldots & X _ { k 1 } \\1 & X _ { 1 n } \ldots & X _ { k n }\end{array}$ = $\left(\begin{array} { l } x _ { 1 } ^ { \prime } \\x _ { 2 } ^ { \prime } \\x _ { n } ^ { \prime }\end{array}\right)$ , and ? = $\begin{array} { l } \beta _ { 0 } \\\beta _ { 1 } \\\beta _ { k }\end{array}$ How would the X matrix change if you added two binary variables, D1 and B1? Demonstrate that in this case the columns of the 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, $\hat \beta$ = ( $X ^ { \prime }$ X)^-1
$X ^ { \prime }$ Y, why does perfect multicollinearity create a problem?