(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?

The Fixed Effects Regression Model

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