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

question 13

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?

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Definitions:

Adrenal Cortex

The outer region of the adrenal glands responsible for producing steroid hormones like cortisol and aldosterone.

Pineal Gland

A small endocrine gland in the brain that produces and regulates some hormones, including melatonin.

Parathyroid Glands

Small endocrine glands in the neck that produce parathyroid hormone (PTH), which regulates calcium levels in the blood and bone metabolism.

Adult Thyroid

The thyroid gland in its mature state, responsible for regulating metabolism, energy generation, and hormonal balance in the body.