(Requires Matrix Algebra)The Population Multiple Regression Model Can Be Written $$\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$$

Essay

(Requires Matrix Algebra)The population multiple regression model can be written in
matrix form as $$\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$$ Where
$$\boldsymbol { Y } = \left( \begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\\vdots \\Y _ { n }\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l } u _ { 1 } \\u _ { 2 } \\\vdots \\u _ { n }\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c c c c c c c } 1 & X _ { 11 } & \cdots & X _ { k 1 } & W _ { 11 } & \cdots & W _ { r 1 } \\1 & X _ { 12 } & \cdots & X _ { k 2 } & W _ { 12 } & \cdots & W _ { r 2 } \\\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\1 & X _ { 1 n } & \cdots & X _ { k n } & W _ { 1 n } & \cdots & W _ { r n }\end{array} \right) \text { and } \beta = \left( \begin{array} { l } \beta _ { 0 } \\\beta _ { 1 } \\\vdots \\\beta _ { k }\end{array} \right)$$ Note that the X matrix contains both k endogenous regressors and (r +1)included
exogenous regressors (the constant is obviously exogenous).
The instrumental variable estimator for the overidentified case is $$\hat { \beta } ^ { V } = \left[ X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } X \right] ^ { - 1 } X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } Y ,$$
where $\boldsymbol { Z }$ is a matrix, which contains two types of variables: first the $r$ included exogenous regressors plus the constant, and second, $m$ instrumental variables.
$$Z = \left( \begin{array} { c c c c c c c } 1 & Z _ { 11 } & \cdots & Z _ { m 1 } & W _ { 11 } & \cdots & W _ { r 1 } \\1 & Z _ { 12 } & \cdots & Z _ { m 2 } & W _ { 12 } & \cdots & W _ { r 2 } \\\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\1 & Z _ { 1 n } & \cdots & Z _ { m n } & W _ { 1 n } & \cdots & W _ { m }\end{array} \right)$$
It is of order $\mathrm { n } \times ( \mathrm { m } + \mathrm { r } + 1 )$ .
For this estimator to exist, both $\left( Z ^ { \prime } Z \right)$ and $\left[ X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } X \right]$ must be invertible. State the conditions under which this will be the case and relate them to the degree of overidentification.