(Requires Matrix Algebra)The Population Multiple Regression Model Can Be Written $\left(\begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\O \\Y _ { n }\end{array}\right)$

Essay

(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as
Y = Xβ + U
where
Y = $\left(\begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\O \\Y _ { n }\end{array}\right)$ , U = $\left( \begin{array} { c } u _ { 1 } \\u _ { 2 } \\\mathrm { O } \\u _ { n }\end{array} \right)$ , X = $\left(\begin{array} { l } 1 &X_{11}& \mathrm{~N} &X_{k 1}& W_{11} &\mathrm{~N} &W_{r 1}\\1 &X_{12}& \mathrm{~N}& X_{k 2} &W_{12}& \mathrm{~N}& W_{r 2}\\O&O&R&O&O&R&O\\1 &X_{1 n}& \mathrm{~N}& X_{k n} &W_{1 n} &\mathrm{~N}& W_{m}\\\end{array}\right)$ , and β = $\left( \begin{array} { l } \beta _ { 0 } \\\beta _ { 1 } \\\mathrm { O } \\\beta _ { k }\end{array} \right)$ 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 $\hat { \beta } ^ { I V } = \left[ X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } X \right] ^ { - 1 } X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } Y,$ where 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 = $\left(\begin{array} { l } 1 &Z_{11} &\mathrm{~N}& Z_{m 1}& W_{11}& \mathrm{~N}& W_{r 1}\\1 &Z_{12}& \mathrm{~N}& Z_{m 2} &W_{12}& \mathrm{~N} &W_{r 2}\\O&O&R&O &O& R &O \\1 &Z_{1 n }&\mathrm{~N}& Z_{mn }& W_{1 n}& \mathrm{~N} &W_{m}\end{array}\right)$ It is of order n × (m+r+1).
For this estimator to exist, both ( $Z ^ { \prime }$ Z)and [ $X ^ { \prime }$ Z( $Z ^ { \prime }$ Z)^-1
$Z ^ { \prime }$ X] must be invertible. State the conditions under which this will be the case and relate them to the degree of overidentification.