For the OLS estimator $\hat { \beta}$ = ( $X ^ { \prime }$ X)^-¹
$X ^ { \prime }$ Y to exist, X'X must be invertible. This is the case when X has full rank. What is the rank of a matrix? What is the rank of the product of two matrices? Is it possible that X could have rank n? What would be the rank of X'X in the case n<(k+1)? Explain intuitively why the OLS estimator does not exist in that situation.

Question

For the OLS estimator

\hat { \beta}

= (

X ^ { \prime }

X)^-¹

X ^ { \prime }

Y to exist, X'X must be invertible. This is the case when X has full rank. What is the rank of a matrix? What is the rank of the product of two matrices? Is it possible that X could have rank n? What would be the rank of X'X in the case n<(k+1)? Explain intuitively why the OLS estimator does not exist in that situation.

Quizplus · Accepted Answer

The Answer of For the OLS estimator $\hat { \beta}$ = ( $X ^ { \prime }$ X)^-¹ $X ^ { \prime }$ Y to exist, X'X must be invertible. This is the case when X has full rank. What is the rank of a matrix? What is the rank of the product of two matrices? Is it possible that X could have rank n? What would be the rank of X'X in the case n<(k+1)? Explain intuitively why the OLS estimator does not exist in that situation.

For the OLS Estimator β^\hat { \beta}β^​ = X′X ^ { \prime }X′

For the OLS Estimator $\hat { \beta}$ = $X ^ { \prime }$