For the OLS Estimator To Exist $X ^ { \prime } X ^ { \prime }$

question 26

Essay

For the OLS estimator $\hat { \boldsymbol { \beta } } = \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { Y }$ to exist, $X ^ { \prime } X ^ { \prime }$ 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 $X ^ { \prime } X$ in the case $n < ( k + 1 ) \text { ? }$ Explain intuitively why the OLS estimator does not exist in that situation.

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