The $F$ -Statistic With $q = 2$ Restrictions When Testing for the Restrictions

Essay

The $F$ -statistic with $q = 2$ restrictions when testing for the restrictions $\beta _ { 1 } = 0$ and $\beta _ { 2 } = 0$ is given by the following formula:
$$F = \frac { 1 } { 2 } \left( \frac { t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } - 2 \hat { \rho } _ { t _ { 1 } , t _ { 2 } } t _ { 1 } t } { 1 - \hat { \rho } _ { t _ { 1 } , t _ { 2 } } ^ { 2 } } \right)$$ Discuss how this formula can be understood intuitively.