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

question 6

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.

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Definitions:

Stopping Distance

The total distance a vehicle travels from the point when the driver first perceives a need to stop, to when the vehicle comes to a complete halt.

Inversely Proportional

A relationship between two variables where their product is constant. When one variable increases, the other decreases proportionally.

Daily Demand

The total quantity of a good or service that consumers want to purchase over the course of a single day.

Radical Expression

An expression involving roots, such as square roots or cube roots, of numbers.