In a linear regression $Y = X\beta + \epsilon$ with unknown error variance $\sigma^2$, we test $H_0: L\beta = q$ where $L$ is a $k \times p$ matrix. What is the distribution of the test statistic $F = \frac{(L\hat{\beta}-q)^T [L(X^T X)^{-1} L^T]^{-1} (L\hat{\beta}-q)}{k \hat{\sigma}^2}$?