How would one design a regression where the dependent variable is measured in degrees on a circle?
The dependent variable is on the range [0, 360), and the independent variables are demographic characteristics on each observation. The goal is to see how (or if) the independent variables affect the direction. The basic problem is that 355 is closer to 5, than 5 is to 30.
You might think about orthogonal projection of the response vector $$ y=\begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} $$ onto the column space of the "design" matrix $$ X = \begin{bmatrix} 1 & \cos\theta_1 & \sin\theta_1 \\ \vdots & \vdots & \vdots \\ 1 & \cos\theta_n & \sin\theta_n \end{bmatrix}. $$ The projection, which is the vector of fitted values, is $\hat y = X(X^TX)^{-1}X^T y$. The coefficients being estimated are the three components of the $3\times1$ matrix $\hat\beta=(X^T X)^{-1}X^T y$.
The most natural hypothesis test to do would probably be of the null hypothesis that the second and third components of $\beta$ are $0$. I probably wouldn't even consider the two components separately, but only together. The F-test would have $2$ degrees of freedom in the numerator and $n-3$ degrees of freedom in the denominator. The F statistic would be $$ F = \frac{\|\hat y - \bar y\mathbf 1_n\|^2/2}{\|y-\hat y\|^2/(n-3)} $$ (where $\mathbf 1_n$ is an $n\times 1$ matrix in which every entry is $1$).
However, I my emend any of this after knowing something about your data.