How do I obtain a correlation between two variables? Can I differentiate a quadratic regression model formula with the form
$$Y_i=\beta_0+\beta_1 X_{1i} +\beta_2 X_{1i}^2+\epsilon_i$$
(added to this formula would be other control variables e.g dummy variables) to obtain the correlation between X and Y holding everything else constant? Also, if I were to add a female dummy variable, how would the correlation of X and Y differ between men and women? Or would the correlation be the same?
For the above equation is differentiable, and hence the partial derivatives will exist:
$\frac{\partial Y}{\partial X_2}= 2\beta_2X_2$.
Adding a dummy variable should not affect the above derivative for all practical purposes - the function will be differentiable a.e.
(note that you can't differentiate with respect to a dummy variable though)
Edit:
After the question was modified, the new expression of the partial derivative would be:
$\frac{\partial Y}{\partial X_1}=\beta_1+ 2\beta_2X_1$.