How do you use a Log-linear model when you have negative Xs? I am trying to us a Log-linear model to derive an elasticity. However, some of my Xs are negative numbers. Being as the model relies on the natural log of the Ys and the Xs, how can this model work when the natural log can only be taken of X>0?
The elasticity of Y with respect to X is often estimated by running a regression like this:
$$ \ln Y = \alpha + \beta \ln X + error$$

However, this isn't always applicable because it is perfectly fine to ask what is the percent change in Y to a percent change in X even if X or Y are negative. There are other issues too, like the fact that the elasticity may not be constant over X. 

Let's go back to the textbook (Varian (1992), p. 235) for the definition of the elasticity of $Y$ with respect to $X$:
$$ \epsilon(Y,X) =\frac{\partial Y}{\partial X} \cdot \frac{X}{Y} $$
Which equals another common definition of the elasticity:
$$=\frac{\partial \ln Y}{\partial \ln X}$$
which we often use when estimating the elasticity with linear regressions. 

However, the advantage of the first form is that you don't have to use the log function to evaluate it. All you need is a local estimate of the derivative, X, and Y. 
One strategy would be just to do that. 

However, what you are doing could well be nonsensical because percentage changes are weird when working with negative values. For example, consider X going from -11 to -10. This is a 9.1 percent decrease in X ((-10-11)/-11). However, this is is an increase in X! Going from 11 to 10 is also a 9.1 percent decrease in X. What would be the economic conditions where we think going from -11 to -10 and +11 to +10 would have the same effect on demand?