How to test hypothesis on a fraction of two coefficients?

Question

Say you have an equation:

$$y = b_0 + b_1X_1 +b_2X_2$$

How would you test the hypothesis that $\frac{b1}{b2} = q$?

Answer 1

You can linearize your hypothesis and conduct an $F(r,n-3)$ test, where $r$ is the number of restrictions (in your case it's 1) and $(n-3)$ is the degrees of freedom (which is the number of observations less 3 variables ($x_1$, $x_2$, and constant). You can rewrite your hypothesis as $b_1-b_2q=0$. Then you test that null. For example, in Stata with $q=2$:

reg y x1 x2    
test x1-q*x2

This will give you the $F$-statistic and its $p$-value. You reject the null hypothesis when the p-value is less than the significance level, usually .05 or .01.

Alternatively, you can test the non-linear hypothesis using a Wald test. This has dubious small sample properties and can be sensitive to the way the restrictions are written. In Stata, this will be

reg y x1 x2    
testnl -b[x1]/_b[x2]=2

I would do both, and see if they agree.