Bootstrap hypothesis test for ratio of regression coefficients

I am interested in estimating the ratio of two regression coefficients: $$\beta=\frac{\beta_{Y|G}}{\beta_{X|G}}$$

The numerator is the linear regression coefficient of a variable G on Y and the denominator is the linear regression coefficient of a the same variable G on X.

I estimated this parameter by simply taking the ratio of the point estimates of the regression coefficients and I have used the bootstrap to compute 95% confidence intervals using the percentile method. Using this approach, I am able to gain some insight on the uncertainty of my estimate, but I would like to formulate a formal hypothesis test (and get a p-value).

Specifically, I am interested in computing a p-value for the hypothesis of $H_0:\beta=0$ and I would like to achieve this using the bootstrap. I have trouble understanding how to define a valid statistic whose null distribution I can sample from using my bootstrap samples.

• I assume you mean $\beta = 1$, as $\beta = 0$ is, with a notably rare exception, the same as $\beta_{Y|G} = 0$?. – jbowman Apr 12 '17 at 19:30
• Can you clarify: Here $X$ and $Y$ are two different response variables in two different regression equations, and $G$ is a common predictor variable? Are all variables measured on exactly the same units? – kjetil b halvorsen Apr 12 '17 at 19:31
• In this case, the $\beta$ is the "ratio estimate" of the effect of X on Y using G as an instrumental variable. In other words, G is constructed so that it is a strong predictor of X and it is then used to infer the causal effect of X on Y. So the null hypothesis is truly of $\beta = 0$ even though this seems unintuitive. – legaultmarc Apr 12 '17 at 19:41
• @kjetilbhalvorsen's understanding of the context is correct: I am considering two distinct regression models that share a common predictor (G). In my case, Y and X are not in the same units. More background can be found here, in section 3.1. – legaultmarc Apr 12 '17 at 19:41