# Inference on ratio of coefficients

Question: How to construct standard errors and test statistics for a ratio of regression coefficients?

Background: I am running the same regression on two related outcomes $$Y_1 = \alpha_1 + \theta_1 X_1 + \beta 'X +\epsilon_1$$

$$Y_2 = \alpha_1 + \theta_2 X_1 + \beta 'X +\epsilon_2$$ (where $$\epsilon_1, \epsilon_2$$ are independent).

I am interested in the ratio of the coefficients from these two regressions. In particular, I want to test the following $$H_0: \frac{\theta_1}{\theta_2}=1, H_A: \frac{\theta_1}{\theta_2}\neq1$$ and also the one-sided version of this test for $$\frac{\theta_1}{\theta_2}>1$$.

• If, say, you know $\theta_2$ is positive, then your test can be re-expressed in terms of the difference $\theta_1-\theta_2,$ thereby reducing the question to an ordinary regression of $Y_1-Y_2$ against $(X_1,X)$ (presumably without intercept, since you posit the same intercept in both models). Do we need to be concerned, then, with the possibility that $\theta_2\le 0$?
– whuber
Mar 27, 2022 at 14:05
• Theoretically either $\theta_1 < 0$ and $\theta_2 < 0$ or $\theta_1> 0$ and $\theta_2 > 0$, so, yes the reformulation in terms of differences would work in my application. Mar 28, 2022 at 9:35

Assuming (as you do in a comment) that the $$\theta_i$$ have the same sign and $$\theta_2$$ is nonzero, the null hypothesis is algebraically equivalent to

$$H_0: \theta_1 = \theta_2$$

while the two alternative hypotheses are equivalent to

$$H_A:\theta_1\ne\theta_2\quad\text{and}\quad H_A^\prime:\theta_1\lt\theta_2\lt 0\text{ or } 0\lt \theta_2\lt\theta_1.$$

Since all the $$\epsilon_1$$ are independent of the $$\epsilon_2,$$ all the $$Y_1$$ responses are independent of the $$Y_2$$ responses and so (presuming the estimates are separately computed, one set for the $$Y_1$$ data and another set for the $$Y_2$$ data) the parameter estimates $$\hat\theta_i$$ are independent.

How you proceed depends on circumstances. To sketch the general approach, let's suppose you would use a $$t$$ test or $$Z$$ test in either regression alone. This means the combined information of assumptions and data is strong enough to suggest the sampling distributions of the $$\hat\theta_i$$ are approximately Normal with estimated variances $$\sigma_i^2$$ respectively. Consequently, the test statistic

$$\hat\theta = \hat\theta_1 - \hat\theta_2$$

is approximately Normal with sampling variance

$$\operatorname{Var}(\hat\theta) = \operatorname{Var}(\hat\theta_1) + \operatorname{Var}(\hat\theta_2) = \sigma_1^2 + \sigma_2^2.$$

You would therefore refer the test statistic

$$Z = \frac{\hat\theta_2 - \hat\theta_2}{\sqrt{\sigma_1^2 + \sigma_2^2}}$$

to the Standard Normal distribution with distribution function as $$\Phi.$$ The critical region for testing $$H_0$$ against $$H_A$$ for a test with confidence $$1-\alpha$$ therefore is

$$\mathcal{C}(\alpha) = (-\infty, \Phi^{-1}(\alpha/2)]\ \cup\ [\Phi^{-1}(1-\alpha/2),\infty).$$

Unless you're pretty sure what the common sign of the $$\theta_i$$ is, $$\mathcal{C}(\alpha)$$ would be the critical region for $$H_A^\prime,$$ too.
When both the coefficient estimates $$\hat\theta_i$$ indicate they are significantly different from zero with the same sign, for testing $$H_A^\prime$$ you could instead replace $$\mathcal C(\alpha)$$ by the one-sided region
$$\mathcal C^\prime(\alpha) = (-\infty, \Phi^{-1}(\alpha)]$$
(for a common negative estimate) or its negative (for a common positive estimate), thereby achieving greater power. Although the confidence of this test would be slightly less than $$1-\alpha,$$ the difference shouldn't matter. A quick simulation study adapted to data like yours would remove any doubts.
When one (or both) of the datasets is "small" (around $$20$$ or less, approximately), consider a Welch t-test instead of a Z test.