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Consider the following linear regression $$Y = \beta_1X_1 + \beta_2X_2 + \epsilon$$

Can I compute a confidence interval for the estimate of the quantity ? $$\frac{\beta_1}{\beta_2}$$

# Load the mtcars dataset
data(mtcars)

Fit a linear regression model with two predictor variables (e.g., mpg and hp)


lm_model <- lm(mpg ~ 0 + hp + wt, data = mtcars)


Summarize the model


summary(lm_model)


Coefficients:
   Estimate Std. Error t value Pr(>|t|)

hp -0.03394    0.03940  -0.861   0.3959

wt  6.84045    1.89425   3.611   0.0011 **

I don't want to resort to the Delta method or bootstrap.

Julien
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    Do you mean confidence interval for $\beta_1 / \beta_2$ rather than $\hat{\beta}_1 / \hat{\beta}_2$? Also the example regression has an intercept, so it's $E(Y) = \beta_0 + \beta_1 x_1 + \beta_2 x_2$. Usually it makes sense to include the intercept term. – dipetkov Sep 02 '23 at 11:28
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    This is a minor variation of your previous question and can be addressed in the same manner described there. The question of how to estimate the ratio $\beta_1/\beta_2$ was recently posed at https://stats.stackexchange.com/questions/625150/estimating-ratio-of-regression-coefficients. For closely related threads (which likely already have answers) see this site search. – whuber Sep 02 '23 at 14:20
  • Why not use the Delta method or the bootstrap? – jbowman Sep 02 '23 at 17:53
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    @jbowman: If $\beta_2$ is close to zero, that can be imprecise, just use the profile likelihood as outlined in the post linked by whuber (and answered by me) – kjetil b halvorsen Sep 02 '23 at 18:42
  • I don't want an intercept here – Julien Sep 02 '23 at 18:46
  • The intercept is not an issue. When $X_1$ is constant, this is a model with a single explanatory variable $X_2$ and an intercept. – whuber Sep 02 '23 at 21:36
  • @kjetilbhalvorsen Jbowman suggested the Delta method and I don't know why you think this method is not suitable. You said that if $\beta_2$ close to zeto, the Delta method is imprecise, but this argument can apply also for any other methods (as $\beta_2 \to 0, \frac{\beta_1}{\beta_2} \to \infty$ and the confident interval becomes inevitably larger. By consequence, Delta method is still the most suitable methods. – NN2 Sep 03 '23 at 09:23
  • @Julien Why don't you want to use the Delta method? Perhaps because of the argument of kjetilbhalvorsen ? – NN2 Sep 03 '23 at 09:25
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    The delta method leads to symmetric confidence intervals which are not realistic here. That would cause at least one of the tails to not have accurate non-coverage probability (e.g., 0.1 when you hope for 0.025 for a two-sided 0.95 CI). If you fit a Bayesian regression you simply examine all the posterior draws of the ratio and construct an exact (aside from simulation error) uncertainty interval (highest posterior density interval or credible interval). – Frank Harrell Sep 03 '23 at 11:57
  • @NN2: The only way to find out is to try both ... (will try in an answer) – kjetil b halvorsen Sep 03 '23 at 12:44

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