# Confidence interval for a variable which is function of the coefficients of a linear regression, $\frac{\hat{\beta}_1}{\hat{\beta}_2}$

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.

• 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. Commented Sep 2, 2023 at 11:28
• 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 stats.stackexchange.com/questions/625150/…. For closely related threads (which likely already have answers) see this site search.
– whuber
Commented Sep 2, 2023 at 14:20
• Why not use the Delta method or the bootstrap? Commented Sep 2, 2023 at 17:53
• @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) Commented Sep 2, 2023 at 18:42
• 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). Commented Sep 3, 2023 at 11:57