Confidence Intervals for Elasticity in Simple Linear Regression

Question

I'm pretty sure this is a very simple question that I am missing something obvious. I have a simple linear regression with multiple independent variables. I want to calculate the elasticity (no problem - $\frac{\partial y}{\partial x} \cdot \frac{x}{y}$), but also report the confidence interval for the elasticity. I know it's straightforward to compute the confidence interval for $\beta$ ($\hat{\beta} \pm t^{*}_{\alpha} \cdot \sigma_{\beta}$ ). But for the confidence interval for the elasticity, would it be the same range? Or do I need to account for the transformation ($\beta * x/y$)?

Answer 1

If I understand correctly, what you are proposing is estimating: $$y = \gamma_0 + \gamma_1 \cdot x + \xi $$ and then using the confidence interval of $\gamma_1$ to estimate the range of elasticities: $$ [\frac{\gamma_1 - 2 \cdot \sigma{\gamma_1}}{y} \cdot x, \frac{\gamma_1 + 2 \cdot \sigma{\gamma_1}}{y} \cdot x] $$

This is wrong, essentially because of Jensen's inequality ($E[f(x)]\neq f(E[x])$). You should absolutely take into account the non-linear uncertainty when calculating the confidence interval. Fortunately, this is easy. If you run the regression estimate of $$\ln(y) = \beta_0 + \beta_1 \cdot \ln(x) + \epsilon$$ then the confidence interval of $\beta_1$ is the confidence interval of the elasticity.

In follow up discussion, the questioner asks if it is possible to do this with the linear regression coefficient ($\gamma_1$), along with $\bar{x},\bar{y}$ and the associated standard errors? Why would you want to know the elasticity at the average values instead of the average implied elasticity over all x-y pairs? The beta parameter is estimated on the full population so why not the elasticity too? Because all we only know the average values and the linear regression coefficient.

In principle, you can do a Monte Carlo simulation like the following: $$ \hat{elasticity}= (\hat{\gamma} + SE_{\gamma} * N(0,1)) * (\bar{x}+ SE_{\bar{x}} * N(0,1)) / (\bar{y}+ SE_{\bar{y}} * N(0,1))$$ do a bunch of simulations, and use this to make a confidence interval around the elasticity estimate.

In simulations that I ran, this did result in 95% standard errors that included the true value, but that might simply be a function of the parameters I tried. Unsurprisingly, the resulting standard errors on the elasticity are much larger than in log specification.