The content of this vignette requires version 0.7.0 of the marginaleffects package.

In some contexts, it is useful to interpret the results of a regression model in terms of elasticity or semi-elasticity. One strategy to achieve that is to estimate a log-log or a semilog model, where the left and/or right-hand side variables are logged. Another approach is to note that \(\frac{\partial ln(x)}{\partial x}=\frac{1}{x}\), and to post-process the marginal effects to transform them into elasticities or semi-elasticities.

For example, say we estimate a linear model of this form:

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \varepsilon\]

Let \(\hat{y}\) be the adjusted prediction made by the model for some combination of covariates \(x_1\) and \(x_2\). The slope with respect to \(x_1\) (or “marginal effect”) is:

\[\frac{\partial \hat{y}}{\partial x_1}\]

We can estimate the “eyex”, “eydx”, and “dyex” (semi-)elasticities with respect to \(x_1\) as follows:

\[ \eta_1=\frac{\partial \hat{y}}{\partial x_1}\cdot \frac{x_1}{\hat{y}}\\ \eta_2=\frac{\partial \hat{y}}{\partial x_1}\cdot \frac{1}{\hat{y}} \\ \eta_3=\frac{\partial \hat{y}}{\partial x_1}\cdot x_1, \]

with interpretations roughly as follows:

  1. A percentage point increase in \(x_1\) is associated to a \(\eta_1\) percentage points increase in \(y\).
  2. A unit increase in \(x_1\) is associated to a \(\eta_2\) percentage points increase in \(y\).
  3. A percentage point increase in \(x_1\) is associated to a \(\eta_3\) units increase in \(y\).

For further intuition, consider the ratio of change in \(y\) to change in \(x\): \(\frac{\Delta y}{\Delta x}\). We can turn this ratio into a ratio between relative changes by dividing both the numerator and the denominator: \(\frac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}\). This is of course linked to the expression for the \(\eta_1\) elasticity above.

With the marginaleffects package, these quantities are easy to compute:

library(marginaleffects)
mod <- lm(mpg ~ hp + wt, data = mtcars)

marginaleffects(mod) |> summary()
#>   Term   Effect Std. Error z value   Pr(>|z|)    2.5 %   97.5 %
#> 1   hp -0.03177    0.00903  -3.519 0.00043365 -0.04947 -0.01408
#> 2   wt -3.87783    0.63273  -6.129 8.8603e-10 -5.11797 -2.63770
#> 
#> Model type:  lm 
#> Prediction type:  response

marginaleffects(mod, slope = "eyex") |> summary()
#>   Term Contrast  Effect Std. Error z value   Pr(>|z|)   2.5 %  97.5 %
#> 1   hp    eY/eX -0.2855    0.08533  -3.345 0.00082157 -0.4527 -0.1182
#> 2   wt    eY/eX -0.7464    0.14190  -5.260 1.4436e-07 -1.0245 -0.4682
#> 
#> Model type:  lm 
#> Prediction type:  response

marginaleffects(mod, slope = "eydx") |> summary()
#>   Term Contrast    Effect Std. Error z value   Pr(>|z|)     2.5 %     97.5 %
#> 1   hp    eY/dX -0.001734  0.0005007  -3.463 0.00053334 -0.002715 -0.0007528
#> 2   wt    eY/dX -0.211647  0.0378683  -5.589 2.2834e-08 -0.285868 -0.1374268
#> 
#> Model type:  lm 
#> Prediction type:  response

marginaleffects(mod, slope = "dyex") |> summary()
#>   Term Contrast  Effect Std. Error z value   Pr(>|z|)   2.5 % 97.5 %
#> 1   hp    dY/eX  -4.661      1.325  -3.519 0.00043365  -7.257 -2.065
#> 2   wt    dY/eX -12.476      2.036  -6.129 8.8603e-10 -16.466 -8.486
#> 
#> Model type:  lm 
#> Prediction type:  response