Elasticity

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)

avg_slopes(mod)
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)    S   2.5 %  97.5 %
#>    hp  -0.0318    0.00903 -3.52   <0.001 11.2 -0.0495 -0.0141
#>    wt  -3.8778    0.63274 -6.13   <0.001 30.1 -5.1180 -2.6377
#> 
#> Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
#> Type:  response

avg_slopes(mod, slope = "eyex")
#> 
#>  Term Contrast Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
#>    hp    eY/eX   -0.285     0.0855 -3.34   <0.001 10.2 -0.453 -0.118
#>    wt    eY/eX   -0.746     0.1418 -5.26   <0.001 22.7 -1.024 -0.468
#> 
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
#> Type:  response

avg_slopes(mod, slope = "eydx")
#> 
#>  Term Contrast Estimate Std. Error     z Pr(>|z|)    S    2.5 %    97.5 %
#>    hp    eY/dX -0.00173   0.000502 -3.46   <0.001 10.8 -0.00272 -0.000751
#>    wt    eY/dX -0.21165   0.037850 -5.59   <0.001 25.4 -0.28583 -0.137464
#> 
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
#> Type:  response

avg_slopes(mod, slope = "dyex")
#> 
#>  Term Contrast Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
#>    hp    dY/eX    -4.66       1.32 -3.52   <0.001 11.2  -7.26  -2.06
#>    wt    dY/eX   -12.48       2.04 -6.13   <0.001 30.1 -16.47  -8.49
#> 
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
#> Type:  response