Solved – Regression Elasticities with categorical variables

I'm regressing a continuous variable (y) on a categorical (x=0,1,2) variable.
Is it possible (or even make sense?) to derive an elasicity of y on x?

In general, the elasticity is defined as a ratio of percentages ($Y$ is the dependent variable and $X$ is the regressor)

$$varepsilon_{Y,X} = frac {text {% change in } Y}{text {% change in } X}$$

So the percentage change must be meaningful in order for the elasticity to be meaningful. When $X$ is categorical, then its percentage change may be meaningful, may be not: if $X$ measures some discrete quantity (like "number of children", "number of deaths", in general "number of…") then its percentage change makes perfect sense. But if it represents an ordinal and not cardinal ranking (e.g. 1 = good, 2 = very good), or if it is a mapping of qualitative states (like e.g. $0$ = no color, $1$ = red, $2$ = blue) then the percentage change has no meaning, and so neither does the elasticity.

If the "meaningfulness criterion" is satisfied, then in general the concept of elasticity appropriate for discrete-valued data is the Arc elasticity

$$varepsilon_{Y,X} = frac {(y_2-y_1)(x_2+x_1)}{(y_2+y_1)(x_2-x_1)}$$

where the subscripts denote specific points, and where we essentially use the mid points of the intervals involved (but the two divisions by $2$ cancel out). In a linear regression setting where we have estimated a constant marginal effect, $hat beta$, this marginal effect substitutes for the two difference-terms in the above expression and the elasticity becomes

$$varepsilon_{Y,X} = hat betafrac {(x_2+x_1)}{(y_2+y_1)}$$

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