Question

Elasticities are defined as the percentage change in a Y variable corresponding to a percentage change in an X variable. How do elasticities relate to regression coefficients in the following three regression cases?

a) A regression of Y (in levels) against X (in levels).
b) A regression of Y (in logs) against X (in logs).
c) A regression of Y (in logs) against X (in levels).

Answer 1

Elasticities measure the responsiveness of one variable to changes in another, and their relationship to regression coefficients varies by the model specification: absolute change, direct elasticity, or percent change based on whether variables are in levels or logs.

Step-by-step explanation:

Elasticities represent the responsiveness of one variable to changes in another variable and are prevalent in various economic contexts. When discussing how elasticities relate to regression coefficients in econometrics, we look at different specifications of regression models:

1. For a regression of Y (in levels) against X (in levels), the regression coefficient represents the absolute change in Y for a one-unit change in X. To convert this coefficient to elasticity, you need to multiply it by the ratio of the average of X to the average of Y.

2. For a regression of Y (in logs) against X (in logs), the regression coefficient directly represents the elasticity of Y with respect to X. This is because the percentage changes are implicitly captured by the log transformation, making the coefficient an elasticity.

3. For a regression of Y (in logs) against X (in levels), the coefficient represents the percent change in Y resulting from a one-unit change in X.

To summarize, elasticities are closely related to regression coefficients; the relationship depends on whether the dependent and independent variables are in levels or logged forms. Real-world examples include the income elasticity of demand, cross-price elasticity of demand, and elasticity of savings with respect to interest rates.