Reading a log–log regression coefficient

Question

You estimate \ln(	ext{volume}) = a + b\,\ln(	ext{spread}) + \varepsilon and get b = -0.8. Interpret b precisely, and contrast it with the meaning of the slope in log–level and level–log regressions.

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The log–log case is an elasticity. Because both variables are in logs, the coefficient is the ratio of proportional changes: b = \frac{d\ln y}{d\ln x} = \frac{dy/y}{dx/x} = 	ext{elasticity}. So b = -0.8 means: a 1% increase in spread is associated with roughly a 0.8% decrease in volume. This holds approximately for small changes and does not depend on the units of either variable, a major reason economists love log–log forms. Contrast with the other functional forms. - Level–level (y = a + bx): b is the change in y (in y-units) per one-unit change in x. A plain slope. - Log–level (\ln y = a …

Reading a log–log regression coefficient

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