Reading a log-log market-impact coefficient as an elasticity

Question

You fit a market-impact model in logs: \log(	ext{impact}) = \beta0 + \beta1 \log(	ext{size}) + \varepsilon, and estimate \hat\beta1 = 0.5. Interpret \beta1 precisely. What does \hat\beta1 = 0.5 say numerically, and how would the interpretation differ in a log-level or level-log model?

Accepted Answer

A log-log slope is an elasticity. In \log y = \beta0 + \beta1 \log x + \varepsilon, the coefficient \beta1 is the percent change in y per percent change in x: \beta1 = \frac{d\log y}{d\log x} = \frac{\partial y / y}{\partial x / x}. It is unit-free, the same whether size is in shares or dollars, which is why elasticities travel so well. What 0.5 means. A 1\% larger order produces about a 0.5\% larger price impact. More usefully, the model says 	ext{impact} \propto 	ext{size}^{0.5}, so impact grows with the square root of size: quadruple the order and impact roughly doubles (4^{0.5} = 2); mu…

Reading a log-log market-impact coefficient as an elasticity

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