Logs with Zeros? Some Problems and Solutions

Jiafeng Chen, Jonathan Roth – 2024

When estimating average treatment effects (ATE) for positive-only outcomes, using log-like transformation that aren’t the log itself can cause issues. These include $\log (1+Y)$, $\text{arcsinh}(Y)$ and so on.

Unfortunately, ATEs for these transformations are scale dependent, and thus should not be interpreted as percentage effects, as arbitrary changes to the scale will affect the estimated effect.

These are mostly used to avoid taking the log of zero. Unfortunately there’s no free lunch — there’s no treatment effect parameter that you can estimate after taking these transformations that is point identified.

Instead, the authors recommend using Poisson regression to estimate the ATE in levels as a percentage: ${\theta }_{ATE\%}=\frac{E[Y(1)-Y(0)]}{E[Y(0)]}$

$e^{\beta }-1={\theta }_{ATE\%}$

So you read off the $\beta$ from the Poisson regression, run it through the quick equation above, and et voila you have your percentage ATE!

Note this method completely abstracts from extensive (zero vs non-zero) vs intensive margin (non-zero vs non-zero) changes.

Also note that this is not literally estimating a percentage effect, it’s really estimating a level effect and then rescaling. This can be an important distinction depending on the application, so tread carefully.

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References

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