Cautions when normalizing the dependent variable in a regression as a z‐score

Jeffrey Penney

Economic Inquiry, 2023, vol. 61, issue 2, 402-412

Abstract: It is common in empirical analysis to facilitate inference by transforming the dependent variable to follow a standard normal distribution. In this paper, I show that using this transformation results in the estimated treatment effects being systematically attenuated toward zero and bounded in magnitude. The level of attenuation can be empirically relevant. I propose an alternative normalization wherein the dependent variable is divided by the square root of its within variation, which corrects these issues. I show that, in a simple linear regression, the method produces an estimated treatment effect that is numerically identical to Cohen's d.

Date: 2023
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