Testing for dummy-variable effects in semi-logarithmic regressions

ABSTRACT

Applied economists often use a non-linear function to estimate percentage-change effects of dummy variables in semi-logarithmic models. Delta-method-based inference on these marginal effects is questionable, especially as the dummy variable can be arbitrarily defined to increase the suggestion of evidence of an impact. Inference should instead be based on the untransformed coefficient.

KEYWORDS:

• Semi-log models
• dummy variables
• delta method
• hypothesis test invariance

JEL CLASSIFICATION:

• C12
• C25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ This article considers the case of homoskedasticity only. As noted in Blackburn (2007), heteroskedasticity in the error terms related to d would lead to $(e^{\hat{\beta }}-1)$ being an inconsistent estimator for the percentage effect on E(y|d,x). This is known as ‘retransformation bias,’ which would be a separate but unrelated reason for why the t-test applied to the exponentiated coefficient would be biased under the null hypothesis of no effect from d. An alternative estimation approach would be appropriate in this case.

² This uses $V(\sqrt{n}\stackrel{⌢}{\beta })\approx \frac{{\sigma }_{\epsilon }^2}{{\sigma }_d^2(1-{\rho }_{xd}^2)}=\frac{{\sigma }_{\epsilon }^2}{.25\left(1-.25\right)}$, so that for n = 100 a choice of ${\sigma }_{\epsilon }^2=0.1875$ is required.

³ This involves estimating the model using $d_i$ first, examining if $\hat{\beta }$ is positive or negative, and then using $d_i^{\ast }=1-d_i$ instead if $\hat{\beta }<0$.

⁴ The Kolmogorov–Smirnov test is the maximum difference (in absolute value) between the empirical distribution function of the standardized estimators and the distribution function of a standard normal random variable.

⁵ The variance ${\sigma }^2$ is changed so that $\sqrt{{\sigma }^2/(n-3)}$ stays constant