# Regression Coefficient Inference with Non-normal Error Distribution

I have a continuous dependent variable that is non-normally distributed, skewed to the right (long right tail). I have fit two OLS regression models to this variable.

Model 1 - Regression model of un-transformed dependent variable regressed on the independent variable of interest, and other covariates.

Model 2 - Regression model of log-transformed (normally distributed) dependent variable regressed on the independent variable of interest, and other covariates.

As expected, the residuals of Model 1 are not normally distributed and their variance is heteroscedastic, while the residuals of Model 2 are normally distributed and variance is homoscedastic. However, the coefficient of my variable of interest in Model 2 is more difficult to interpret because of the dependent variable transformation. My understanding of the normal error distribution assumption of a regression model is that if it is violated, coefficient p-values may be biased.

My question - Are there any alternative inference options available for me to use Model 1 with interpretable coefficients, while still getting appropriate p-values for coefficient inference? Is bootstrapping an option here? Does residual variance heteroscedasticity come into play?

• I would stick on the Model 2. Homogeneous normally distributed residual. This kind of good situation is not common in practice. Could you describe your difficulty on interpretation? – user158565 May 1 '17 at 14:46
• The interpretation of the coefficient goes from the variable's "effect on costs" in Model 1 to "effect on logged costs" in Model 2. I would much rather talk about effects on costs directly, rather than an effect on logged costs, if possible. – APK May 1 '17 at 17:48
• So in model 1, $\beta$ (regression coefficient for covariate $X$) means the costs when $X$ increases 1 unit. In model 2, $e^{\beta}$ means how many times the cost increases when $X$ increases 1 unit. Maybe your cost increase along the $X$ is not a constant. instead, the ratio of costs is constant. – user158565 May 1 '17 at 18:17

Changing the variable changes the interpretation of effects. Using a change of variable which gives normal errors (by visual inspection or Box-Cox transformation) often leads to a second problem of interpreting effects. Non-rational powers are nonsensical, and log transforms shouldn't be applied unless a multiplicative difference is the desired scale of effect: to explain more, when you apply a log transform for an outcome, an effect $\exp(\hat{\beta})$ describes a percentage difference where 1 is the null and 1.1 reflects a 10% difference, so that 10 to 11 and 100 to 110 are equidistant.
Normally distributed errors is not a necessary assumption except for finite (read: small) sample inference (using an F test). If you have a reasonably large sample size (>40), the regression coefficient has an approximately normal distribution due to the central limit theorem, (so using a $\chi^2$ approximation).
With heteroscedasticity, the analysis isn't biased but it's inefficient because the more variable points are lumped together with the less variable ones. You can use robust standard errors, also called sandwich errors or Huber White standard errors, to downweight observations with bigger residuals in the calculation of standard error. This ensures the 95% CIs and $p$-values are correct.