A robust test for linear and log-linear models against Box-Cox alternatives

The Box-Cox transformation, first introduced by Box and Cox (1964), is a popular approach for testing the linear and log-linear functional forms, because both are special cases of the transformation. The usual approach is to estimate the Box-Cox model by maximum likelihood, assuming normally distributed homoskedastic errors, and test the restrictions on the transformation parameter, which lead to linear and log-linear specifications using a Wald or likelihood ratio test.

Despite the popularity of this approach, the estimator of the transformation parameter is not just restricted to the search for nonlinearity but also to one that leads to more normal errors, with constant variance. This can result in an estimate that favors log-linearity over linearity even though the true model is linear with non-normal or heteroskedastic errors. These issues are resolved by xtloglin because the GMM estimator is consistent under less restrictive distributional assumptions.

Box, G. E., and Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society: Series B (Methodological), 26(2), 211–243.

Savin, N. E., and Würtz, A. H. (2005). Testing the semiparametric Box–Cox model with the bootstrap. Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, 322–354.