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

The purpose of this presentation is to describe a new command xtloglin, which tests the suitability of the linear and log-linear regression models against Box-Cox alternatives. The command uses a GMM-based Lagrange Multiplier test, which is robust to non-normality and heteroskedasticity of the errors and extends the analysis by Savin and Würtz (2005) to panel data regressions after xtreg. The Box-Cox transformation, first introduced by Box and Cox (1964), is a popular approach for testing the linear and log-linear functional forms, as 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, that 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 non-linearity, but also to one that leads to more normal errors, with constant variance. This can result in an estimate that favours log-linearity over linearity even though the true model is linear with non-normal or heteroskedastic errors. These issues are resolved by xtloglin, as the GMM-estimator is consistent under less restrictive distributional assumptions.