Transformations for smooth regression models with multiplicative errors

We consider whether one should transform to estimate nonparametrically a regression curve sampled from data with a constant coefficient of variation, i.e. with multiplicative errors. Kernel-based smoothing methods are used to provide curve estimates from the data both in the original units and after transformation. Comparisons are based on the mean-squared error (MSE) or mean integrated squared error (MISE), calculated in the original units. Even when the data are generated by the simplest multiplicative error model, the asymptotically optimal MSE (or MISE) is surprisingly not always obtained by smoothing transformed data, but in many cases by directly smoothing the original data. Which method is optimal depends on both the regression curve and the distribution of the errors. Data-based procedures which could be useful in choosing between transforming and not transforming a particular data set are discussed. The results are illustrated on simulated and real data.