Least Squares Percentage Regression

Percentage error (relative to the observed value) is often felt to be more meaningful than the absolute error in isolation. The mean absolute percentage error (MAPE) is widely used in forecasting as a basis of comparison, and regression models can be fitted which minimize this criterion. Unfortunately, no formula exists for the coefficients, and models for a given data set may not be unique. We instead explore least squares regression based on the percentage error. We are able to derive exact expressions for the regression coefficients when the model is linear in these coefficients. Another advantage of this approach over MAPE is that the solution is unique. Furthermore, it has been shown that this approach provides strongly consistent coefficient estimates, and is superior to ordinary least squares when the data does not possess constant variance. For the practitioner we demonstrate that percentage regression models can easily be fitted using ordinary regression software, as well as by spreadsheets, using simple transformation of the data. Finally, we show that when the relative error is normally distributed, least squares percentage regression provides maximum likelihood estimates.