RMSE of Weighted Nonlinear Regression

I fit a nonlinear model with weight and without weight. Whilst there is an improvement in R-squared in the weighted model, its RMSE value is, unexpectedly, higher than that of the unweighted model. I did try to run the example code from Matlab click here, and realised that it also has the same 'problem'.

The Stats of Weighted Model, as given in the example:
Root Mean Squared Error: 24 
R-Squared: 0.908,  Adjusted R-Squared 0.885

The Stats of Unweighted Mode, which I found from the same set of data:
Root Mean Squared Error: 17.1
R-Squared: 0.88,  Adjusted R-Squared 0.851

Can anyone explain the reason behind this?

Answer 1

[Mukul Rao]

Hi,

I believe the reason is that traditional weighted regression computes R-square based on transformed variables, the better R-square is indicative of weighting addressing the heteroskedastic nature of the data. However, ordinary least squares does give you the best possible RMSE, which is computed based on the original variables.

In fact, this paper talks about the very same issue.

A cautionary note on R^2

Please read the bottom left paragraph on page 237 ( Inserted as an image here ).