Chosing a linear model based on error or independence of residuals I build two linear regression models. The first model has a low error (RMSE) but the residuals are not independent of each other.
The second model has a higher error but the residuals are less dependent of each other.
How can I chose between the two models?
Is a model with highly dependent residuals generally useless?
 A: Models with dependent residuals are not "generally" useless, but the dependence could indicate a trend or feature of the data that your model is failing to capture. If so, the model could give biased predictions or generalize poorly in situations where that missing feature is more significant. Whether that occurs, and whether it has significance to your work, depends on what you are trying to do with the model. The model with the higher RMSE could have the same problem - it may just be less noticeable because there are additional sources of error mixed in.
Another possibility is your data is subject to a correlated noise component, such as additive Gaussian process with a non-identity correlation matrix. Similar to the issue above, this could bias your regression coefficients or make your model generalize less well. This page and this page provide some examples of how correlated noise can affect your data and what you can do about it. Again, the model with the higher RMSE could just be concealing this problem.
How are you making the judgment about the residuals' correlation? Using graphs or some statistical test? Can we see the graphs or the test results? And what is your goal - are you making a predictive model, trying to infer a regression coefficient, or something else?
