Is there any information we can extract from the residual
If your estimate is based on a linear time invariant model then you could use the following measures for the quality of your estimate: cross-correlation between the estimate and residual, and the autocorrelation of the residual. Namely the cross-correlation should be close to zero at all time, otherwise there would still be some residual dynamics that could be modeled. The autocorrelation should be close to zero at all time except at zero, where it should be close to one. If this is the case then this would mean that the residual would be white noise, which can't be predicted/modeled.