Why Would Pre Filtering Measurement Data Affect the Least Squares Estimate? In estimating parameters in a discrete time model I've often seen the use of filters applied to the input data, before its applied to least squares processing. I've been told that the filters are useful if one wishes to remove bias or high frequency noise that's not considered a part of the model.

But I'm currently working with a problem where I'm using the Moore Penrose pseudoinverse to calculate the least squares. I'm filtering the input data with a lowpass filter to remove high frequency noise well beyond the model dynamics, but the choice of the filter pole seems to have a significant affect on the outcome of the estimate. And in any case the residues are near zero (they don't seem to be better or worse according to pole selection).

How should one chose to filter input data, and should it affect the estimate?
I'm not sure what's you model is.

Let's say it is something like:

$$ y = H x + n $$

Now, using the Least Squares model is optimal (In the MSE sense) when $ n $ is AWGN (It is the linear optimal estimator if the noise is white).

So unless the noise in your model is colored, no gain by filtering the data before applying the Least Squares method.

Now, what if the noise isn't IID White Noise?


If it is white just with different statistics, then use the Weighted Least Squares.
If it is colored you should apply Whitening filter on the data.


So, if the Whitening filter is given by $ W $ then you have:

$$ z = W y = W H x + W n = A x + v $$

Now you have classic LS model, where your model matrix is $ A = W H $ and your white noise is $ v = W n $.
Now you can just apply the regular Least Squares.