Estimating AR1 Signal

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Lets take AR1 signal with p=0.77, driving noise variance q=0.01 . Assume we have noisy observations (additive Gaussian, var 4/12).

Observations can be 'cleaned', as we know the signal generating process. Standard least squares methods will do, and they provide error estimates as well :

The gain of the signal model is more evident if we set p=0.99:

There is one problem: how do we know the p and q of the signal? Underestimating p leads to severe problems. Let's simulate a new data set, and keep the p=0.99:

What happens if we use assumption of p=0.77 in the estimation? This:

How do we find p ? It would not be fair to use a reconstruction that already assumes something about p. (For the measurements above, the nred
method gives 0.06, the estimate without differencing will be 0.12)
Continues..