Hi: see andrew harvey's books for the detailed discussion. His earlier one
( I forget the
title names ) is more comprehensive. But I bet they both talk about it.

what you have is an "almost" random walk with noise model but the
coefficient on the ar(term) is not 1. you can estimate that using the DLM
package. or any of the other kalman filtering packages.

there's also a way to write it  as an ARIMA but I'm not sure if Harvey
discusses that. If you want to go that route, email me and I'll figure out
how to re-write it. doing that mostly just a lot of algebra. but I'm
leaving now and won't be back till very late tonight. also, a better
list for that type of question is R-Sig-Finance.











On Thu, Jun 20, 2013 at 6:45 PM, ivo welch <ivo.welch@anderson.ucla.edu>wrote:

> dear R and stats wizards:  I would like to estimate an AR1 model with
> constant and measurement noise:
>
>   true[t] = a + b*true[t-1] + noise1[t]
>   observed[t] = true[t] + noise2[t]
>
> (true is never observed.)  I am very interested in forecasting
> observed[t+1]., and modestly interested in inferring b and true[t].  I have
> a lot of data.  in truth, I really have a panel with thousands of
> individuals, so I don't get the usual strong AR1 bias when b is close to 1.
>
> my intuition is that a good forecast of observed[t+1] (and thus of true[t])
> is a historical weighted average of past observed[t] values, with more
> weights on more recent observeds, and in effect shrunk towards the long-run
> mean.  by simulating the model, I can observe how the auto-corrollelogram
> looks like, and fit it. however, both of these are amateurish---this
> problem seems so canonical that it has probably been solved a gazillion
> times.
>
> could someone please point me to some simple textbook = howto treatments of
> this problem and/or R packages that implement this?  feel free to point out
> your own work...this way I can cite it.
>
> regards,
>
> /iaw
> ----
> Ivo Welch (ivo.welch@gmail.com)
>
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