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I have a very highly sampled time series that I would like to fit an autoregressive model (AM) to (~3 million samples). From knowing what they represent, I have believe there should be unique information out to ~1k sample points in the past. Thus I'm considering modifying a more standard AM.

This is what I'm calling the "standard" AM (from Wikipedia): $$X_t = c + \sum_{i=1}^p \varphi_i X_{t-i} + \epsilon_t$$

The problem with this for my purposes is that to go out 1k samples require $p=1000$, which is a lot of parameters and I worry about over fitting. What I'd like to do is use exponentially spaced history points to do the fit: $$X_t = c + \sum_{i=0}^{p-1} \varphi_i X_{t-2^i} + \epsilon_t$$

So with $p=11$ I would be considering points up to 1024 samples in the past. My question is, is there something wrong with this approach? Are there frequencies that can fall in between the powers of two that I can't fit with this model? If it does work, why doesn't everyone do it like this?

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I think the best way is to think your model as a long memory model (fractional integrated). You should use for your purpose an ARFIMA.

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  • $\begingroup$ It's not really clear to me from this what you want me to do. Wikipedia's ARFIMA page wasn't very helpful either. How does using a more complex model help things? $\endgroup$ – Mimshot Dec 8 '12 at 19:30
  • $\begingroup$ sorry for my delay. In your model your saying that a shock at time $t$ has a long impact in your time series. Your ACF decays very slowly. As you know you prefer a parsimonious model, in your model you have 1000 parameters to estimate! in ARFIMA model you have for the long memory only one: $d$. $\endgroup$ – Marco Jan 15 '13 at 10:38

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