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Note:

There is another post with very similar title, but it is really on the author's specific problem with submarines. Mine is more general.

I have asked a similar question before but turns out it wasn't a good question, so I am asking a modified one. (a practice recommended by the meta site I think)

Question starts:

I have time series $X_t$ and $Y_t$, $t=0,...,N$. I want to develop a model to use $X_t$ to predict $Y_t$. This is not a forecasting question, because in the remote future I won't observe $Y_t$. So I won't be using the history of $Y_t$ to predict its future.

So I thought the easiest approach is to use linear regression lm(Y~x), possibly with lagged covariates, except the residuals $\epsilon_t$ are very clearly auto-correlated. So i thought I could try Rob Hyndman's OLS with ARIMA residuals. But then this is where I got stuck. Say you have trained your model now and the residuals follow an AR(1). When you end up using it to predict at the remote future $t=N+M$, $M$ large, you can easily get the fixed part $X_{N+M}\hat{\beta}$, but

  1. How do you get $\hat{\epsilon}_{N+M}$? Do you set it to be 0? Or do you do the crazy thing of forecasting it $M$ timestep from $t=N$?
  2. Say you set it to 0, what about $\hat{\epsilon}_{N+M+1}$? Do you start using your AR(1) developed with a 0 for $\hat{\epsilon}_{N+M}$?
  3. On that note, what if the residuals are ARIMA? Do you set the first difference to be 0?
  4. Is there a better approach? How about nlme:gls with corStruct=corARMA() (although it seems to be doing the same thing)?

[edited: this is realy a general methodology question. But if you want something more concrete, say both $X_t$ and $Y_t$ are daily data over 4 years. So maybe seasonality is there.]

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  • $\begingroup$ First I would like to take a look at your data to see what model we can use. $\endgroup$ Commented Feb 26, 2014 at 0:36
  • $\begingroup$ @Imorin updated the question. It is really a general question, if you want to, let X and Y be daily data over 4 years. There will be seasonal behaviour. $\endgroup$
    – qoheleth
    Commented Feb 26, 2014 at 0:57
  • $\begingroup$ without more information on X,Y , your question is too broad. There are infinite ways of modelling X ,Y and the dependencies between X and Y. $\endgroup$ Commented Feb 26, 2014 at 1:34
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    $\begingroup$ @Imorin can you suggest a few? $\endgroup$
    – qoheleth
    Commented Feb 26, 2014 at 1:49

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