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Context: I have $K$ timeseries over the interval $[0,T]$ and $N$ timeseries over the interval $[S,T]$, and would like to backcast the $N$ timeseries over the interval $[0,S]$.

I am quite new to this kind of problem and have a few questions. When backcasting, is possible to simply treat the problem by reversing the timeseries and forecasting the missing values, or is it a completely different problem entirely? If so, why is this allowed? What are some general approaches to backcasting, from both the classical ARMA and the more modern deep learning perspectives Are there any recommended papers to read for this problem?

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    $\begingroup$ See otexts.com/fpp3/backcasting.html $\endgroup$
    – Rob Hyndman
    Commented Apr 22, 2021 at 5:20
  • $\begingroup$ @RobHyndman hi thanks for the reply. Why does this work though? For example, with a normal model, we have $X_{t+1}=f(X_t)$, which makes sense since we are conditioning on past information, but with backcasting, $X_t=f(X_{t+1})$ which means conditioning on the future? $\endgroup$
    – user107224
    Commented Apr 22, 2021 at 12:34
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    $\begingroup$ It works because the correlation of variable 1 with variable 2 is the same as the correlation of variable 2 with variable 1. It may be easier not to think of this as time, but as distance, which is clearly reversible, e.g. the correlation of temperature of two points a mile apart is the same no matter which direction you are considering. $\endgroup$
    – zbicyclist
    Commented Apr 22, 2021 at 12:42
  • $\begingroup$ @zbicyclist I thought particularly get this. When you talk about correlation between variables are you referring to the autocorrelation between two points on the series? $\endgroup$
    – user107224
    Commented Apr 23, 2021 at 13:14
  • $\begingroup$ Yes, the autocorrelation between the two variables. Variable 1 is the unlagged series, Variable 2 is he lagged series. $\endgroup$
    – zbicyclist
    Commented Apr 23, 2021 at 16:10

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