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Suppose I have a well-performing time-series regression such as:

$$ x_{2020} = \beta_0 + \beta_{2019}x_{2019} + \beta_{2018}x_{2018} + \beta_{2017}x_{2017} $$

Having fit this regression to $x_{2020}$, I'd now like to use it to predict $x_{2021}$. How could I now use $x_{2020}$ as an independent variable? My initial thought was to shift the variables so that $x_{2019}$ would get $\beta_{2018}$'s weight, but I don't think it's that simple since my predictions for $x_{2021}$ look a bit odd.

Thanks.

P.S: Sorry if this is a duplicate; the closest I could find are Time series regression coefficient interpretation with differenced independent variable and How to account for the recency of the observations in a regression problem? but I'm not sure they're what I'm looking for.

dmn
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  • Another possible duplicate is https://stats.stackexchange.com/questions/205232/how-to-down-weight-older-data-in-time-series-regression but I don't know that I want to down-weight older data... – dmn May 18 '21 at 21:31
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    This is called an autoregression. See https://otexts.com/fpp3/AR.html. – Rob Hyndman May 18 '21 at 22:32
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    Probably want to learn about lag operators and difference operators. – Alexis May 19 '21 at 05:28
  • Thanks @RobHyndman and Alexis, but doesn't AR require an observation every time period? Like sometimes I have data for 2018 but not always, and it's not missing at random. – dmn May 19 '21 at 15:41
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    No, you can fit an autoregression with missing observations. – Rob Hyndman May 20 '21 at 03:21
  • Wow, I definitely need to read more about this. Your book has some good examples that help me think about how to work with my data. Thank you :) – dmn May 21 '21 at 12:44

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