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I have 2 time series, $X$ and $Y$, and I'm trying to find the best lag range that correlates $X$ to $Y$ (find the amount(s) of lag of $X$ that best correlate to the target variable $Y$).

For instance, if the best lag range is between $t = 8$ and $t = 10$, then the final equation would be $Y_t = \alpha_1 X_{t-8} + \alpha_2 X_{t-9} + \alpha_3 X_{t-10} + \alpha_4$.

Since the value of $Y$ could depend not only on some specific lag of $X$, but rather on a range of lags, I can't just find the correlation coefficient between $Y$ and different time lags of $X$ individually, and can't just run a regression model for several lags of $X$ as independent variables, since there are huge colinearities between those lags (series $X$ is a slow-changing time series).

Is there a way to find what are the best lags of $X$ to be used to predict my variable $Y$?

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  • $\begingroup$ Cross-correlograms $\endgroup$
    – Dayne
    Jan 16, 2022 at 18:00
  • $\begingroup$ " the correlation coefficient between Y and different time lags of X individually" Would you elaborate? Since the X and Y is function of t, shouldn't one delay to entire Y signal applied to each point at each t? $\endgroup$
    – Cloud Cho
    Jun 16 at 18:09

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This is commonly called cross-correlation, lagged regression, or distributed lag.

One commonly applied algorithm is ARMAX model.

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