# How to find lagged cross correlation between time series?

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 $$Y$$?

• Cross-correlograms Commented Jan 16, 2022 at 18:00
• " 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? Commented Jun 16, 2023 at 18:09