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I have a time series dataset that contains variable x and y per month (180 rows). x represents incidents that happen infrequently and y is a continuous variable. However, when there is some incident y sees a decline for k months afterwards.

Based on a linear regression model, I have found that x has a significant effect on y if I use a time lag of n months. But in this case I have set the number of lags myself and this is not very useful.

Therefore, I am interested in finding the value of k empirically from the data. That is to find how long the effect of x on y may last?

I would really appreciate any kind of help, especially if you could indicate what type of test would be useful here.

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