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I have a time series $Y_t$ (example:university applications received in a certain month) which I want to forecast. I have another time series $X_t$ and I know that $Y_t$ is related to past lags of $X_t$ (example: high school graduates per month), meaning that a certain amount of $X_t$ at month $t$ will determine the amount of $Y_t$, say, 12 months later. I also know that the probability applying at the university is decreasing with every month once graduated. A lot of students will apply right after graduating but only a small percentage will apply, say, 6 month later. $X_t$ has a seasonal pattern and $Y_t$ has several shocks because of law changes.

My questions are: Is a dynamic regression model/ARIMAX appropriate to forecast $Y_t$ with $X_{t-k}$? How do I deal with the shocks in $Y_t$ knowing that they hide the underlying structure of this time series?

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