# Time series model for demand forecasting?

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?