*Bounty: 50*

*Bounty: 50*

X and Y are time series of length T. X is the predictor and Y is the response. A linear model is fitted as follows:

$$hat{Y_t}=alpha+sum_{i=1}^{N}{beta_iX_{t-i}}$$

where $beta$‘s and $alpha$ are such that they minimise squared errors between $Y$ and $hat{Y}$.

Now I want to know **"How sensitive is $hat{Y}$ to X?"**

In an ordinary linear regression (without the temporally lagged quantities on the right), the answer would just be $beta$, but here I have $N$ different $beta$‘s. Are there ways in which I can condense the$N$ different $beta$‘s into a scalar quantity? Or any other method to answer **"How sensitive is $hat{Y}$ to X?"**

Potentially relevant information but ignore if not needed:

- X and Y vectors are highly auto-correlated. For example, X is daily temperature, and Y is daily ice cream sales.
- When I say "How sensitive is $hat{Y}$ to X?", I mean how much is Y affected for changes in X. For example, ice cream sales would likely be very sensitive to daily lagged temperature, but laptop sales would probably be insensitive to daily lagged temperature.