How to quantify sensitivity in time series model? 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.

 A: To answer your question we should cast your equation (I am using different notation)
$$y_t = w_0 + \sum_{n=1}^Nw_nx_{t-n}$$
into a different formulation.
Let us introduce $\mathbf{w} = [w_1, w_2, \ldots, w_N]^T$ and $\mathbf{x}_t = [x_{t-1}, x_{t-2},\ldots, x_{t-N}]^T$.
Hence, we obtain
$$y_t = w_0 + \mathbf{w}^T\mathbf{x}_t.$$
We have the dot product between $\mathbf{w}$ and $\mathbf{x}_t$. We know that this will be zero when $\mathbf{w}$ and $\mathbf{x}_t$ are orthogonal.
Hence, we note that the sensitivity of $y_t$ due to changes in $\mathbf{x}_t$ is depending on the weights $\mathbf{w}$.
For orthogonal $\mathbf{w}$ and $\mathbf{x}_t$ the effect on $y_t$ will be $0$.
If $\mathbf{x}_t$ is colinear to $\mathbf{w}$ we get greatest change in $y_t$.
In order to visualize what is happening. Let us simplify to a two-dimensional case. We also assume that $\mathbf{x}_t$ has unit length (see picture $\hat{\mathbf{x}}$). Scaling $\mathbf{x}_t$ will only scale the sensitivity by the same amount.

The red arrow is the projection of $\hat{\mathbf{x}}_t$ onto $\mathbf{w}$ is a measure of (unit) sensitivity.
A: The response of $Y_t$ to a unit shock in $X_{t-1}$ is $\beta_1$ and so on for the rest of the lags present. You could consider the cumulative effect on $Y_t$. But this, of course, reflects the impact of an event (an isolated unit shock at some moment in the past) which is highly unlikely, since you tell us that $X$ is highly autocorrelated.
That's the reason why response functions are usually computed for unit shocks in the innovations.
