## 4.1 Objectives

There are two ways one can think of the objectives in this kind of problem posed above. One is that we observe $$x_t$$ and we know $$\beta_j$$ and we want to know how the series $$x_t$$ is affected by convolving $$x_t$$ with the series $$\beta_j$$. Then the $$y_t$$ series represents the filtered version of $$x_t$$. Describing and characterizing the behavior of such a linear system is typical of the domain of engineering and physics. In these applications it might be assumed that $$\varepsilon_t=0$$ for all $$t$$, so that we perfectly observe the input and output.

Another way to think about this setting is that we observe $$x_t$$ and $$y_t$$ and want to estimate the values $$\beta_j$$. This is more of a traditional statistical problem here want to infer the impulse-response function and make statements about its true values. We will focus here on the latter problem as statisticians are typically faced with a system where the impulse-response is unknown and must be inferred. Furthermore, often that system will not be under our direct control, and therefore we must deal with issues like confounding.