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.