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.