Statistical Modeling and Inference for Social Science

In Chapter 6, we saw a variety of examples of data-generating processes (DGPs) that are common in statistical models, which consist of a DGP for the data and a link from its parameters to explanatory factors. Although the parametric family of the DGP may be assumed, the specific parameters of the DGP that gave rise to data we have available are generally not known. The whole point of empirical analysis is to use observable data to learn something about those parameters. A theory may assert that the conditional mean of Y increases as X increases, for instance, but in empirical analysis, we wish to determine how credible this assertion actually is. That is the point of statistical inference, which is the subject of the rest of this book.

To understand how we can use data to inform ourselves about parameters of DGPs when they are unknown, it is useful to see what happens when we pretend these parameters are known. If we have data drawn from a known DGP with known parameters, we can see how summaries and statistics computed from that observed data are related to those parameters. That is the subject of the present chapter. Here we study how summaries and statistics computed exclusively from observable data are related to the DGP, given a collection of observations from the DGP.