INTRODUCTION

As discussed in Chapter 1, the fundamental problem of causal inference is the presence of missing data – for each unit we can observe at most one of the potential outcomes. A key component in a causal analysis is, therefore, what we call the assignment mechanism: the process that determines which units receive which treatments, hence which potential outcomes are realized and thus can be observed, and, conversely, which potential outcomes are missing. In this chapter we introduce a taxonomy of assignment mechanisms that will serve as the organizing principle for this text. Formally, the assignment mechanism describes, as a function of all covariates and of all potential outcomes, the probability of any vector of assignments. We consider three basic restrictions on assignment mechanisms:

1. Individualistic assignment: This limits the dependence of a particular unit's assignment probability on the values of covariates and potential outcomes for other units.

2. Probabilistic assignment: This requires the assignment mechanism to imply a non-zero probability for each treatment value, for every unit.

3. Unconfounded assignment: This disallows dependence of the assignment mechanism on the potential outcomes.

Following Cochran (1965), we also make a distinction between experiments, where the assignment mechanism is both known and controlled by the researcher, and observational studies, where the assignment mechanism is not known to, or not under the control of, the researcher.

We consider three classes of assignment mechanisms, covered in Parts II, III, IV, V, and VI of this book. The first class, studied in Part II, corresponds to what we call classical randomized experiments. Here the assignment mechanism satisfies all three restrictions on the assignment process, and, moreover, the researcher knows and controls the functional form of the assignment mechanism. Such designs are well understood, and in such settings causal effects are often relatively straightforward to estimate, and, moreover, it is often possible to do finite sample inference.

We refer to the second class of assignment mechanisms, studied in Parts III and IV of this text, as regular assignment mechanisms.