In his 2009 book titled Causality: Models, Reasoning, and Inference, Judea Pearl lays out a powerful and extensive graphical theory of causality. Pearl's work provides a language and a framework for thinking about causality that differs from the potential outcome model presented in Chapter 2. Beyond the alternative terminology and notation, Pearl (2009, section 7.3) shows that the fundamental concepts underlying the potential outcome perspective and his causal graph perspective are equivalent, primarily because they both encode counterfactual causal states to define causality. Yet, each framework has value in elucidating different features of causal analysis, and we will explain these differences in this and subsequent chapters, aiming to convince the reader that these are complementary perspectives on the same fundamental issues.

Even though we have shown in the last chapter that the potential outcome model is simple and has great conceptual value, Pearl has shown that graphs nonetheless provide a direct and powerful way of thinking about full causal systems and the strategies that can be used to estimate the effects within them. Some of the advantage of the causal graph framework is precisely that it permits suppression of what could be a dizzying amount of notation to reference all patterns of potential outcomes for a system of causal relationships. In this sense, Pearl's perspective is a reaffirmation of the utility of graphical models in general, and its appeal to us is similar to the appeal of traditional path diagrams in an earlier era of social science research. Indeed, to readers familiar with path models, the directed graphs that we will present in this chapter will look familiar.

Regression models are perhaps the most common form of data analysis used to evaluate alternative explanations for outcomes of interest to quantitatively oriented social scientists. In the past 50 years, a remarkable variety of regression models have been developed by statisticians. Accordingly, most major data analysis software packages allow for regression estimation of the relationships between interval and categorical variables, in cross sections and longitudinal panels, and in nested and multilevel patterns. In this chapter, however, we restrict our attention to ordinary least squares (OLS) regression, focusing mostly on the regression of an interval-scaled variable on a binary causal variable. As we will show, the issues are complicated enough for these models, and it is our knowledge of how least squares models work that allows us to explain this complexity. In addition, nearly all of the insight that can be gained from a deep examination of OLS models carries over to more complex regression models because the identification and heterogeneity issues that generate the complexity apply in analogous fashion to all regression-type models.

In this chapter, we present least squares regression from three different perspectives: (1) regression as a descriptive modeling tool, (2) regression as a parametric adjustment technique for estimating causal effects, and (3) regression as a matching estimator of causal effects. We give more attention to the third of these three perspectives on regression than is customary in methodological texts because this perspective allows one to understand the others from a counterfactual perspective.

In this chapter, we present the basic conditioning strategy for the estimation of causal effects. We first provide an account of the two basic implementations of conditioning – balancing the determinants of the cause of interest and adjusting for other causes of the outcome – using the language of “back-door paths.” After explaining the unique role that collider variables play in systems of causal relationships, we present what has become known as the back-door criterion for sufficient conditioning to identify a causal effect. To bring the back-door criterion into alignment with related guidance based on the potential outcome model, we then present models of causal exposure, introducing the treatment assignment and treatment selection literature from statistics and econometrics. We conclude with a discussion of the identification and estimation of conditional average causal effects by conditioning.

Conditioning and Directed Graphs

In Section 1.5, we introduced the three most common approaches for the estimation of causal effects, using language from the directed graph literature: (1) conditioning on variables that block all back-door paths from the causal variable to the outcome variable, (2) using exogenous variation in an appropriate instrumental variable to isolate covariation in the causal variable and the outcome variable, and (3) establishing the exhaustive and isolated mechanism that intercepts the effect of the causal variable on the outcome variable and then calculating the causal effect as it propagates through the mechanism.

Do charter schools increase the test scores of elementary school students? If so, how large are the gains in comparison to those that could be realized by implementing alternative educational reforms? Does obtaining a college degree increase an individual's labor market earnings? If so, is this particular effect large relative to the earnings gains that could be achieved only through on-the-job training? Did the use of a butterfly ballot in some Florida counties in the 2000 presidential election cost Al Gore votes? If so, was the number of miscast votes sufficiently large to have altered the election outcome?

At their core, these types of questions are simple cause-and-effect questions of the form, Does X cause Y? If X causes Y, how large is the effect of X on Y? Is the size of this effect large relative to the effects of other causes of Y?

Simple cause-and-effect questions are the motivation for much research in the social, demographic, and health sciences, even though definitive answers to cause-and-effect questions may not always be possible to formulate given the constraints that researchers face in collecting data and evaluating alternative explanations. Even so, there is reason for optimism about our current and future abilities to effectively address cause-and-effect questions. Over the past four decades, a counterfactual model of causality has been developed and refined, and as a result a unified framework for the prosecution of causal questions is now available.

In this chapter, we introduce the foundational components of the potential outcome model. We first discuss causal states, the relationship between potential and observed outcome variables, and the usage of the label “counterfactual” to refer to unobserved potential outcomes. We introduce average causal effects and then discuss the assumption of causal effect stability, which is maintained explicitly in most applications that use the potential outcome model. We discuss simple estimation techniques and demonstrate the importance of considering the relationship between the potential outcomes and the process of causal exposure. We conclude by extending our presentation to over-time potential outcome variables for one or more units of analysis, as well as causal variables that take on more than two values.

Defining the Causal States

The counterfactual framework for observational data analysis presupposes the existence of well-defined causal states to which all members of the population of interest could be exposed. As we will show in the next section, causal effects are then defined based on comparisons of outcomes that would result from exposure to alternative causal states. For a binary cause, the two states are usually labeled treatment and control. When a many-valued cause is analyzed, the convention is to refer to the alternative states as alternative treatments.

Although these labels are simple, the assumed underlying states must be very carefully defined so that the contribution of an empirical analysis based upon them is clear.