Causal Modeling

In the analysis of social science data, researchers to a great extent rely on methods to infer causal relationships between concepts represented by measured variables—notably, those between dependent variables and independent variables. Although the issue of causality can be rather philosophical, a number of analytical approaches exist to assist the researcher in understanding and estimating causal relationships. Causal thinking underlies the basic design of a range of methods as diverse as experiments, and models based on the Markov chain. One thing that distinguishes experimental from nonexperimental design is the fact that the independent variables are actually set at fixed values, or manipulated, rather than merely observed. Thus, with nonexperimental or observational design, causal inference is much weaker. Given an observational design, certain conditions can strengthen the argument that the finding of a relationship between, say, X and Y is causal rather than spurious. Of particular importance is temporality (i.e., a demonstration that X occurred before Y in time). Also important is meeting the assumptions necessary for causal inference. In classical regression analysis, these are sometimes referred to as the Gauss-Markov assumptions.

Methods for analyzing observational data also include all kinds of regression-type models subsumed [Page 102]under general linear models or generalized linear models, especially graphic modeling, path analysis, or structural equation modeling (SEM), which allow for multiequation or simultaneous equation systems. SEM, which can include latent variables to better represent concepts, can be estimated with software such as AMOS, EQS, LISREL, or M-PLUS.

A typical simultaneous equation system may contain reciprocal relationships (e.g., X causes Y and Y causes X), which renders them nonrecursive. Here is an example of such a system, which may be called a causal model:

The phrase causal model is taken to be a synonym for a system of simultaneous, or structural, equations. Sometimes, too, a sketched arrow diagram, commonly called a path diagram, accompanies the presentation of the system of equations itself. Before estimation proceeds, the identification problem must be solved through examination of the relative number of exogenous and endogenous variables to see if the order and rank conditions are met. In this causal model, the X variables are exogenous, the Y variables endogenous. As it turns out, when each equation in this system is exactly identified, the entire system is identified, and its parameters can be estimated. By convention, exogenous variables are labeled X and endogenous variables labeled Y, but it must be emphasized that simply labeling them such does not automatically make them so. Instead, the assumptions of exogeneity and endogeneity must be met.

Once the model is identified, estimation usually goes forward with some instrumental variables method, such as two-stage least squares. A three-stage least squares procedure also exists, but it has not generally been shown to offer any real improvement over the two-stage technique. The method known as indirect least squares can be employed if each equation in the system is exactly identified. Ordinary least squares cannot be used for it will generally produce biased and inconsistent estimates unless the system is recursive, whereby causality is one-way and the error terms are uncorrelated across equations. There are maximum likelihood estimation analogs to two-stage and three-stage least squares: limitedinformation maximum likelihood and full-information maximum likelihood, respectively.

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