Simultaneous Equation Modeling

Ordinary least squares (OLS), along with its cousins such as probit, is the workhorse method of empirical political science. Starting with, as usual, the linear model,

OLS is a fine way of estimating β and θ as long as x and z are exogenous. An explanatory variable is exogenous if whatever statistical process determines it does not depend on the statistical process that determines either the dependent variable or the error term. If an explanatory variable is not exogenous (and so is said to be endogenous), then OLS has severe problems. These problems are typically worse than for other forms of misspecification and do not disappear as the sample size grows. Endogeneity issues cannot be fixed by the usual tweaks to OLS (generalized least squares); different estimation methods are required.

Predetermined socioeconomic characteristics are, by definition, exogenous. Beyond a few simple variables (physical characteristics of people or [Page 2408]countries), the argument that a variable is exogenous must be made theoretically and must be made relative to the dependent variable of interest. Thus, for example, if we believe that democracy is exogenous to economic development, a regression of economic development on democracy yields meaningful results; but we need some theory to know that democracy is exogenous—that is, that it is not the case that increased wealth leads to more democracy. Empirical results are conditional on this assumption; the assumption of exogeneity cannot be tested empirically.

Suppose, for example, that we are interested in explaining attitudes toward political candidates. Let y be a measure of how much one likes a political candidate and x be a measure of how close the candidate is to the voter on various issues. It may be the case that voters who like a candidate more are more likely to perceive the candidate as being close to them on issues, so x may not be exogenous. If x is endogenous to y, it is well known that OLS can be badly biased, and these problems persist even with huge sample sizes (i.e., OLS is not even consistent). The solution, worked out in the 1940s by various Nobel Prize-winning econometricians associated with the Cowles Foundation, is to estimate a series of simultaneous equations for both x and y.

Thus, for two equations, if x determines y but y also determines x, we can write

where z and w are exogenous and ε and ζ are error terms that may well be correlated with each other (but separately satisfy the usual Gauss-Markov conditions). One issue is what method is best for estimating the various model parameters; but a more fundamental issue is whether it is even possible to estimate these parameters. The latter is called the identification problem.