Structural equation models in econometrics vs psychology, political science, etc

Can anyone tell me if the sort of the sort of simultaneous equation/structural equation modeling of economic relationships that that was championed and to some extent developed out of the Cowles Commission (later Foundation) is technically the same under the hood as the "structural equation models" (SEM) framework common in, for example, psychology & political science? I note that Wikipedia states that SEM models are "not to be confused with" econometric simultaneous equation models, but does not actually spell out the difference.

The short descriptions of what they do seem similar -- so similar that I can imagine building an causal economic model using SEM software. But the SEM framework pays a lot of attention to various various concerns that don't seem relevant to or are not commonly used by econometric modelers. For instance, unobservable variables play a key role in SEM modelling, with substantial apparatus devoted to confirmatory factor analysis and measurement models associated with them. Economics usually bases its models primarily on quantities and prices that are, at least in principle, observable, while unobservable quantities (level of efficiency? total social welfare?) are usually assumed to be firmly grounded in economic theory, with changes, if not absolute levels, indirectly observable in levels and flows of economic goods. I think most economists would regard efficiency and welfare as more concrete than SEM's latent variables (e.g. introversion, legitimacy).

I've had more exposure to economics than psychology. but despite this I don't feel like I can list distinctive concerns or capacities of the economic approach to structural equation modeling that are not present in SEM. I have a somewhat inchoate impression that SEM is more concerned with validating model constructs and establishing the flow of causation, while economic structural modeling is more concerned with establishing that the equations which fit together into an economy are estimated consistently with one another, and that the equations and data together provide a uniquely determined ("identified") solution.

But I don't know if these differences in emphasis imply that these are different techniques, or if people are actually wielding largely identical tools to solve their respective problems. For instance, I know that path analysis, common in SEM models, is infrequent in econometric work, but I don't know if that is because the usual econometric implementations of structural models is incompatible with a path analysis approach.

• My understanding is that they are related, but they are not the same thing. SEM with latent variables can be thought of as simultaneous equation models + factor analysis. John Antonakis has written on the two procedures, e.g. here: datascienceassn.org/sites/default/files/… – Jeremy Miles Jul 23 '19 at 22:33

The key distinction in my view between SEM as psychologists use the term and structural estimation/modeling as economists use the term is economists' focus on specifying a full economic model, solving for its equilibrium, and using restrictions implied by the equilibrium to estimate the parameters. (FWIW, my impression is that modern practice is somewhat different from Cowles Commission-style econometrics done in the 1940s-60s.)

In economics, structural estimation often refers to estimating the "deep" (i.e. structural) parameters of an economic model — things like elasticities, risk aversion, the distribution of value that bidders in an auction place on the good, etc. In practice, this often takes the form of postulating some model in which agents optimize (e.g. a game theoretic or microeconomic model) and solving for the equilibrium. The assumption is that the observed data are equilibrium outcomes of the model. The equilibrium conditions often imply restrictions on the data that can yield estimators for the parameters of the model using (for example) generalized method of moments or maximum likelihood estimation.

For an example, in urban and regional economics, many papers employ the Rosen-Roback spatial equilibrium framework. These models assume that each city/region has its own labor and housing markets, and that perfectly mobile workers decide where to live in order to maximize their utility. The model specifies labor supply and demand curves, which might depend on features of the city like amenities, and housing supply and demand curves. The equilibrium condition implies that utility across all cities must be equalized. If people could obtain a higher utility by moving, they'd move to the city in which they get higher utility. But by moving, they'd drive wages down (through increased labor supply) and housing prices up (through increased demand for housing). This process would continue until utility across regions is equalized.

This equilibrium condition implies a set of restrictions on the data that can yield an estimator of the parameters of the model. E.g., if we specify utility that people can obtain in city $$c$$ as $$U_{c} = f(Z_{c})+ \epsilon_{c}$$, in equilibrium we have a set of moment conditions $$U_{c} = U_{c'} = \bar{v}$$ for all $$c, c'$$ and some (arbitrary) constant $$\bar{v}$$. These moment conditions can be used to estimate parameters of $$f$$.

In political science, structural models are much less common, but one place they do show up is in ideal point estimation. The idea is that we want to infer a political actor's ideology based on observable behavior such as roll-call votes or campaign donations. We then specify a model relating unobserved parameters of the decision problem (e.g. the utility the legislator with a given ideal point would get from passing a bill vs. the utility she gets from the status quo) to observable behavior (e.g. whether the member of Congress voted for or against the bill). Given some parametric assumptions about idiosyncratic error terms in the actor's utility function, we can derive a likelihood for the (unobservable) ideal points. This problem turns out to be nearly identical to the psychometrics problem of inferring "ability" through educational tests, and is often viewed as an Item Response Theory model.

I am much less familiar with the psychology literature on SEM, but my impressions of it are similar to yours — a focus on measurement (as in the IRT models) and latent variable modeling.

Caveat: I am a political scientist, not an economist.

• So, are you saying that the difference is that SEM models as used by political scientists (and maybe psychologists) usually do not rely on model restrictions imposed by theory for identification? – andrewH Jul 30 '19 at 3:25
• I can't speak to the way that psychologists think about the relationship between theory and the statistical model, but I don't think your statement is accurate for the political science ideal point models I mentioned above. The theory implies restrictions on the data (namely that the legislator votes for the policy that's "closer" to her in policy space). The difference is that it's a decision-theoretic model, rather than a game-theoretic model. Clinton, Jackman, and Rivers (2003) is a good primer cs.princeton.edu/courses/archive/fall09/cos597A/papers/… – Will Marble Aug 7 '19 at 18:28