Do you know what SEM is? (If you know what SEM is, read an overview of Stata’s SEM capabilities.)

SEM stands for structural equation modeling. SEM is

Stata’s sem command implements linear structural equation models.

For those of you unfamiliar with SEM, it is worth your time to learn about it if you ever fit linear regressions, multivariate linear regressions, seemingly unrelated regressions, or simultaneous systems, or if you are interested in generalized method of moments (GMM).

As you may have figured out, SEM is based on the linear model. What it brings to the table is flexible specification—nearly anything can be allowed to be correlated or constrained to be uncorrelated—and unobserved (latent) variables which can be treated (almost) as if they were observed.

SEM fits the first and second moments of the distribution of observed variables—means, variances, and covariances—rather than fitting the observed values themselves. Both maximum likelihood and GMM methods are available; sem uses a weighting matrix corresponding to asymptotic distribution free estimation in the SEM literature.

You still think of the model in the same way as usual, but in a model like

let’s now call ej the error. Reserve the word residual for the true residuals of the SEMs, which are the differences between the observed and predicted moments.

When SEM is used to fit models that can be fit by the other linear estimators, results are the same, asymptotically the same—by which we mean different in finite samples, and there is no theoretical reason to prefer one set of estimated results to the other—or the SEM results are asymptotically the same and the SEM results should be better in finite samples because of theoretical reasons.

Individual structural equation models are usually described using path diagrams, such as

This diagram is composed of

Thus the above figure corresponds to the equations

There’s a third way of writing this model, namely