# Difference Between Simultaneous Equation Model and Structural Equation Model

Can anybody please help me to understand the differences between simultaneous equations models and structural equation models (SEM)? It will be great if somebody can provide me some literature on it.

Also, is there any literature where SEM has been used in time series contexts? The literature I'm getting mostly explains SEM in a cross-sectional data context.

• Not sure about time series, but the SEM framework has been widely used in latent growth curve modeling, for example; see work by Bengt O. Muthén and coll., and references on the Mplus homepage.
– chl
Commented Jul 5, 2013 at 9:26

Simultaneous equation models (let's call them SIM to separate the two types of models), are models where you have some simultaneity. For example,

$$y=\alpha+\beta x + u_y\\ x=\gamma+\delta y + u_x$$

As you can see, the two equations form a system of equations. These are widely used in econometrics and applied economics, but it is not guaranteed that they have a reasonable (economic) interpretation.

Furthermore, to make things even more complicated, SIMs can be written in both structural and reduced form. So you can speak of a simultaneous equation model in a structural form, without referring to what is traditionally known as structural equation modeling (SEM)! If you want a reference, Econometric Analysis of Cross Section and Panel Data by Wooldridge is pretty good.

In the SEM universe you try to estimate causal relationships and things you cannot observe. For example, IQ is impossible to observe, but you may exploit relationships between related (observable) variables to study it. Factor analysis is a common SEM method.

For applications of SEM on time series, you may want to have a look at dynamic factor analysis.

• Thanks Sebastian! Your answer seems correct. But waiting for some other relies as well. :)
– Beta
Commented Jul 5, 2013 at 9:36
• @RichardHardy: I actually forgot to marked it answered :) I generally don't do that! Thanks for reminding me.
– Beta
Commented Oct 9, 2017 at 9:17
• I agree with @hejseb . Just an additional point that Structural Equation Models are used in Econometrics, but most statisticians don't use them or like them too much. The problem is that they make many strong assumptions about the data and its form. Usually stats folks like to estimate these relationships from the data itself. Commented Mar 30, 2019 at 18:01
• @hejseb; I’m not disagree with your explanation, at least mathematically it make sense to me. However I have doubts about interpretation. If "structural form" may not have his traditional (=causal) meaning in SIM, what meaning it have there? Commented Mar 17, 2021 at 9:56

It seems me that the interpretation of structural equation models (SEM) in econometrics is matters of debate. Judea Pearl strongly defends the causal interpretation of SEM, and of its parameters. For example you can read: The Causal Foundations of Structural Equation Modeling - Pearl (2012).

The distinction between SIM (simultaneous equation model) and SEM is artificial even if, literally speaking, SEM can be considered a broader class.

The term SIM is widely used in economics but unfortunately the role of such a model is ambiguous. In Pearl's opinion, SIM is a terminological strategy for remove/obscure causal meaning at SEM (page 3). In his opinion, SEM must be carried out such that it always has a clear causal meaning.

SIM are not thinked for forecasting, Indeed VAR (vector autoregression) born precisely with the idea of a system, useful for forecasting, that avoid economic/structural/causal thinking. SIM bring us to speak about structural form and reduced form. Please, if you know one econometrics textbook or serious article that speak about SIM without this distinction let me know. Reduced form, per se, carried out only correlational/regression meaning but via identification we achieve structural one. Surely structural meaning go beyond correlational one (in broad sense, not necessarily linear) but if the structural meaning is not causal It is not clear what it is. Moreover SIM share his math and logic with SVAR (structural VAR). Even if the in the SVAR some lags are usually added we can consider SVAR like a restyling/extension for SIM; but the mentioned ambiguities remain.

Time series context is also related, see my question here: About the meaning of ARMA parameters

• You misread Pearl. He did not say the two are synonyms. He gave the phrase "simultaneous equations" as an example of "causality-free nomenclature" that you might see in a SEM textbook. He also lists "covariance structure" and "regression analysis" as other examples. None of these examples are synonyms with structural equation modeling (SEM).
– wmay
Commented Jan 30 at 0:18
• No friend, you misinterpret Pearl article. Him denunce the “steady erosion of the basic understanding of SEMs [its causal meaning]”. Simultaneous equation model (SIM) is a name that since the origin was used as synonym of structural equation model (SEM). Indeed in Pearl refers strongly to Haavelmo (1943) article, “The statistical implications of a system of simultaneous equations” as the most influential contribution for SEM! Commented Jan 30 at 7:56
• Moreover even the Authors criticized by Pearl refers on the same article. Indeed many econometrics books the term SIM bring us always to speak about “structural” and reduced form … even if causal meaning are sometimes permitted and sometimes not and sometimes it is not clear. Note that for the same argument even the term SEM are sometimes used. They are extensively used as synonyms. Commented Jan 30 at 7:56
• The problem is precisely the loss of causal understanding of SIM/SEM. The causal-free concepts/terminology that you listed are largely exploit for avoid causal meaning. SIM and SEM are far from to be a clear cut different concepts in literature. The question of the asker come from precisely from this conflation and for the same reason do not admit trivial reply. Commented Jan 30 at 7:56
• Probably you focused too much on the term “synonym” and misunderstood my message. I quote myself: "In Pearl's opinion, the last [SIM] is a terminological strategy for remove/obscure causal meaning at SEM (page 3)". I edited my answer for clarify better. Commented Feb 1 at 7:24

To add up to the former answer, I would say it is not different at all; in fact they have different point of view. Simultaneous equation term focuses on simultaneity, so according to the concept it is recommended to use techniques other than simple OLS to estimate parameters. On the other hand, structural equation term focuses on structure itself, so it may include latent variables etc. In fact, there are numerous ways to model structural equations.

• Can you elaborate on the similarity? I thought it's different because simultaneous equations can have feedback loop but SEM can't as far as i know Commented May 13, 2017 at 16:15