# Difference Between Simultaneous Equation Model and Structural Equation Model

Can anybody please help me to understand what are the differences between Simultaneous Equation Model and Structural Equation Model (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 timeseries context? The literatures I'm getting are mostly explained SEM in cross-sectional data context.

Thank You!

• 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 Jul 5 '13 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 Jul 5 '13 at 9:36
• @RichardHardy: I actually forgot to marked it answered :) I generally don't do that! Thanks for reminding me. – Beta Oct 9 '17 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. – krishnab Mar 30 '19 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? – markowitz Mar 17 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).

He consider terms like simultaneous equation model (SIM) as synonym of SEM. In Pearl's opinion, the last 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.

Surely in the context of SIM and SEM, there are always structural forms and reduced forms, where reduced are achieved by identification. Please, if you know one econometrics textbook or serious article that speak about SIM/SEM without these distinctions let me know. Reduced form, per se, carried out only correlational/regression meaning but via identification we achieve causal one. Surely structural meaning go beyond correlational (in broad sense, not necessarily linear) one but if the structural meaning is not causal I don't know what it is.

Time series context is also related, see my question here: Structural equation and causal model in economics

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 – KH Kim May 13 '17 at 16:15