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.
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.
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
Structural equation modeling (SEM) represents the structure - the causal relationships between variables - using equations, graphs, and other tools. As a framework for causal inference, SEM encompasses a diverse set of methods applied across disciplines: covariance structure analysis in psychology, composite SEM in marketing and business research, and structural causal models (SCM) or nonparametric SEM (originating from Pearl’s work) in epidemiology, computer science, and medicine. For more details, see Kline (2023), page 10.
The Wikipedia page for SEM, which you cited, states:
Different (from SEM) yet mathematically related modeling approaches developed in psychology, sociology, and economics. Early Cowles Commission (econometrics pioneers) work on simultaneous equations estimation ...
In econometrics, simultaneous equations models, whose foundation was established by the Cowles Commission in the 1940s and 1950s, share similarities with SEM, with one key distinction: the emphasis on simultaneity.:
simultaneous equations models are a system of equations describing a set of economic assumptions, in which the dependent variables are functions of other dependent variables, rather than just independent variables.
as explained in the Wikipedia page for simultaneous equations model, which you also cited. A classic example is the supply and demand system, where both price and quantity are dependent variables, each influencing the other. Like SEM, simultaneous equation models reflect an underlying structure - in this case, economic theory.
When the system of simultaneous equations is transformed to express the dependent variables solely as functions of independent variables, the result is known as the reduced form. The original system, representing the theoretical structure, is called the structural form.
Today, the usage of these terms in economics literature has become more flexible. A reduced form does not necessarily have to be derived from simultaneous equations, and structural models are not limited to systems of simultaneous equations; they can involve single equations representing a theoretical structure. Similarly, a reduced form refers to any model that does not explicitly represent a structural relationship, regardless of its origin. From this perspective, structural models in econometrics are more closely aligned with SEM.
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.
