The equations in SEM are mathematical and statistical properties that are implied by the model and its structural features, and then estimated with statistical algorithms (usually based on matrix algebra and generalized linear models) run on experimental or observational data. ![]() The causal structures imply that specific patterns of connections should appear among the values of the variables, and the observed connections between the variables’ values are used to estimate the magnitudes of the causal effects, and to test whether or not the observed data are consistent with the postulated causal structuring. The postulated causal structuring is often depicted with arrows representing causal connections between variables (as in Figures 1 and 2) but these causal connections can be equivalently represented as equations. The structural aspect of the model implies theoretical associations between variables that represent the phenomenon under investigation. SEM involves the construction of a model, to represent how various aspects of an observable or theoretical phenomenon are thought to be causally structurally related to one another. It is used most in the social and behavioral sciences. Structural equation modeling ( SEM) is a label for a diverse set of methods used by scientists in both experimental and observational research across the sciences, business, and other fields. The unlabeled arrow pointing to academic performance acknowledges that things other than intelligence can also influence academic performance. This model postulates that separate measurement errors influence each of the two indicators of latent intelligence, and each indicator of latent achievement. It is hoped a good indicator has been chosen for each latent, but the 1.0 values do not signal perfect measurement because this model also postulates that there are other unspecified entities causally impacting the observed indicator measurements, thereby introducing measurement error. The 1.0 effect connecting a latent to an indicator specifies that each real unit increase or decrease in the latent variable’s value results in a corresponding unit increase or decrease in the indicator’s value. Because intelligence and academic performance are merely imagined or theory-postulated variables, their precise scale values are unknown, though the model specifies that each latent variable’s values must fall somewhere along the observable scale possessed by one of the indicators. Similar to Figure 1 but without standardized values and fewer items. An example structural equation model before estimation. This situation prompted the development, spearheaded by the Cowles Commission in the 1940s and 1950s, of various techniques that estimate each equation in the model seriatim, most notably limited information maximum likelihood and two-stage least squares.Figure 2. And while it would be natural to estimate all simultaneous equations at once, this often leads to a computationally costly non-linear optimization problem even for the simplest system of linear equations. Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand and then set the price. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables.
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