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There are valuable resources on the lexicon of types of variables quickly accessible, such as here. However, some of these concepts appear side-by-side often enough to make them confusing.

For example, Wikipedia mentions two causes for endogeneity:

  1. Uncontrolled confounder (omitted variable bias);
  2. Loops of causality between dependent and independent variables (simultaneity).

This is in line with the Encyclopedia of Health Economics formulation

$$Y=X_1β_1+X_2β_2+C_oβ_o+C_uβ_u+\varepsilon\tag 1$$

where the generating true model in the equation is opaque to the researcher because of $C_u$ - an unobservable confounding variable which, unlike $C_o,$ may be suspected, but cannot be controlled for, resulting in a flawed $Y=X_1\beta_1+X_2β_2+C_o\beta_o+e$ model. In this case, if $X_1$ was correlated with $C_u,$ we could say that $X_1$ is endogenous because of the presence of an unobservable confounder.

Based on this, should confounding variables be considered a type of endogenous variable? Endogenous > Confounding?


But, then in this Stata blog it seems to have a different mathematical formulation as compared to confounders and OVB. In this linked document $X_1$ would be endogenous if

$$\begin{align} Y &= X_1\beta_1 + X_2\beta_2+\varepsilon\\[2ex] &\mathbb E(\varepsilon \vert X_1)\neq0\\[2ex] &\mathbb E(\varepsilon \vert X_2)=0 \end{align}$$

It is evident that this may just be a different mathematical formulation of the same concept in equation (1): the omitted unobservable confounder manifests itself through residual lack of independence of $X_1$ and $\varepsilon,$ a rather theoretical mathematical equation since $\varepsilon$ cannot be directly measured, and the actual residuals are orthogonal to the model matrix by construction.

According to this blog, confounders and endogenous variables are two separate issues.


Perhaps the true understanding of the differences comes from graphical causal models:

one only needs to consider the three elementary causal structures from which all DAGs can be constructed: chains $A\to C\to B$ (and its contraction $A\to B$), forks $A\leftarrow C\to B,$ and inverted forks $A\to C \leftarrow B.$ Conveniently, these structures correspond exactly to causation, confounding, and endogenous selection.

Here, confounder = common cause while endogenous selection = common outcome.


There may be some field dependency. From J Epidemiol Community Health 2008;62;858-861 F Imlach Gunasekara, K Carter and T Blakely:

Endogenous explanatory variable: An explanatory variable in a multiple regression model that is correlated with the error term, either because of an omitted variable, measurement error, or simultaneity.

...

When endogeneity is discussed in biostatistics texts with respect to longitudinal data it is mainly a factor of reverse causation. An endogenous exposure variable is a predictor of the outcome at time $t$ and is also predicted by the outcome at time $t–1.$ This can be controlled for by adding time-lagged variables to the model.


QUESTION:

Could we precisely define the differences and overlapping features of these two types of variables with emphasis on any hierarchical structure? Can we formulate their respective definitions in a precise (linear) algebraic equation, or through directed acyclic graphs (DAGs)?

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The question is not totally clear, but I will attempt to give you some guidance.

To answer your first questions, confounding variables are not a type of endogenous variable. We do not observe nor are we interested in the confounding variables, which means they are not endogenous variables in our model.

You later give the correct definition of an endogenous variable "An explanatory variable in a multiple regression model that is correlated with the error term". Hence, there can be several reasons why a variable would be endogenous, but they all have to do with correlation with the error term.

An endogenous variable is one, which cannot have a causal interpretation in a regression model. When something leads to a variable being endogenous, we call this a source of endogeneity.

As you mention, an important source of endogeneity are omitted variables. If we omit a variable, which affects the outcome variable, then the omitted variable is captured by the error term in our model. If the omitted variable is correlated with an explanatory variable, then that explanatory variable becomes endogenous, because it is now correlated with the error term.

Other important sources of endogeneity are reverse causality and simultaneity, as you mention. For reverse causality, we have that the explanatory variabe is supposed to influence the outcome variable, but if the outcome variable also affects the explanatory variable, then in our regression model, this latter channel is necessarily captured by the error term. This is because such a channel determines the relationship between the outcome and explanatory variable, but we cannot capture it in any variable. Putting the explanatory variable on the left side instead and using the outcome variable as an explanatory variable does not help either, because we then have the same problem. Then the explanatory variable is correlated with the error term, which makes the explanatory variable endogenous.

So overlapping features are that all sources of endogeneity stem from a correlation of an explanatory variable with the error term. Distinctions are the source of this correlation and consequently how to deal with it. For example, omitted variable bias may be dealt with by observing and adding the omitted variable to the regression. This strategy does not work when trying to address reverse causality.

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  • $\begingroup$ Thank you for your elaborate answer, which helps clarify the defining idea of correlation with the error term. I did follow you explanation well, except for the fact that I don't know what you mean by "We do not observe nor are we interested in the confounding variables." What does the "Uncontrolled confounder" in Wikipedia mean? $\endgroup$ – Toni May 9 '18 at 23:20
  • $\begingroup$ That means an omitted variable, i.e. one that is correlated with the explanatory variable and affects the outcome variable, but which is not controlled for. Suppose you are interested in the effect of education on earning. Education is endogenous, because we do not observe the innate ability of students, i.e. smarter students tend to get more education, but would earn more than less smart people, regardless of schooling. Ability is the uncontrolled confounder. $\endgroup$ – BB King May 10 '18 at 8:01

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