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:
- Uncontrolled confounder (omitted variable bias);
- 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)?