# Endogeneity versus unobserved heterogeneity

What is the difference between endogeneity and unobserved heterogeneity? I know that endogeneity comes for example from omitted variables? But as far as I understand, unobserved heterogeneity causes the same problem. But where exactly lays the difference between these two notions?

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unobserved heterogeneity can have different interpretations (google does not give a unique definition for example), can you please cite some reference, or give the precise definition you want to be explained. –  mpiktas May 17 '11 at 10:21
@mpiktas: I'm analyzing the problem of omitted variables in a regression. The omitting of variables is causes problems with consistency of the estimator. There are a bunch of other problems that causes inconsisteny (simultaneaus causality and measurement errors for example). All these problems are known as Endogenity. But in this context you often hear the word unobserved heterogenity. And I'm not sure if this is a synonym for Endogenity? Sorry I can't give you more information, because I don't have any (References are in Polish, you wouldn't understand :)) –  MarkDollar May 18 '11 at 7:25
try me, I know Russian, and formulas are the same for all languages. –  mpiktas May 24 '11 at 13:16

The terms endogeneity and unobserved heterogeneity often refer to the same thing but usage varies somewhat, even within economics, the discipline I most associate with the terms.

In a regression equation, an explanatory variable is endogenous if it is correlated with the error term.

Endogeneity is often described as having three sources: omitted variables, measurement error, and simultaneity. Though it is often helpful to mention these "sources" separately, confusion sometimes arises because they are not truly distinct. Imagine a regression predicting the effect of education on wages. Perhaps our measure of education is simply the number of years someone spent in formal education, regardless of the type of education. If I have a clear idea of what type of education affects wages, I might describe this situation as measurement error in the education variable. Alternatively, I could describe the situation as an omitted variables problem (the variables indicating type of education).

Perhaps wages also affect education decisions. If wages and education are measured at the same time this is an example of simultaneity, but it too, might be reframed in terms of omitted variables.

Unobserved heterogeneity is simply variation/differences among cases which are not measured. If you understand endogeneity, I think you understand the implications of unobserved heterogeneity in a regression context.

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I would also include autoregression with autocorrelated errors and sample selection as additional ways endogeneity can arise. –  Dimitriy V. Masterov Sep 7 '12 at 21:20
@DimitriyV.Masterov, Thanks for mentioning those concepts, I think they extend the point I was making. Couldn't, for example, a given case of autoregression with autocorrelated errors or sample selection be reframed as a problem of omitted variables? I know I'm not teaching you anything. I just want learners to think about how these terms are related and realize that the same statistical issue can be conceptualized in many ways. –  Michael Bishop Sep 8 '12 at 18:54

I agree with @Michael's description of endogeneity---this is about a problem with the variables that you include and their relationship to the variables that you do not (i.e., the stuff in the error term).

Unobserved heterogeneity is typically about unobservable componenents of the effects that you are estimating. Continuing with @Michael's education example, unobserved heterogeneity might be that some people have higher returns (e.g., increases in wages) from going to school than others. Let the returns for person $i$ be $\beta + b_i$ with $\mathbb{E}(b_i) = 0$. We have $$\begin{equation*} y_i = x_i (\beta + b_i) + w^\prime_i \gamma + \epsilon_i, \end{equation*}$$ where $y_i$ is (typically, log) income, $x_i$ is years of education, and $w_i$ is a set of other controls. An example of endogeneity is when $x_i$ is correlated with $\epsilon_i$ (e.g., education is correlated with IQ, which is not among our other predictors).

If we estimate a single coefficient, we have $$\begin{equation*} y_i = x_i \beta + w^\prime_i \gamma + (\epsilon_i + b x_i) = x_i \beta + w^\prime_i \gamma + \tilde{\epsilon}_i \end{equation*}$$ See that the included variable $x_i$ is correlated with the error term $\tilde{\epsilon}_i$, inducing the same problems as the case of endogeneity.

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To wrap it up:

• Unobserved heterogeneity is one possible cause of endogeneity.
• Endogeneity is therefore the broader term.
• Unobserved heterogeneity implies endogeneity but not the other way around.
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