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I I would like to ask about demographic variables. In general, we usually include those variables as independent variables when we run regressions such as wage equation or test score equation. What I wonder is that why we take them as exogenous in general? Intuitively, I understand that those are something given to agents when they need to make a decision, so in that regard it makes sense that we say they are exogenous. However, mathematically E[Xe]=0 is the definition of exogoneity, and so I guess demographic variables can be still correlated with unobserved error terms. For example, if we consider the following equation Y_i = X_i*b_i + e_i where Y_i is test score, X_i are demographic covariates (race, gender, age ...). Then, isn't it possible those variables are correlated with innate abilities or whatever in e_i ? But, I do not find any convincing answers why (nearly) every people take those as exogenous....

Any answers for this?

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You are completely correct that mathematically endogeneity is defined $\text{cov(X,e)}\neq0$, which is true not only when it’s correlated with omitted variable but also with measurement error or reverse causation/simultaneity.

However, every single variable in economics can be subject to correlation with omitted variables or measurement error, so if you would want to use the word for any variable that potentially has this problem then it would be meaningless. So the word is reserved for those cases where this is actually a serious issue.

Furthermore, often endogeneity is used more narrowly as to referring just to simultaneity or reverse causality - variables determined by the model as opposes to variables outside the model - because that’s where the term originated.

Also some demographic variables can be definitely subject to endogeneity, but I would also say most of them aren’t. Consider race for example in a regression where you try to estimate let’s say how it affects the employment. Well you can’t pick your race (at least to my understanding technology is not there yet), so there is no possibility for simultaneity. Next race would be simple categorical variable so there is not much space for measurement error and also I am not biologist but from my understanding biologically there is only human race so social construct of race should be orthogonal on unobservables like innate ability. Similar story would hold also for sex (although you can have sex change that is usually not done on a whim but when person actually experiences gender dysmorphia so it can be treated as predetermined at birth) and age.

An example of endogenous demographics variable could be number of children in family that could have simultaneous relationship between wages/wealth and could be also correlated with some unobservables.

Sometimes it can be also context dependent. In model where you regress Y on X there can be endogeneity there but in a model where you regress S on the same X it might not be there because conditions are not satisfied. In your research you should always think hard about this issue and decide on case by case basis. But in general demographic variables just happen to not be endogenous often when comparing to other types of variables.

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