Two exogenous regressors, endogenous to each other

Question

Given the following regression:

$y_i = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon $

If $x_1$ and $x_2$ are exogenous regressors to $y_i$, but are endogenous to each other, or determined simultaneously, what effect does this have on the $\beta$s.

For example, assuming that education and wages are exogenous to health, but we know they themselves are both caused by ability:

$health_i = \alpha + \beta_1 education_{i} + \beta_2 wages_{i} + \text{other variables}+ \epsilon $

Note: I know this may not be a great example, and this is not a question that I am posing for my research (I am not studying the effect of wages and education on health) but just the best example I could think of to illustrate my question.

Answer 1

You will very often have explanatory variables that are related to each other. As far as I know, we reserve the term endogeneity for covariates that are correlated with the error term of the regression and I wouldn't use the term in a situation like yours.

Depending on what you are trying to accomplish with your regression analysis, the consequences of related explanatory variables are somewhere between no problem at all - problematic.

Lets focus on two aims of a regression analysis:

1. Fitting values for the dependent variable (predicting $y$)
2. Estimate the partial effect of covariate1 on $y$ (estimate coefficients)

In the first case, collinearity is no problem. In the latter case, however, you may run into trouble, depending on how strong the relationship between $x_2$ and $x_1$ is. The reason is that the regression model will not be able to distinguish the effect that $x_1$ has on $y$, from the effect that $x_2$ has. This shows in larger standard errors for the coefficients, which is the regression's way of saying: I'm not sure I should link $y$'s variation to $x_1$ or to $x_2$, because they always move together. Ultimately this will lead to insignificant coefficients and you will not be able to say something (as) useful about the partial effects of $x_1$.

You can measure the effect of multicollinearity with the variance inflation factor.

Answer 2

Endogenous coregressor is a not so well-developed concept in econometrics since as above mentioned. "endogenous" is more associated with the concept of the correlation or relationship with the error term from an exogenous variables, and not exactly the case where exogenous variables maybe endogenous among each other, because if one changes, the other changes. But basically, you're implying that your linear model of the form of

$$ Y = X_{1} \beta _{1} + X_{2} \beta _{2} + E $$

When you derive the OLS estimator from this for $\beta _{1}$ you'll notice you'll get:

$$ \beta _{1} = (X'_{1}X_{1})^{-1}X'_{1}Y - (X'_{1}X_{1})^{-1}X'_{1}X_{2}\beta_{2} $$

In this point, if you want to get an unbiased estimator you may assume that $X_{1}$ and $X_{2}$ have a linear relationship, and that will not bias the result. But that is quite different from having a variable which is not only linearly related but rather endogenous determinated among the regressors, that is, when $X_{1}$ or $X_{2}$ changes, the counter-part endogenous regressor also does. Commonly expressed the case where $Cov(X_{1},X_{2}) \neq 0$ and this derives in a biased estimator.

For some reason, this topic is never discussed, but it can bias the estimates harmfully than single multicollinearity does ! (recall, multicollinearity is just a linear association or relationship, but itself it is not implying that covariances are changing together between your explanatory variables)

To get an unbiased estimator for $\beta_{1}$ you need to argue that $Cov(X_{1},X_{2}) = 0$, which is something that hardly people do today in empirical literature, since they put all relevant variables without considering the possibility of an endogenous co-regressor!