# Endogenously determined variables in regression analysis

I am working on a project with a former professor, and we are considering using two-stage least squares (2SLS) regression to deal with some endogeneity we suspect in our model. To make sure I'm up to speed on 2SLS, I have been reviewing the procedure and the conditions that require it. As I understand it, endogeneity describes a situation where one or more of your regressors are correlated with the error term, such that the regressor(s) can be predicted by the error term.

This has made me wonder if endogeneity can be considered a special case of under-specification--specifically, a case of under-specification in which an omitted variable is correlated with an included variable. In a fully specified model, would endogeneity arise?

Yes, "endogeneity" in this context includes the case of an omitted variable that is correlated with an included regressor. The term simply an artifact of the literature.

In the econometrics literature, an "endogenous" variable originally referred to a variable in a system of regression equations that was a dependent variable in one of the equations. In other words, a variable that was endogenous to an economic agent's decision. For example:

$$P = \beta_1 Q + \beta_2 M + \varepsilon$$ $$Q = \alpha_1 P + \alpha_2 I + \nu$$ where $P$ is price of the good, $Q$ is quantity demanded or supplied, $M$ is material cost, and $I$ is income. The first equation can be thought of as the supplier's equation, where the price the supplier charges is a function of the quantity demanded and the price of input goods. The second equation can be thought of as the demander's equation, with quantity purchased being a function of the price and the person's income. These models would come from some model of firm and buyer behavior, where $M$ and $I$ are taken as given (exogenous) while the agents choose $P$ or $Q$ to maximize their respective objective function (hence "endogenous").

Substituting $P$ in the first equation into the second yields

$$Q(1 - \alpha_1\beta_1) = \alpha_1\beta_2 M + \alpha_2 I + \alpha_1\varepsilon + \nu$$

So $Q$ depends on $\varepsilon$ and OLS of equation 1 will be biased for $\beta_1$.

The resulting bias from naive estimation with such a regressor is referred to as "simultaneity" or "reverse causality" bias. See here for a randomly Googled example. However, the actual statistical problem is that the "endogenous" regressor is correlated with the error term. Over time, "endogenous regressor" has come to refer to any regressor that is correlated with the error term for any reason, whether the context was a multi-equation model or not. This includes simulteneity bias, omitted/lurking variable bias, measurement error bias, etc. This is true in some circles, at least. Not every uses "endogeneity" as a short-hand for everything.

Edit: In summary, "endogenous" regressor is generally a catch-all for any regressor correlated with the error term, so an omitted variable is actual a special case of an endogenous variable, not vice versa. In a fully specified model, you could still get bias this general "endogeneity", e.g., from measurement error.

• Thank you, dmsul. Is there a better/more specific term for what I'm talking about than "endogeneity"? – Jon Boyette Sep 16 '16 at 18:54
• "Endogeneity" is the catch-all in the econometrics lit (can't speak to pure statistics people), so it's fine. Bias from an omitted variable that is correlated with an included is just "omitted variable bias". If the omitted variable is a jointly driving the regressor and the outcome some fields will call the omitted variable a "lurking variable". On re-reading your question, I realize I didn't entirely address what you were asking, so I've edited my answer a bit. – dmsul Sep 16 '16 at 19:01