# How do endogenous variables relate with the error term?

Are endogenous variables stochastic or non-stochastic? If they are stochastic,can we say they are uncorrelated or correlated with the error term?

I read this in Basic Econometrics by Gujarati (5th edition) page 594, section 16.3, "It is assumed that the explanatory variables are non-stochastic. If they are stochastic, they are uncorrelated with the error term. Sometimes it is assumed that the explanatory variables are strictly exogenous. A variable is said to be strictly exogenous if it does not depend on current, past, and future values of the error term u."

I know the explanatory variables are considered stochastic if they are random. The regressors will be random if they are correlated with the error term and non-random if otherwise. The book clearly contradicts the statement I just made. Can someone clarify this and make clear whether endogenous variables are stochastic and how they relate with the error term?

## 2 Answers

The phrasing is unfortunate in the book, but not necessarily incorrect in this case:

It is assumed that the explanatory variables are non-stochastic. If they are stochastic, they are uncorrelated with the error term.

He doesn't claim that stochastic regressors are always exogenous in this passage. He states that when the regressors are stochastic then it is assumed that they are uncorrelated with errors, i.e. exogenous.

Stochastic regressors can be endogenous, which creates problems for OLS analysis. Therefore, exogeneity assumption is imposed on OLS.

The regressors will be random if they are correlated with the error term and non-random if otherwise

This is not true. Regressors can be random even if are uncorrelated with error term. More, if intercept is included, regressors and error are uncorrelated by construction.

Unfortunately the econometrics book of Gujarati is unreliable. His concept of non stocastic regressors is unfortunate. Worse, many times it conflate statistical and causal concepts. So the concept of exogeneity/endogeneity used therein is ambiguous. Moreover, several contradictions can be find in this book.

Unfortunately problems like those are quite common in econometrics literature, read here: How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?

This my post and links therein can help you: Under which assumptions a regression can be interpreted causally?

Moreover, before to read what I suggest, note that your question:

How do endogenous variables relate with the error term?

can be properly understood only if causal inference is the goal.

• if intercept is included, regressors and error are uncorrelated by construction: I think that depends on how you define the error term. Commented May 25, 2022 at 15:41
• Under what definition my statement seems you not true? Commented May 25, 2022 at 20:47
• Under the definitions used in most econometric textbooks. I think you could call them structural errors. It is not a coincidence econometric textbooks discuss the case of errors being correlated to regressors e.g. due to an omitted variable bias. Commented May 26, 2022 at 6:32
• Indeed this point is the core of ambiguities. Primarily for this reason most econometric book should be rewritten, in my opinion. Moreover speaking about "regressors" in the definition is the worst case. In this way become evident that authors have in mind regression and/or suggest to think about it. Unfortunately this definition is inconsistent with that of conditional expectation ... this fact produce any sort of contradictions. Commented May 26, 2022 at 7:12
• I am aware of the problems, and you are right to point them out. Commented May 26, 2022 at 7:15