Correlation with Residuals and Ceteris Paribus Assumption

[PhamPhuongGiang Nguyen]

This is not a particularly a STATA question, rather a general Econometrics question, but...

Does the correlation of an independent variable with the residuals (the misspecified model: an omitted variable, that is correlated with the existing independent variable in the model, is considered in the residuals) automatically mean that the ceteris paribus assumption does not hold?

From my understanding, when there is a correlation between the stated explanatory variable (x₁) and the omitted variable (u_x), then as x₁ changes, the residuals will also change, which means that the ceteris paribus assumption can not be held. Do I understand those concepts correctly, or are there is any other theory that I have overlooked?

Thank you.

[Andrew Musau]

The problem is referred to as endogeneity. See https://en.wikipedia.org/wiki/Endoge...(econometrics). Note that these are separate variables, where one is omitted from the specification. So the variables can change independently (it's not an issue of the variables being the same). Therefore, this violates the exogeneity assumption of OLS.

[George Ford]

Perhaps I'm speaking over my head here, but I think you are correct that the ceteris paribus assumption is violated because ux is not held constant, under your assumptions. But, holding ux constant would not be an issue if ux was uncorrelated with x1, as the coefficient on x1 is unbiased. Thus, thinking about the way you are may not be productive.

What is held constant is what is in the regression, but whether things left out of the regression matters for the results depends on the correlation of variables.

[Carlo Lazzaro]

PhamPhuongGiang:
welcome to this forum.
An as aside to Andrew and George's helpful advice, you may want to take a look at econometrics - Ceteris paribus relationship and parameters in linear regression models - Cross Validated (stackexchange.com).

[Jeff Wooldridge]

I'll just add a bit more. One can always do the ceteris paribus thought experiment -- for example, it makes sense to think of someone receiving one more year of schooling without changing anything else about them. The problem is that, when we collect retrospective data, the level of schooling people choose is correlated with other variables -- many of which we cannot observe. Our inability to "hold those factors fixed" results in omitted variable bias.

I prefer to distinguish between "error" and "residual." If residuals are from the estimation, by construction they are uncorrelated with any included variable in an OLS regression.