Greene [1] and Wooldridge [2] emphasize that in the standard multiple linear regression model $${\bf y}=X{\bf b}+{\bf e}$$ a key assumption is that $$E[{\bf e}|X]=E[{\bf e}].$$ Or, in other words, $X$ provide no information about the expected value of ${\bf e}$. Provided that we include an intercept in the model, this assumption will be equivalent to $$E[{\bf e}|X]=E[{\bf e}]={\bf 0}.$$ Wooldridge call it zero conditional mean assumption.

On the other hand, Gelman and Hill [3 (p. 46 kindle edition)] list the assumptions of linear regression in order of importance as

  1. Validity
  2. Additivity and linearity
  3. Independence of errors
  4. Equal variance of errors (for the efficiency of the estimator)
  5. Normality of errors (for small sample inference of variance of the estimator).

Note that, at this point in the book Gelmen and Hill have not discussed the use of linear regression as a tool for causal inference yet. In the discussion of causal inference, in chapter 9, they introduce ignorability assumption that $$y_0, y_1 ⊥ T | X.$$ The ignorability assumption seems to be closely related to zero conditional mean assumption.

The Questions

  1. Why Gelman and Hill did not include zero conditional mean at the begining?
  2. How can we interprete OLS estimator of ${\bf b}$ when zero conditional mean is violated?


[1] Greene, William, 2008, Econometric Analysis, 6th Edition, Pearson.

[2] Wooldridge, Jeffery, 2015, Introductory Econometrics, 6th Edition, Cengage Learning.

[3] Gelman, Andrew, and Jennifer Hill, 2007, Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press.


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