Question about violating the zero conditional mean assumption

Question

Question for y'all with a background in econometrics.

Does the zero conditional mean assumption rely on complete randomness in a trial when doing regression analysis?

For example, if I was testing the effect of a free training program on wage, and the men in the trial were randomly given x amount of months of training, but the women were not assigned randomly, would that be a violation of a ZCM? I am struggling to make the connection between randomness and ZCM at all, but it was pointed out to me by someone that it is in fact influential.

Answer 1

If you introduce any sort of correlation between the explanatory variables $X$ and the error $\epsilon$ then the zero conditional mean may be violated. In your example, if you were to assign women their training time according to their age, then the exogeneity assumption breaks.

More formally, this last condition means

$$ \mathbb{E}[\epsilon|X] = 0 $$

If, wlog we assume $\mathbb{E}[\epsilon] = 0$, and there's a non-vanishing correlation between $X$ and $\epsilon$ then

$$ 0\not= \mathbb{C}{\rm ov}[X,\epsilon] = \mathbb{E}[X^t\epsilon] + \mathbb{E}[X] \mathbb{E}[\epsilon] = \mathbb{E}[X^t\epsilon] $$

therefore $\mathbb{E}[\epsilon|X]\not = 0$

Answer 2

Recall the assumptions behind the Multiple linear regression model assumptions 1-4:

MLR.1: Linear parameters
MLR.2: Random sample
MLR.3: No perfect Colinearity
MLR.4: Zero condtional mean $\mathbb{E}(\epsilon|x_i,...,x_n)=0$

In order to have unbiased estimates you require that all of these conditions hold.

In your case it is very much possible that you can have a non-random sample that does not violate the fourth assumption.

However your estimates will be off because of the non-random sampling , so though you dont have any problems which like endogeneity when MLR.4 is violated, you will end up with estimates which do not accurately represent the influence of variables on the subject in question (because of the violation of MLR.2).

In your example you will end up with accurate estimates for the men but not for the women based on the fulfillment of MLR.2.

Answer 3

Perhaps you are having trouble because the meaning of "nonrandom assignment" is not specific enough. Just saying "random" is not good enough. You need to ask "random (or nonrandom) in what sense".

If the assignment is random in the sense that it is independent of any factors whatsoever, i.e., of all the factors that affect $y$, then it is also independent of the regression error (with zero unconditional mean), and ZCM holds (because independence implies ZCM).

In other cases (nonrandom assignment), whether ZCM holds depends on what kind of factors the assignment depends on. If the assignment is nonrandom in the sense that it depends on other explanatory variables (such as $age$), ZCM still holds. If the assignment is nonrandom in the sense that it depends on the error term (the factors that affect $y$ other than the included control variables), then ZCM is likely violated.