OVB with endogenous predictor vs exogenous predictor

Question

I have a question regarding Omitted Variable Bias (OVB):

I know that endogenous predictor can be an issue, as the association between the dep. variable and the indep. variable could be confounded by another variable, which is not controlled for in the model.

I'm wondering though whether the exogeneity of a predictor is a sufficient requirement for not having OVB in a regression. I read many times that we need exogenity for an unbiased estimator. but this is not conclusive for me.

Let's say we have a lottery, where 50% wins so much money, that they never have to work again while the other half wins nothing at all.

After 2 months, we measure their happiness levels and find out, that the winner group is on average significantly happier than the losers of the lottery. Would you now conclude that this effect is causal? Clearly winning the lottery is exogenous...

I did not find an answer to this question on the internet, that's why I'm a bit confused. But in my opinion, we cannot conclude a causal relationship because there could be mediating variables that also get influenced by the predictor.

Let's say we control for time spent with family. In that case, the coefficient for the predictor could become small and insignificant. It's not the lottery that caused the happiness increase, but the increased time spent with the family, which was caused by the lottery. We have a significant association between lottery winning and time spent with family and between time spent with family and happiness.

So the indirect effect over the mediator (time spent with family) is significant while the direct effect from the predictor (lottery) to the dep. variable (happiness) is not.

If we did not include time spent with family, we would have a positive bias in the lottery coefficient ie. OVB.

Is this reasoning correct or what am I doing wrong?

Thanks for your help :)

Answer 1

Whenever someone talks about bias, you need to ask: "bias with respect to what"?

In order to decide whether you should or should not include a variable in a regression equation, you need to first be able to define what your target of inference is (e.g, a total effect, a direct effect, etc),

For instance, in your example, let $Y$ be happiness and $X$ be "winning a lof of money." Clearly, $E[Y|do(x)] = E[Y|x]$, and the total effect of winning money on happiness is identified in your setup, without adjusting for anything else, so long as that the lottery was indeed randomized.

Now let $Z$ be time spent with family. Suppose you want to know: does money have a direct effect on happiness, not through its effect on time spent with family? Now your target query is different, it is not the total effect anymore. For instance, to be more precise, suppose your study question is, "how much does happiness increase if someone wins the lottery, but we hold their time spent with family constant?" This question is given by the controlled direct effect $E[Y|do(x), do(z)]- E[Y|do(x'), do(z)]$.

Obviously, in general, $E[Y|do(x), do(z)]\neq E[Y|do(x)]$, that is, the average total effect in general will be different from a controlled direct effect. Thus to estimate the direct effect, you indeed need to "block" the mediation path due to $Z$.

However you cannot simply blindly add $Z$ to your regression, even when $X$ is randomized. Since $Z$ was not randomized, adjusting for $Z$ can actually induce bias, because there could be unmeasured confounders of $Z$ and $Y$. If that is the case, after including $Z$ in your regression you do not have a valid estimate neither of the total effect nor of the direct effect.

For details and more discussion on all this, read the Crash Course in Good and Bad Controls, especially Model 11 (reproduced below) which is very related to your question.