There is "a general phenomenon known as Berkson’s paradox (Berkson, 1946), whereby observations on a common consequence of two independent causes render those causes dependent. For example, the outcomes of two independent coins are rendered dependent by the testimony that at least one of them is a tail" (Pearl, 2009).
$$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon $$
If (a) we make a linear regression model as above in which $x_2$ is a common consequence of independent causes $y$ and $x_1$ and (b) these latter two are rendered dependent by observations on $x_2$:
- Is $\hat\beta_1 \ne 0$? Why?
- "Yes, because that is what Pearl must mean by "rendered dependent."
- Is the true value of $\beta_1 = 0$? Why?
- "Yes, because that is what Pearl must mean by "independent causes."
*Or, do we not have enough information?
*Or, is Berkson's paradox not applicable (e.g., to a linear regression model with a continuous $y$)?
Berkson, J. (1946). Limitations of the application of fourfold table analysis to hospital data. Biometrics Bulletin 2, 47–53.