Does adjusting for superfluous variables bias OLS estimates? The usual textbook treatment of adjusting for superfluous variables in OLS states that the estimator is still unbiased, but may have larger variance (see, for example, Greene, Econometric Analysis, 7th ed., p. 58). 
The other day I stumbled upon Judea Pearl's treatment of Simpson's Paradox and a nice webpage that simulates how "stepwise inclusion of control variables into a regression model switches the sign of an estimated causal association in every step". To me, this somehow contradicts the above statement. I feel this could be a very subtle (though incredibly important) problem, so any pointer to further literature would be very helpful. What especially strikes me is that Greene claims he has a proof for his assessment.
 A: Consider a postulated linear regression model
$$y_i = b_0 + b_1X_{1i} + b_2X_{2i} + u_i,\;\; i=1,...,n$$
As a matter of algebra (and not any stochastic assumptions), the OLS estimator in matrix notation is
$$\hat b = b + \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf u$$
Its expected value conditional on the regressor matrix is therefore
$$E\left(\hat b\mid \mathbf X\right) = b + \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'E\left(\mathbf u\mid\mathbf X \right)$$
So:
If "strict exogeneity" of the regressors with respect to the error term holds, or, in other words, if all error terms are mean-independent from all regressors,past present and future, (which is the benchmark assumption in the Classical Linear Regression model), i.e. if $E\left(\mathbf u\mid\mathbf X \right)=\mathbf 0$, we will have  
$$E\left(\hat b\mid \mathbf X\right) = b + \mathbf 0 \Rightarrow E(\hat b) = b$$
using also the law of iterated expectations.  
Given all the above, what "superfluous variable" means? I take it, it means "unrelated" to the dependent variable. But "unrelated" should be translated as "stochastically independent". But if it is independent from the dependent variable, it is necessarily independent from the error term (and so also strictly exogenous with respect to it), so all the above hold for any superfluous variable also, and the OLS estimator is unbiased even if, say, the variable $X_2$ is "superfluous" and the true model does not contain it.  
This is how econometricians understand the issue. Now, in a more general setting, "superfluous" could mean that say, $X_2$ is independent of $y$ conditional on the presence of $X_1$ (which I suspect is more close to what Pearl has in mind).  Still, as long as $X_2$ is strictly exogenous to the error term, the unbiasedness result holds.
A: There's no contradiction. 
The first paragraph there talks about superfluous variables.
If Simpson's paradox applies, the variables are not superfluous.
