Mean dependence

In probability theory, a random variable $$Y$$ is said to be mean independent of random variable $$X$$ if and only if its conditional mean $$E(Y \mid X = x)$$ equals its (unconditional) mean $$E(Y)$$ for all $$x$$ such that the probability density/mass of $$X$$ at $$x$$, $$f_X(x)$$, is not zero. Otherwise, $$Y$$ is said to be mean dependent on $$X$$.

Stochastic independence implies mean independence, but the converse is not true. ; moreover, mean independence implies uncorrelatedness while the converse is not true. Unlike stochastic independence and uncorrelatedness, mean independence is not symmetric: it is possible for $$Y$$ to be mean-independent of $$X$$ even though $$X$$ is mean-dependent on $$Y$$.

The concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables ($$X_1 \perp X_2$$) and the weak assumption of uncorrelated random variables $$(\operatorname{Cov}(X_1, X_2) = 0).$$