Mean dependence

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

Stochastic independence implies mean independence, but the converse is not true.[1][2]; 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 ${\displaystyle Y}$ to be mean-independent of ${\displaystyle X}$ even though ${\displaystyle X}$ is mean-dependent on ${\displaystyle Y}$.

The concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables (${\displaystyle X_{1}\perp X_{2}}$) and the weak assumption of uncorrelated random variables ${\displaystyle (\operatorname {Cov} (X_{1},X_{2})=0).}$