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 $(\mathrm{Cov}(X_1,X_2)=0).$

Further reading

• Cameron, A. Colin; Trivedi, Pravin K. (2009). Microeconometrics: Methods and Applications (8th ed.). New York: Cambridge University Press. ISBN 9780521848053. 
• Wooldridge, Jeffrey M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). London: The MIT Press. ISBN 9780262232586.