Mean dependence

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

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

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.