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 | X=x) equals its (unconditional) mean E(Y) for all x such that the probability that X = x is not zero. Y is said to be mean dependent on X if E(Y | X=x) is not constant for all x for which the probability is non-zero.

According to Cameron and Trivedi (2009, p. 23) and Wooldridge (2010, pp. 54, 907), stochastic independence implies mean independence, but the converse is not true.

Moreover, mean independence implies uncorrelatedness while the converse is not true.

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 $(\operatorname {Cov} (X_{1},X_{2})=0).$

References

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.

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