Law of total covariance

In probability theory, the law of total covariance,[1] covariance decomposition formula, or ECCE states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then

$\operatorname{cov}(X,Y)=\operatorname{E}(\operatorname{cov}(X,Y \mid Z))+\operatorname{cov}(\operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z)).\,$

The nomenclature in this article's title parallels the phrase law of total variance. Some writers on probability call this the "conditional covariance formula"[2] or use other names.

(The conditional expected values E( X | Z ) and E( Y | Z ) are random variables in their own right, whose values depends on the value of Z. Notice that the conditional expected value of X given the event Z = z is a function of z (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is just g(Z). Similar comments apply to the conditional covariance.)

Proof

The law of total covariance can be proved using the law of total expectation: First,

$\operatorname{cov}[X,Y] = \operatorname{E}[XY] - \operatorname{E}[X]\operatorname{E}[Y]$

from the definition of covariance. Then we apply the law of total expectation by conditioning on the random variable Z:

$= \operatorname{E}[\operatorname{E}[XY\mid Z]] - \operatorname{E}[\operatorname{E}[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]]$

Now we rewrite the term inside the first expectation using the definition of covariance:

$= \operatorname{E}\!\left[\operatorname{cov}[X,Y\mid Z] + \operatorname{E}[X\mid Z]\operatorname{E}[Y\mid Z]\right] - \operatorname{E}[\operatorname{E}[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]]$

Since expectation of a sum is the sum of expectations, we can regroup the terms:

$= \operatorname{E}\!\left[\operatorname{cov}[X,Y\mid Z]] + \operatorname{E}[\operatorname{E}[X\mid Z]\operatorname{E}[Y\mid Z]\right] - \operatorname{E}[\operatorname{E}[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]]$

Finally, we recognize the final two terms as the covariance of the conditional expectations E[X|Z] and E[Y|Z]:

$= \operatorname{E}(\operatorname{cov}(X,Y \mid Z))+\operatorname{cov}(\operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z))$

Notes and references

1. ^ Matthew R. Rudary, On Predictive Linear Gaussian Models, ProQuest, 2009, page 121.
2. ^ Sheldon M. Ross, A First Course in Probability, sixth edition, Prentice Hall, 2002, page 392.