Law of total covariance

In probability theory, the law of total covariance,^[1] covariance decomposition formula, or conditional covariance formula 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

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

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.

Note: The conditional expected values E( X | Z ) and E( Y | Z ) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is 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,

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

from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z:

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

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

${\displaystyle =\operatorname {E} \!{\big [}\operatorname {cov} (X,Y\mid Z)+\operatorname {E} [X\mid Z]\operatorname {E} [Y\mid Z]{\big ]}-\operatorname {E} {\big [}\operatorname {E} [X\mid Z]{\big ]}\operatorname {E} {\big [}\operatorname {E} [Y\mid Z]{\big ]}}$

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

${\displaystyle =\operatorname {E} \!{\big [}\operatorname {cov} (X,Y\mid Z){\big ]}+\operatorname {E} {\big [}\operatorname {E} [X\mid Z]\operatorname {E} [Y\mid Z]{\big ]}-\operatorname {E} {\big [}\operatorname {E} [X\mid Z]{\big ]}\operatorname {E} {\big [}\operatorname {E} [Y\mid Z]{\big ]}}$

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

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

See also

• Law of total variance, a special case corresponding to X = Y.
• Law of total cumulance, of which the law of total covariance is a special case.

Notes and references