Conditional variance

In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of ARCH models.

The conditional variance of a random variable Y given that the value of a random variable X takes the value x is

$\operatorname {Var} (Y|X=x)\equiv \operatorname {E} ((Y-\operatorname {E} (Y\mid X=x))^{2}\mid X=x),$

where E is the expectation operator. An alternative notation for this is :$\operatorname {Var} _{Y\mid X}(Y|x).$

The law of total variance says

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

where, for example, $\operatorname {Var} (Y|X)$ is understood to mean that the value x at which the conditional variance would is evaluated is allowed to be a random variable, X. In this "law", the inner expectation or variance is taken with respect to Y conditional on X, while the outer expectation or variance is taken with respect to X.