# Conditional variance

In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. That is, it is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of autoregressive conditional heteroskedasticity (ARCH) models.

## Definition

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

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

where E is the expectation operator with respect to the conditional distribution of Y given that the X takes the value x. An alternative notation for this is :$\operatorname{Var}_{Y\mid X}(Y|x).$

The above may be stated in the alternative form that, based on the conditional distribution of Y given that the X takes the value x, the conditional variance is the variance of this probability distribution.

## Components of variance

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 be 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. This expression represents the overall variance of Y as the sum of two components, involving a prediction of Y based on X. Specifically, let the predictor be the least-mean-squares prediction based on X, which is the conditional expectation of Y given X. Then the two components are:

• the average of the variance of Y about the prediction based on X, as X varies;
• the variance of the prediction based on X, as X varies.