Conditional variance

The conditional variance describes in probability theory and statistics, the variance of a random variable under the condition that additional information about the outcome of the underlying random experiment are available. It is defined as the conditional expected value of the square deviation of the random variables of its conditional expectation. As in this example, the condition may be that it is known whether a certain event has occurred or which values ​​a further random variable has accepted; abstract the additional information can be seen as sub-space of the underlying event space.

An important application is the variance decomposition, a formula can be represented by the variances of conditional variances and conditional expectations and also plays a role in the regression analysis. Time series models such as ARCH or GARCH generalization whose use conditional variances to target to model stochastic dependencies in processes as they occur mainly in mathematical questions.

Definition

Let and be two real random variable on a probability space, then that means

The conditional variance of given (or variance of conditional on ).

Similar to the conditional expectation one considers the conditional variances

  • Given an event
  • Given that takes the value,

And generally

  • Been a part - σ - algebra.

For this purpose, in the definition of the two expected values ​​of each are on, or conditionally.

In the following, all formulas are given only for the condition to another random variable, for the other cases they shall apply accordingly. However, it is to note that, and non-negative real numbers (or ), while it is in, and to the random variable. All of the following equations and inequalities for the latter are due to the non-uniqueness of the conditional expectation values ​​to be understood as -almost surely, without this being explicitly stated.

Simple calculation rules

From the ( unconditional ) variance analogous definition is the result together with the rules for computing conditional expectations that the rules for computing variances continue to apply mutatis mutandis. In particular, we have:

  • Non-negativity:
  • Affine transformations for all
  • Displacement law:

Variance decomposition

An important statement in connection with the conditional variance is the variance decomposition ( also set called of the total variance), after which the (unconditional ) variance of a random variable is the sum of the expected value of its conditional variance, and the variance of its conditional expected value:

Autoregressive conditional heteroskedasticity Mathematical finance
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