2
Jan 17

## Conditional variance properties

### Preliminaries

Review Properties of conditional expectation, especially the summary, where I introduce a new notation for conditional expectation. Everywhere I use the notation $E_Y\pi$ for expectation of $\pi$ conditional on $Y$, instead of $E(\pi|Y)$.

This post and the previous one on conditional expectation show that conditioning is a pretty advanced notion. Many introductory books use the condition $E_xu=0$ (the expected value of the error term $u=0$ conditional on the regressor $x$ is zero). Because of the complexity of conditioning, I think it's better to avoid this kind of assumption as much as possible.

### Conditional variance properties

Replacing usual expectations by their conditional counterparts in the definition of variance, we obtain the definition of conditional variance:

(1) $Var_Y(X)=E_Y(X-E_YX)^2.$

Property 1. If $X,Y$ are independent, then $X-EX$ and $Y$ are also independent and conditioning doesn't change variance:

$Var_Y(X)=E_Y(X-EX)^2=E(X-EX)^2=Var(X),$

Property 2. Generalized homogeneity of degree 2: if $a$ is a deterministic function, then $a^2(Y)$ can be pulled out:

$Var_Y(a(Y)X)=E_Y[a(Y)X-E_Y(a(Y)X)]^2=E_Y[a(Y)X-a(Y)E_YX]^2$ $=E_Y[a^2(Y)(X-E_YX)^2]=a^2(Y)E_Y(X-E_YX)^2=a^2(Y)Var_Y(X).$

Property 3. Shortcut for conditional variance:

(2) $Var_Y(X)=E_Y(X^2)-(E_YX)^2.$

Proof.

$Var_Y(X)=E_Y(X-E_YX)^2=E_Y[X^2-2XE_YX+(E_YX)^2]$

(distributing conditional expectation)

$=E_YX^2-2E_Y(XE_YX)+E_Y(E_YX)^2$

(applying Properties 2 and 6 from this Summary with $a(Y)=E_YX$)

$=E_YX^2-2(E_YX)^2+(E_YX)^2=E_YX^2-(E_YX)^2.$

Property 4The law of total variance:

(3) $Var(X)=Var(E_YX)+E[Var_Y(X)].$

Proof. By the shortcut for usual variance and the law of iterated expectations

$Var(X)=EX^2-(EX)^2=E[E_Y(X^2)]-[E(E_YX)]^2$

(replacing $E_Y(X^2)$ from (2))

$=E[Var_Y(X)]+E(E_YX)^2-[E(E_YX)]^2$

(the last two terms give the shortcut for variance of $E_YX$)

$=E[Var_Y(X)]+Var(E_YX).$

Before we move further we need to define conditional covariance by

$Cov_Y(S,T) = E_Y(S - E_YS)(T - E_YT)$

(everywhere usual expectations are replaced by conditional ones). We say that random variables $S,T$ are conditionally uncorrelated if $Cov_Y(S,T) = 0$.

Property 5. Conditional variance of a linear combination. For any random variables $S,T$ and functions $a(Y),b(Y)$ one has

$Var_Y(a(Y)S + b(Y)T)=a^2(Y)Var_Y(S)+2a(Y)b(Y)Cov_Y(S,T)+b^2(Y)Var_Y(T).$

The proof is quite similar to that in case of usual variances, so we leave it to the reader. In particular, if $S,T$ are conditionally uncorrelated, then the interaction terms disappears:

$Var_Y(a(Y)S + b(Y)T)=a^2(Y)Var_Y(S)+b^2(Y)Var_Y(T).$