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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),

see Conditioning in case of independence.

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).

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