Raising the bar

Law of iterated expectations: geometric aspect

There will be a separate post on projectors. In the meantime, we'll have a look at simple examples that explain a lot about conditional expectations.

Examples of projectors

The name "projector" is almost self-explanatory. Imagine a point and a plane in the three-dimensional space. Draw a perpendicular from the point to the plane. The intersection of the perpendicular with the plane is the points's projection onto that plane. Note that if the point already belongs to the plane, its projection equals the point itself. Besides, instead of projecting onto a plane we can project onto a straight line.

The above description translates into the following equations. For any $latex x\in R^3$ define

(1) $latex P_2x=(x_1,x_2,0)$ and $latex P_1x=(x_1,0,0).$

$latex P_2$ projects $latex R^3$ onto the plane $latex L_2=\{(x_1,x_2,0):x_1,x_2\in R\}$ (which is two-dimensional) and $latex P_1$ projects $latex R^3$ onto the straight line $latex L_1=\{(x_1,0,0):x_1\in R\}$ (which is one-dimensional).

Property 1. Double application of a projector amounts to single application.

Proof. We do this just for one of the projectors. Using (1) three times we get

(1) $latex P_2[P_2x]=P_2(x_1,x_2,0)=(x_1,x_2,0)=P_2x.$

Property 2. A successive application of two projectors yields the projection onto a subspace of a smaller dimension.

Proof. If we apply first $latex P_2$ and then $latex P_1$, the result is

(2) $latex P_1[P_2x]=P_1(x_1,x_2,0)=(x_1,0,0)=P_1x.$

If we change the order of projectors, we have

(3) $latex P_2[P_1x]=P_2(x_1,0,0)=(x_1,0,0)=P_1x.$

Exercise 1. Show that both projectors are linear.

Exercise 2. Like any other linear operator in a Euclidean space, these projectors are given by some matrices. What are they?

The simple truth about conditional expectation

In the time series setup, we have a sequence of information sets $latex ... \subset I_t \subset I_{t+1} \subset...$ (it's natural to assume that with time the amount of available information increases). Denote

$latex E_tX=E(X|I_t)$

the expectation of $latex X$ conditional on $latex I_t$. For each $latex t$,

Property 1. Double application of conditional expectation gives the same result as single application:

(4) $latex E_t(E_tX)=E_tX$

($latex E_tX$ is already a function of $latex I_t$, so conditioning it on $latex I_t$ doesn't change it).

Property 2. A successive conditioning on two different information sets is the same as conditioning on the smaller one:

(5) $latex E_tE_{t+1}X=E_tX,$

(6) $latex E_{t+1}E_tX=E_tX.$

Property 3. Conditional expectation is a linear operator: for any variables $latex X,Y$ and numbers $latex a,b$

$latex E_t(aX+bY)=aE_tX+bE_tY.$

It's easy to see that (4)-(6) are similar to (1)-(3), respectively, but I prefer to use different names for (4)-(6). I call (4) a projector property. (5) is known as the Law of Iterated Expectations, see my post on the informational aspect for more intuition. (6) holds simply because at time $latex t+1$ the expectation $latex E_tX$ is known and behaves like a constant.

Summary. (4)-(6) are easy to remember as one property. The smaller information set wins: $latex E_sE_tX=E_{\min\{s,t\}}X.$