18
Oct 18

## 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 $x\in R^3$ define

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

$P_2$ projects $R^3$ onto the plane $L_2=\{(x_1,x_2,0):x_1,x_2\in R\}$ (which is two-dimensional) and $P_1$ projects $R^3$ onto the straight line $L_1=\{(x_1,0,0):x_1\in R\}$ (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) $P_2[P_2x]=P_2(x_1,x_2,0)=(x_1,x_2,0)=P_2x.$

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

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

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

If we change the order of projectors, we have

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

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 $...\subset I_t\subset I_{t+1}\subset...$ (it's natural to assume that with time the amount of available information increases). Denote

$E_tX=E(X|I_t)$

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

 $E_t$$E_t$ is a projector onto the space of random functions that depend only on the information set $I_t$$I_t$.

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

(4) $E_t(E_tX)=E_tX$

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

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

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

(6) $E_{t+1}E_tX=E_tX.$

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

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

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 $t+1$ the expectation $E_tX$ is known and behaves like a constant.

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