7 - Conditional Expectation

Summary

Introduction

The concept of conditional expectation is fundamental in econometrics because regression models are statistical models of conditional expectation. We consider here conditional expectation with respect to a random vector, but it would be possible to define it more generally with respect to a σ-algebra.

We place ourselves in the Hilbert space L² of square integrable variables relative to a reference probability distribution: if z is a real random variable belonging to this space, then E(z²) < ∞. Recall that L² is a normed vector space on ℝ with the norm defined by ∥z∥ = [E(z²)]^1/2. If z₁ and z₂ are two elements of this space, then we can write their inner product as E(z₁z₂); these variables are said to be orthogonal in the sense of L² if E(z₁z₂) = 0. Moreover, we say that a sequence z_n in this space converges to a random variable z if ∥z_n − z∥ → 0 when n → +∞. This notion of orthogonality and of mean square convergence (we also say “in the sense of L²” or “in quadratic norm”) will allow us in the following to use the notions of orthogonal projection and best approximations in terms of least squares.

The usual demand on rigorousness requires to distinguish carefully between equality of random variables and almost sure equality. Here, we will not make this distinction, the mathematical readers can naturally add the necessary P^θ – a.s.