What is meant by $E(\epsilon_{i} \mid \boldsymbol{X}_{i})=0$?

Question

I'm studying OLS estimator in econometrics and have a problem as follows:

Here $\epsilon_i, \textbf X_i$ are random variables.

Usually, I see $E(\epsilon_{i} \mid X_{i} = x)$. which is the expectation of $\epsilon_{i}$, provided that $X_{i} = x$. As such, the sentence $E(\epsilon_{i} \mid X_{i})$ is a random function of $X_i$ makes sense to me.

If we have $E(\epsilon_{i} \mid X_{i} = x)$ is a constant for all $x$, then should it be said that "we require this function to be a constant function of value zeros" rather than "we require this function to be a constant of value zero".

Could you please elaborate more on this point? Thank you so much!

Answer 1

Subjective answer.

My formulation would be: "we require this function to be constant, taking value $0$ everywhere".

Secondly I would not say that $\mathbb E(\epsilon_i\mid X_i)$ is a random function of $X_i$, but that $\mathbb E(\epsilon_i\mid X_i)$ is a random variable. This with the special property that we can write it as $f(X_i)$ where $f:\mathbb R\to\mathbb R$ is a Borel-measurable function.

A random variable is by definition a function, but not a random function in my terminology.

Random functions are a chapter apart and I would speak of a random function if we are dealing with e.g. a function $\mathbb R\times\Omega:\to\mathbb R$ or $[0,\infty)\times\Omega:\to\mathbb R$ as in Poisson processes.

To be discerned are two functions.

• 1) the function $\Omega\to\mathbb R$ prescribed by $\omega\mapsto X_i(\omega)$. This function is a random variable.
• 2) the function $\mathbb R\to\mathbb R$ prescribed by $x\mapsto \mathbb E(\epsilon_i\mid X_i=x)$. This function is not a random variable.

The second one is a constant function in this context and is actually prescribed by $x\mapsto0$.