This is another question from my self-study of Hayashi's Econometrics.
How do we show in mathematical proof that given:
$X = \begin{bmatrix}x_{1}' \\x_{2}' \\\vdots \\x_{n}'\end{bmatrix}$ where $x_{i} = \begin{bmatrix}x_{i1} \\x_{i2} \\\vdots \\x_{ik} \end{bmatrix} $
and $y = \begin{bmatrix}y_{1}' \\y_{2}' \\\vdots \\y_{n}'\end{bmatrix}$
If $(y, X)$ is a random sample, where $(y_{i},x_{i})$ and $(y_{j},x_{j})$ are i.i.d. for $i\neq j$, and $\epsilon_{i}$ is a function of $(y_{i},x_{i})$, then the assumption:
$E[\epsilon_{i}|X]$ = $E[\epsilon_{i}|x_{i}]$
and
$E[\epsilon_{i}^2|X]$ = $E[\epsilon_{i}^2|x_{i}]$ ?
I mean it looks intuitively correct, but I checked the probability text and couldn't find a matching theorem corresponding to this.
The original text from the book Econometrics Hayashi is quoted:
Thank you very much.
Update of the proof from my cousin:
$E[\epsilon_{i}|X]$ = $E[\epsilon_{i}|x_{i}]$
$E[\epsilon_{i}|x_{1},x_{2}, \ldots,x_{n}] = \int_{-\infty}^{\infty} \epsilon_{i} \cdot f_{\epsilon_{i}|X}(\epsilon_{i}|x_{1},x_{2}, \ldots,x_{n})d\epsilon_{i}$
$= \int_{-\infty}^{\infty} \epsilon_{i} \cdot \dfrac{f(\epsilon_{i},x_{1},x_{2}, \ldots,x_{n})}{f(x_{1},x_{2}, \ldots,x_{n})} d\epsilon_{i}$
$= \int_{-\infty}^{\infty} \epsilon_{i} \cdot \dfrac{f(\epsilon_{i},x_{1},x_{2}, \ldots,x_{n})}{f(x_{1})f(x_{2}) \cdots f(x_{n})} d\epsilon_{i}$
$= \int_{-\infty}^{\infty} \epsilon_{i} \cdot \dfrac{f(\epsilon_{i},x_{i})f(x_{1})f(x_{2}) \cdots f(x_{n})}{f(x_{i})f(x_{1})f(x_{2}) \cdots f(x_{i-1})f(x_{i+1})f(x_{n})} d\epsilon_{i}$
$= \int_{-\infty}^{\infty} \epsilon_{i} \cdot \dfrac{f(\epsilon_{i},x_{i})}{f(x_{i})} d\epsilon_{i}$
$= \int_{-\infty}^{\infty} \epsilon_{i} \cdot f(\epsilon_{i}|x_{i}) d\epsilon_{i}$
$= E[\epsilon_{i}|x_{i}]$
QED
I really appreciate the help from StackExchange Mathematics. This question is more related to econometrics in Cross Validation section. I post the same question there but got downvoted for no reason and was held on hold. Maybe my questions are naiive to some of you, but I really appreciate it your tolerance and kindness in helping me understand some of these theorems/proofs.