Let me write the model as $y=\beta_0+\beta_1x+\dots+\beta_px_p+e$, or $y=\mathbf{x}^T\boldsymbol{\beta}+e$, $\mathbf{x}=(1,x_1,\dots,x_p)$, $\boldsymbol{\beta}=(\beta_0,\dots,\beta_p)$.
$E[e\mathbf{x}]$ follows from $E[e\mid \mathbf{x}]=0$. In general, if $E[e\mid \mathbf{x}]=0$, then
- $E[e]=0$ by the law of total expectation:
$$E[e]=E[E[e\mid \mathbf{x}]]=E[0]=0$$
- $E[f(\mathbf{x})e]=0$, where $f(\mathbf{x})$ is an arbitrary finite valued function, by the same law: $E[f(\mathbf{x})e]=E[E[f(\mathbf{x})e\mid\mathbf{x}]]$, but when $\mathbf{x}$ is given, $f(\mathbf{x})$ is given too, so:
$$E[f(\mathbf{x})e]=E[E[f(\mathbf{x})e\mid\mathbf{x}]]=E[f(\mathbf{x})E[e\mid\mathbf{x}]]=0$$
- $E[\mathbf{x}e]=0$: let $f$ be the identity function.
- $E[y\mid\mathbf{x}]=\mathbf{x}^T\boldsymbol{\beta}$.
If $E[e]=0$ and $E[\mathbf{x}e]=0$, then $\mathbf{x}$ and $e$ are uncorrelated:
$$\text{Cov}(\mathbf{x},e)=E[\mathbf{x}e]-E[\mathbf{x}]E[e]=0$$
What does it mean to talk about the expectation of the product of the
error term and an independent variable? Like, why do we even need to
mention E(eiXik)? What is it actually describing or what is the
intuition behind it?
In other words: what happens if $\mathbf{x}$ and $e$ are not uncorrelated?
Let's say that the 'true' model is:
$$y=\beta_0+\beta_1x_1+\beta_2x_2+e$$
but your model is:
$$y=\beta_0+\beta_1x_1+u$$
where $u=\beta_2x_2+e$.
If $x_1$ and $x_2$ are correlated, say $x_2=cx_1$, then:
- $E[u]=E[\beta_2x_2+e]=E[\beta_2cx_1+e]=\beta_2cE[x_1]\ne 0$
- $E[y\mid x_1]\ne \beta_0+\beta_1x_1$
- $\hat\beta_0$ and $\hat\beta_1$ will be biased and inconsistent.
