I refer here to a classical linear regression whose true representation is given by the equation: $y_i=x_i'\beta+u_i$,
where as usual $x_i$ is a $K\times 1$ vector of independent explanatory variables, $\beta$ is a $K\times 1$ vector of parameters and $u_i$ the error term of the ith observation and by construction of the true representation $u_i$ is considered to be independent of $x_i$.
My first question on that is basic: If we assume independence between $x_i$ and $u_i$ does this imply $E[u_i^d|x_i]=E[u_i^d]$ for $d=1,2,..$ - so to speak, can we generalize the Law of iterated expectations consequently to all moments?
Secondly, what is the basis of many textbooks to claim that $x_i$ and $u_i$ are independent in the classical regression model? By independence, I don't mean uncorrelatedness, rather $f(u,x)=f(u)f(x)$. I am aware that the two key standard assumptions on which this small sample properties model is based on is strict exogenity $E[u_i|X]=0$, and normally distributed error terms $u_i|x_i \thicksim N(0, \sigma^2)$. Where does the claim of the independence between $x_i$ and $u_i$ originates from in textbooks? Is it implied by these two statements jointly together or is this a separate assumption? If the latter, why is it not mentioned exclusively in the set of assumptions for classical linear regression models which only assumes strict exogenity and normally distributed errors?