In a regression model, can we use $E(u)=0$ as the moment condition? Consider the linear model with stochastic regressors:
$$y_t = \beta_0^\prime x_t + u_t, \quad E(u_t | x_t)= 0$$
So that $E(u_t | x_t)= \beta_0^\prime x_t $. Using the e Law of Iterated Expectations (LIE) we find:
$$E(x_t u_t ) = E( x_t (y_t - \beta_0^\prime x_t)) = 0$$
that are the moment conditions for this model.
By similiar argument using LIE, we have
$$E(u_t) = E(y_t - \beta_0^\prime x_t) =0 $$
So my question is: Why we can't use this equation as the moment condition?
 A: As you have demonstrated here, the conditional moment condition is stronger than the marginal moment condition.  The conditional moment condition implies the marginal moment condition, but the converse is not true.  If you were to use the (weaker) marginal moment condition then it would only mean that the error terms average out to zero only when taken across the range of explanatory variables (under a stipulated distribution for these).  This is a very weak assumption which would effectively remove the "linearity" assumption in the regression; it would not adequately constrain the conditional moments of the error term in the model.
To give an example of how silly this might look, consider the following model:
$$y_t = \beta x_t + u_t
\quad \quad \quad 
x_t \sim \text{U}(0, 2\pi)
\quad \quad \quad 
u_t = \alpha \sin(x_t).$$
In this regression model the "error term" $u_t$ is a deterministic function of the explanatory variable $x_t$, but it also clearly has marginal expectation $\mathbb{E}(u_t)=0$ (when averaged over the distribution of this explanatory variable).
