Suppose I have the following strucutural equation in time-series:
$$y_t=\beta_0+\beta_1x_{1t}+\beta_2x_{t2}+\zeta w_t+\epsilon_t \quad (1)$$
In which both $x_{t1}$ and $x_{t2}$ are endogenous variables and $w_t$ is an exogenous variable. Suppose that $z_{1t}$ and $z_{2t}$ are the elected instruments for a 2SLS estimation strategy, but you can only theoretically address one instrument to one endogenous variable, that is, $x_{t1}$ should be instrumented by $z_{t1}$ and $x_{t2}$ with $z_{t2}$. For example, that situation would arise if you're trying to estimate a demand with cross-price elasticities, in which one would have costs from different companies to instrument the different prices. One particular company cost should explain (instrument) the price of the same company.
In such a situation, a strategy in which you estimate a system of equations in the first stage, could be as follows:
$$x_{t1}=b_0+b_1z_{t1}+b_2w_t+e_t \quad(2)$$
$$x_{t2}=a_0+a_1z_{t2}+a_2w_t+u_t \quad(3)$$
By this system of equations, I believe that I cannot guarantee that the instruments won't be correlated with the residuals of both equations -- so, using the predicted values $\hat{x_{t1}}$ and $\hat{x_{t2}}$ in the structural equation would not be adequate. So, if this situations arises, what is the correct and more up-to-date approach?