If we assume we have 2 equations and each equation contains the other dependent variable.
$y_1 = \beta_0 + \beta_1 y_2 + \beta_2 z_1 + u_1$
$y_2 = \alpha_0 + \alpha_1 y_1 + \alpha_2 z_2 + u_2$
For exact identification, we would need two instruments. Let's say these instruments are $z_3$ and $z_4$.
The first stage of TSLS is equal to the regression of the assumed endogenous variables $y_1$ and $y_2$ on the matrix of instruments $W$ which contains all the instruments and exogenous variables. So the columns of the matrix of instruments would look like
$W = [z_1, z_2, z_3, z_4]$
and the first stage of TSLS like
$y_1 = W\pi_1 + v_1$
$y_2 = W\pi_2 + v_2$
For identification, I need the significance of exact one of those instruments, i.e., of $z_3$ and $z_4$ in each of those reduced form equations.
My question is: does matrix $W$ indeed contain both instruments or do we regress onto $W_1=[z_1, z_2, z_3]$ and $W_2=[z_1, z_2, z_4]$ if we take $z_3$ as a instrument for $y_1$ and the instrument variable $z_4$ for the endogenous variable $z_4$?
Is that right that we would need at least 3 instruments if we observe something like?
$y_1 = \beta_0 + \beta_1 y_2 + \beta_2 y_3 + \beta_3 z_1 + u_1$
$y_2 = \alpha_0 + \alpha_1 y_1 + \alpha_2 y_3 + \alpha_3 z_2 + u_2$
$y_3 = \gamma_0 + \gamma_1 y_1 + \gamma_2 y_2 + \gamma_3 z_3 + u_3$
so that the matrix of instruments would look like
$W = [z_1, z_2, z_3, z_4, z_5, z_6]$
if $z_4, z_5$ and $z_6$ are our instruments.