To obtain the method of moment estimate for instrument variables, we use the moment condition $z'\varepsilon=0$ in the exact identified case (number of endogenous variables = number of instruments) or $\hat{x}'\varepsilon=0$ in the overidentified case where the hat represents ttted value of the endogenous variable in a first stage regression on the instrumets. We use the sample counterpart and replace the true errors by the residuals. In the former case, we obtain $\hat{\beta}_{IV}=(z'x)^{-1}(z'y)$ and in the latter $\hat{\beta}_{2\mathrm{SLS}}=(\hat{x}'\hat{x})^{-1}(\hat{x}'y)$. The intuition for the usual Sargan test of overidentifying restrictions is that we regress the endogenous variable on the instruments in the first stage, obtain the predicted value of the endogenous variables, use these predicted values in the second stage on the RHS and obtain the (possibly) unbiased and conistent estimate of our parameters. We then use this estimate of the paramaters to obtain unbiased estimates of the true errors (the residuals in the second stage) and run the following regression: $e=\alpha_0+\alpha_1 z_1+\cdots+\alpha_k z_k+u_k$ where $e$ represent the residuals (unbiased estimates of the true errors) and $z$ represent the various instruments.
We then run a usual $NR^2$ test to test the joint significance of the instruments. If instruments are valid, then we should not be able to reject the null of orthogonality. My question is that shouldnt this always be the case by construction? After all, we use this specific moment condition in the first place! Any help is much appreciated.
