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How does one check the validity of instruments for a just-identified model, since the Sargan test can only be performed if the model is overidentified?

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In a just-identified model you cannot statistically test the validity of your instruments. You are forced to rely on your subject-matter judgment regarding the exclusion restriction. As you point out, the Sargan test statistic is exactly 0 under just-identification, so useless as a test statistic.

The situation is in some sense similar to OLS - OLS is such that residuals and regressors are by construction orthogonal to each other (i.e., $X'\hat{u}=0$), so you cannot use that lack of correlation to argue that the assumption required for causal interpretation of OLS coefficients, namely $E(X'u)=0$, does hold.

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