Prediction of the 1st stage in 2SLS not important? My question was sparked by the discussion Why is the functional form of the 1st stage in 2SLS not important?
I understand that in the first-stage we aim to predict our endogenous variable ($\hat{x}$), which is now supposed to be uncorrelated with the error term of the second stage. However, do we care how accurate the prediction of $\hat{x}$ actually is? Do the predicted values in the first stage have any interpretation? Can the predicted value $\hat{x}$ be evaluated against the actual values of the instrumented variable?
Is there any literature that compares instruments based on predicted values in the first stage to determine whether the instrument is strong/weak?

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*Do we care how accurate the prediction of x^ actually is?

Yes, we definitely do. The instruments must be valid for the model to be considered acceptable. Any unnecessary instruments will not predict the x^ accurately. To check if the instruments are valid (predicting well), you can apply the Sargan Test. This test will also indicate if any of the instruments belong in the second stage (actual equation to be estimated).


*Do the predicted values in the first stage have any interpretation?

*Can the predicted value x^ be evaluated against the actual values of the instrumented variable?

You could say that the first stage removes the bias from the model caused due to endogeneity. The first stage attempts to capture the exogenous effects and remove any endogenous effects from the instrumented variable. If the instruments are valid, you could consider it as the exogenous effects of 'x' on the model equation. So, the x^ used in the second stage is supposed to represent the exogenous effects.
I hope this helps give you some intuition into the 2SLS stages.
