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Weak instruments refers to a scenario where the instrumental variables are only weakly correlated with the endogenous explanatory variables. Why would that be a problem? So the weak instruments case here it concerns this correlation between X and Z. If X and Z are not strongly correlated then instrumental variable estimation can be very imprecise and it can be biased. To understand why that is the case lets take a look at the equation for calculating the instrumental variable estimate for beta here. So thats the equations. We can see that beta is this correlation between Z and Y divided by correlation X and Z and why thats the case? Well if you consider what is the model implied correlation between Z and Y it is the Z X correlation times this beta regression coefficient here and from there you can solve the beta. So why would it be a problem with this correlation is very small? Its a problem because then the sampling variation of that correlation - whatev