What does the "relevance" condition for a valid instrumental variable
𝑧
(used for an endogenous variable
𝑥
) require?

Question

What does the "relevance" condition for a valid instrumental variable 
𝑧
 (used for an endogenous variable 
𝑥
) require?

BlackTom AI · Accepted Answer

To start, let's consider what the relevance condition of an instrumental variable means in the context of IV estimation. It requires that the instrument z be related in some way to the endogenous regressor x, so that z can help explain variation in x that is not captured by other exogenous factors.
Option 1: 'The instrument z must be uncorrelated with the dependent variable y.' This describes a lack of direct association with the outcome, which is not the relevance criterion. In fact, an instrument can be correlated with y only through x, but the key requirement is the z–x relationship, not a blanket statement about y.
Option 2: 'The instrument z must be partially correlated with the endogenous variable x ( Cov(z, x | other exogenous) ≠ 0 ).' This aligns with the standard relevance condition: there must be some nonzero covariance between z and x after accounting for other exogenous controls, ensuring z carries information about x.
Option 3: 'The instrument z must be uncorrelated with the error term u ( Cov(z, u) = 0 ).' While this describes exogeneity, it is the exogeneity condition, not the relevance condition. An instrument should be exogenous to avoid bias, but this statement does not capture the required link to x that makes z useful.
Option 4: 'The instrument z must be uncorrelated with any other instruments used.' There is no general requirement that instruments be uncorrelated with each other. Some correlation between instruments can occur, and in some designs it is even addressed during estimation; the core relevance criterion does not hinge on inter-instrument correlations.
By evaluating each option, we see that the essential and correctly framed requirement is the partial correlation between z and the endogenous x, conditional on other exogenous variables. This is exactly what option 2 states.

What does the "relevance" condition for a valid instrumental variable 𝑧 (used for an endogenous variable 𝑥 ) require?

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