On the relevance of weaker instruments

ABSTRACT

We study the asymptotic properties of the standard GMM estimator when additional moment restrictions, weaker than the original ones, are available. We provide conditions under which these additional weaker restrictions improve the efficiency of the GMM estimator. To detect “spurious” identification that may come from invalid moments, we rely on the Hansen J-test that assesses the compatibility between existing restrictions and additional ones. Our simulations reveal that the J-test has good power properties and that its power increases with the weakness of the additional restrictions. Our theoretical characterization of the J-test provides some intuition for why that is.

KEYWORDS:

• GMM
• J-test
• local misspecification
• redundancy
• weak IV

JEL CLASSIFICATION:

• C32
• C12
• C13
• C51

Notes

¹See Phillips and Moon (1999) for a general discussion of sequential aymptotics.

²The terminology “identification strength” was introduced by Kleibergen and Mavroeidis (2007).

³Regularity assumptions are the same as the ones needed for the validity of the J-test; see the discussion after Theorem 1 above and Theorem 4.4 in Antoine and Renault (2012).

⁴The intuition is that the J-test has more power when well focused on the cause of misspecification. This case is strikingly similar to the predictive test of structural change of Ghysels and Hall (1990), which looks like an overidentification test in the postbreak period but using the GMM estimator of the prebreak period.

⁵In this homoskedastic framework, the GMM estimators correspond to the two-stage least square estimators.

⁶We reproduce here some of the results of Hahn et al. (2011): see their Tables 1–3.