Estimation and inference with weak instruments

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


This thesis contributes to the theory of estimation and inference of endogenous models with weak instruments. In Chapter 1, we discuss overidentification testing with weak instruments and heteroskedasticity. We derive limiting distributions of the J-test (Hansen, 1982) and the KP-test (Kleibergen & Paap, 2006), interpretable as 2SLS-based and LIML-based robust score tests respectively. This is the first application of the KP-test to overidentification testing. We find that KP typically outperforms J and can be considered a more reliable test under heteroskedastic weak instruments. Staiger and Stock (1997) suggest LIML-based inference in homoskedastic weak-instrument settings, and we recommend the same under heteroskedasticity. In Chapter 2, we apply the results of Chapter 1 to the macroeconomic problem of estimating the elasticity of intertemporal substitution (EIS) in consumption lifecycle models. We suggest that J frequently erroneously rejects the overidentifying restrictions, whereas KP does not suffer from this problem. This suggests that instrument invalidity is not a likely cause of the lack of consensus regarding estimates of sensible values of the EIS seen in the empirical macroeconomics literature. In Chapter 3, we develop a new characterisation of weak instruments in a nonparametric setting. With this characterisation, we derive rates of convergence of the Tikhonov-regularised and series truncation estimators, and derive the minimally impermissible rate of localisation - the instrument weakness bound for which estimators are not consistent. The degree to which instruments can be weak is dependent on the degree of ill-posedness and the smoothness of the structural function. With mildly weak instruments, researchers can still consistently estimate the structural function. We also find that series truncation is more robust to weak instruments than Tikhonov regularisation, in that the estimator is consistent in a wider range of models than Tikhonov when the joint density of the endogenous regressor and instrument is known.
Date of Award18 Jun 2024
Original languageEnglish
Awarding Institution
  • University of Bristol
SupervisorStefan Hubner (Supervisor) & Senay Sokullu (Supervisor)


  • instrumental variables
  • overidentification testing
  • weak identification
  • nonparametric estimation
  • causal inference

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