A weak instrument [Formula: see text]-test in linear IV models with multiple endogenous variables

Area of Science:

• Econometrics
• Statistical Inference

Background:

• Weak instruments pose challenges in econometrics, particularly with multiple endogenous variables.
• Existing methods like Stock and Yogo (2005) address rank deficiency, but a near rank reduction of one requires specific attention.

Purpose of the Study:

• To develop a robust method for testing weak instruments in models with multiple endogenous variables.
• To investigate the properties of conditional first-stage F-statistics in the presence of weak instruments.

Main Methods:

• Analysis of weak instrument asymptotics for near rank reduction of one.
• Adaptation of the conditional first-stage F-statistic proposed by Angrist and Pischke (2009).
• Comparison with the Cragg and Donald (1993) minimum eigenvalue rank test.

Main Results:

• The variance in the denominator of the Angrist-Pischke F-statistic requires adjustment for correct asymptotic distribution when testing specific hypotheses.
• A corrected conditional F-statistic is shown to be equivalent to the Cragg-Donald statistic.
• The corrected statistic provides insights into the bias of 2SLS estimators and size distortions of Wald tests.

Conclusions:

• The corrected conditional first-stage F-statistic is a valuable tool for weak instrument testing.
• In cases with more than two endogenous variables or specific parameter structures, these statistics offer additional information beyond the Cragg-Donald statistic.