A Robust Test for Weak Instruments for 2SLS with Multiple Endogenous Regressors

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a test for instrument strength based on the bias of two-stage least squares (2SLS) that: (1) generalizes the Stock-Yogo (2005) and Sanderson-Windmeijer (2016) tests to be robust to heteroskedasticity and autocorrelation (HAC), and (2) extends the Montiel Olea-Pflueger (2013) robust test from models with a single endogenous regressor to models with multiple endogenous regressors—the important remaining gap identified by Andrews et al. (2019). The test is based on a weighted quadratic loss in the asymptotic bias of 2SLS and can use either the Stock-Yogo absolute bias criterion or the 2SLS bias relative to Montiel Olea-Pflueger’s worst-case benchmark. Extensions are developed to test whether instruments are weak for individual 2SLS coefficients. In simulations, the test controls size and is powerful, and the authors provide efficient code packages. The test is applied to state-dependent fiscal multipliers (Ramey-Zubairy 2018).

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the key gap in the existing weak instrument testing literature that this paper fills?

The key gap is the absence of a test for weak instruments that is both HAC robust and applicable to models with multiple endogenous regressors. Stock-Yogo (2005) requires conditionally homoskedastic and serially uncorrelated (CHSU) errors. Montiel Olea-Pflueger (2013) introduced a HAC-robust effective F-statistic for a single endogenous regressor but their test does not extend to multiple regressors. Sanderson-Windmeijer (2016) addressed multiple endogenous regressors but retained the CHSU assumption. This paper combines HAC robustness with multiple-regressor generality, filling the gap Andrews et al. (2019) identify as the most important remaining open problem in the literature.

Q2. What is the test statistic and what are its two bias criteria?

The test statistic is based on a weighted quadratic loss in the asymptotic bias of the 2SLS estimates when first-stage coefficients are close to zero, with two criteria: (i) the absolute bias criterion of Stock-Yogo (2005)—the 2SLS bias relative to the maximum OLS bias; and (ii) the 2SLS bias relative to Montiel Olea-Pflueger’s (2013) worst-case benchmark. The test accommodates both the Stock-Yogo setting (instruments weak because the first-stage coefficient matrix is near rank zero) and the Sanderson-Windmeijer setting (instruments weak because the first-stage coefficient matrix is near having a rank reduction of one rather than near rank zero).

Q3. What extensions are provided for individual coefficient testing?

Extensions are developed to test whether instruments are weak for individual 2SLS coefficients, by applying the test to a transformed regression that isolates the coefficient of interest, accommodating the Sanderson-Windmeijer (2016) setting in which one regressor is locally under-identified while others may not be. This is important in practice because researchers with multiple endogenous regressors often care about whether instruments are weak for each coefficient separately, not just for the system as a whole; the extension provides a formal basis for this common applied practice.

Q4. What does the empirical application show?

The paper demonstrates the testing procedures in the context of estimating state-dependent fiscal multipliers as in Ramey and Zubairy (2018), where the two endogenous regressors are lagged spending interacted with a state variable (recession/expansion indicator), illustrating both the implementation of the test and how inference differs from relying on CHSU-based critical values. In simulations, the test controls size accurately and is powerful against alternatives where instruments are strong, providing a reliable and practically useful tool with efficient code packages distributed for applied researchers.

Key concepts

weak instruments test : a test assessing whether the first-stage regression is sufficiently strong to make 2SLS inference reliable; based on the maximum bias of 2SLS relative to a benchmark; weak instruments cause 2SLS to inherit the bias of OLS. HAC robustness : robustness to heteroskedasticity and autocorrelation; absent from Stock-Yogo (2005), meaning researchers who use their critical values while allowing for HAC errors in second-stage inference apply mismatched validity assumptions. effective F-statistic : the statistic introduced by Montiel Olea and Pflueger (2013) for HAC-robust weak instruments testing with a single endogenous regressor; generalized in this paper to the multiple-regressor setting. absolute bias criterion : the criterion that the 2SLS relative bias (standardized absolute bias) is below a threshold; equivalently, the 2SLS bias as a proportion of the maximum OLS bias; defined by Stock-Yogo (2005) and generalized here to the HAC-robust multi-instrument setting.