Weak-instrument-robust subvector inference in instrumental variables regression: A subvector Lagrange multiplier test and properties of subvector Anderson-Rubin confidence sets

Abstract

We propose a weak-instrument-robust subvector Lagrange multiplier test for instrumental variables regression. We show that it is asymptotically size-correct under a technical condition or as the number of instruments grows to infinity. This is the first weak-instrument-robust subvector test for instrumental variables regression to recover the degrees of freedom of the commonly used non-weak-instrument-robust Wald test. Additionally, we provide a closed-form solution for subvector confidence sets obtained by inverting the subvector Anderson-Rubin test. We show that they are centered around a k-class estimator. We show that the subvector confidence sets for single coefficients of the causal parameter are jointly bounded if and only if Anderson's likelihood-ratio test rejects the null hypothesis that the first-stage regression parameter is of reduced rank, that is, that the causal parameter is not identified. Finally, we show that if a confidence set obtained by inverting the Anderson-Rubin test is bounded and nonempty, it is equal to a Wald-based confidence set with a data-dependent confidence level. We explicitly compute this Wald-based confidence set and its confidence level.

Figures (15)

• Figure 1: The data was sampled from a Gaussian causal model with $n=100$, $k=3$, and ${{m_x}} = {{m_w}} = 1$. The $p$-value of anderson1951estimating's anderson1951estimating test of reduced rank is $0.07$. The multivariate confidence sets for $\beta = (\beta_1, \beta_2)$ are bounded for $1 - \alpha = 0.8$ and unbounded for $1 - \alpha = 0.9, 0.95$. The subvector confidence sets for the individual coefficients are jointly bounded for $1 - \alpha = 0.8, 0.9$ and unbounded for $1 - \alpha = 0.95$. Notably, for $1 - \alpha = 0.9$, the confidence set for $\beta = (\beta_1, \beta_2)$ is unbounded, while the subvector confidence sets for the individual coefficients are bounded. In all cases, the confidence sets for the individual coefficients are substantially smaller than projection-based confidence sets dufour2005projection.
• Figure 2: Power curves of various weak-instrument-robust subvector tests, based on 10'000 simulations from the data-generating process proposed by guggenberger2012asymptotic with $n=1000, k=10$, but with $\Pi_W$ scaled such that $\sqrt{n} \| \Pi_W \| = 10$. AR (GKM) uses the critical values of guggenberger2019more.
• Figure 3: Power curves for the causal effect of an included exogenous regressor based on 10'000 simulations from the data-generating process proposed by guggenberger2012asymptotic with $n=1000, k=10$, but with $\Pi_W$ scaled such that $\sqrt{n} \| \Pi_W \| = 10$. AR (GKM) uses the critical values of guggenberger2019more. We apply the CLR test by including the exogenous regressor $D$ into both the set of endogenous covariates of interest $X$ and instruments $Z$.
• Figure 4: Empirical maximal rejection frequency for $\alpha=0.05$ over 25 linearly spaced values of $\tau$ in $[0, \pi)$ for various $\lambda_1, \lambda_2$. This uses the data-generating process proposed by kleibergen2021efficient with $k=100$ and $\Omega = \mathrm{Id}_3$, based on 2500 simulations per $(\lambda_1, \lambda_2, \tau)$. As we are taking the maximum over $\tau$, values up to around $0.06$ are expected for size-correct tests. Blue indicates conservativeness, green indicates near-nominal level, yellow indicates slightly above nominal level. AR (GKM) uses the critical values of guggenberger2019more.
• Figure 5: Causal graphs visualizing \ref{['model:1']}. On the left, ${{m_w}}={{m_d}}=0$. On the right, we split the endogenous variables into endogenous variables of interest $X$ and endogenous variables not of interest $W$, and split the exogenous variables into exogenous variables of interest $D$ and exogenous variables not of interest $C$. First-stage parameters are not labeled on the right to avoid clutter.

Theorems & Definitions (2)

• Definition 1
• Definition 2