Anderson-Rubin Confidence Intervals

Author

Bas Machielsen

Published

October 25, 2025

Introduction

Anderson-Rubin (AR) confidence intervals offer a robust method for inference in the context of instrumental variable (IV) regression, particularly when the instruments are weak. Unlike traditional confidence intervals based on Wald statistics, which can be unreliable in the presence of weak instruments, AR confidence intervals maintain correct coverage probabilities regardless of the instrument strength. This exposition provides a mathematical breakdown of the AR confidence interval.

The Instrumental Variables Model

We begin with a standard linear instrumental variables model. The primary equation of interest, often called the structural equation, is:

\[y = X\beta + \epsilon\]

where:

  • \(y\) is an \(n \times 1\) vector of observations of the dependent variable.
  • \(X\) is an \(n \times k\) matrix of endogenous regressors. For simplicity, we’ll consider the case with one endogenous regressor, so \(X\) is \(n \times 1\).
  • \(\beta\) is the scalar coefficient of interest.
  • \(\epsilon\) is an \(n \times 1\) vector of error terms.

The endogeneity of \(X\) arises from the correlation between \(X\) and \(\epsilon\). To address this, we introduce a set of instrumental variables.

The relationship between the endogenous regressor and the instruments is given by the first-stage regression:

\[X = Z\Pi + \nu\]

where:

  • \(Z\) is an \(n \times l\) matrix of instrumental variables (where \(l \ge k\)).
  • \(\Pi\) is an \(l \times k\) matrix of first-stage coefficients. With one endogenous regressor, \(\Pi\) is \(l \times 1\).
  • \(\nu\) is an \(n \times 1\) vector of error terms.

The key assumptions of the IV model are:

  1. Instrument Relevance: The instruments are correlated with the endogenous regressor(s), meaning \(\Pi\) has full column rank. A violation of this assumption, where \(\Pi\) is close to zero, leads to the “weak instrument” problem.
  2. Instrument Exogeneity: The instruments are uncorrelated with the structural error term, i.e., \(\text{Cov}(Z, \epsilon) = 0\).

The Anderson-Rubin Test

The Anderson-Rubin confidence interval is constructed by “inverting” the Anderson-Rubin test. The AR test evaluates the null hypothesis \(H_0: \beta = \beta_0\) for a specific value \(\beta_0\).

Under the null hypothesis that \(\beta = \beta_0\), the structural equation can be rewritten as:

\[y - X\beta_0 = \epsilon\]

If the null hypothesis is true and the instruments \(Z\) are exogenous, then \(Z\) should be uncorrelated with \(y - X\beta_0\). This is the fundamental insight behind the AR test.

To formalize this, we can estimate the following regression:

\[y - X\beta_0 = Z\gamma + u\]

Under the null hypothesis, the coefficients \(\gamma\) should be jointly equal to zero. The AR test is essentially an F-test of this hypothesis.

The Anderson-Rubin statistic is given by:

\[AR(\beta_0) = \frac{ (y - X\beta_0)'Z(Z'Z)^{-1}Z'(y - X\beta_0) / l }{ (y - X\beta_0)'(I - Z(Z'Z)^{-1}Z')(y - X\beta_0) / (n - l) }\]

where:

  • \(n\) is the number of observations.
  • \(l\) is the number of instruments.

Under the null hypothesis and the assumption of homoskedasticity, the \(AR(\beta_0)\) statistic follows an \(F\)-distribution with \(l\) and \(n-l\) degrees of freedom.

Constructing the Anderson-Rubin Confidence Interval

A confidence interval is a set of parameter values that are plausible given the data. A classical approach to constructing a confidence interval is to invert a hypothesis test. Specifically, a \((1-\alpha)\) confidence interval for \(\beta\) is the set of all values \(\beta_0\) for which the null hypothesis \(H_0: \beta = \beta_0\) is not rejected at the \(\alpha\) significance level.

Therefore, the \((1-\alpha)\) Anderson-Rubin confidence interval for \(\beta\) is defined as:

\[CI_{AR} = \{ \beta_0 : AR(\beta_0) \le F_{l, n-l}(1-\alpha) \}\]

where \(F_{l, n-l}(1-\alpha)\) is the \((1-\alpha)\) quantile of the \(F\)-distribution with \(l\) and \(n-l\) degrees of freedom.

This means we test every possible value of \(\beta_0\), and the confidence interval consists of all the values for which the AR statistic is less than the critical F-value. In practice, this is a quadratic inequality in \(\beta_0\), which can be solved analytically.

Robustness to Weak Instruments

The robustness of the Anderson-Rubin confidence interval to weak instruments stems from the construction of the AR test itself. The distribution of the AR statistic under the null hypothesis does not depend on the value of \(\Pi\), the matrix of first-stage coefficients.

Here’s a more detailed explanation:

The AR test directly examines the exogeneity condition by testing the correlation between the instruments \(Z\) and the transformed residual \(y - X\beta_0\). Under the null hypothesis, this transformed residual is equal to the true structural error \(\epsilon\). The exogeneity assumption states that \(Z\) is uncorrelated with \(\epsilon\), regardless of the strength of the relationship between \(Z\) and \(X\) (i.e., the value of \(\Pi\)).

Therefore, the validity of the F-test for the significance of \(\gamma\) in the regression of \(y - X\beta_0\) on \(Z\) holds even if \(\Pi = 0\) (the case of completely irrelevant instruments). Because the null distribution of the test statistic is unaffected by the strength of the instruments, the resulting confidence intervals maintain their correct coverage probability.

In contrast, Wald-based confidence intervals, which are centered around the two-stage least squares (2SLS) estimator, are highly sensitive to weak instruments. The distribution of the 2SLS estimator and its standard error depend heavily on \(\Pi\). When \(\Pi\) is close to zero, the 2SLS estimator is biased and its distribution is non-normal, leading to incorrect inference from Wald statistics.

Properties and Limitations

The Anderson-Rubin confidence interval has several notable properties:

  • Exact Coverage: Under the assumption of normally distributed errors, the AR confidence interval has exact finite-sample coverage.
  • Robustness: As discussed, it is robust to weak instruments.
  • Potential for Unboundedness: A significant drawback is that the AR confidence interval can be unbounded (i.e., the entire real line) if the instruments are very weak. This occurs when the F-statistic for the significance of the instruments in the first-stage regression is small.
  • Can be the Empty Set: In some cases, the quadratic inequality may have no real solution, resulting in an empty confidence set.
  • Informativeness: While an unbounded interval correctly reflects the lack of information provided by weak instruments, it is not practically informative for pinning down the value of \(\beta\).

In conclusion, the Anderson-Rubin confidence interval provides a reliable method for inference in instrumental variables models, especially when there is concern about weak instruments. Its construction by inverting a test that is insensitive to the strength of the first-stage relationship ensures valid coverage probabilities. However, the potential for unbounded or empty sets highlights the inherent uncertainty that arises when instruments are not strongly correlated with the endogenous regressors.