Anderson-Rubin Tests Under Weak Identification

Why Test a Structural Coefficient This Way?

In instrumental-variables regression, a conventional test of a structural coefficient can behave poorly when the instruments only weakly predict the endogenous regressor. The Anderson–Rubin (AR) test takes a different route: for a proposed coefficient value, it asks whether the outcome remaining after subtracting that proposed effect is related to the excluded instruments. It does not estimate the structural coefficient by dividing one noisy instrument-related quantity by another.

This construction makes the test useful when first-stage strength is uncertain. Its validity does not require the first-stage coefficient to be large or even nonzero. The test can nevertheless have limited power when the instruments provide little information, so a non-rejection is not evidence that the proposed coefficient has been established.

Model and Hypothesis

For observations \(i=1,\ldots,n\), consider one endogenous regressor \(x_i\), an outcome \(y_i\), a vector of included exogenous controls \(W_i\), and a vector of \(q\) excluded instruments \(Z_i\). Write the structural equation as

where \(\beta\) is the structural coefficient of interest, \(\gamma\) is a vector of control coefficients, and \(u_i\) is the structural disturbance. The instruments are excluded from this structural equation: conditional on \(x_i\) and \(W_i\), they have no separate effect on \(y_i\). For the instruments to support valid inference, they must also be exogenous with respect to the structural disturbance, conditional on the controls.

A convenient first-stage representation is

where \(\pi\) describes the instruments’ predictive relationship with \(x_i\), and \(v_i\) is the first-stage disturbance. The test concerns \(H_0:\beta=\beta_0\), for a specified value \(\beta_0\). The vector \(\gamma\) remains unrestricted under the null.

Why the Null Becomes a Test of Instrument Coefficients

Under \(H_0\), subtracting \(\beta_0x_i\) from both sides of the structural equation gives

Thus, under the null, the transformed outcome is explained by the included controls and has no additional relationship with the excluded instruments. This is the central AR restriction. In particular, the disturbance in this transformed regression under the null is \(u_i\), not \(u_i-\beta_0v_i\). The first-stage disturbance does not enter the null regression because the hypothesized structural effect has been subtracted from the outcome.

The same restriction can be seen through reduced-form regressions. Regress \(y_i\) and \(x_i\) separately on the same set of controls and instruments. Let \(\rho\) denote the vector of excluded-instrument coefficients in the outcome regression and \(\pi\) the corresponding vector in the regressor regression. Ordinary least squares is linear in the dependent variable when the regressors are held fixed. Therefore the excluded-instrument coefficient vector in the regression of \(y_i-\beta_0x_i\) is

The AR test is equivalently a test that this vector equals zero. At the population level, under instrument exogeneity and the structural model, the corresponding reduced-form coefficients satisfy \(\rho=\beta\pi\). Consequently, under \(H_0\), \(\rho=\beta_0\pi\). Unlike a ratio-based estimate, this equality remains a well-defined restriction when \(\pi\) is zero or close to zero.

The Conventional AR F Statistic

Let \(k\) be the rank of the control matrix \(W\), including an intercept if one is used. Assume the full regressor matrix formed by \(W\) and \(Z\) has rank \(k+q\), with \(n>k+q\). Fit two ordinary least-squares regressions of the transformed outcome \(y-\beta_0x\): a restricted regression on \(W\) alone and an unrestricted regression on \(W\) and \(Z\). Denote their residual sums of squares by \(RSS_R\) and \(RSS_U\), respectively. The AR statistic is

The numerator measures the reduction in residual variation from adding the \(q\) excluded instruments, per restriction. The denominator is the unrestricted regression’s residual variance estimate, with \(n-k-q\) residual degrees of freedom. Because the unrestricted regression includes every regressor in the restricted regression, \(RSS_R-RSS_U\) cannot be negative, apart from numerical rounding.

For an exact finite-sample \(F\) reference distribution under the null, impose the following conditional error assumptions: given \(W\) and \(Z\), the observations \(u_i\) are independent, normally distributed, have conditional mean zero, and have a common conditional variance \(\sigma^2\). Under these conditions, the null regression is \(y-\beta_0x=W\gamma+u\), and

These assumptions concern \(u\), the structural disturbance in the null regression. No normality or independence assumption on \(v_i\) is needed for this exact null reference distribution: \(x_i\) cancels from the null equation. Instrument exogeneity and exclusion are also essential to the IV interpretation of the test. The exact \(F\) claim, however, specifically uses the stated conditional distributional assumptions on \(u\).

Justification of the F Reference Distribution

The exact result follows from the nested-regression geometry. Conditional on \(W,Z\), let \(P_W\) be the projection matrix onto the columns of \(W\), and \(P_{WZ}\) the projection matrix onto the columns of the full regressor matrix \([W,Z]\). Under the null, the transformed outcome is \(W\gamma+u\). Since both regressions include \(W\), their residual sums of squares can be written using \(u\):

Subtracting these expressions gives

The matrix \(P_{WZ}-P_W\) projects onto the \(q\)-dimensional part of the full regressor space that is added by \(Z\). Conditional normality and the common variance therefore give \((RSS_R-RSS_U)/\sigma^2\sim\chi^2_q\). The unrestricted residual projection has rank \(n-k-q\), so \(RSS_U/\sigma^2\sim\chi^2_{n-k-q}\). These projections are orthogonal; under conditional normality their quadratic forms are independent. Dividing each chi-square quantity by its degrees of freedom yields the stated \(F_{q,n-k-q}\) distribution. This argument also shows why the null disturbance used in the derivation must be \(u\).

Worked Examples Through Reduced-Form Coefficients

Interpretation, Power, and a Common Pitfall

For each candidate \(\beta_0\), the AR test asks whether the excluded instruments jointly explain the transformed outcome after the controls are included. A small p-value is evidence against that candidate under the maintained structural and instrument-validity assumptions. A large p-value does not give the probability that \(H_0\) is true and is not proof that the instruments are informative. When the first stage is weak, the test can be valid yet have low power: many candidate values may not be rejected. Inverting the test over candidate values can therefore produce a wide or unbounded confidence set.

Common mistake: A tempting but incorrect argument is that the null-regression error is \(u_i-\beta_0v_i\), so assumptions about that combination justify the usual AR \(F\) distribution. This substitutes the wrong disturbance. Under \(H_0:\beta=\beta_0\), direct subtraction gives \(y_i-\beta_0x_i=W_i'\gamma+u_i\); the null-regression disturbance is \(u_i\). The exact \(F\) result therefore requires the specified conditional assumptions on \(u_i\), not merely on \(u_i-\beta_0v_i\).

A second pitfall is to confuse the AR test with a test based on a 2SLS estimate divided by its standard error. The AR test instead evaluates a restriction on reduced-form coefficients, \(\rho=\beta_0\pi\), through a regression of \(y-\beta_0x\). This distinction is precisely why its null calibration does not rely on a strong first stage. Its usefulness still depends on credible instrument exclusion and exogeneity; weak-instrument robustness does not protect against invalid instruments.