The Anderson-Rubin Test

From a Structural Hypothesis to a Reduced-Form Test

Instrumental-variable estimates can behave poorly when instruments have little predictive power for an endogenous regressor. The Anderson–Rubin test takes a different route: for each hypothesized structural coefficient, it tests whether the instruments predict the outcome after that coefficient is imposed. Its key advantage is that, under its stated assumptions, the null distribution does not require a strong first stage.

Let \(y\) and \(x\) be \(n\)-vectors, let \(W\) be an \(n\times p\) matrix of included controls with full column rank, and let \(Z\) be an \(n\times k\) matrix of excluded instruments. Consider the structural and first-stage equations

Here \(\beta\) is the structural coefficient of interest, and \(\pi\) describes the instruments’ contribution to the first stage. We test \(H_0:\beta=\beta_0\), treating \(\beta_0\) as a specified value.

To see why this is the relevant restriction, substitute the first-stage equation into the structural equation. The coefficient of \(Z\) in the reduced form for \(y\) is \(\beta\pi\), whereas the coefficient of \(Z\) in the reduced form for \(x\) is \(\pi\). Thus the coefficient of \(Z\) in \(y-\beta_0x\) is \((\beta-\beta_0)\pi\). Under \(H_0\), that coefficient is zero. This restriction remains meaningful even if \(\pi\) is small or zero.

Constructing the F Statistic

Define the projection onto the columns of \(W\) and its residual-maker by

The number \(N\) is the dimension left after removing the controls. Let \(A=M_WZ\) be the instruments after partialling out \(W\). Assume \(\operatorname{rank}(A)=k\) and \(N>k\). Define the projection onto the columns of \(A\) as \(P_A=A(A'A)^{-1}A'\), and let \(q=y-\beta_0x\). The test statistic is

Both quadratic forms use the outcome residualized with respect to \(W\): because \(A\) lies in the range of \(M_W\), \(P_Aq=P_AM_Wq\). The numerator measures the additional sum of squares explained by the excluded instruments; the denominator is the residual sum of squares after fitting both \(W\) and \(Z\), divided by its residual degrees of freedom. Reject at level \(\alpha\) when \(F_{\mathrm{AR}}>F_{k,N-k;1-\alpha}\), where \(F_{k,N-k;1-\alpha}\) is the \(1-\alpha\) quantile of the central F distribution.

Exact Null Distribution and Proof

The exact result uses a Gaussian, homoskedastic error assumption. Treat \(W\) and \(Z\) as fixed, or condition on them, and assume \(u\mid W,Z\sim N(0,\sigma_u^2 I_n)\) for some \(\sigma_u^2>0\). No assumption that the first stage is strong is needed.

Proof. Under the null, the structural equation gives \(y-\beta_0x=W\gamma+u\). Thus \(M_Wq=M_Wu\). In particular, the tested error after removing the controls is \(u\), not \(u-\beta_0v\). The projections \(P_A\) and \(M_W-P_A\) are orthogonal: \(P_A\) projects onto a \(k\)-dimensional subspace of the range of \(M_W\), while \(M_W-P_A\) projects onto its orthogonal complement within that range, of dimension \(N-k\).

The Gaussian vector \(u\) is spherical with covariance \(\sigma_u^2I_n\). For any orthogonal projection \(P\) of rank \(r\), \(u'Pu/\sigma_u^2\) has a chi-square distribution with \(r\) degrees of freedom. Applying this fact to the two projections gives

These two quadratic forms are independent: the projected Gaussian vectors have zero cross-covariance because \(P_A(M_W-P_A)=0\); jointly Gaussian vectors with zero cross-covariance are independent. Their ratio, with each quadratic form divided by its degrees of freedom, is therefore \(F_{k,N-k}\). The common factor \(\sigma_u^2\) cancels, proving the stated conditional distribution. Its conditional rejection probability is \(\alpha\) for every admissible \(W,Z\), so averaging over \(W,Z\), if they are random, also gives size \(\alpha\). \(\square\)

The argument identifies precisely why weak first-stage prediction does not distort null size: under \(H_0\), \(q=W\gamma+u\), so the null distribution of the test statistic involves \(u\) and the instrument design, but not \(\pi\) or \(v\). The test does not need to estimate or divide by the first-stage coefficient.

Power Can Still Depend on Instrument Strength

Robust null size is not the same as high power. For a simple alternative, write \(\beta=\beta_0+\Delta\), where \(\Delta\ne0\). Then

Using \(x=W\delta+Z\pi+v\), the coefficient of \(Z\) in this regression is \(\Delta\pi\). If \(u\) and \(v\) are jointly Gaussian, independent across observations, and independent of the fixed design, the alternative shifts the tested regression in the instrument direction. Its signal is proportional to \(\Delta\pi\); a weak first stage can therefore make alternatives harder to distinguish from the null.

Common Mistake and Practical Significance

A common error is to substitute reduced-form errors carelessly and conclude that the tested error is \(u-\beta_0v\). Under the null, the structural equation itself gives \(y-\beta_0x=W\gamma+u\), so after removing \(W\) the error is \(M_Wu\). Equivalently, the reduced-form error in \(y\) contains \(\beta v+u\), while that in \(\beta_0x\) contains \(\beta_0v\); under \(\beta=\beta_0\), their difference is \(u\), not \(u-\beta_0v\). The variance in the null proof is therefore \(\sigma_u^2\).

Another important qualification is that the exact result is not automatic under heteroskedasticity or non-Gaussian errors. The theorem relies on spherical Gaussian errors to obtain independent chi-square quadratic forms. The Anderson–Rubin idea—testing the reduced-form restriction—still motivates inference under broader assumptions, but the exact F calibration proved here requires the conditions stated above.