Identification-robust inference, such as the generalized Anderson-Rubin (GAR) statistic, gives valid confidence regions even under weak identification. When the moment variance matrix is singular, however, the standard chi-square result can break down, and removing the singularity can make two-step GMM less efficient, the topic of the author's research.
How to read these notes
This is an advanced note aimed at strong master's and PhD students, and anyone wanting an accessible route into the author's research. It assumes you are comfortable with GMM and the weak-instrument problem. The aim is to explain, in words and a little notation, two related research themes: identification-robust inference, and the complications that arise when the moment variance matrix is (almost) singular. It is a guided overview of ideas developed in Grant (2018) and Grant & Smith (2025), not a substitute for those papers.
1. Recap: why weak identification breaks standard inference
Standard GMM and 2SLS inference rests on a point estimate that is consistent and asymptotically normal at the root-\(T\) rate. As the weak-instrument note showed, that picture collapses under weak identification: the estimator can have a non-normal distribution, large or even infinite variance, and t-statistics built on it can be badly size-distorted. Confidence intervals of the form "estimate plus or minus two standard errors" are then untrustworthy, because both the centring and the scaling are unreliable.
The natural response is to stop relying on the point estimate altogether and test a hypothesised parameter value directly through the moment conditions. This is the idea behind identification-robust inference.
2. The generalized Anderson-Rubin statistic
The Anderson-Rubin (AR) statistic, generalised to the GMM setting, tests a candidate value \(\theta_0\) of the parameter by asking whether the sample moments evaluated at \(\theta_0\) are jointly close enough to zero. In the GMM notation, with sample moments \(\hat g_T(\theta)\) and moment variance \(\Omega\), the statistic is essentially the efficient criterion function evaluated at the hypothesised value,
Crucially, GAR is evaluated at the hypothesised value \(\theta_0\), not at an estimate. It therefore never has to trust a point estimate that weak identification may have corrupted. Under the null, and under standard regularity conditions, the statistic has a chi-square limiting distribution, so a confidence region is formed by collecting all values \(\theta_0\) that the GAR test does not reject — inverting the test rather than inverting a variance matrix around an estimate.
This gives a confidence region with correct coverage whether identification is strong, weak or absent. The work of Grant & Smith (2025) studies the construction of exactly such a region — one with pointwise asymptotically correct size for the true parameter — built on the GAR statistic. The subtlety they confront is what happens when the moment variance matrix \(\Omega\), which sits inside the statistic as \(\Omega^{-1}\), is singular.
3. What "singular moment variance" means
The moment variance matrix \(\Omega\) is the asymptotic variance of the (scaled) sample moments. Standard theory assumes it is positive definite, so that \(\Omega^{-1}\) exists and the moments carry information in every direction. Singularity means \(\Omega\) has a zero eigenvalue: some linear combination of the moment conditions has zero variance at the true parameter.
In the author's research, the variance is written through its eigenvalue decomposition, separating the eigenvalues bounded away from zero from those that converge to (or equal) zero. The "almost singular" case allows some eigenvalues to shrink toward zero at a rate \(T^{-\delta}\) as the sample grows, capturing variance matrices that are near-singular in finite samples even if technically full rank.
Why would the variance of a moment be zero? Grant (2018) catalogues several routine sources, so this is not a pathological curiosity:
| Source of singularity | Where it arises |
|---|---|
| Identification failure in non-linear models | The moment loses sensitivity to the parameter, collapsing its variance. |
| Simultaneous-equation and dynamic panel models | Arellano-Bond style moment sets can produce (near-)singular variance. |
| Parameter-restriction tests under the null | The variance of certain moments degenerates exactly on the null hypothesis. |
| Common shocks interacted with instruments | In linear simultaneous IV models, shared unobservables can make moment combinations perfectly correlated. |
4. The consequences for the GAR statistic and two-step GMM
Singularity has sharp consequences for both inference and estimation. Two results from the research are worth stating in words.
The GAR statistic
Grant & Smith (2025) analyse the GAR statistic near points of singularity using a Laurent series expansion of the variance matrix. They identify a condition — termed first-order moment singularity — under which the GAR statistic retains its usual chi-square limiting distribution along parameter sequences approaching the truth. When that condition is violated, however, the GAR statistic becomes asymptotically unbounded: the \(\Omega^{-1}\) inside it magnifies the near-zero-variance direction without limit. The paper then shows how to discretise the parameter space appropriately to implement a feasible GAR-based confidence region that still attains correct size.
Two-step GMM
Grant (2018) studies the efficient two-step GMM estimator when \(\Omega\) is (asymptotically) singular at the true parameter but non-singular for small perturbations away from it. The estimator is shown to have a highly non-standard limiting distribution. Strikingly, it can converge at the faster rate \(T\) in certain directions — rather than the usual \(\sqrt{T}\) — when the null space of the moment variance is not contained in that of the expected moment derivative.
Applied researchers routinely remove singularities — by regularising the weight matrix, by adding or deleting instruments, or by adding noise (Ruge-Murcia's "stochastic singularity"). Grant (2018) shows that doing so generally increases the asymptotic variance and can slow the rate of convergence of two-step GMM. The convenience of removing the singularity to validate standard normal inference is therefore paid for in efficiency; using the singularity-robust distribution can give both valid inference and a more efficient estimator.
5. The takeaways
Two threads tie this material together. First, when identification is in doubt, shift from estimate-based inference to identification-robust inference, of which the GAR statistic is the canonical example: it tests parameter values directly and inverts to a confidence region. Second, the building block that all of this relies on — the moment variance matrix — cannot always be assumed well behaved. Singular and near-singular moment variance arises in mainstream models, and naively inverting it, or mechanically removing it, can invalidate inference or sacrifice efficiency.
For a research student, this is the frontier where GMM, weak identification and matrix asymptotics meet. The eigenvalue and Laurent-series tools used to handle it — drawing on perturbation-theory results for matrices — are exactly the technical machinery developed in the papers referenced here.
PhD econometrics tuition
This is research-level material. For 1-1 help with GMM asymptotics, weak identification, identification-robust inference or a dissertation on these topics, see PhD econometrics tuition or econometrics tuition.