Subvector AR Test Statistic

• Subvector AR test statistic is a generalized Anderson-Rubin method that profiles out nuisance parameters to focus on parameters of interest.
• The method minimizes a quadratic form over nuisance parameters, ensuring valid size control even under weak identification and heteroskedasticity.
• Enhanced with data-dependent critical values and regularization techniques, the approach yields reliable confidence intervals in both IV and time series models.

A subvector Anderson-Rubin (AR) test statistic is a set of procedures for valid hypothesis testing and confidence set construction on one or more components of a parameter vector in models with nuisance parameters, weak identification, and potential nonlinearity or heteroskedasticity. Subvector AR methodology generalizes the classical AR test by minimizing or profiling out nuisance parameters, yielding tests and intervals with guaranteed size control, often under minimal assumptions and for a wide array of identification regimes.

1. Core Definitions and Framework

The canonical subvector AR test addresses the hypotheses

$$H_0: \psi = \psi_0 \quad \text{vs} \quad H_1: \psi \neq \psi_0$$

where $\psi \in \mathbb{R}^q$ is a subvector of interest from the full parameter vector $\theta = (\psi', \lambda')'$, with $\lambda \in \mathbb{R}^{p-q}$ comprising nuisance parameters (Lounis, 2013). In time series AR($m$) models,

$Y_t = \sum_{j=1}^m \theta_j Y_{t-j} + \epsilon_t,$

one partitions $\theta$ accordingly and exploits the local asymptotic normality (LAN) expansion to form the test.

In linear instrumental variable (IV) models,

$$y = X\beta + W\gamma + \varepsilon, \qquad X = Z\Pi_X + V_X, \qquad W = Z\Pi_W + V_W,$$

where $\beta$ are the parameters of interest and $\gamma$ the nuisance parameters, the AR null tested is $H_0: \beta = \beta_0$.

The subvector AR statistic operates by minimizing a criterion (typically a quadratic or generalized quadratic form in appropriate moments) over the nuisance parameter space, delivering a test statistic and confidence set that profile out the nuisance parameter and avoid reliance on possibly unidentified or weakly identified nuisance components (Londschien et al., 2024, Inoue et al., 1 Jul 2025).

2. Mathematical Formulation of the Subvector AR Statistic

Time Series Models

For AR($m$) processes, the subvector test statistic is constructed as

$$W_n = [S_n^{\mathrm{eff}}]' I_{\psi|\lambda}^{-1} S_n^{\mathrm{eff}},$$

where $S_n^{\mathrm{eff}}$ is the efficient score for $\psi$, obtained by removing the effect of the nuisance score (via regression and the block-Fisher information matrix $I(\theta_0)$) (Lounis, 2013):

$$S_n^{\rm eff} = \Delta_{1n} - I_{12}I_{22}^{-1}\Delta_{2n},$$

with

$$I_{\psi|\lambda} = I_{11} - I_{12}I_{22}^{-1}I_{21}.$$

Under $H_0$, $W_n \rightarrow_d \chi^2_q$.

Linear IV Model

For a model

$y = X\beta + W\gamma + \varepsilon,$

with $Z$ as instruments, the subvector AR statistic (profiled in $\gamma$) is (Londschien et al., 2024):

$$\operatorname{AR}(\beta) = \min_{\gamma \in \mathbb{R}^{m_w}} \frac{n-k}{k-m_w} \frac{u(\beta, \gamma)' P_Z u(\beta, \gamma)}{u(\beta, \gamma)' M_Z u(\beta, \gamma)},$$

where $u(\beta, \gamma) = y - X\beta - W\gamma$, $P_Z = Z(Z'Z)^{-1}Z'$, and $M_Z = I_n - P_Z$. The minimizer $\hat\gamma_{\mathrm{LIML}}(\beta)$ corresponds to the LIML-type fit; plugging it in yields a closed-form test statistic.

In the most general GMM/M-estimation setting, the subvector AR statistic can be written as

$$\operatorname{AR}_C(\beta_0) = \min_{\gamma} Q_n(\beta_0, \gamma)$$

with

$$Q_n(\beta, \gamma) = n \, \hat g_n(\beta, \gamma)' \, \widehat\Omega_n(\beta, \gamma)^{-1} \, \hat g_n(\beta, \gamma),$$

where $\hat g_n$ is the sample mean of the moment conditions and $\widehat\Omega_n$ a HAC or robust covariance estimator (Inoue et al., 1 Jul 2025).

3. Distributional Theory and Critical Values

Across models and implementations, the subvector AR test is designed such that, under regularity conditions and $H_0$ (regardless of identification strength),

$$W_n \rightsquigarrow \chi^2_q, \qquad (k-m_w) \operatorname{AR}(\beta_0) \leq \chi^2_{k-m_w}, \qquad \operatorname{AR}_C(\beta_0) \leq \chi^2_{d-d_\gamma}$$

with $q$ the dimension of the subvector of interest, $k-m_w$ the "effective" degrees of freedom after profiling out $m_w$ nuisance parameters, and $d_\gamma$ the dimension of $\gamma$.

A critical innovation is the use of data-dependent critical values, particularly in the IV setting with potentially weak instruments. Guggenberger, Kleibergen and Mavroeidis (GKM) propose conditional critical values—tabulated as $c_{1-\alpha}(z, k-m_w)$—functioning of a conditioning variable $z$, typically the largest or second-smallest eigenvalue of a matrix $M$ formed from the data (Hoekstra et al., 25 Jan 2026). When $m_w > 1$, conditioning on the second-smallest eigenvalue yields strictly higher power while maintaining exact or conservative size:

$$\text{Reject if} \quad S = \lambda_{\min}(M) > c_{1-\alpha}(\kappa_{p-1}, k-m_w).$$

In the general GMM/CUE setting, no data-dependent critical values or bootstrapping are required: one uses the nominal $\chi^2_{d-d_\gamma}$ cut-off for uniformly valid subvector inference (Inoue et al., 1 Jul 2025).

4. Robustness to Identification, Heteroskedasticity, and High Dimensionality

The subvector AR approach is explicitly designed to be robust in settings of weak, partial, or non-identification of nuisance parameters:

• In linear IV models, the AR test achieves weak-instrument-robust size under minimal moment and rank conditions (Londschien et al., 2024).
• In GMM and nonlinear moment models, uniform validity is achieved by perturbing the first-order CUE conditions, employing truncated SVD for regularization, and ensuring all identification regimes are covered (Inoue et al., 1 Jul 2025).
• Heteroskedasticity-robust variants use kernel or block-diagonal (AKP) covariance estimators, maintaining validity and typically improving power when the structure is appropriately captured (Guggenberger et al., 2021, Hoekstra et al., 25 Jan 2026).

Modern enhancements deploy ridge regularization and jackknife bias correction (RJAR), extending subvector AR tests to cases where the number of instruments exceeds sample size $(k > n)$ or covariance matrices are ill-conditioned, while maintaining weak-IV and heteroskedasticity robustness (Dovì et al., 2022).

5. Power Properties and Optimality

Subvector AR tests exhibit several notable optimality and power properties:

• For local alternatives, the efficient-score-based AR test in LAN models is locally most powerful invariant and attains the Le Cam optimality bound (Lounis, 2013).
• Conditioning critical values on the second-smallest eigenvalue (instead of the largest) results in uniformly higher power for $m_w > 1$, since the rejection region more accurately adapts to the weakest direction of identification (Hoekstra et al., 25 Jan 2026).
• In the heteroskedastic AKP setting, model selection ensures that validity is always maintained, and power gains accrue when the covariance exhibits approximate Kronecker structure (Guggenberger et al., 2021).
• Simulations confirm that ridge-regularized and jackknifed AR tests outperform their nonregularized analogs in high dimensional, many-instrument regimes (Dovì et al., 2022).

6. Inference: Confidence Regions and Implementation

Confidence sets are formed by inversion of the AR test, either as the set of subvector parameter values for which the test statistic does not exceed the critical value, or, for scalar subvector problems, by searching for roots of the AR test equation. In the linear IV framework, the confidence set can be written explicitly as a (possibly unbounded) ellipsoid centered at a k-class estimator, with precise closed-form dependence on the data and critical value (Londschien et al., 2024).

Implementation for both linear and nonlinear models involves minimizing the AR criterion over nuisance parameters, computation of HAC or block-diagonal covariance matrices as applicable, and lookup or calculation of critical values. No bootstrapping is required for nominal size control, provided the model-specific distributional results hold (Inoue et al., 1 Jul 2025, Hoekstra et al., 25 Jan 2026). For high dimensional or weakly identified settings, special care in optimization and estimation (jackknife, ridge, Kronecker estimates) is warranted (Dovì et al., 2022, Guggenberger et al., 2021).

7. Extensions and Practical Recommendations

• The methodology extends to general time series (including nonstationary, nonlinear, and heteroskedastic processes), cross-sectional models, and cases with more instruments than observations.
• When the covariance structure closely follows a Kronecker product, AKP-based tests are recommended for power; when this structure fails, fully robust AR procedures are preferred for exactness (Guggenberger et al., 2021, Hoekstra et al., 25 Jan 2026).
• For multiple nuisance parameters, conditioning on the second-smallest eigenvalue is recommended for test power and size (Hoekstra et al., 25 Jan 2026).
• Ridge regularization is essential in "many instrument" settings to ensure well-posedness and valid inference (Dovì et al., 2022).

Summary Table: Subvector AR Statistic Variants

Model Context	Test Statistic / Conditioning	Robustness
AR($m$) time series	Efficient score quadratic form	LAN, optimal, reparam.-inv
Linear IV, homoskedastic	$\lambda_{\min}(M)$, cond. on $\kappa_{p-1}$	Weak IV, strong IV
Linear IV, heteroskedastic	AKP-AR, switching to AR/AR	Arbitrary heteroskedasticity
Nonlinear GMM/CUE	Prof. $Q_n(\beta,\gamma)$ over $\gamma$	Weak/partial/non-ID, HAC
High-dimensional IV	Ridge-jackknife AR	$k > n$, weak IV, het.

All methods aim to preserve correct asymptotic size, provide uniform validity over identification regimes, and exploit conditioning or regularization to maximize power, particularly in models with many or weak instruments and/or heteroskedasticity (Lounis, 2013, Londschien et al., 2024, Hoekstra et al., 25 Jan 2026, Inoue et al., 1 Jul 2025, Dovì et al., 2022, Guggenberger et al., 2021).