RobustSIRR: Robust VAR and SUR Methods

• RobustSIRR is a family of robust inference and shrinkage methods for high-dimensional VAR and SUR models that mitigate contamination and model misspecification.
• The methodology uses MM estimators, robust covariance techniques, and hyperparameter selection to enhance stability and estimation efficiency in noisy data.
• Empirical evidence shows RobustSIRR improves R² and risk reduction, outperforming classical estimators in the presence of heavy-tailed errors and outliers.

RobustSIRR refers to a family of robust inference, estimation, and selection methods for high-dimensional multivariate time series and regression models, particularly for impulse response function (IRF) estimation in vector autoregressions (VAR), and for Seemingly Unrelated Regressions (SUR) under contamination. These methods aim to provide high statistical efficiency and robustness against model misspecification or data contamination, and to deliver principled hyperparameter selection in settings with heavy-tailed distributions, outliers (row-wise and/or cell-wise), or dynamic model deviations.

1. Model Classes and Problem Statement

RobustSIRR methodology encompasses two primary classes of models:

• Seemingly Unrelated Regression (SUR) Models: Here, multiple linear regression equations for $m$ outcomes across $n$ observations are estimated jointly, exploiting potential contemporaneous correlation structure in the error terms. Standard form is $y_{ik} = x_{ik}^\top \beta_i + \epsilon_{ik}$, $i = 1,..,m$, $k = 1,..,n$, with errors for each $k$ correlated via $m \times m$ covariance $\Sigma$, i.e., $\operatorname{Cov}(\epsilon|X)=\Sigma \otimes I_n$ (Saraceno et al., 2021, Peremans et al., 2018).
• VAR and IRF Estimation under Misspecification: Multivariate time series are modeled as VAR with possible local misspecification, e.g., the DGP is $Y_t = F Y_{t-1} + M \epsilon_t + O(T^{-1/2})$, and structural IRFs are estimated at horizon $h$ for impact vector $\Xi$ as $$\operatorname{IRF}_{\text{true}} = M^\prime F^h M \Xi$$. RobustSIRR here targets shrinkage and lag-selection in IRF estimation when the VAR may be misspecified (González-Casasús et al., 6 Feb 2025).

In both problem domains, classical estimators (GLS, MLE, OLS, Bayesian mean) are highly sensitive to data contamination or model misspecification, motivating the need for robust approaches in both point and interval estimation.

2. RobustSIRR Estimation Objectives and Risk Criteria

The central statistical objective in RobustSIRR for VAR/IRF is minimization of a weighted quadratic loss measuring IRF estimation error:

$$\operatorname{Loss}(\operatorname{IRF}_\text{est}) = \| \operatorname{IRF}_\text{est} - \operatorname{IRF}_\text{true} \|_W^2 = \operatorname{tr}\left( [\operatorname{IRF}_\text{est} - \operatorname{IRF}_\text{true}]^\prime W [\operatorname{IRF}_\text{est} - \operatorname{IRF}_\text{true}] \right)$$

This is normalized to the “SIRR” risk as $$\mathcal{R}_{\text{IRF}}(\iota, \lambda, p) = \lim_{T \to \infty} T \, \mathbb{E}[\operatorname{Loss}(\operatorname{IRF}_\text{est}(\iota, \lambda, p))]$$, with $\iota$ indicating estimator class (MLE, LFE), $\lambda$ shrinkage, $p$ lag length (González-Casasús et al., 6 Feb 2025).

For SUR, the estimator is constructed to minimize the robustified FGLS criterion

$$\hat \beta = \arg\min_{\beta} (y - X\beta)^\top W (\hat \Sigma_1^{-1} \otimes I_n) W (y - X\beta)$$

with $W$ a block-diagonal weight matrix, and $\hat\Sigma_1$ a robust covariance estimate of first-stage residuals (Saraceno et al., 2021).

3. Algorithmic Framework

3.1. RobustSIRR for VAR/IRF

The procedure for RobustSIRR (termed IRFC in (González-Casasús et al., 6 Feb 2025)) combines robust risk estimation and joint hyperparameter selection:

1. Candidate Estimator Generation: For each tuple $(\iota, \lambda, p)$, compute the restricted Bayes (shrinkage) posterior mean $\overline{\Psi}_T(\iota, \lambda, p)$.
2. Empirical Reference Estimator: Use the local projection (LP) estimator with no shrinkage and maximal lag $q$, $\overline{\Psi}_T(\text{lfe}, 0, q)$, as risk baseline.
3. Asymptotically Unbiased Risk Estimation: For each candidate,

$$\operatorname{IRFC}_T(\iota, \lambda, p) = T \| M^\prime [\overline{\Psi}_T(\iota, \lambda, p) - \overline{\Psi}_T(\text{lfe}, 0, q)] M \Xi \|_W^2 + 2 \hat{\mathrm{Cov}}$$

with $\hat{\mathrm{Cov}}$ a consistent plug-in estimator of the variance cross-term (González-Casasús et al., 6 Feb 2025).

1. Hyperparameter Selection: Minimize $\operatorname{IRFC}_T$ over $(\iota, \lambda, p)$ to select estimator class, shrinkage, and lag length.
2. Reporting: Return the selected IRF estimate, lag, shrinkage, and estimator type.

3.2. RobustSIRR for Seemingly Unrelated Regression

The methodology (Saraceno et al., 2021, Peremans et al., 2018) consists of:

1. Equation-Wise MM Regression: For each equation $i=1,..,m$, compute robust MM-estimates of coefficients and scales for residuals.
2. Robust Covariance Estimation: Stack first-stage residuals and estimate a robust scatter $\hat\Sigma_1$ via a 2SGS estimator.
3. Weight Construction: Compute cell-wise weights $w_{ik}$ using an M-function of the normalized residuals; $W = \operatorname{diag}(W_1,\ldots,W_m)$.
4. Two-Stage Weighted FGLS: Solve the above robustified GLS for $\hat\beta$.
5. Optional Refinement: Recompute residuals and robust scatter, optionally repeating the weighted FGLS update.

The fast-and-robust bootstrap (FRB) enables robust interval estimation and hypothesis testing (Peremans et al., 2018).

4. Theoretical Properties

RobustSIRR procedures under both paradigms possess several formal guarantees:

• Consistency: Under standard moment and design conditions and moderate contamination, the estimator is consistent.
• Breakdown Point: For SUR, the MM-estimation and S-step provide breakdown points up to $0.5$ for both row-wise (THCM) and maintain positive breakdown under cell-wise (ICM) contamination, unlike classical and Fast-SUR approaches whose robustness collapses as $m$ increases (Saraceno et al., 2021).
• Asymptotic Efficiency: MM estimators attain high efficiency relative to MLE in clean cases, with asymptotic relative efficiency tuned by the choice of $\rho$-function (Peremans et al., 2018).
• Oracle-Optimal Selection: For VAR/IRF, under mild regularity, the minimizer of the empirical IRF criterion converges to the oracle minimizer of the (unknown) true IRF risk (González-Casasús et al., 6 Feb 2025).
• Robust Interval and Hypothesis Testing: FRB yields bootstrap confidence intervals with nominal coverage under both clean and contaminated data, as well as valid robust tests for both parameter restrictions and diagonality of $\Sigma$ (Peremans et al., 2018).

5. Empirical and Simulation Evidence

RobustSIRR methods are validated via both Monte Carlo experiments and real-data applications.

• SUR Monte Carlo (Saraceno et al., 2021):
  • Under THCM contamination up to $\sim 10\%$, RobustSIRR and Fast-SUR outperform classical SUR; at higher contamination only RobustSIRR remains stable.
  • Under ICM, only RobustSIRR shows stable performance as $m$ grows.
  • Computation time: robust but computationally tractable ($5$–$11$ s/run vs $220$ s/run for Fast-SUR at $m=10, p_i=5$).
• Fiscal Expenditure Data (Saraceno et al., 2021):
  • RobustSIRR downweights approximately $2.3\%$ of cells and $8.8\%$ of rows.
  • System-wide $R^2$ for RobustSIRR exceeds classical SUR ($0.457$ vs $0.325$).
• VAR/IRF Monte Carlo (González-Casasús et al., 6 Feb 2025):
  • IRFC-based shrinkage selection closely tracks oracle risk minima; joint selection nearly attains oracle finite-sample risk.
  • Under misspecification, IRFC-based selection yields up to $100\%$ risk reduction over marginal data density (MDD) hyperparameter selection.
• FRED-QD Macro Data Application (González-Casasús et al., 6 Feb 2025):
  • IRFC selects local projection estimators in $60-85\%$ of samples, with lag and shrinkage configuration systematically adapted to horizon and misspecification.

6. Comparative Table of RobustSIRR Variants

Model Context	Main RobustSIRR Mechanisms	Distinguished Properties
SUR	MM/FGLS + 2SGS scatter, cell weighting	Robust to row & cell outliers, efficient, fast bootstrap
VAR/IRF with misspec	Unbiased IRF risk, shrinkage & lag selection	Asymptotically oracle-optimal, joint selection

7. Connections, Limitations, and Recommendations

RobustSIRR succeeds under conditions where classical estimators (including affine equivariant M-estimators or simple Bayesian approaches) break down—especially with heavy-tailed errors or complex outlier patterns. In high-dimensional or contaminated data, robustified selection and inference mitigate risk explosion and retain validity for scientific inference. A plausible implication is that misspecification-robust shrinkage is essential for reliable VAR analysis in practical macroeconometric settings.

For applications where either row-wise or cell-wise contamination is plausible (financial panels, healthcare records, macroeconomic or sensor time series with structural breaks or regime changes), RobustSIRR constitutes a theoretically justified and computationally practical solution (González-Casasús et al., 6 Feb 2025, Saraceno et al., 2021, Peremans et al., 2018).