arxiv: 2604.21734 · v1 · submitted 2026-04-23 · 💱 q-fin.RM · math.PR· q-fin.GN

Recognition: unknown

Modeling dependency between operational risk losses and macroeconomic variables using Hidden Markov Models

Cl\'ement Fernandes, Dorinel Bastide, Nikeethan Selvaratnam, Wojciech Pieczynski

Pith reviewed 2026-05-08 12:56 UTC · model grok-4.3

classification 💱 q-fin.RM math.PRq-fin.GN

keywords operational riskhidden Markov modelsmacroeconomic variablesexpectation-maximization algorithmtime series modelingrisk event typesstress testing

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The pith

An extended hidden Markov model uses an auxiliary variable to model how operational risk losses depend on macroeconomic variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the challenge of predicting operational risk losses, which exhibit heterogeneous and time-dependent structures, especially their relationship to economic conditions for stress testing. It proposes extending hidden Markov models to multivariate observations by adding a third auxiliary variable that incorporates macroeconomic covariates into the time series. The model is calibrated using the expectation-maximization algorithm, and results are shown for different risk event types to evaluate the contribution of the economic factors. A sympathetic reader would care because improved capture of these dependencies could support better risk forecasting and regulatory stress tests in financial firms.

Core claim

The authors introduce an extension of hidden Markov models for multivariate time-series data that adds an auxiliary variable to account for economic covariates. This allows the model to represent dependencies between operational risk losses and macroeconomic variables. The calibration is performed with the EM algorithm, and the results for various risk-event types demonstrate the relevance of the macroeconomic covariates.

What carries the argument

An extended hidden Markov model with a third auxiliary variable that accommodates economic covariates in the time-series observations.

If this is right

The model can represent the heterogeneous and time-dependent dependencies in operational risk data.
Calibration via the EM algorithm yields parameters suitable for different risk event types.
Including macroeconomic covariates enhances the model's ability to support stress test exercises.
The approach provides a way to analyze the relationship between losses and economic conditions over time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar extensions could apply to other risk types like credit risk where macro factors play a role.
Testing the model on out-of-sample data would verify its predictive power for future losses.
Integration with existing bank risk systems might allow real-time stress scenario analysis.

Load-bearing premise

The extended hidden Markov model structure with the auxiliary variable can adequately capture the heterogeneous, time-dependent dependencies between operational risk losses and macroeconomic variables, with the EM algorithm producing reliable parameter estimates.

What would settle it

Observing that the fitted model fails to improve the likelihood or predictive accuracy on new operational risk data compared to a standard hidden Markov model without the auxiliary variable would indicate the extension does not add value.

Figures

Figures reproduced from arXiv: 2604.21734 by Cl\'ement Fernandes, Dorinel Bastide, Nikeethan Selvaratnam, Wojciech Pieczynski.

**Figure 1.** Figure 1: Graphical model of a bivariate Gaussian HMM - the discrete latent variables view at source ↗

**Figure 2.** Figure 2: Quarterly 90th–percentile predictions for 2-state HMMs with and without the VS view at source ↗

**Figure 3.** Figure 3: Weekly 90th–percentile predictions for 2-state HMMs with and without the VSTOXX view at source ↗

read the original abstract

Predicting future operational risk losses gives rise to a significant challenge due to the heterogeneous and time-dependent structures present in real-world data. Furthermore, stress test exercises require examining the relationship with operational losses. To capture such relationship, we propose to use an extension of Hidden Markov Models to multivariate observations. This model introduces a third auxiliary variable designed to accommodate the economic covariates in the time-series data. We detail the unique aspects of operational risk data and describe how model calibration is achieved via the Expectation-Maximization (EM) algorithm. Additionally, we provide the calibration results for the various risk-event types and analyze the relevance of the inclusion of the macroeconomic covariates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends HMMs with an auxiliary variable to tie operational risk losses to macro covariates, calibrates via EM, and shows parameter results per event type, but offers no baseline comparisons or out-of-sample checks.

read the letter

The paper extends Hidden Markov Models by adding an auxiliary variable to handle macroeconomic covariates when modeling dependencies with operational risk losses. They calibrate the model using the EM algorithm and present results for different risk-event types, including parameter estimates and some discussion of how the covariates influence transitions. This setup is new in its specific application to operational risk, where data heterogeneity and time-dependence are key issues. The description of the model and the calibration process is clear, and they do address the practical aspects of op-risk data like varying event types. The calibration results provide concrete numbers on transition probabilities and emission parameters across categories, which could be useful for seeing macro effects. That said, the support for the central idea—that this auxiliary variable better captures the heterogeneous, time-dependent dependencies—relies on in-sample analysis alone. There are no out-of-sample tests, no predictive accuracy metrics like log-likelihood on held-out data, and no head-to-head comparisons with a standard HMM without the auxiliary or with direct time-series regressions. For sparse loss data like operational risk, which often has many zeros and sudden events, this raises the possibility that the extra variable is fitting noise or creating artificial persistence rather than revealing true relationships. The approach avoids circular reasoning and uses standard methods, so the technical foundation is solid. This work would interest risk managers at financial institutions who deal with stress testing and scenario analysis for operational risks. It gives a structured way to incorporate macro variables into regime-based modeling. A serious editor should send it to peer review. Referees can push for the missing validation steps, but the core modeling contribution is worth evaluating.

Referee Report

2 major / 2 minor

Summary. The paper proposes an extension of Hidden Markov Models to multivariate observations for capturing dependencies between operational risk losses and macroeconomic variables. It introduces a third auxiliary variable to accommodate economic covariates in the time-series data, details unique aspects of operational risk data, calibrates the model via the Expectation-Maximization algorithm, and presents calibration results for various risk-event types along with an analysis of the relevance of the macroeconomic covariates.

Significance. If the auxiliary-variable construction demonstrably improves representation of heterogeneous, time-dependent loss-macro relationships, the work would offer a structured approach to incorporating macroeconomic drivers into operational risk modeling, with direct relevance to stress-testing exercises. The use of standard EM calibration on real risk-event types provides a concrete starting point, though the absence of comparative benchmarks currently limits the assessed impact.

major comments (2)

[Calibration results and parameter analysis] The central claim that the auxiliary variable produces a faithful representation of time-varying, heterogeneous dependencies requires evidence beyond in-sample parameter examination. The calibration results section reports EM fits and transition probabilities across risk-event types but provides no out-of-sample log-likelihood, no predictive scores, and no direct comparison against a baseline multivariate HMM without the auxiliary variable; this leaves open whether the auxiliary state improves fit or merely absorbs noise.
[Model description] The model construction in the HMM extension section defines the auxiliary variable to accommodate economic covariates, yet the manuscript does not derive or test why this third variable is necessary rather than extending the emission distribution directly or using a standard multivariate HMM; without such justification or ablation, the load-bearing role of the auxiliary variable for the dependency claim remains unestablished.

minor comments (2)

[Model description] Notation for the auxiliary variable and its integration into the emission and transition probabilities should be clarified with explicit equations to avoid ambiguity in the multivariate observation model.
[Abstract] The abstract and introduction would benefit from a brief statement of the data sources, time span, and number of observations used in the calibration to allow readers to assess applicability to typical operational risk series.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the validation needs for our proposed HMM extension. We address each major comment below, agreeing that additional evidence would strengthen the claims, and outline the planned revisions.

read point-by-point responses

Referee: The central claim that the auxiliary variable produces a faithful representation of time-varying, heterogeneous dependencies requires evidence beyond in-sample parameter examination. The calibration results section reports EM fits and transition probabilities across risk-event types but provides no out-of-sample log-likelihood, no predictive scores, and no direct comparison against a baseline multivariate HMM without the auxiliary variable; this leaves open whether the auxiliary state improves fit or merely absorbs noise.

Authors: We agree that the manuscript's current focus on in-sample EM calibration and parameter analysis leaves the improvement attributable to the auxiliary variable unquantified. In the revised version we will add out-of-sample log-likelihood comparisons, predictive scores on held-out periods, and a direct benchmark against a standard multivariate HMM without the auxiliary variable. These additions will be placed in a new subsection of the calibration results. revision: yes
Referee: The model construction in the HMM extension section defines the auxiliary variable to accommodate economic covariates, yet the manuscript does not derive or test why this third variable is necessary rather than extending the emission distribution directly or using a standard multivariate HMM; without such justification or ablation, the load-bearing role of the auxiliary variable for the dependency claim remains unestablished.

Authors: The auxiliary variable is introduced to let the shared hidden chain govern regime-dependent dependence between losses and covariates while keeping their conditional distributions separate; a direct multivariate emission or standard HMM does not isolate this regime-switching dependence in the same way. We will insert a short derivation subsection explaining this choice and include an ablation study comparing the full model to the two alternatives mentioned. revision: yes

Circularity Check

0 steps flagged

No significant circularity: standard HMM extension and EM calibration

full rationale

The manuscript proposes an extension of Hidden Markov Models that adds an auxiliary variable to incorporate macroeconomic covariates, then applies the standard Expectation-Maximization algorithm for calibration on operational-risk time series. Calibration results and parameter analysis for different risk-event types are reported, but no derivation step reduces a claimed prediction or uniqueness result to a fitted quantity by construction, nor does any load-bearing premise rest on a self-citation chain. The model structure is introduced explicitly rather than defined in terms of its own outputs, and EM is invoked as the conventional estimation procedure without renaming fitted parameters as independent predictions. The paper therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard HMM assumptions plus the new auxiliary variable; parameters are fitted via EM without independent verification of the extension's effectiveness.

free parameters (1)

transition and emission parameters
Fitted via EM algorithm to historical loss and covariate data; specific values not reported in abstract.

axioms (2)

domain assumption Hidden states follow a first-order Markov chain
Standard assumption for HMMs invoked to model time-dependent risk processes.
domain assumption Observations are conditionally independent given hidden states and auxiliary variable
Core HMM conditional independence used to incorporate multivariate risk and macro data.

invented entities (1)

auxiliary variable for economic covariates no independent evidence
purpose: To accommodate macroeconomic variables within the multivariate HMM observation model
Newly introduced component whose relevance is analyzed but lacks independent falsifiable evidence outside the model fit.

pith-pipeline@v0.9.0 · 5420 in / 1398 out tokens · 24960 ms · 2026-05-08T12:56:38.717958+00:00 · methodology

discussion (0)

Reference graph

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