Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

A Bayesian Gaussian Process Dynamic Factor Model

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Bayesian dynamic factor model with Gaussian-process observation equations is computationally feasible and forecasts four US macro series more accurately than the standard linear DFM, with gains concentrated in COVID and zero-lower-bound e

desk verdict A genuinely useful nonlinear DFM with an honest forecasting exercise, but the spectral truncation boundary and missing replication leave the headline gains provisional. read the letter →

arxiv 2509.04928 v1 pith:F75ACQWX submitted 2025-09-05 econ.EM

Tony Chernis , Niko Hauzenberger , Haroon Mumtaz , Michael Pfarrhofer This is my paper · ORCID
classification econ.EM
keywords GaussianprocessdynamicfactormodelnonlinearstatespacemodelsBayesianMCMCparticleGibbswithancestorsamplingspectralapproximationmacroeconomicforecastingstochasticvolatilityglobalinflationdynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper's aim is to make the workhorse linear dynamic factor model (DFM) nonlinear — with the link between latent factors and every observed series learned from data instead of assumed — while keeping estimation computationally feasible. It places a Gaussian process prior on each observation equation, keeps the factor dynamics a linear VAR, and approximates the GP by a truncated spectral basis so the whole model can be estimated jointly with a Bayesian MCMC sampler. Forecasting four key US series from FRED-QD, the GP-DFM with two factors and stochastic volatility beats a linear DFM benchmark by 6–14% in joint predictive scores, with gains concentrated in the COVID and zero-lower-bound episodes and driven by tighter predictive densities. In a 61-country inflation application, world and regional factors affect national inflation in sign- and size-dependent ways that a linear DFM cannot express. If right, the payoff is a practical upgrade path: nonlinearity added to a standard policy tool at modest computational cost, with full uncertainty quantification and with structural tools (impulse responses, variance decompositions) still available because factor dynamics stay linear.

What carries the argument

The load-bearing device is the truncated spectral (Hilbert space) approximation of the GP kernels. On the bounded domain Ω = [-L,L]^D, each squared-exponential kernel expands in sine eigenfunctions of the Laplacian, weighted by its spectral density; truncating at M terms gives g_i(f_t) ≈ Σ_m φ_m(f_t)c_im, making the measurement equation linear (y_t = CΦ(f_t) + v_t) and cutting GP cost from O(T³) to O((T+1)M) per equation. Two kernels are used: multiplicative (M = M̃^D basis functions, allowing factor interactions) and additive (M = D × M̃, no interactions, scaling linearly in the number of factors). The latent factor path is sampled with Particle Gibbs with Ancestor Sampling to counter path

What would settle it

Posterior draws of the latent factors in the recursive forecasting exercise can be checked against the fixed box [-L, L] (L set once from the initial window): the sine eigenfunctions are pinned toward zero at the boundary, so any non-negligible share of out-of-sample factor draws outside the box means the forecast likelihood is evaluated outside its support. A direct robustness variant is to re-run the exercise with L doubled, halved, or re-computed from each expanding window and see whether the reported 6–14% energy-score gains survive; the paper tunes the number of basis functions but report

Watch

Extended reading notes

Core claim

Claim: a dynamic factor model with a nonparametric observation equation — Gaussian process priors on the unknown factor-to-variable links — is computationally feasible, with factor dynamics kept linear. The enabling device is a reduced-rank spectral approximation: stationary kernels are expanded in Laplacian eigenfunctions on a bounded domain and truncated, making the measurement equation linear in basis functions (y_t = CΦ(f_t) + v_t), sampled via Particle Gibbs with Ancestor Sampling. Evidence: two factors plus stochastic volatility cut energy-score losses by 6–14% versus linear DFMs, and in 61-country CPI data, factor contributions to inflation variance depend on shock sign and size.

Load-bearing premise

The load-bearing premise is that the truncated spectral approximation of the Gaussian-process kernels stays accurate on the fixed box Ω = [-L,L]^D, with L chosen once as 1.2 times the largest observed principal component. When the model is used to forecast recursively, latent factors are extrapolated beyond the estimation window; nothing keeps the drawn factor paths inside the box, and outside it the approximate likelihood and forecasts can distort. The paper reports tuning o

Editorial extensions

If this is right

  • Nonlinear DFMs become a practical option: the estimation algorithm runs on standard macro datasets (N around 100, D = 2–4) and delivers forecast gains, giving institutions that use linear DFMs a feasible upgrade path.
  • Fewer factors suffice: a two-factor GP-DFM outperforms linear DFMs with four to eight factors, implying that nonlinear factor extraction packs more information per component.
  • Forecast gains concentrate in turbulent episodes (COVID, the zero lower bound) and come from tighter predictive densities around realized outcomes rather than from point forecasts alone.
  • The additive kernel performs about as well as the multiplicative one, so interactions between factors are not a key source of forecast-relevant nonlinearity in this dataset.
  • Structural analysis becomes state-dependent: linear factor dynamics keep standard impulse-response and variance-decomposition tools valid at the factor level, while the nonlinear measurement map produces shock responses for observables that depend on the sign and size of the shock — demonstrated for global inflation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The box size L is the fragile point of the construction: the paper fixes L from the initial estimation window, but in an expanding-window forecast the factors are extrapolated, and beyond [-L,L] the sine eigenfunctions are pinned toward zero by the boundary conditions, so the implied function is data-free. Re-estimating or widening L each recursion is a direct robustness check that would either co
  • The near-parity of the additive kernel with the multiplicative one suggests macroeconomic nonlinearity is mostly factor-wise rather than interaction-driven; a natural testable extension is a hybrid kernel that adds interaction terms only for selected factor pairs, recovering flexibility at multiplicative cost only where the data demand it.
  • Because the state equation stays linear, all conventional VAR tools transfer to the factors and the GP map transfers them — nonlinearly — to observables. This makes the model a ready instrument for 'at-risk' quantities (the distribution of output growth or inflation conditional on factor shocks), where asymmetric sign- and size-dependence is exactly the object of interest.
  • The spectral approximation extends periodically outside the box: the sine basis continues the function beyond [-L,L] rather than stopping it, so long-horizon forecasts and stress scenarios that push factors outward inherit function values that are artifacts of the truncation, not of the data. Sensitivity of scenario analyses to L would be a worthwhile check in applied use.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Gaussian Process Dynamic Factor Model (GP-DFM) in which the observation equation maps latent factors to observables through unknown, potentially nonlinear functions with Gaussian process priors. Factor dynamics follow a linear VAR, optionally with stochastic volatility. To make estimation feasible, the GP is approximated by a truncated spectral (Hilbert space) representation on a bounded domain, and posterior inference is conducted via Particle Gibbs with Ancestor Sampling. Two empirical exercises are reported: (i) recursive out-of-sample forecasting of four FRED-QD variables relative to linear DFM benchmarks, with the main claim that GP-DFMs with SV improve predictive accuracy, particularly during COVID and near the zero lower bound; and (ii) a semi-structural decomposition of global inflation into world, developed, and EMDE factors with state-dependent forecast error variance decompositions.

Significance. If the results hold, the paper makes a practical contribution by showing that a nonlinear DFM with a nonparametric observation equation can be estimated with a feasible MCMC algorithm and can improve forecast accuracy and structural inference relative to linear DFMs. The additive kernel is a sensible computational device that appears to sacrifice little accuracy, and the out-of-sample evaluation is honest in reporting that gains are concentrated in turbulent episodes. The structural application illustrates a useful way to obtain state-dependent factor decompositions. However, the central forecast claim rests on the validity of the truncated spectral approximation out-of-sample, and the paper provides no direct evidence on this point. The absence of approximation-error checks and of an explicit identification discussion means that the headline results, while plausible, are not yet fully established.

major comments (4)
  1. [2.3, 3.1] The spectral approximation in Eqs. (7)–(8) is valid only on the bounded domain Omega=[-L,L]^D, with L set in Section 3.1 as 1.2 times the maximum absolute value of the first D principal components of the estimation data. The factor VAR in Eq. (2) has Gaussian innovations with unbounded support, and forecasts are iterated up to eight quarters ahead. Out-of-sample factor draws are therefore almost guaranteed to leave Omega at some recursion, especially during the COVID and ZLB episodes where the reported gains are largest. No sensitivity analysis for L is reported. The authors should provide evidence that the approximate measurement equation remains faithful along the out-of-sample factor paths: e.g., report the share of posterior factor draws falling outside Omega, re-run the forecasting exercise with L inflated by a factor of 1.5 or 2.0 (or with L updated on an expanding window), and sho
  2. [2.3, 3.1] The paper chooses M-tilde=8 basis functions per dimension for the additive kernel and M-tilde=8 or 4 for the multiplicative kernel, with the remark in footnote 10 that 'we have empirically tested different M-tilde values' but no numerical results are reported. For a claim that the GP-DFM outperforms linear benchmarks, it is load-bearing that the truncated basis actually approximates the intended squared-exponential kernel over the relevant input region. The authors should report, for the selected M and L, a concrete approximation error measure (e.g., the maximum absolute or relative error between K_M and the exact squared-exponential kernel over Omega, or the resulting KL divergence between the implied GP priors) and show that the forecast rankings are insensitive to doubling M-tilde.
  3. [2.1, 4] The model in Eqs. (1)–(2) with unknown functions g_i has a substantial rotational and scaling indeterminacy: any invertible transformation of the factors can be absorbed into the functions g_i and the factor VAR. The paper does not discuss how the factors are identified, normalized, or labeled, beyond a brief statement in Section 4 that a tight prior is placed on the first row of loadings using a principal-component initialization. This matters for both the forecast application (where factor draws are fed into the nonlinear measurement equation) and the structural application in Section 4 (where the GIRFs and GFEVDs are interpreted as responses to structural shocks to specific factors). The authors should state the identifying assumptions explicitly and explain how the posterior sampler avoids rotational mixing or label switching, or at least provide evidence that the reported factor pat
  4. [3.2, Figures 1–3] Statistical significance is assessed with one-sided Diebold–Mariano tests, but the paper compares 16 specifications across multiple horizons, evaluation windows, and four target variables. The practice of highlighting the best-performing specification in bold, combined with one-sided tests without any multiple-testing control, overstates the strength of the evidence for the claim that 'the GP-DFM with SV outperforms linear versions of the model' (Section 5). The qualitative conclusion may survive, but the authors should either apply a multiple-testing correction (e.g., FDR control) or pre-specify a limited set of key comparisons and report unadjusted and adjusted p-values together.
minor comments (5)
  1. [3.2] Typo: 'Forth' should be 'Fourth' in the list of main takeaways.
  2. [2.3] Duplicate phrase 'we introduce introduce' in the first paragraph of the additive squared-exponential kernel subsection.
  3. [Figures 1–3] Several cells in the reported tables appear corrupted, e.g., '0.331', '0.338', '0.005', and '0.003'. If these are not actual numbers, the figures should be regenerated; if they are extreme values, they require a note or explanation.
  4. [5] The conclusion refers to 'CPU inflation rates'; this should be 'CPI inflation rates'.
  5. [A.2] The ancestor-sampling step uses the approximation with tau1=5, but no sensitivity analysis is given for this tuning choice. A brief robustness check would be useful, though I do not view this as central to the paper's claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GP-DFM results are genuine out-of-sample forecasts from a self-contained Bayesian model.

full rationale

The paper's central claims are empirical: the GP-DFM is estimated on expanding windows and evaluated on genuine out-of-sample forecast periods, so the reported gains are not in-sample fits relabeled as predictions. The model is defined by explicit equations (measurement equation (1), factor VAR (2), GP priors (3)-(4)) and estimated with a fully specified MCMC/PGAS algorithm; no parameter is fitted to the target forecast and then reported as a prediction. The spectral approximation (7)-(8) is a standard reduced-rank eigenfunction expansion imported from Solin & Sarkka (2020) and Riutort-Mayol et al. (2023), not a result derived from the paper's own output. The self-citations (e.g., Hauzenberger et al. 2025, Pfarrhofer and Stelzer 2025) are used as methodological context or for computational tools, not as load-bearing justification for the paper's novelty or empirical conclusions. The choice of boundary L as 1.2 times the maximum absolute value of the first D principal components is an approximation-domain choice; concerns about out-of-sample factor paths leaving that domain are robustness/validity issues, not circularity, because the forecasts are still computed from the stated model rather than constructed to match the evaluation data.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The model introduces no new physical or theoretical entities. The central claim rests on the accuracy of the truncated spectral GP approximation, the validity of PGAS for this state space model, and the absence of severe factor identification problems.

free parameters (4)
  • number of basis functions per dimension (M-tilde) = 8 for additive kernels (all D) and multiplicative D=2; 4 for multiplicative D=4
    Chosen by empirical testing on the same data, not by a formal rule (Section 3.1, footnote 10).
  • boundary L of the truncated GP domain = 1.2 times the maximum absolute value of the first D principal components
    Semi-automatic, data-dependent choice (Section 3.1). No sensitivity analysis reported.
  • number of particles H in PGAS = not reported
    PGAS requires H to be specified; the paper omits this value, affecting mixing and runtime.
  • tau1, lags used in ancestor sampling approximation = 5
    Set to 5 in the approximation formula for ancestor sampling weights (Appendix A.2).
assumptions (3)
  • standard math Eigenexpansion of a stationary covariance function on a bounded domain Omega is a valid approximation to the GP kernel.
    The Hilbert space approximation in Section 2.3 follows Solin and Sarkka (2020) and assumes the GP is restricted to Omega = [-L,L]^D.
  • domain assumption Particle Gibbs with Ancestor Sampling produces a Markov kernel with the correct invariant distribution for this nonlinear state space model.
    Used in Section 2.5 and Appendix A.2; relies on regularity conditions in Lindsten et al. (2014) that are not verified for this model.
  • domain assumption Latent factors are identified enough for interpretation through priors, without explicit rotation or normalization restrictions for the forecasting model.
    No identification restrictions are imposed in Sections 2.1-2.4; the tight prior on first-row loadings appears only in the inflation application (Section 4).

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Bayesian Gaussian Process Dynamic Factor Model." pith.science (2026). https://pith.science/paper/F75ACQWX

@misc{pith2026250904928,
  author       = {Pith},
  title        = {Pith review of: A Bayesian Gaussian Process Dynamic Factor Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F75ACQWX}},
  note         = {Machine review of arXiv:2509.04928}
}
read the original abstract

We propose a dynamic factor model (DFM) where the latent factors are linked to observed variables with unknown and potentially nonlinear functions. The key novelty and source of flexibility of our approach is a nonparametric observation equation, specified via Gaussian Process (GP) priors for each series. Factor dynamics are modeled with a standard vector autoregression (VAR), which facilitates computation and interpretation. We discuss a computationally efficient estimation algorithm and consider two empirical applications. First, we forecast key series from the FRED-QD dataset and show that the model yields improvements in predictive accuracy relative to linear benchmarks. Second, we extract driving factors of global inflation dynamics with the GP-DFM, which allows for capturing international asymmetries.

Figures

Figures reproduced from arXiv: 2509.04928 by the authors.

Figure 5
Figure 5. plots the estimated posterior distribution of the three factors. The global factor has a distinct peaks in the earlier part of the sample in 1974Q1 correspond to the aftermath of the oil shock. After 1990, the global factor displays a declining trend consistent with the Great Moderation. This came to an end with a sharp rise in 2022Q1, possibly associated with supply disruptions in the post-COVID period [PITH_FULL_… view at source ↗
Figure 6
Figure 6. Forecast error variance decomposition of country-specific inflation series. (a) Developed economies (b) EMDEs Note: The figure displays the contribution to the forecast error variance of inflation at the one-year horizon of shocks to the global factor (red), developed country factor (blue) for different shock sizes. The contributions are averaged across initial conditions. The solid lines are medians while the shade… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Clustered Local Projections for Short and Ultra-Short Time Series -- A Hierarchical Bayesian Framework

    econ.EM 2026-08 conditional novelty 6.0 of 10

    A hierarchical Bayesian sparse-mixture model pools local-projection impulse responses across an unbalanced panel, improving short-sample estimates and revealing heterogeneous price responses to supply-chain and oil shocks.

Reference graph

Works this paper leans on

93 extracted references · 80 canonical work pages · cited by 1 Pith paper

  1. [1]

    Adrian T, Boyarchenko N, and Giannone D (2019), Vulnerable Growth, American Economic Review 109(4), 1263--1289

    2019

  2. [2]

    Aguilar O, and West M (2000), Bayesian dynamic factor models and portfolio allocation, Journal of Business & Economic Statistics 18(3), 338--357

    2000

  3. [3]

    Andreini P, Izzo C, and Ricco G (2023), Deep Dynamic Factor Models, arXiv 2007.11887

    work page Pith review arXiv 2023

  4. [4]

    Andrieu C, Doucet A, and Holenstein R (2010), Particle Markov chain Monte Carlo methods , Journal of the Royal Statistical Society Series B 72(3), 269--342

    2010

  5. [5]

    Babii A, Ghysels E, and Striaukas J (2022), Machine learning time series regressions with an application to nowcasting, Journal of Business & Economic Statistics 40(3), 1094--1106

    2022

  6. [6]

    Bai J, and Ng S (2008), Forecasting economic time series using targeted predictors, Journal of Econometrics 146(2), 304--317

    2008

  7. [7]

    Inflation During the COVID Era, Working Paper 30613, National Bureau of Economic Research

    Ball LM, Leigh D, and Mishra P (2022), Understanding U.S. Inflation During the COVID Era, Working Paper 30613, National Bureau of Economic Research

    2022

  8. [8]

    Ba \'n bura M, Giannone D, and Reichlin L (2010), Large Bayesian vector auto regressions, Journal of Applied Econometrics 25(1), 71--92

    2010

Show all 93 references
  1. [9]

    Bassetti F, Casarin R, and Leisen F (2014), Beta-product dependent Pitman--Yor processes for Bayesian inference, Journal of Econometrics 180(1), 49--72

    2014

  2. [10]

    Beaudry P, Hou C, and Portier F (2024), The Dominant Role of Expectations and Broad-Based Supply Shocks in Driving Inflation, Working Paper 32322, National Bureau of Economic Research

    2024

  3. [11]

    Benigno P, and Eggertsson GB (2023), It's Baaack: The Surge in Inflation in the 2020s and the Return of the Non-Linear Phillips Curve, Working Paper 31197, National Bureau of Economic Research

    2023

  4. [12]

    Bhattacharya A, Pati D, and Dunson D (2014), Anisotropic function estimation using multi-bandwidth Gaussian processes, Annals of Statistics 42(1), 352

    2014

  5. [13]

    Bok B, Caratelli D, Giannone D, Sbordone AM, and Tambalotti A (2018), Macroeconomic nowcasting and forecasting with big data, Annual Review of Economics 10(1), 615--643

    2018

  6. [14]

    Brunnermeier MK, and Sannikov Y (2014), A Macroeconomic Model with a Financial Sector, American Economic Review 104(2), 379--421

    2014

  7. [15]

    Camacho M, Perez-Quiros G, and Poncela P (2018), Markov-switching dynamic factor models in real time, International Journal of Forecasting 34(4), 598--611

    2018

  8. [16]

    Large Bayesian vector autoregressions with stochastic volatility and non-conjugate priors

    Carriero A, Chan J, Clark TE, and Marcellino M (2022), Corrigendum to "Large Bayesian vector autoregressions with stochastic volatility and non-conjugate priors"[J. Econometrics 212 (1)(2019) 137--154], Journal of Econometrics 227(2), 506--512

    2022

  9. [17]

    Carter CK, and Kohn R (1994), On Gibbs sampling for state space models, Biometrika 81(3), 541--553

    1994

  10. [18]

    Carvalho CM, Polson NG, and Scott JG (2010), The horseshoe estimator for sparse signals, Biometrika 465--480

    2010

  11. [19]

    Chan JC, and Jeliazkov I (2009), Efficient simulation and integrated likelihood estimation in state space models, International Journal of Mathematical Modelling and Numerical Optimisation 1(1-2), 101--120

    2009

  12. [20]

    Chauvet M (1998), An Econometric Characterization of Business Cycle Dynamics with Factor Structure and Regime Switching, International Economic Review 39(4), 969--996

    1998

  13. [21]

    Chauvet M, and Senyuz Z (2016), A dynamic factor model of the yield curve components as a predictor of the economy, International Journal of Forecasting 32(2), 324--343

    2016

  14. [22]

    Cheng MY, and Wu Ht (2013), Local linear regression on manifolds and its geometric interpretation, Journal of the American Statistical Association 108(504), 1421--1434

    2013

  15. [23]

    Chernis T, and Sekkel R (2017), A dynamic factor model for nowcasting Canadian GDP growth , Empirical Economics 53(1), 217--234

    2017

  16. [24]

    Chronopoulos I, Raftapostolos A, and Kapetanios G (2024), Forecasting value-at-risk using deep neural network quantile regression, Journal of Financial Econometrics 22(3), 636--669

    2024

  17. [25]

    Clark TE (2011), Real-time density forecasts from Bayesian vector autoregressions with stochastic volatility, Journal of Business & Economic Statistics 29(3), 327--341

    2011

  18. [26]

    Clark TE, Huber F, and Koop G (2025), A Nonparametric Approach to Augmenting a Bayesian VAR with Nonlinear Factors, arXiv 2508.13972

    2025 arXiv

  19. [27]

    Clark TE, Huber F, Koop G, and Marcellino M (2024 a ), Forecasting US inflation using Bayesian nonparametric models, The Annals of Applied Statistics 18(2), 1421--1444

    2024

  20. [28]

    Clark TE, Huber F, Koop G, Marcellino M, and Pfarrhofer M (2024 b ), Investigating growth-at-risk using a multicountry nonparametric quantile factor model, Journal of Business & Economic Statistics 42(4), 1302--1317

    2024

  21. [29]

    Clark TE, and Ravazzolo F (2015), Macroeconomic forecasting performance under alternative specifications of time-varying volatility, Journal of Applied Econometrics 30(4), 551--575

    2015

  22. [30]

    Cogley T, and Sargent TJ (2005), Drifts and volatilities: monetary policies and outcomes in the post WWII US, Review of Economic Dynamics 8(2), 262--302

    2005

  23. [31]

    Del Negro M, and Otrok C (2008), Dynamic factor models with time-varying parameters: measuring changes in international business cycles, Staff Report 326, Federal Reserve Bank of New York

    2008

  24. [32]

    Diebold FX, and Mariano RS (1995), Comparing predictive accuracy, Journal of Business & Economic Statistics 13(3), 253--263

    1995

  25. [33]

    Diebold FX, Shin M, and Zhang B (2023), On the aggregation of probability assessments: Regularized mixtures of predictive densities for Eurozone inflation and real interest rates, Journal of Econometrics 237(2, Part C), 105321

    2023

  26. [34]

    Doz C, and Fuleky P (2020), Dynamic factor models, Macroeconomic Forecasting in the Era of Big Data: Theory and Practice 27--64

    2020

  27. [35]

    Farrell MH, Liang T, and Misra S (2021), Deep neural networks for estimation and inference, Econometrica 89(1), 181--213

    2021

  28. [36]

    Forni M, Hallin M, Lippi M, and Reichlin L (2000), The generalized dynamic-factor model: Identification and estimation, Review of Economics and statistics 82(4), 540--554

    2000

  29. [37]

    Frigola R, Lindsten F, Sch\" o n TB, and Rasmussen CE (2013), Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC, in C Burges, L Bottou, M Welling, Z Ghahramani, and K Weinberger (eds.) Advances in Neural Information Processing Systems, v...

    2013

  30. [38]

    Fr \"u hwirth-Schnatter S (1994), Data augmentation and dynamic linear models, Journal of Time Series Analysis 15(2), 183--202

    1994

  31. [39]

    Fr \"u hwirth-Schnatter S, Hosszejni D, and Lopes HF (2025), Sparse Bayesian factor analysis when the number of factors is unknown (with discussion), Bayesian Analysis 20(1), 213--344

    2025

  32. [40]

    Geweke J, and Amisano G (2010), Comparing and evaluating Bayesian predictive distributions of asset returns, International Journal of Forecasting 26(2), 216--230

    2010

  33. [41]

    Giannone D, Lenza M, and Primiceri GE (2021), Economic predictions with big data: The illusion of sparsity, Econometrica 89(5), 2409--2437

    2021

  34. [42]

    Giannone D, Reichlin L, and Small D (2008), Nowcasting: The real-time informational content of macroeconomic data, Journal of Monetary Economics 55(4), 665--676

    2008

  35. [43]

    Gneiting T, and Raftery AE (2007), Strictly proper scoring rules, prediction, and estimation, Journal of the American statistical Association 102(477), 359--378

    2007

  36. [44]

    Goulet Coulombe P (2024), The macroeconomy as a random forest, Journal of Applied Econometrics 39(3), 401--421

    2024

  37. [45]

    Goulet Coulombe P, Leroux M, Stevanovic D, and Surprenant S (2022), How is machine learning useful for macroeconomic forecasting? Journal of Applied Econometrics 37(5), 920--964

    2022

  38. [46]

    Guerr \'o n-Quintana P, Khazanov A, and Zhong M (2023), Financial and Macroeconomic Data Through the Lens of a Nonlinear Dynamic Factor Model, FEDS Working Paper 2023-27

    2023

  39. [47]

    Ha J, Kose MA, and Ohnsorge F (2023), One-stop source: A global database of inflation, Journal of International Money and Finance 137, 102896

    2023

  40. [48]

    Hamilton JD (1989), A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 357--384

    1989

  41. [49]

    --- --- --- (2001), A parametric approach to flexible nonlinear inference, Econometrica 69(3), 537--573

    2001

  42. [50]

    Hauzenberger N, Huber F, and Klieber K (2023), Real-time inflation forecasting using non-linear dimension reduction techniques, International Journal of Forecasting 39(2), 901--921

    2023

  43. [51]

    Hauzenberger N, Huber F, Klieber K, and Marcellino M (2024 a ), Bayesian neural networks for macroeconomic analysis, Journal of Econometrics 105843

    2024

  44. [52]

    Hauzenberger N, Huber F, Marcellino M, and Petz N (2025), Gaussian process vector autoregressions and macroeconomic uncertainty, Journal of Business & Economic Statistics 43(1), 27--43

    2025

  45. [53]

    Hauzenberger N, Marcellino M, Pfarrhofer M, and Stelzer A (2024 b ), Bayesian Nowcasting with mixed frequency data using gaussian processes, arXiv 2402.10574

    2024 arXiv

  46. [54]

    Huber F, and Feldkircher M (2019), Adaptive shrinkage in Bayesian vector autoregressive models, Journal of Business & Economic Statistics 37(1), 27--39

    2019

  47. [55]

    Huber F, Koop G, Onorante L, Pfarrhofer M, and Schreiner J (2023), Nowcasting in a pandemic using non-parametric mixed frequency VARs, Journal of Econometrics 232(1), 52--69

    2023

  48. [56]

    Jin X, Maheu JM, and Yang Q (2022), Infinite Markov pooling of predictive distributions, Journal of Econometrics 228(2), 302--321

    2022

  49. [57]

    Kalli M, and Griffin JE (2018), Bayesian nonparametric vector autoregressive models, Journal of Econometrics 203(2), 267--282

    2018

  50. [58]

    Kastner G, and Fr \"u hwirth-Schnatter S (2014), Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models, Computational Statistics & Data Analysis 76, 408--423

    2014

  51. [59]

    Kaufmann S, and Schumacher C (2019), Bayesian estimation of sparse dynamic factor models with order-independent and ex-post mode identification, Journal of Econometrics 210(1), 116--134

    2019

  52. [60]

    Klieber K (2024), Non-linear dimension reduction in factor-augmented vector autoregressions, Journal of Economic Dynamics and Control 159, 104800

    2024

  53. [61]

    Koop G, and Korobilis D (2013), Large time-varying parameter VARs, Journal of Econometrics 177(2), 185--198

    2013

  54. [62]

    Koop G, Pesaran MH, and Potter SM (1996), Impulse response analysis in nonlinear multivariate models, Journal of Econometrics 74(1), 119--147

    1996

  55. [63]

    Korobilis D (2013), Assessing the transmission of monetary policy using time-varying parameter dynamic factor models, Oxford Bulletin of Economics and Statistics 75(2), 157--179

    2013

  56. [64]

    Kose MA, Otrok C, and Whiteman CH (2003), International business cycles: World, region, and country-specific factors, American Economic Review 93(4), 1216--1239

    2003

  57. [65]

    Lanne M, and Nyberg H (2016), Generalized Forecast Error Variance Decomposition for Linear and Nonlinear Multivariate Models, Oxford Bulletin of Economics and Statistics 78(4), 595--603

    2016

  58. [66]

    Lee J, Bahri Y, Novak R, Schoenholz SS, Pennington J, and Sohl-Dickstein J (2018), Deep Neural Networks as Gaussian Processes,

    2018

  59. [67]

    Lindsten F, Jordan MI, and Sch \"o n TB (2014), Particle Gibbs with Ancestor Sampling, Journal of Machine Learning Research 15(63), 2145--2184

    2014

  60. [68]

    Luo Y, Paige B, and Griffin J (2025), Time-varying Factor Augmented Vector Autoregression with Grouped Sparse Autoencoder, arXiv 2503.04386

    2025 arXiv

  61. [69]

    Makalic E, and Schmidt DF (2015), A simple sampler for the horseshoe estimator, IEEE Signal Processing Letters 23(1), 179--182

    2015

  62. [70]

    Marcellino M, and Pfarrhofer M (2024), Bayesian nonparametric methods for macroeconomic forecasting, in MP Clements, and AB Galv\ a o (eds.) Handbook of Research Methods and Applications in Macroeconomic Forecasting, chapter 5, 90--125, Cheltenham, UK/Northampton, USA: Edward ...

    2024

  63. [71]

    McCracken MW, and Ng S (2016), FRED-MD: A monthly database for macroeconomic research, Journal of Business & Economic Statistics 34(4), 574--589

    2016

  64. [72]

    --- --- --- (2021), FRED-QD: A Quarterly Database for Macroeconomic Research , Review 103(1), 1--44

    2021

  65. [73]

    Medeiros MC, Vasconcelos GF, Veiga \'A , and Zilberman E (2021), Forecasting inflation in a data-rich environment: the benefits of machine learning methods, Journal of Business & Economic Statistics 39(1), 98--119

    2021

  66. [74]

    Mumtaz H, and Surico P (2012), Evolving international inflation dynamics: world and country-specific factors, Journal of the European Economic Association 10(4), 716--734

    2012

  67. [75]

    Neal RM (1996), Priors for Infinite Networks, 29--53, New York, NY: Springer New York

    1996

  68. [76]

    Pfarrhofer M, and Stelzer A (2025), Scenario analysis with multivariate Bayesian machine learning models, arXiv 2502.08440

    2025

  69. [77]

    Primiceri GE (2005), Time varying structural vector autoregressions and monetary policy, The Review of Economic Studies 72(3), 821--852

    2005

  70. [78]

    Riutort-Mayol G, B \"u rkner PC, Andersen MR, Solin A, and Vehtari A (2023), Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming, Statistics and Computing 33(1), 17

    2023

  71. [79]

    Sims CA, and Zha T (2006), Were there regime switches in US monetary policy? American Economic Review 96(1), 54--81

    2006

  72. [80]

    Snellman O (2024), Non-Linear Dynamic Factor Analysis With a Transformer Network, SSRN 5000165

    2024

  73. [81]

    Solin A, and S \"a rkk \"a S (2020), Hilbert space methods for reduced-rank Gaussian process regression, Statistics and Computing 30(2), 419--446

    2020

  74. [82]

    Stock J, and Watson M (2016), Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics , in JB Taylor, and H Uhlig (eds.) Handbook of Macroeconomics , Handbook of Macroeconomics, volume 2, chapter 0, 415--525, Elsevier

    2016

  75. [83]

    Stock JH, and Watson MW (2002), Macroeconomic forecasting using diffusion indexes, Journal of Business & Economic Statistics 20(2), 147--162

    2002

  76. [84]

    --- --- --- (2012), Disentangling the Channels of the 2007-2009 Recession, Technical report, National Bureau of Economic Research

    2012

  77. [85]

    a rkk \"a S, and Sch \

    Svensson A, Solin A, S \"a rkk \"a S, and Sch \"o n T (2016), Computationally efficient Bayesian learning of Gaussian process state space models, in Artificial Intelligence and Statistics, 213--221, PMLR

    2016

  78. [86]

    Ter \"a svirta T (1994), Specification, estimation, and evaluation of smooth transition autoregressive models, Journal of the american Statistical association 89(425), 208--218

    1994

  79. [87]

    Turner R, Deisenroth M, and Rasmussen C (2010), State-Space Inference and Learning with Gaussian Processes, in YW Teh, and M Titterington (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Re...

    2010

  80. [88]

    van der Vaart AW, and van Zanten JH (2008), Rates of contraction of posterior distributions based on Gaussian process priors , The Annals of Statistics 36(3), 1435 -- 1463

    2008

  81. [89]

    Essays on methods to cast nonlinear unobserved components

    Velasco S (2024), Let the Tree Decide: FABART A Non-Parametric Factor Model, Phd thesis, Queen Mary University of London, chapter 3 of "Essays on methods to cast nonlinear unobserved components"

    2024

  82. [90]

    Williams CK, and Rasmussen CE (2006), Gaussian processes for machine learning, volume 2, Cambridge, MA, USA: MIT Press

    2006

  83. [91]

    Zhou X, Nakajima J, and West M (2014), Bayesian forecasting and portfolio decisions using dynamic dependent sparse factor models, International Journal of Forecasting 30(4), 963--980

    2014

  84. [92]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor eid howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.senten...

  85. [93]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.