Large Bayesian VARs for Binary and Censored Variables

Joshua C.C. Chan; Michael Pfarrhofer

REVIEW 4 major objections 4 minor 47 references

Large Bayesian VARs for Binary and Censored Variables

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper extends the vector autoregression to jointly model binary, censored, and continuous variables, and shows that the latent states have a truncated Gaussian posterior with a banded precision matrix, a structure that makes…

desk verdict A genuinely useful unified probit-tobit VAR with a real sampling contribution, but the 'scales well' claim is under-evidenced and the forecast evaluation lacks uncertainty quantification. read the letter →

arxiv 2506.01422 v1 pith:I7NVLMVS submitted 2025-06-02 econ.EM stat.CO

Joshua C.C. Chan , Michael Pfarrhofer This is my paper

classification econ.EMstat.CO MSC 62F1562M1091B84

keywords BayesianVARbinaryvariablescensoreddataaugmentationHamiltonianMonteCarlotruncatedGaussianrecessionforecastingeffectivelowerbound

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

Vector autoregressions (VARs) are the workhorse of macroeconomic forecasting and structural analysis, but they treat every variable as continuous, which is wrong for indicators such as recessions (binary) and interest rates near their effective lower bound (censored). This paper extends the VAR to jointly model binary, censored, and continuous variables by treating discrete outcomes as arising from latent continuous processes. The paper's central methodological claim is that the joint posterior of all latent variables is a truncated Gaussian whose precision matrix is banded, a sparsity pattern that allows Hamiltonian Monte Carlo to sample the high-dimensional latent states efficiently. If true, this makes mixed-outcome VARs practical at the scale of datasets with dozens of variables. In out-of-sample exercises, models that exploit this structure forecast recessions and short-term interest rates well, especially when the dataset is large and heteroskedasticity is accommodated.

What carries the argument

The load-bearing object is the precision matrix $V_l^{-1} = H_l' \Omega^{-1} H_l$ of the joint conditional posterior of the latent variables, where $H_l$ encodes the VAR dynamics for the latent block and $\Omega$ is the error covariance matrix. Because the VAR is finite-order, this matrix is banded with non-zero entries confined to a diagonal band, so iterative linear algebra and Hamiltonian Monte Carlo updates scale with the bandwidth rather than the full dimension squared. The paper combines this banded structure with a truncated Gaussian target and uses Zigzag Hamiltonian Monte Carlo, which requires only the precision matrix, to generate proposals that respect the inequality restrictions on binary and censored observations.

What would settle it

Simulate a system in which one variable (say the interest rate) has a volatility break that the others do not share, estimate the proposed ProToVAR on the simulated data, and check whether the posterior credible sets for the latent recession indicator achieve their nominal coverage; systematic under-coverage would show that the scalar volatility assumption is not innocuous.

Watch

Extended reading notes

Core claim

The discovery is that the data-augmented VAR for a mix of binary, censored, and continuous variables can be estimated at scale because the latent states, conditional on parameters and observed data, follow a truncated Gaussian distribution with a banded precision matrix. The paper derives the moments of this distribution by stacking the VAR into a single banded system and conditioning on observed values, and then samples from it using Zigzag Hamiltonian Monte Carlo, which needs only the precision matrix rather than its inverse. The authors further show that the framework accommodates heteroskedasticity through a scalar time-varying error multiplier, handles missing data through the same latent-variable mechanism, and supports conditional forecasting and generalized impulse response analysis. Empirical results indicate that larger datasets improve recession forecasts, that heteroskedastic models outperform homoskedastic ones after 2000, and that treating interest rates as censored improves short-rate forecasts.

Load-bearing premise

The time-varying covariance is restricted to a common scalar multiplier $\Sigma_t = o_t \Sigma$, so all variables must share the same proportional volatility dynamics; if one variable has its own volatility regime, the model is misspecified.

Editorial extensions

If this is right

Recession probability forecasts improve when the information set grows from a handful of core variables to about 25 financial, labor, and price variables, especially at horizons up to one year.
Treating interest rates as censored at the effective lower bound yields better short-rate forecasts than treating them as continuous, because iterative forecasts are guaranteed to stay above the bound.
Allowing for heteroskedasticity via an outlier component improves forecast performance, particularly in the post-2000 sample.
Generalized impulse responses show that a financial shock has larger effects on recession probabilities when the business cycle is near a turning point, and the effects are roughly symmetric across shock signs.
The same latent-variable machinery supports conditional forecasting with hard and soft constraints, as illustrated with Federal Reserve stress-test and December 2023 projection scenarios.

Reading between the lines

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

The banded-precision sampler could be carried over to ordinal and count variables without new conceptual machinery, since those also reduce to latent Gaussian states with inequality restrictions.
The shadow-rate estimates produced by the model offer a continuous measure of monetary policy stance at the effective lower bound, which researchers could use as a regressor or outcome in other settings.
If the scalar volatility assumption is too tight, a block-diagonal or factor-based $o_t$ would preserve the banded structure and could capture idiosyncratic volatility regimes.
The time-varying generalized impulse responses suggest that state-dependence of shock effects is identifiable in other binary-outcome VAR settings, such as sovereign defaults or financial crisis flags.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core of the paper is real: they extend the VAR to handle multiple binary and censored variables jointly, and they show that the latent states have a joint truncated Gaussian posterior with a banded precision matrix. That banded structure, combined with Zigzag HMC, is a sensible way to sample the high-dimensional latent block, and the derivation in Section 2.2 is clean. The paper is the first to put these pieces together in a large-VAR setting, and the empirical work is a genuine step beyond existing QualVAR and shadow-rate VAR applications.

What the paper does well: the forecasting design respects the NBER release calendar and ragged edges, the scenario analysis is a nice illustration of hard and soft constraints, and the GIRFs for recession probabilities show the framework can deliver economically interpretable results. The adaptation of Chan et al. (2023) to binary and censored variables is substantial, not a copy-and-paste.

The soft spots, in order of importance. First, the scalability claim is asserted, not demonstrated. The paper says the method 'scales well to high-dimensional settings,' but the banded precision matrix has bandwidth that grows with the number of variables, so a direct banded solve costs O(T P^2 n^3). That is cubic in n. The largest empirical model has n=25, and the simulation study uses n=6. No timing or scaling experiments are reported. The method is likely a real improvement over naive sampling, but 'high-dimensional' needs to be qualified. Second, the forecast evaluation uses AUC and CRPS as point estimates with no confidence intervals or significance tests, so it is hard to know whether the reported differences are meaningful. Third, no replication code is provided. Fourth, the covariance specification Sigma_t = o_t * Sigma restricts heteroskedasticity to a common scaling factor; variable-specific volatility dynamics, which could matter at the effective lower bound, are ruled out. The paper acknowledges this as a deliberate trade-off, but it is still a limitation for the ELB applications. None of these are fatal, and the central methodological claim holds up. The paper is a solid contribution that deserves a serious referee. A revision should add scaling experiments, uncertainty quantification for forecast comparisons, and ideally code. I would cite it if I worked on large Bayesian VARs or ELB modeling.

Referee Report

4 major / 4 minor

Summary. This paper extends the standard vector autoregression (VAR) to jointly model binary, censored, and continuous variables via latent data augmentation. The core methodological contribution is an efficient MCMC algorithm that samples the latent variables from their joint conditional posterior, which is shown to be truncated Gaussian with a banded precision matrix, using Hamiltonian Monte Carlo (specifically Zigzag HMC). The paper presents an out-of-sample forecasting exercise for U.S. recessions and interest rates across several model sizes and volatility specifications, as well as applications to conditional forecasting and structural impulse response analysis of a financial shock. The main claims are that the proposed ProToVAR framework scales well to high-dimensional settings and that models incorporating censoring and heteroskedasticity improve forecast performance.

Significance. If the scalability claim holds, this paper provides a practically useful and unified approach for handling mixed discrete-continuous data in large Bayesian VARs, with direct applications to recession forecasting, shadow-rate estimation, and structural analysis. The methodological derivation in Section 2.2 is sound and builds appropriately on Chan et al. (2023). The forecasting exercise is comprehensive and the empirical applications are relevant. However, the central scalability claim is not substantiated by formal complexity analysis or runtime scaling experiments, and the forecast comparisons lack uncertainty quantification. These issues currently limit the strength of the paper's conclusions.

major comments (4)

[Section 2.2, Eq. (7)] The claim that the banded precision matrix 'can be exploited to vastly speed up the computations' is not supported by a formal complexity analysis. The precision matrix V_l^{-1} = H_l' Ω^{-1} H_l is block-banded with half-bandwidth O(P n), because each non-zero block is a dense n×n matrix. Computing the conditional mean μ_l requires solving a banded linear system of dimension O(T n) with bandwidth O(P n), which costs O(T P^2 n^3) per MCMC sweep using standard banded solvers. For n=100 or more this becomes prohibitive. The paper reports no timing or scaling experiments beyond n=25 in the empirical applications and n=6 in the simulation appendix, so the claim that the approach 'scales well to high-dimensional settings' is not established. Please provide runtime scaling evidence for increasing n, or temper the scalability claim accordingly.
[Section 3.1, Figure 1 and Table 2] The forecasting evaluation relies on point estimates of AUC and CRPS ratios without any accompanying measures of uncertainty. For example, the differences between ProToVAR-het and ProVAR-hom at a given horizon could be within sampling variability, especially given the limited number of recession episodes in the evaluation sample. Please report posterior intervals or standard errors for the forecast metrics (e.g., via a bootstrap or a Bayesian posterior over the metrics), or at least discuss the magnitude of the reported differences relative to sampling uncertainty.
[Section 2.1, Σ_t = o_t Σ] The assumption that time-varying volatility is captured by a single common scalar o_t is restrictive: it cannot accommodate idiosyncratic volatility changes in individual variables, such as a regime shift in interest-rate volatility at the effective lower bound that is not shared by the rest of the system. Given that the paper emphasizes the empirical relevance of heteroskedasticity, please discuss the implications of this restriction for the recession probability forecasts and impulse responses, and provide robustness evidence with a more flexible volatility specification if possible.
[Appendix A] The simulation study only considers n=6 with a single sample size and does not report computation times or effective sample sizes as n increases. A more extensive simulation with n=25, 50, and 100 would directly address the scalability question and help readers assess the practical feasibility of the approach for the high-dimensional settings emphasized in the introduction.

minor comments (4)

[Abstract] The abstract contains a typo: 'wide rage' should be 'wide range'.
[Section 2.2] The notation bdiag(•) is used without definition; please define it when first introduced.
[Section 4.1, Figure 4] The text states the shadow rate reaches about -2% in 2021, but the first ZLB episode shown in Figure 4 begins in 2008 and the text elsewhere describes the evolution over 2008-2016; this appears to be a typo and should be corrected.
[Figure 2] The caption of Figure 2 does not specify that the top and bottom panels correspond to the core and large information sets; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: out-of-sample forecasts and model-implied GIRFs are not fitted inputs, and the banded-latent result rests on an independently published derivation.

full rationale

The paper's central claims are (i) that the latent states in the mixed binary/censored/continuous VAR have a truncated Gaussian joint posterior with banded precision, enabling HMC sampling; (ii) that the model forecasts recessions and interest rates well out of sample; and (iii) that financial shocks raise recession probabilities in a state-dependent way. None of these reduces to the model's inputs by construction. The forecasting evaluation is genuinely out-of-sample: parameters are estimated on expanding windows ending before each forecast origin, and the reported AUC/CRPS values compare model outputs to realized NBER classifications and interest rates, not to fitted values. The GIRFs and conditional forecasts are posterior/predictive functionals of the estimated model driven by externally specified shocks or Fed scenarios; they are not tuned to match the recession outcomes they describe. The banded-precision result is a mathematical consequence of the VAR transition structure and the block-diagonal error precision, and it is imported from Chan et al. (2023), a published, parameter-free derivation that does not contain the present paper's empirical targets; the adaptation here adds inequality restrictions and an HMC sampler, which constitute new content. The main risk flagged by the skeptic concerns the asymptotic cost of solving Eq. (7) as n grows and the absence of scaling experiments beyond n=25; that is a computational-complexity/correctness concern, not a circularity. No fitted parameter is relabeled as a prediction, and no load-bearing argument reduces to a self-citation chain.

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

The method relies on standard latent-variable data augmentation and on several modeling choices (lag length, ELB threshold, outlier scaling, prior degrees of freedom). None of these are fit to the forecasting targets; they are set a priori from prior practice.

free parameters (4)

Lag length P = 6
Chosen for monthly data; not estimated from the data.
ELB threshold for censoring = 0.25 percentage points
Taken from Carriero et al. (2025); observations below this threshold are treated as censored.
Outlier scaling bounds = U(2,10)
Chosen support for the common volatility outlier component o_t.
Inverse Wishart prior degrees of freedom = n+2
Chosen to make the prior proper in the parameter expansion sampler.

assumptions (4)

domain assumption Binary and censored observations are generated by latent Gaussian variables via probit and tobit links (Eqs. 1 and 2).
Imposes linear Gaussian dynamics on the latent process underlying discrete outcomes.
domain assumption The latent vector follows the linear VAR in Eq. (3) with error covariance Σt = otΣ.
The common scalar volatility is restrictive; variable-specific volatility changes are excluded.
standard math The precision matrix of the conditional latent posterior is banded (Eq. 7).
Follows from the banded structure of M and block-diagonality of Ω; underpins the fast sampler.
standard math Zigzag HMC (Nishimura et al., 2020) samples from the truncated normal in Eq. (8) correctly.
Assumes the sampler converges and its implementation is exact, which is supported by the authors' simulation but not proven formally.

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Cite this review

Pith. "Pith review of Large Bayesian VARs for Binary and Censored Variables." pith.science (2026). https://pith.science/paper/I7NVLMVS

@misc{pith2026250601422,
  author       = {Pith},
  title        = {Pith review of: Large Bayesian VARs for Binary and Censored Variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7NVLMVS}},
  note         = {Machine review of arXiv:2506.01422}
}

read the original abstract

We extend the standard VAR to jointly model the dynamics of binary, censored and continuous variables, and develop an efficient estimation approach that scales well to high-dimensional settings. In an out-of-sample forecasting exercise, we show that the proposed VARs forecast recessions and short-term interest rates well. We demonstrate the utility of the proposed framework using a wide rage of empirical applications, including conditional forecasting and a structural analysis that examines the dynamic effects of a financial shock on recession probabilities.

Figures

Figures reproduced from arXiv: 2506.01422 by the authors.

**Figure 1.** Area under ROC curve (AUC). Higher values indicate superior performance of the respective specification. Rows refer to subsamples, colored squares indicate the best performing specification by row. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗

**Figure 2.** Recession probabilities from ProToVAR-het for horizons h = −6, 0, 12 estimated using the core (upper panel) and large datasets (lower panel). Predictive median alongside 68/90 percent credible sets. comparison, we include two models that omit the binary recession indicator. The first is a standard VAR which treats all variables as continuous, which we denote as VAR. The second model treats the interest rates as cens… view at source ↗

**Figure 3.** presents the dynamic evolution of the latent business cycle indicator and the corresponding recession probabilities. Note that our implementation yields a countercyclical indicator. A pro-cyclical indicator can be readily obtained by re-defining the binary observed variable or adjusting Eq. (1) to ˜yb,it = I(yb,it < 0). As discussed in more detail in Dueker and Assenmacher-Wesche (2010), the distance of the latent … view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Estimates of the partially latent states associated with the interest rates subject to truncation at the ELB from ProToVAR-het-L. term interest rates hit the ELB. The first episode begins during the Great Recession and lasts till around 2016; the second corresponds to …

**Figure 5.** Figure 5: Conditional forecasts and restrictions for the 2019-12-01 forecast origin. Solid dots indicate realized values. In our second set of conditional forecasts, we consider the scenario based on the Federal Open Market Committee Projections, particularly the “Economic Proje…

**Figure 6.** Figure 6: Conditional forecasts and restrictions for the 2023-12-01 forecast origin. Solid dots indicate realized values. 4.3. Financial shocks in the US For this application, we investigate the dynamic effects of a financial shock on the recession probability. In contrast to pr…

**Figure 7.** Figure 7: Impulse response functions for selected latent and observed variables. Cumulated responses for differenced variables. Posterior median (solid black) alongside 68/90 percent posterior credible sets (gray shades) [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]

**Figure 8.** Figure 8: Peak generalized impulse response functions for recession probabilities over time, peak horizon h in shades of blue and red. Dots mark the posterior median alongside 68 percent credible set error bars. Gray shades indicate NBER recessions. (see [PITH_FULL_IMAGE:figure…

**Figure 9.** Figure 9: Generalized impulse response functions for recession probabilities. The left panels plot the posterior median response for each period (“Period medians”). The panel “Periods” shows the responses for selected periods alongside 68 percent credible sets, the rightmost pan…

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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.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

, " * 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.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

[2] [2]

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 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

[3] [3]

Albert J, and Chib S (1993), Bayesian Analysis of Binary and Polychotomous Response Data, Journal of the American Statistical Association 88, 669--679

work page 1993

[4] [4]

Anceschi N, Fasano A, Durante D, and Zanella G (2023), Bayesian conjugacy in probit, tobit, multinomial probit and extensions: a review and new results, Journal of the American Statistical Association 118(542), 1451--1469

work page 2023

[5] [5]

Antolin-Diaz J, Petrella I, and Rubio-Ram \' rez JF (2021), Structural scenario analysis with SVARs, Journal of Monetary Economics 117, 798--815

work page 2021

[6] [6]

Ba \'n bura M, Giannone D, and Lenza M (2015), Conditional forecasts and scenario analysis with vector autoregressions for large cross-sections, International Journal of forecasting 31(3), 739--756

work page 2015

[7] [7]

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

work page 2010

[8] [8]

Botev ZI (2017), The normal law under linear restrictions: simulation and estimation via minimax tilting, Journal of the Royal Statistical Society Series B: Statistical Methodology 79(1), 125--148

work page 2017