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REVIEW 5 major objections 4 minor 73 references

This paper claims that a two-dimensional Markov chain with power-law intensity can reproduce the long-memory behavior of a Hawkes process while remaining finite-dimensional and provably stable.

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2026-08-01 09:13 UTC pith:V4LKDD4L

load-bearing objection Interesting model, but the central stability proof rests on a false tail bound; this needs refereeing, not acceptance as-is. the 5 major comments →

arxiv 2607.20838 v1 pith:V4LKDD4L submitted 2026-07-23 math.PR stat.ME

Long-memory Markov chains with power-law intensities

Kyungsub Lee This is my paper
classification math.PR stat.ME MSC 60J0560G55
keywords self-exciting point processpower-law Hawkes processlong memoryMarkov chainHarris recurrencestability conditioninverse-integrated intensityinter-arrival times
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to show that a two-dimensional Markov chain (X_n, c_n) can capture the essential dynamics of a power-law Hawkes process while remaining finite-dimensional. The conditional intensity between events is μ + X_n(1 + t/c_n)^{-p}, and at each event both the intensity and its slope jump by amounts matching the power-law Hawkes update, with no dependence on the full history. Under an explicit parameter inequality, the author proves the embedded chain is positive Harris recurrent, meaning a unique stationary distribution exists, and that it is ψ-irreducible, aperiodic, and a T-chain under mild noise conditions. Simulations at the stability boundary give frequency-domain estimates of the memory parameter close to 0.5, indicating long-range dependence in event durations. The key proof step assumes a tail-decay property of the residual distribution that is not guaranteed by a finite second moment.

Core claim

At each arrival, the state (X_n,c_n) is updated as X_n = X_{n-1} r_n^{-p} + ξ and X_n/c_n = (X_{n-1}/c_{n-1}) r_n^{-(p+1)} + ξ/γ, where r_n = 1 + τ_n/c_{n-1}; these relations reproduce, step by step, the local jumps in intensity and slope of a power-law Hawkes process. The central claim is that this two-dimensional Markov chain is ψ-irreducible, aperiodic and a T-chain whenever the residual distribution has a density and finite first two moments, and is positive Harris recurrent under the stability condition ξ < p/(2γ) together with the companion bound in Assumption 1. The proof uses a drift function V(x,c)=x+wc: outside a compact set the drift is negative, which yields the unique invariant

What carries the argument

The central object is the two-dimensional latent state (X_n, c_n), encoding the current intensity level and its power-law time scale, together with the decay ratio r_n = 1 + τ_n/c_{n-1}. The update equations above are the engine: the factors r_n^{-p} and r_n^{-(p+1)} contract the state toward (ξ,γ), the constant ξ/γ provides excitation, and the inverse-integrated-intensity map Φ(t;x,c)=μt + (xc/(p-1))(1-(1+t/c)^{1-p}) turns an i.i.d. residual ε_n into the waiting time τ_n. The stability proof uses the linear combination V(x,c)=x+wc as a Lyapunov function; showing its conditional drift is negative outside a compact set is what yields positive Harris recurrence.

Load-bearing premise

The negative-drift proof for X_n assumes that the tail integral ∫_K^∞(u−K)P(ε>u)du decays faster than 1/K whenever the residual ε has finite second moment; heavy-tailed residuals violate this, so the drift bound O(x^{-1}) can fail.

What would settle it

Take a Pareto residual distribution with density ~ u^{-3.5}, compute the tail integral numerically, and confirm it decays like K^{-1/2}, not o(K^{-1}); then simulate the chain with these residuals under Assumption 1 and check whether the empirical mean of X_n over long horizons stays bounded or diverges.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Under the stability condition, the chain has a unique invariant probability measure, so long-run averages of any bounded function of the state converge to a fixed expectation.
  • The inverse-integrated-intensity representation gives an exact simulation and likelihood recipe: draw ε_n, set τ_n = Φ^{-1}(ε_n; X_{n-1}, c_{n-1}), and update the state.
  • At the stability boundary γ*, the estimated memory parameter is close to 0.5, the maximal value compatible with stationarity, so the model can generate near-critical persistence in event times.
  • The irreducibility, aperiodicity and T-chain results hold for a broad class of residual distributions, so the structural conclusions are not tied to exponential waiting times.
  • The explicit stability region in terms of p, ξ and γ allows a practitioner to tune excitation strength and decay exponent while keeping the process stationary.

Where Pith is reading between the lines

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

  • If the tail-decay gap is filled (for instance, by requiring a residual moment of order > 2), the same Lyapunov argument likely goes through; a simple check is whether the drift function can be replaced by x^a + w c^a with a > 1.
  • Long memory is currently demonstrated by simulation; a proof that the stationary spectral density diverges at frequency zero would turn the numerical evidence into a theorem.
  • The two-dimensional construction suggests a template for other power-law kernels: adding auxiliary slope states increases the Markovian dimension, and the controllability rank condition in the paper gives a criterion for when the extra states are needed.
  • For empirical work on financial durations, the model offers a Markovian competitor to fractional ACD models, with the apparent advantage that near-criticality emerges from a stability boundary rather than from a fractional integration parameter.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper introduces a two-dimensional Markov chain (X_n,c_n) designed to mimic the local update structure of power-law Hawkes processes while remaining finite-dimensional. Inter-arrival times are generated by inverse transform through an integrated rate function Φ(t;x,c), driven by an i.i.d. sequence ε_n. The authors claim that, under a stated stability condition (Assumption 1), the chain is ψ-irreducible, aperiodic, a T-chain and positive Harris recurrent, with a unique invariant distribution; simulations of log inter-arrival times using the local Whittle estimator are claimed to show long memory near the stability boundary. The main proof strategy is a control-system argument for irreducibility and a Foster–Lyapunov drift calculation for Harris recurrence.

Significance. If the main theorem were correct, the construction would be a useful finite-dimensional Markovian model for power-law self-excitation, with a tractable inverse-transform simulation scheme. The paper is clear about the model and gives explicit drift inequalities, which are valuable. However, the central stability proof relies on a false tail-order implication, and the control-theoretic irreducibility argument has a reachability gap. The simulation evidence is suggestive but not statistically documented. No machine-checked proofs or reproducible code are supplied. Given these issues, the paper as written does not support its main claims, although the model idea may be salvageable with stronger assumptions.

major comments (5)
  1. [Section 3.1, Prop. 4, Eqs. (36)–(37)] The proof claims that E[ε^2]<∞ implies ∫_K^∞(u−K)F̄_ε(u)du = o(K^{-1}). This is false. Finiteness of the second moment only gives I(K)=∫_K^∞F̄_ε(u)du=o(K^{-1}); the stronger statement fails, e.g. for F̄_ε(u)∼c u^{-2.5}, for which E[ε^2]<∞ but the tail integral is ∼ const·K^{-1/2}. With K_x=c(μ+k_p x), Eq. (36) then contributes x·O(x^{-1/2})=O(x^{1/2}) after multiplication by x in Eq. (37), not O(x^{-1}). Hence the negative drift in Eq. (38) is not established. Since Theorem 12 uses Proposition 4 to control the X-component, positive Harris recurrence is unproved under the stated assumptions.
  2. [Section 4.1, Lemma 6 and Definitions 3–7] Lemma 6 only shows that feedback controls with r_n→∞ make (X_n,c_n) converge to (ξ,γ). But by the update (10), X_n=X_{n-1}r^{-p}+ξ>ξ for every finite n, so (ξ,γ) is not exactly attainable in finite time from any state with X>ξ. Definition 7 and Theorem 5 require exact reachability x*∈A+(y), not merely limit-point reachability. Therefore the conclusion that CM(Ψ) is M-irreducible, and the subsequent ψ-irreducibility/T-chain conclusion in Theorem 9, are not supported by the argument given.
  3. [Section 4.1, Lemma 8] The determinant formula in Lemma 8 contains a factor (γ−c) and vanishes at c=γ. The attempted fix—'take u1>0... the first step moves the chain to a state with c1>ξ'—is not justified in general (e.g. when γ<ξ, c1>ξ need not hold), and no determinant computation after a positive u1 is supplied. Forward accessibility therefore remains unproved for initial states with c=γ, which are included in the state space. This is a further load-bearing gap for the T-chain property.
  4. [Assumption 1, Eq. (39) versus Prop. 3, Eq. (26)] The threshold C_2 in Proposition 3 (Eq. (26)) depends on μ_ε, but the version used in Assumption 1 (Eq. (39)) omits μ_ε. Unless μ_ε=1 is silently assumed by a scaling convention, the stated stability condition and the boundary values used in Table 1 are not correct for the general ε distributions admitted in Definition 2. This affects the explicit stability criterion central to Theorem 12.
  5. [Section 5.2, Table 1] The text says that for fixed p the estimated d̂ increases with ξ and that boundary estimates are close to 0.5. The table contradicts this: for p=1.5 the boundary estimates are 0.3024, 0.3899, 0.3242, and for p=2.0 they decrease from 0.2933 to 0.2096. No standard errors, confidence intervals, or number of independent replications are reported for the single sample size n=50,000. This weakens the empirical claim of long memory at the boundary.
minor comments (4)
  1. [Section 3.1, Prop. 4] The phrase 'If c<2γ, or equivalently c<p/ξ' is not an equivalence. Under ξ<p/(2γ), c<2γ implies c<p/ξ, but the converse need not hold.
  2. [Appendix A, Lemma 13] Typo: 'evaluated derivatives of of X_2' should be 'of X_2'.
  3. [Section 5.1] The stability plateau claim is supported only by Figure 1 for one parameter set; the text says plateaus are observed across bandwidth choices but no other cases are shown.
  4. [Section 5.2] The table header uses 'γ=γ*' but the parameter is γ; the star is decorative and could confuse the stability-boundary notation γ*.

Circularity Check

0 steps flagged

No significant circularity; the stability proof is self-contained and no prediction reduces to its inputs.

full rationale

The derivation is self-contained: the model is defined explicitly via Definition 2 and the inverse-integrated intensity transform, and the stability claims are proved from drift conditions (Prop. 3, Prop. 4) and standard Meyn–Tweedie criteria (Thm 11), not from the target conclusions. The only self-references (Lee 2025, 2026) supply the general inverse-integrated intensity framework and are not used as evidence for irreducibility, recurrence, or long memory; no uniqueness theorem or ansatz is imported from those papers. The simulation section is an empirical demonstration, not a fitted prediction—parameters are chosen and local Whittle estimates are reported, so there is no fitted input being renamed as a prediction. The skeptic's counterexample to Prop. 4's tail bound (E[ε^2]<∞ does not imply ∫_K^∞(u−K)F̄(u)du=o(K^{-1})) is a correctness gap in the proof, not circularity, since the proof does not assume the negative-drift conclusion. Therefore no load-bearing circular step is present.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No parameters are fitted to data; ξ, γ, p, and μ are explicit model inputs. The load-bearing hidden assumption is the false tail-order bound in Prop. 4. The general stability condition also silently assumes μ_epsilon=1, which is not stated in Assumption 1.

axioms (4)
  • domain assumption The inter-arrival distribution F_epsilon has F_epsilon(0)=0, E[epsilon]<∞, and E[epsilon^2]<∞.
    Section 3, Definition 2; used for moment bounds in Lemma 2 and Prop. 4.
  • domain assumption The disturbance tau has a continuous density with open support O_tau.
    Section 4.1, Theorem 9; needed to pass from controllability to T-chain and psi-irreducibility.
  • domain assumption Stability condition Assumption 1: ξ < p/(2γ) and ξ < p/C2 with C2 computed as if μ_epsilon=1.
    Section 3.1, Eq. (39); the paper's Harris-recurrence theorem is conditional on this. The mean μ_epsilon is dropped, so for general F_epsilon the condition is not well-specified.
  • ad hoc to paper E[epsilon^2] < ∞ implies ∫_K^∞ (u-K) F̄_epsilon(u) du = o(K^{-1}).
    Section 3.1, Prop. 4 proof, Eq. (36)-(37). This is false in general (e.g., Pareto tail u^{-3.5}); a stronger tail decay assumption is needed but not stated.

pith-pipeline@v1.3.0-alltime-deepseek · 15647 in / 25604 out tokens · 207268 ms · 2026-08-01T09:13:34.151840+00:00 · methodology

0 comments
read the original abstract

We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.

Figures

Figures reproduced from arXiv: 2607.20838 by Kyungsub Lee.

Figure 1
Figure 1. Figure 1: Stability plot of the Local Whittle estimates [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

73 extracted references

  1. [1]

    , TITLE =

    Hawkes, Alan G. , TITLE =. J. Roy. Statist. Soc. Ser. B , FJOURNAL =. 1971 , PAGES =

    1971

  2. [2]

    Biometrika , volume=

    Spectra of some self-exciting and mutually exciting point processes , author=. Biometrika , volume=. 1971 , publisher=

    1971

  3. [3]

    Estimation of slowly decreasing

    Bacry, Emmanuel and Jaisson, Thibault and Muzy, Jean--Fran. Estimation of slowly decreasing. Quantitative Finance , volume=. 2016 , publisher=

    2016

  4. [4]

    Journal of the Operational Research Society , volume=

    Hawkes-based models for high frequency financial data , author=. Journal of the Operational Research Society , volume=. 2022 , publisher=

    2022

  5. [5]

    Modeling high frequency data using

    Zhang, Changyong , journal=. Modeling high frequency data using. 2016 , publisher=

    2016

  6. [6]

    Non-parametric kernel estimation for symmetric

    Bacry, Emmanuel and Dayri, Khalil and Muzy, Jean-Fran. Non-parametric kernel estimation for symmetric. The European Physical Journal B , volume=. 2012 , publisher=

    2012

  7. [7]

    Nonparametric estimation of

    Bonnet, Anna and Sangnier, Maxime , booktitle=. Nonparametric estimation of. 2025 , publisher=

    2025

  8. [8]

    (Almost) complete characterization of the stability of a discrete-time

    Costa, Manon and Maillard, Pascal and Muraro, Anthony , journal=. (Almost) complete characterization of the stability of a discrete-time. 2024 , doi=

    2024

  9. [9]

    Advances in Applied Probability , volume=

    Hawkes processes with variable length memory and an infinite number of components , author=. Advances in Applied Probability , volume=. 2017 , publisher=

    2017

  10. [10]

    Hawkes Processes with Variable Length Memory:

    Sacha Quayle and Anna Bonnet and Maxime Sangnier , year=. Hawkes Processes with Variable Length Memory:

  11. [11]

    Efficient Non-parametric

    Rui Zhang and Christian Walder and Marian-Andrei Rizoiu and Lexing Xie , year=. Efficient Non-parametric

  12. [12]

    Markov approximation for controlled

    Mahmoud Khabou and Mehdi Talbi , year=. Markov approximation for controlled

  13. [13]

    Mean-field limit of age and leaky memory dependent

    Schmutz, Valentin , journal =. Mean-field limit of age and leaky memory dependent. 2022 , doi =

    2022

  14. [14]

    Nonlinear Poisson autoregression and nonlinear

    Huang, Lorick and Khabou, Mahmoud , journal =. Nonlinear Poisson autoregression and nonlinear. 2023 , doi =

    2023

  15. [15]

    Estimation, diagnostics, and extensions of nonparametric

    Zhuang, Jiancang , journal =. Estimation, diagnostics, and extensions of nonparametric. 2020 , doi =

    2020

  16. [16]

    Journal of Applied Probability , volume=

    A cluster process representation of a self-exciting process , author=. Journal of Applied Probability , volume=. 1974 , publisher=

    1974

  17. [17]

    Engle and Jeffrey R

    Robert F. Engle and Jeffrey R. Russell , journal =. Autoregressive Conditional Duration:

  18. [18]

    2007 , publisher=

    An Introduction to the Theory of Point Processes: volume II: General Theory and Structure , author=. 2007 , publisher=

    2007

  19. [19]

    2020 , publisher=

    Point Process Calculus in Time and Space , author=. 2020 , publisher=

    2020

  20. [20]

    Self and Mutually Exciting Point Process Embedding Flexible Residuals and Intensity with Discretely

    Lee, Kyungsub , journal=. Self and Mutually Exciting Point Process Embedding Flexible Residuals and Intensity with Discretely. 2025 , publisher=

    2025

  21. [21]

    I , author=

    An Introduction to Probability Theory and Its Applications, Vol. I , author=. 1950 , publisher=

    1950

  22. [22]

    2011 , address =

    Econometrics of Financial High-frequency Data , author=. 2011 , address =

    2011

  23. [23]

    2009 , publisher=

    Markov Chains and Stochastic Stability , author=. 2009 , publisher=

    2009

  24. [24]

    Stochastic Processes and their Applications , volume =

    Geometric ergodicity of affine processes on cones , author =. Stochastic Processes and their Applications , volume =. 2020 , doi =

    2020

  25. [25]

    Stochastic Processes and their Applications , volume =

    Ergodicity and stability of hybrid systems with piecewise constant type state-dependent switching , author =. Stochastic Processes and their Applications , volume =. 2023 , doi =

    2023

  26. [26]

    Exact asymptotics of ruin probabilities with linear

    Palmowski, Zbigniew and Pojer, Simon and Thonhauser, Stefan , journal =. Exact asymptotics of ruin probabilities with linear. 2025 , articleno =

    2025

  27. [27]

    Kwan, Tsz-Kit Jeffrey and Chen, Feng and Dunsmuir, William T. M. , journal =. Ergodic properties of the

  28. [28]

    Stability of

    Meyn, Sean P and Tweedie, Richard L , journal=. Stability of. 1992 , publisher=

    1992

  29. [29]

    The existence of stationary measures for certain

    Harris, Theodore E , booktitle=. The existence of stationary measures for certain

  30. [30]

    2018 , publisher=

    Markov Chains , author=. 2018 , publisher=

    2018

  31. [31]

    The Annals of mathematical statistics , volume=

    Cumulative frequency functions , author=. The Annals of mathematical statistics , volume=. 1942 , publisher=

    1942

  32. [32]

    The Econometrics Journal , volume=

    Non-monotonic hazard functions and the autoregressive conditional duration model , author=. The Econometrics Journal , volume=. 2000 , publisher=

    2000

  33. [33]

    The Annals of Mathematical Statistics , volume=

    On the stochastic matrices associated with certain queuing processes , author=. The Annals of Mathematical Statistics , volume=. 1953 , publisher=

    1953

  34. [34]

    International Journal of Forecasting , volume=

    A comparison of financial duration models via density forecasts , author=. International Journal of Forecasting , volume=. 2004 , publisher=

    2004

  35. [35]

    Journal of Financial Econometrics , volume=

    Volatility estimation and forecasts based on price durations , author=. Journal of Financial Econometrics , volume=. 2023 , publisher=

    2023

  36. [36]

    Handbook of financial time series , pages=

    Modelling financial high frequency data using point processes , author=. Handbook of financial time series , pages=. 2009 , publisher=

    2009

  37. [37]

    Journal of Financial Econometrics , volume=

    A stochastic price duration model for estimating high-frequency volatility , author=. Journal of Financial Econometrics , volume=. 2024 , publisher=

    2024

  38. [38]

    Annales d'

    Bauwens, Luc and Giot, Pierre , title =. Annales d'

  39. [39]

    Journal of Econometrics , volume=

    A family of autoregressive conditional duration models , author=. Journal of Econometrics , volume=. 2006 , publisher=

    2006

  40. [40]

    Journal of Business & Economic Statistics , volume=

    Evaluating models of autoregressive conditional duration , author=. Journal of Business & Economic Statistics , volume=. 2006 , publisher=

    2006

  41. [41]

    Critical reflexivity in financial markets: a

    Hardiman, Stephen J and Bercot, Nicolas and Bouchaud, Jean-Philippe , journal=. Critical reflexivity in financial markets: a. 2013 , publisher=

    2013

  42. [42]

    Ubiquitous power law scaling in nonlinear self-excited

    Kanazawa, Kiyoshi and Sornette, Didier , journal=. Ubiquitous power law scaling in nonlinear self-excited. 2021 , publisher=

    2021

  43. [43]

    Journal of Financial Econometrics , pages=

    Assessing the risk of liquidity suppliers on the basis of excess demand intensities , author=. Journal of Financial Econometrics , pages=. 2003 , volume =

    2003

  44. [44]

    Computational Statistics & Data Analysis , volume=

    Regime-switching Pareto distributions for ACD models , author=. Computational Statistics & Data Analysis , volume=. 2006 , publisher=

    2006

  45. [45]

    Journal of Financial Econometrics , volume=

    Time-varying arrival rates of informed and uninformed trades , author=. Journal of Financial Econometrics , volume=. 2008 , publisher=

    2008

  46. [46]

    Journal of Forecasting , volume=

    Forecasting stock return volatility: Realized volatility-type or duration-based estimators , author=. Journal of Forecasting , volume=. 2023 , publisher=

    2023

  47. [47]

    Combining Volatility Forecasts of Duration-Dependent

    Turatti, Douglas Eduardo and Mendes, Fernando Henrique de Paula e Silva and Mazzeu, Jo. Combining Volatility Forecasts of Duration-Dependent. Journal of Forecasting , volume=. 2025 , publisher=

    2025

  48. [48]

    Journal of Forecasting , volume=

    Forecasting realized volatility of Bitcoin: The informative role of price duration , author=. Journal of Forecasting , volume=. 2023 , publisher=

    2023

  49. [49]

    Modeling bid and ask price dynamics with an extended

    Lee, Kyungsub and Seo, Byoung Ki , journal=. Modeling bid and ask price dynamics with an extended. 2023 , publisher=

    2023

  50. [50]

    Journal of Financial Econometrics , volume=

    Modeling price and variance jump clustering using the marked Hawkes process , author=. Journal of Financial Econometrics , volume=. 2024 , publisher=

    2024

  51. [51]

    Finance Research Letters , volume=

    Limit Order Book dynamics and order size modelling using Compound Hawkes Process , author=. Finance Research Letters , volume=. 2024 , publisher=

    2024

  52. [52]

    Journal of Applied Probability , volume=

    Multivariate Hawkes processes: an application to financial data , author=. Journal of Applied Probability , volume=. 2011 , publisher=

    2011

  53. [53]

    Market Microstructure and Liquidity , volume=

    Hawkes processes in finance , author=. Market Microstructure and Liquidity , volume=. 2015 , publisher=

    2015

  54. [54]

    Risk analysis via generalized

    He, Yi and Peng, Liang and Zhang, Dabao and Zhao, Zifeng , journal=. Risk analysis via generalized. 2022 , publisher=

    2022

  55. [55]

    Combining a self-exciting point process with the truncated generalized

    Ji, Jingru and Wang, Donghua and Xu, Dinghai and Xu, Chi , journal=. Combining a self-exciting point process with the truncated generalized. 2020 , publisher=

    2020

  56. [56]

    Lee, Kyungsub , title =

  57. [57]

    The Annals of Statistics , volume=

    Gaussian semiparametric estimation of long range dependence , author=. The Annals of Statistics , volume=. 1995 , publisher=

    1995

  58. [58]

    , title =

    Russell, Jeffrey R. , title =

  59. [59]

    Cox, D. R. , title =. Scandinavian Journal of Statistics , volume =

  60. [60]

    Journal of Financial Econometrics , volume =

    Bauwens, Luc and Hautsch, Nikolaus , title =. Journal of Financial Econometrics , volume =

  61. [61]

    Journal of Economic Dynamics and Control , volume =

    Hautsch, Nikolaus , title =. Journal of Economic Dynamics and Control , volume =

  62. [62]

    and Schorfheide, Frank , title =

    Chen, Fei and Diebold, Francis X. and Schorfheide, Frank , title =. Journal of Econometrics , volume =

  63. [63]

    Journal of Economic Dynamics and Control , volume =

    Li, Yang and Nolte, Ingo and Nolte, Stephan , title =. Journal of Economic Dynamics and Control , volume =. 2021 , pages =

    2021

  64. [64]

    Meyer, Paul-Andr. D. S

  65. [65]

    , title =

    Brown, Mark and Nair, Vijay N. , title =. Journal of Applied Probability , volume =

  66. [66]

    International Journal of Forecasting , volume =

    Feng, Yi and Zhou, Chen , title =. International Journal of Forecasting , volume =

  67. [67]

    , title =

    Karr, Alan F. , title =

  68. [68]

    Advances in Applied Probability , year =

    Costa, Manon and Graham, Carl and Marsalle, Laurence and Tran, Viet Chi , title =. Advances in Applied Probability , year =

  69. [69]

    Advances in Applied Probability , year =

    Daw, Andrew and Pender, Jamol , title =. Advances in Applied Probability , year =

  70. [70]

    Granger, Clive W. J. and Joyeux, Roselyne , title =. Journal of Time Series Analysis , year =

  71. [71]

    , title =

    Deo, Rohit and Hsieh, Mengchen and Hurvich, Clifford M. , title =. Journal of Statistical Planning and Inference , year =

  72. [72]

    International Journal of Forecasting , year =

    Feng, Yuanhua and Zhou, Chen , title =. International Journal of Forecasting , year =

  73. [73]

    and Russell, Jeffrey R

    Engle, Robert F. and Russell, Jeffrey R. , title =. Econometrica , year =