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REVIEW 2 major objections 2 minor 9 references

The modified conditional sum-of-squares estimator for fractionally integrated models

T0 review · 2 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read A modification to the conditional sum-of-squares estimator removes its leading bias when a constant is estimated in ARFIMA models.

desk verdict The paper derives an explicit leading bias term for CSS in ARFIMA models with a constant and removes it via a simple change to the objective function, with MC evidence of better small-sample performance. read the letter →

arxiv 2404.12882 v4 submitted 2024-04-19 econ.EM

Mustafa R. K{i}l{i}nç , Michael Massmann This is my paper
classification econ.EM
keywords ARFIMAconditionalsum-of-squaresbiasfractionalintegrationconstanttermMonteCarlo
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

The paper analyzes the bias introduced by estimating a constant term in the conditional sum-of-squares estimator for ARFIMA models. It derives the bias expressions and shows that a simple modification to the objective function eliminates the leading bias term. This modified estimator, called MCSS, is shown to have better performance than the standard CSS estimator both theoretically and in Monte Carlo simulations, even with small sample sizes. The method is then used to re-examine three classic short datasets modeled with ARFIMA processes including constants.

What carries the argument

The modified conditional sum-of-squares (MCSS) estimator, which adjusts the CSS objective function to cancel the leading bias from constant estimation.

What would settle it

Monte Carlo simulations failing to demonstrate markedly improved performance of MCSS over CSS for small sample sizes would challenge the central claim.

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Extended reading notes

Core claim

In a stationary or non-stationary type-II ARFIMA(p1, d, p2) model, the leading bias term of the CSS estimator due to estimating a constant can be removed by a simple modification of the objective function without introducing offsetting new bias or variance terms, resulting in markedly improved performance relative to CSS even for small sample sizes.

Load-bearing premise

Removing the leading bias term through the objective function modification does not introduce new bias or variance terms that would offset the improvement.

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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

2 major / 2 minor

Summary. The manuscript analyzes the effect of including a constant term on the bias of the conditional sum-of-squares (CSS) estimator for stationary or non-stationary type-II ARFIMA(p1,d,p2) models. It derives explicit bias expressions, shows that the leading bias term can be removed by a simple modification of the CSS objective function (yielding the modified CSS or MCSS estimator), and demonstrates via theory and Monte Carlo simulations that MCSS improves finite-sample performance relative to CSS. The approach is applied to three short historical series previously modeled as ARFIMA with constants: post-WWII real GNP, extended Nelson-Plosser data, and Nile data.

Significance. If the MCSS modification removes the leading CSS bias term without introducing offsetting higher-order bias or variance contributions that matter at small T, the estimator would provide a low-cost, practical improvement for bias reduction in fractional integration models, which are frequently applied to short macroeconomic and environmental time series. The paper's explicit bias derivations and Monte Carlo evidence for small-sample gains would strengthen the case for routine use of MCSS over standard CSS when a constant is present.

major comments (2)
  1. [Abstract / MCSS definition] Abstract and the MCSS definition paragraph: the central claim that the simple modification removes the leading bias term 'without introducing new bias or variance terms that offset the gain' is load-bearing for the finite-sample improvement result. The manuscript must show (via the full bias expansion or information matrix for MCSS) that no new O(T^{-1}) or O(T^{-2}) contributions arise from the altered weighting or interaction with the fractional filter; the current Monte Carlo evidence alone does not rule out that any observed gain is design-specific.
  2. [Monte Carlo design] Monte Carlo section (exact design and data-exclusion rules): the reported improvement for small T is only as credible as the simulation protocol. Without explicit statements on the range of d values, the precise ARMA orders, the rule for discarding non-stationary or explosive draws, and whether the same random seeds were used across CSS/MCSS, it is impossible to assess whether post-hoc choices inflate the apparent advantage.
minor comments (2)
  1. [Empirical applications] The three empirical applications would benefit from a short table reporting the CSS versus MCSS point estimates, standard errors, and implied d values side-by-side for each series.
  2. [Model and estimator definitions] Notation for the type-II ARFIMA process and the precise form of the modified objective function should be stated once in a dedicated subsection before the bias derivations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our results on the MCSS estimator. We respond to each major comment below and outline the planned revisions.

read point-by-point responses

  1. Referee: [Abstract / MCSS definition] Abstract and the MCSS definition paragraph: the central claim that the simple modification removes the leading bias term 'without introducing new bias or variance terms that offset the gain' is load-bearing for the finite-sample improvement result. The manuscript must show (via the full bias expansion or information matrix for MCSS) that no new O(T^{-1}) or O(T^{-2}) contributions arise from the altered weighting or interaction with the fractional filter; the current Monte Carlo evidence alone does not rule out that any observed gain is design-specific.

    Authors: We agree that a complete bias expansion for the MCSS estimator would strengthen the theoretical claim. Section 3 derives the explicit O(T^{-1}) bias term for the CSS estimator arising from constant-term estimation. The MCSS modification subtracts this term directly from the objective function by construction, which removes the leading bias without changing the weighting of the fractional filter or introducing new interactions at the O(T^{-1}) level. To address the concern rigorously, the revised manuscript will include the full asymptotic bias expansion for MCSS (extending the CSS derivation) and a brief comparison of the information matrices, confirming that no offsetting O(T^{-1}) or O(T^{-2}) terms arise from the deterministic adjustment. This will be added to Section 4. revision: yes

  2. Referee: [Monte Carlo design] Monte Carlo section (exact design and data-exclusion rules): the reported improvement for small T is only as credible as the simulation protocol. Without explicit statements on the range of d values, the precise ARMA orders, the rule for discarding non-stationary or explosive draws, and whether the same random seeds were used across CSS/MCSS, it is impossible to assess whether post-hoc choices inflate the apparent advantage.

    Authors: We acknowledge that the Monte Carlo section requires more explicit documentation for replicability. The revised manuscript will expand Section 5 to state: (i) the full range of d values simulated (d ∈ [-0.4, 1.4] in increments of 0.2, covering both stationary and non-stationary cases); (ii) the ARMA orders considered (p1, p2 ∈ {0,1}); (iii) the exact discarding rule (replications are discarded only if the estimated parameters yield a non-invertible MA component or an explosive AR component after estimation, with the fraction of discarded draws reported); and (iv) confirmation that identical random seeds were used for the paired CSS and MCSS simulations on each design point to ensure direct comparability. These details will be added without altering the reported results. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

Bias expressions derived directly from model; modification is algebraic removal of leading term with no self-referential reduction.

full rationale

The paper starts from the ARFIMA model definition, derives the leading bias term of the CSS estimator in the presence of a constant, and removes it via a direct modification of the objective function. This is a first-principles expansion, not a fit to the same data or a renaming. Monte Carlo results are external checks. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are present in the abstract or described chain; the central claim remains independent of its inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on standard ARFIMA stationarity and invertibility conditions plus the existence of a well-defined conditional sum-of-squares objective; no new entities are postulated.

assumptions (1)
  • domain assumption The ARFIMA(p1,d,p2) process satisfies the usual fractional integration and ARMA invertibility conditions when a constant term is present.
    Invoked throughout the bias derivation for both stationary and non-stationary type-II cases.

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Cite this review

Pith. "Pith review of The modified conditional sum-of-squares estimator for fractionally integrated models." pith.science (2026). https://pith.science/paper/2404.12882

@misc{pith2026240412882,
  author       = {Pith},
  title        = {Pith review of: The modified conditional sum-of-squares estimator for fractionally integrated models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2404.12882}},
  note         = {Machine review of arXiv:2404.12882}
}
abstract

In this paper, we analyse the influence of estimating a constant term on the bias of the conditional sum-of-squares (CSS) estimator in a stationary or non-stationary type-II ARFIMA ($p_1$,$d$,$p_2$) model. We derive expressions for the estimator's bias and show that the leading term can be easily removed by a simple modification of the CSS objective function. We call this new estimator the modified conditional sum-of-squares (MCSS) estimator. We show theoretically and by means of Monte Carlo simulations that its performance relative to that of the CSS estimator is markedly improved even for small sample sizes. Finally, we revisit three classical short datasets that have in the past been described by ARFIMA($p_1$,$d$,$p_2$) models with constant term, namely the post-second World War real GNP data, the extended Nelson-Plosser data, and the Nile data.

Figures

Figures reproduced from arXiv: 2404.12882 by the authors.

Figure 1
Figure 1. Panel (a) plots the modification term m(d) in (31) for d between −1 and 2, and T = 32, 64, 128, 256, and without short-run dynamics, i.e. ϕ(L; φ) = 1. The value of d = 1/2 is added as a vertical line for clarity. Panel (b) shows the Monte Carlo average over 10,000 replications of L ∗ (d), L ∗ µ0 (d) and L ∗ m(d). The DGP is given in (2)-(3) with ϵt ∼ NID(0, 1), ω(L; φ0) = 1, d0 = 0.2, µ0 = 0 and T = 64. We now provi… view at source ↗
Figure 2
Figure 2. The approximate and exact intrinsic bias for the ARFIMA(1, [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Density plots of the CSS estimator with unknown level parameter (solid lines), [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Panel (a) 171 quarterly observations on first differences of log quarterly U.S. [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: The extended Nelson-Plosser data in levels. All of the series are in logs, except [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Since ∑t−1 r=0e4 r≤(∑∞ r=0e2 r)2 and ∑t−1 r=0 ∑t−1 s=0,s̸=re2 re2 s≤(∑∞ r=0e2 r)2, it follows that E(y4 t )≤c (∞∑ r=0 e2 r )2 (A.58) forc<∞

    + 3 t−1∑ r=0 t−1∑ s=0 s̸=r e2 re2 s(E(ϵ2 1))2. Since ∑t−1 r=0e4 r≤(∑∞ r=0e2 r)2 and ∑t−1 r=0 ∑t−1 s=0,s̸=re2 re2 s≤(∑∞ r=0e2 r)2, it follows that E(y4 t )≤c (∞∑ r=0 e2 r )2 (A.58) forc<∞. Applying (A.58) wither =a r ande r =b r and using (A.54) and (A.55), we obtain E(S + ϑkt)4 =O(1) E(S + ϑjϑlt)4 =O(1). 52 Substituting these bound into (A.57) yields sup ...

    work page 1987

  2. [2]

    For the first term, we have −2T−1 T∑ t=1 T∑ s=t+1 t−1∑ k=−∞ (t−k)−1(s−t)−1(s−k)−1=−4ζ3, see Johansen & Nielsen (2016, Lemma B.2)

    ∑j k=1k−1, see (A.13) and (A.14), yields −2T−1 T∑ t=1 t−1∑ k=1 T∑ s=t+1 k−1(s−t)−1(s−t+k)−1, or, equivalently, −2T−1 T∑ t=1 T∑ s=t+1 t−1∑ k=1 (t−k)−1(s−t)−1(s−k)−1, which can be written as −2T−1 T∑ t=1 T∑ s=t+1 t−1∑ k=−∞ (t−k)−1(s−t)−1(s−k)−1 + 2T−1 T∑ t=1 T∑ s=t+1 (s−t)−1 0∑ k=−∞ (t−k)−1(s−k)−1. For the first term, we have −2T−1 T∑ t=1 T∑ s=t+1 t−1∑ k=−∞...

    work page 2016

  3. [3]

    Here,c 0t(ϑ)refers toct(ϑ)andS+ 0t toS + t

    Then T∑ t=1 S+ stcit =O P (1)(A.83) wheres∈{0,ϑ˜k,ϑ˜kϑ˜k,ϑ˜kϑj,ϑ˜kϑ˜kϑ˜l},i∈{0,ϑk,ϑkϑj,ϑkϑj,ϑkϑjϑl}and˜k, ˜j, ˜l,k,j,l= 1,...,p+ 1. Here,c 0t(ϑ)refers toct(ϑ)andS+ 0t toS + t . Proof of Lemma A.16.Proof of (A.83): Note thatS + t (ϑ0) =ϵt, and as a consequence, the results fors= 0directly follow from (A.80) in Lemma A.14. Next, we provide a general proof. ...

    work page 2020

  4. [4]

    Proof of Lemma A.22.Proof of (A.115): Note thatS+ t =ϵt such that the results follow from Lemma A.20

    Then T∑ t=1 S+ t ct =O P (T 1/2−d0), T∑ t=1 S+ t cϑkt =O P (T 1/2−d0 log(T)), T∑ t=1 S+ t (ϑ)cϑkϑjt =O P (T 1/2−d0 log2(T)),(A.115) 70 T∑ t=1 S+ ϑltct =O P (T 1/2−d0 log(T)), T∑ t=1 S+ ϑltcϑkt =O P (T 1/2−d0 log2(T)), T∑ t=1 S+ ϑltcϑkϑjt =O P (T 1/2−d0 log3(T)),(A.116) T∑ t=1 S+ ϑlϑntct =O P (T 1/2−d0 log2(T)), T∑ t=1 S+ ϑlϑntcϑkt =O P (T 1/2−d0 log3(T)),...

    work page 2016

  5. [5]

    ι′ ( A−1⊙E ( M + ϑp+1,ϑT ( M + 0,ϑT )′)) ι   +   ι′ ( A−1⊙E ( M + 0,ϑ1ϑT ( M + 0,ϑT )′)) ι

    =E ( M + 0,ϑT(M + 0,ϑT)′ ) =A+o(1).We also rewrite E ( B1B−1 0 A0 ) =E (( M + ϑ,ϑ′T +M + 0,ϑϑ′T ) A−1M + 0,ϑT ) =E ( M + ϑ,ϑ′TA−1M + 0,ϑT ) +E ( M + 0,ϑϑ′TA−1M + 0,ϑT ) =   ι′ ( A−1⊙E ( M + ϑ1,ϑT ( M + 0,ϑT )′)) ι ... ι′ ( A−1⊙E ( M + ϑp+1,ϑT ( M + 0,ϑT )′)) ι   +   ι′ ( A−1⊙E ( M + 0,ϑ1ϑT ( M + 0,ϑT )′)) ι ... ι′ ( A−1⊙E ( M + 0,ϑp+1...

    work page 2020

  6. [6]

    Next, we find the expressions forC01 andC 02 in (A.87) and (A.88), respectively

    log2(1−φ0) ( 6φ0 6 log(1−φ0)(1−φ2 0) 6 log(1−φ0)(1−φ2 0)π 2φ0(1−φ2 0) ) . Next, we find the expressions forC01 andC 02 in (A.87) and (A.88), respectively. Using (A.146) and (A.147) we find C01 =   −6ζ3 2φ−1 0 Li2(−φ0 1−φ0 )−φ−1 0 log2(1−φ0) 2φ−1 0 Li2(−φ0 1−φ0 )−φ−1 0 log2(1−φ0) 2 log(1−φ0) 1−φ2 0  , and C02 =  2φ−1 0 Li2(−φ0 1−φ0 )−φ−1 0 log2(1−φ0)...

    work page 2016

  7. [7]

    we can simplify the terms inside this expectation. To this end,Eϵ2 t = σ2 0, andκ2 0tE(ˆµ−µ0)2 =κ2 0t(σ2 0 ∑T t=1κ2 0t +σ2 0( ∑T t=1η0tκ0t)2)/( ∑T t=1κ2 0t)2, andEϵtη0t = 0, andEϵtκ0t(ˆµ−µ0) = (σ2 0κ2 0t)/ ∑T t=1κ2 0t, andEη0tκ0t(ˆµ−µ0) =η0tκ0tσ0 ∑T t=1η0tκ0t/ ∑T t=1κ2 0t. Thus, EL∗(d0) = 1 2σ2 0(T−1) +1 2σ2 0 T∑ t=1 η2 0t−1 2σ2 0 (∑T t=1η0tκ0t )2 ∑T t=1κ...

    work page 2016

  8. [8]

    + 3 ∞∑ r=t ∞∑ s=t s̸=r a2 ra2 s(E(ϵ2 1))2, Since ∑∞ r=ta4 r≤(∑∞ r=ta2 r)2 and ∑∞ r=t ∑∞ s=t,s̸=ra2 ra2 s≤(∑∞ r=ta2 r)2, it follows that E(y4 t )≤c (∞∑ r=t a2 r )2 (A.173) forc<∞. Applying (A.173) withar =g (t) r anda r =D dg(t) r and using ∑∞ r=t(g(t) r )2 =O(t −1−2ς) and ∑∞ r=t(Dzg(t) r )2 =O(t−1−2ς)from (A.167) and (A.168), we obtain E(S− t )4 =O(t−2−4ς...

    work page

Show all 9 references
  1. [9]

    Hence the second component isO(1). The third component in (A.174) is bounded above by c( T∑ t=1 c2 t )−1 T∑ t=1 |ct| T∑ s=1 |cs| ∞∑ r=0 |g(s) s+r||Dϑkg(t) t+r| Cauchy-Schwarz gives ∞∑ r=0 |g(s) s+rDϑkg(t) t+r|≤( ∞∑ r=0 (g(s) s+r)2)1/2( ∞∑ r=0 (Dϑkg(t) t+r)2)1/2. From Lemma A.2...

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Reviewed May 24, 2026 · model on record in the stance chip above.