A spectral based coefficient of determination for the fit of an MA(q) model

Holger Dette; Sebastian K\"uhnert

A spectral based coefficient of determination for the fit of an MA(q) model

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read A spectral coefficient of determination measures how well MA(q) models approximate a stationary process's spectral density.

desk verdict This paper gives a spectral R^2 for MA(q) fit with periodogram estimators, asymptotic normality, tests, and order selection. read the letter →

arxiv 2606.18445 v1 pith:T55VMYOP submitted 2026-06-16 math.ST stat.MEstat.TH

Holger Dette , Sebastian Kühnert This is my paper · ORCID

classification math.STstat.MEstat.TH

keywords coefficientofdeterminationMA(q)modelspectraldensityperiodogramasymptoticnormalitytimeseriesfitorderselection

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 develops a coefficient of determination based on the spectral density to assess how closely an MA(q) model represents the spectrum of a weakly stationary time series. It introduces periodogram-based estimators, proves their asymptotic normality, and uses them to build tests for the MA(q) hypothesis as well as procedures that select the smallest order q meeting a target approximation level. A sympathetic reader would care because the approach supplies a frequency-domain tool for model evaluation and order choice that can complement time-domain diagnostics. The work targets processes whose continuous spectra admit meaningful MA(q) approximations.

What carries the argument

The spectral coefficient of determination that quantifies the fit between the true spectral density and its projection onto the MA(q) class.

What would settle it

Empirical evidence that the periodogram-based coefficient fails to converge in distribution to the claimed normal limit under the MA(q) hypothesis, or that the order-selection procedure does not achieve the prescribed spectral approximation quality on data generated from an MA(q) process.

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Extended reading notes

Core claim

We develop a spectral based coefficient of determination to measure how well the spectral density of a stationary process is represented by the class of MA(q) models. Using periodogram-based estimators, we establish asymptotic normality, derive tests for the MA(q) hypothesis, and construct procedures for determining the smallest order q achieving a prescribed approximation quality.

Load-bearing premise

The underlying process is weakly stationary with a continuous spectral density that admits a meaningful approximation by the MA(q) class, and the periodogram-based estimators satisfy the regularity conditions needed for asymptotic normality.

Editorial extensions

If this is right

The coefficient supports formal hypothesis tests of whether an MA(q) model adequately represents the spectral density.
Asymptotic normality of the estimators permits construction of tests and confidence statements for the MA(q) hypothesis.
Data-driven procedures identify the minimal q that attains a user-specified approximation quality.
Periodogram-based computation allows direct application to observed time series without parametric assumptions beyond stationarity.

Reading between the lines

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

The same spectral-fit idea could be extended to other parametric spectral families such as AR(p) or ARMA models.
It offers an alternative order-selection criterion focused on spectral approximation rather than one-step prediction error.
The method may improve diagnostics in settings where frequency-domain accuracy is the primary modeling goal.

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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.

Desk Editor's Note

read the letter

The core contribution is a new coefficient that quantifies how well an MA(q) model approximates the spectral density of a weakly stationary process. They define it via the true spectrum versus the parametric MA class, estimate it with periodogram functionals, prove asymptotic normality, and use that to test the MA(q) hypothesis and select the smallest q meeting a target fit level.

This is a clean extension of spectral methods to model assessment. The construction stays non-circular because it directly compares the nonparametric spectrum estimate to the MA(q) family rather than relying on fitted residuals in a tautological way. The inference steps follow the usual pattern in spectral time series work, which keeps the argument internally consistent under standard conditions like continuous spectrum.

The soft spots are modest but real. The abstract supplies no explicit formula for the coefficient itself or the precise regularity conditions behind the normality result, so it is impossible to check whether the asymptotics require stronger mixing or smoothness than is typical. There is also no simulation evidence or comparison to existing spectral distances or information criteria, which leaves the practical gain unclear. These are normal gaps at the abstract stage but would need filling for a full evaluation.

The paper is aimed at time series researchers who already work with spectral diagnostics and MA models. A reader looking for a frequency-domain alternative to residual-based fit measures would find the procedures usable. It is solid enough on its own terms to warrant peer review rather than desk rejection, though the referee would likely ask for the missing derivations and some finite-sample checks.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a spectral-based coefficient of determination to quantify how well the spectral density of a weakly stationary process is approximated by the MA(q) class. Periodogram-based estimators are used to establish asymptotic normality of the coefficient, from which tests for the MA(q) hypothesis are derived and procedures are constructed for selecting the smallest q that achieves a prescribed approximation quality.

Significance. If the central claims hold, the work supplies a new spectral goodness-of-fit measure together with asymptotically justified inference tools for MA order selection. This is a direct, falsifiable extension of classical periodogram methods and supplies reproducible asymptotic results under standard weak-stationarity and continuity conditions on the spectrum.

minor comments (2)

[Abstract] Abstract: the statement that asymptotic normality is established would be strengthened by a one-sentence indication of the principal regularity conditions (e.g., summability of autocovariances or integrability of the spectral density) under which the result is proved.
Notation: the precise definition of the population coefficient (the quantity being estimated by the periodogram functional) should be displayed as a numbered display equation early in the paper so that subsequent asymptotic statements can refer to it directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its contributions to spectral goodness-of-fit measures for MA(q) approximation, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The manuscript constructs a spectral coefficient of determination that directly compares a stationary process's spectral density to its best approximation within the MA(q) parametric family, then derives asymptotic normality of periodogram-based estimators under standard weak-stationarity and continuity assumptions. No equation or procedure reduces by definition to a fitted parameter that is then relabeled as a prediction; the central R^2-type measure is not obtained by fitting to the same data it evaluates. No load-bearing self-citation chain or uniqueness theorem imported from the authors' prior work is invoked to force the result. The derivation chain therefore remains independent of its own outputs and is self-contained against external time-series spectral theory.

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

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. Full text would be needed to audit any regularity conditions on the spectral density or moment assumptions underlying the asymptotic normality claim.

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Cite this review

Pith. "Pith review of A spectral based coefficient of determination for the fit of an MA(q) model." pith.science (2026). https://pith.science/paper/T55VMYOP

@misc{pith2026260618445,
  author       = {Pith},
  title        = {Pith review of: A spectral based coefficient of determination for the fit of an MA(q) model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T55VMYOP}},
  note         = {Machine review of arXiv:2606.18445}
}

abstract

We develop a spectral based coefficient of determination to measure how well the spectral density of a stationary process is represented by the class of MA($q$) models. Using periodogram-based estimators, we establish asymptotic normality, derive tests for the MA($q$) hypothesis, and construct procedures for determining the smallest order $q$ achieving a prescribed approximation quality.

Figures

Figures reproduced from arXiv: 2606.18445 by the authors.

**Figure 1.** displays the empirical rejection probabilities of the test in (2.5) for the processes from Example 2.1 with coefficients ψj = r j , considering the cases q = 1 and q = 4 and various sample sizes. The results are based on 1000 simulation runs. Under the null hypothesis H0 : 1 − R2 q = 0, the empirical rejection probabilities approach the nominal level α = 10%, whereas under the alternative H1 : 1 − R2 q > 0, they ten… view at source ↗

**Figure 2.** Histograms of estimator qˆ in (2.7) for q ∗ in (2.6) based on data-generating process of Example 2.1 for r = 0.45, ν = 0.9 and thus q ∗ = 1 (upper panel), and for r = 0.8, ν = 0.65 and thus q ∗ = 2 (lower panel) at nominal level α = 10%. A Proofs Proof of Theorem 2.1. The proof proceeds in several steps. Step 1. We first show that, as N → ∞, √ N(TˆN,q − µT ,q) d −→ N (0, Σeq), (A.1) where TˆN,q := [PITH_FULL_IMAGE:… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references

[1]

Bartlett, M. S. (1946). On the theoretical specification and sampling properties of autocorrelated time-series. Supplement to the Journal of the Royal Statistical Society: Series B (Methodological)\/ 8\/ (1), 27--41

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Box, G. E. P., G. M. Jenkins, G. C. Reinsel, and G. M. Ljung (2015). Time Series Analysis: Forecasting and Control\/ (5 ed.). Wiley

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Brillinger, D. R. (2001). Time Series: Data Analysis and Theory , Volume 36 of Classics in Applied Mathematics . Philadelphia: Society for Industrial and Applied Mathematics (SIAM)

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Brockwell, P. J. and R. A. Davis (1991). Time Series: Theory and Methods\/ (2nd ed.). Springer Series in Statistics. New York: Springer

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Kinsvater, and M

Dette, H., T. Kinsvater, and M. Vetter (2011). Testing non‐parametric hypotheses for stationary processes by estimating minimal distances. Journal of Time Series Analysis\/ 32\/ (5), 447--461

2011
[6]

Francq, C. and H. Raïssi (2007). Multivariate portmanteau test for autoregressive models with uncorrelated but nonindependent errors. Journal of Time Series Analysis\/ 28\/ (3), 454--470

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[7]

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis\/ (1 ed.). Cambridge: Cambridge University Press

1985
[8]

Ljung, G. M. and G. E. P. Box (1978). On a measure of lack of fit in time series models. Biometrika\/ 65\/ (2), 297--303

1978

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Mahdi, E. and A. Ian McLeod (2012). Improved multivariate portmanteau test. Journal of Time Series Analysis\/ 33\/ (2), 211--222

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Orey, S. (1958). A central limit theorem for m -dependent random variables. Duke Mathematical Journal\/ 25\/ (4), 543--546

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Priestley, M. (1981). Spectral Analysis and Time Series 1 . New York: Academic

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

Bartlett, M. S. (1946). On the theoretical specification and sampling properties of autocorrelated time-series. Supplement to the Journal of the Royal Statistical Society: Series B (Methodological)\/ 8\/ (1), 27--41

1946

[2] [2]

Box, G. E. P., G. M. Jenkins, G. C. Reinsel, and G. M. Ljung (2015). Time Series Analysis: Forecasting and Control\/ (5 ed.). Wiley

2015

[3] [3]

Brillinger, D. R. (2001). Time Series: Data Analysis and Theory , Volume 36 of Classics in Applied Mathematics . Philadelphia: Society for Industrial and Applied Mathematics (SIAM)

2001

[4] [4]

Brockwell, P. J. and R. A. Davis (1991). Time Series: Theory and Methods\/ (2nd ed.). Springer Series in Statistics. New York: Springer

1991

[5] [5]

Kinsvater, and M

Dette, H., T. Kinsvater, and M. Vetter (2011). Testing non‐parametric hypotheses for stationary processes by estimating minimal distances. Journal of Time Series Analysis\/ 32\/ (5), 447--461

2011

[6] [6]

Francq, C. and H. Raïssi (2007). Multivariate portmanteau test for autoregressive models with uncorrelated but nonindependent errors. Journal of Time Series Analysis\/ 28\/ (3), 454--470

2007

[7] [7]

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis\/ (1 ed.). Cambridge: Cambridge University Press

1985

[8] [8]

Ljung, G. M. and G. E. P. Box (1978). On a measure of lack of fit in time series models. Biometrika\/ 65\/ (2), 297--303

1978