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arxiv: 1907.09762 · v1 · pith:FBAMIZS2new · submitted 2019-07-23 · 🧮 math.ST · stat.TH

Consistent model selection criteria and goodness-of-fit test for affine causal processes

Jean-Marc Bardet (SAMM) , Kare Kamila (SAMM) , William Kengne (THEMA) This is my paper

Pith reviewed 2026-05-24 17:17 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords model selectionquasi-likelihoodcausal time seriesconsistencyBICportmanteau testaffine processesGARCH
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The pith

Sufficient conditions on the penalty ensure consistent model selection by quasi-likelihood for affine causal processes, but BIC does not always work.

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

The paper develops a penalized quasi-likelihood method for choosing among a broad family of causal time series models that includes ARMA, GARCH, APARCH, and their combinations. It supplies growth conditions on the penalty that make the selection procedure consistent and that make the quasi-maximum likelihood estimator in the chosen model both consistent and asymptotically normal. The same conditions imply that the usual BIC penalty fails to deliver consistency in some members of the class, for example autoregressive models driven by infinite-order ARCH noise. A portmanteau statistic is introduced to test the fit of the selected model. Simulations and an application to the FTSE index confirm the theoretical claims, including the observed inconsistency of BIC.

Core claim

We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. It appears from these conditions that the Bayesian Information Criterion (BIC) does not always guarantee the consistency. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on the portmanteau Test.

What carries the argument

Penalized quasi-likelihood contrast whose penalty term must satisfy explicit growth conditions to guarantee consistency of selection and of the resulting estimator.

If this is right

  • Model selection is consistent whenever the penalty meets the stated growth conditions.
  • The quasi-maximum likelihood estimator computed on the selected model is consistent and asymptotically normal.
  • BIC can produce inconsistent order selection for AR models with infinite ARCH errors.
  • The portmanteau statistic supplies an asymptotic test of goodness-of-fit after selection.

Where Pith is reading between the lines

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

  • Practitioners fitting financial series should check that their chosen penalty grows faster than log n when error structures are rich.
  • The same penalty conditions could be checked for other contrasts such as least squares or robust likelihoods.
  • The non-consistency result for BIC suggests re-examining default software choices for GARCH-type models on real data.

Load-bearing premise

The data must be generated by an affine causal process for which a quasi-likelihood contrast is well-defined and the penalty satisfies the paper's growth conditions.

What would settle it

A sequence of samples from an AR(p) process with ARCH(∞) errors in which the BIC-selected order fails to converge in probability to the true order as sample size tends to infinity.

Figures

Figures reproduced from arXiv: 1907.09762 by Jean-Marc Bardet (SAMM), Kare Kamila (SAMM), William Kengne (THEMA).

Figure 1
Figure 1. Figure 1: Daily closing FTSE 100 index (January 4th, 2010 to December 31 st, 2018). [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
read the original abstract

This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. It appears from these conditions that the Bayesian Information Criterion (BIC) does not always guarantee the consistency. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on the portmanteau Test. Numerical simulations and an illustrative example on the FTSE index are performed to highlight the obtained asymptotic results, including a numerical evidence of the non consistency of the usual BIC penalty for order selection of an AR(p) models with ARCH($\infty$) errors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a penalized quasi-likelihood procedure for model selection in the broad class of affine causal processes (encompassing ARMA, GARCH, APARCH, ARMA-GARCH and related models). It states sufficient conditions on the penalty term that ensure consistency of the selection procedure together with consistency and asymptotic normality of the post-selection quasi-maximum likelihood estimator. The authors show that the BIC penalty fails to satisfy these conditions in identifiable cases (e.g., AR(p) with ARCH(∞) errors), supply numerical confirmation, and propose a portmanteau test for goodness-of-fit of the selected model, illustrated by simulations and an FTSE-index example.

Significance. If the stated sufficient conditions on the penalty are correctly derived and the BIC counter-example holds, the work supplies a usable theoretical framework for consistent selection and post-selection inference in a large family of dependent processes where standard criteria can fail. The explicit growth conditions, the portmanteau diagnostic, and the reproducible numerical evidence constitute concrete strengths.

minor comments (3)
  1. [Abstract] The abstract and introduction would benefit from an explicit statement (even a high-level one) of the growth-rate conditions imposed on the penalty term, rather than only the claim that such conditions exist.
  2. Notation for the affine causal process and the quasi-likelihood contrast should be introduced once, with a single reference to the defining equation, to avoid repeated re-definition across sections.
  3. The numerical section would be strengthened by reporting the exact sample sizes and replication counts used in the BIC counter-example simulations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which correctly summarizes the main contributions of the paper. The recommendation for minor revision is noted; we will make the corresponding editorial adjustments in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives explicit sufficient conditions on the penalty term that guarantee consistency of the penalized quasi-likelihood procedure and asymptotic normality of the post-selection QMLE for affine causal processes. These conditions are stated directly in the theorems, shown analytically to be violated by BIC in specific cases such as AR(p) with ARCH(∞) errors, and supported by separate numerical simulations. The weakest assumption (DGP belongs to the class where the quasi-likelihood contrast is well-defined) is precisely the setting in which the results are proved, with no reduction of any claimed prediction or consistency result to a fitted parameter, self-citation chain, or definitional equivalence. The portmanteau goodness-of-fit tool is likewise derived from standard residuals without circular input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the data belonging to the affine causal class and on the quasi-likelihood being a valid contrast; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The observed process belongs to the class of affine causal time series models
    The abstract opens by restricting attention to this class that includes ARMA, GARCH, APARCH and hybrids.
  • domain assumption A quasi-likelihood contrast can be defined for every candidate model in the class
    The penalized contrast is built directly on the quasi-likelihood of the model.

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