Specification Testing for Higher-Order GARCH Models

Definitions and mathematical statement

For a family $\mathcal M$ of probability laws for a time series, a specification test addresses the null hypothesis \[ H_0:\quad \text{the law of the observed time series belongs to }\mathcal M. \] Here $\mathcal M$ is only a placeholder: the supplied record does not specify the model families, their parameter spaces, or the observation scheme. The names $\operatorname{GARCH}(1,1)$, higher-order GARCH, and augmented GARCH are retained as source terminology; their defining recursions and admissibility conditions are not supplied.

Problem. Can the proposed specification-testing theory for $\operatorname{GARCH}(1,1)$, including its asymptotic and bootstrap results, be generalized to higher-order GARCH models and augmented GARCH processes?

The requested extension must encompass both the asymptotic results and the bootstrap results of the intended baseline theory. However, the record does not identify the baseline theorem statements, test statistics, estimators, bootstrap procedure, limiting laws, or hypotheses. It also does not specify which higher orders or augmented models are intended, or whether the desired results concern null distributions, consistency against alternatives, local alternatives, or some combination. Consequently, the supplied question cannot be converted into a uniquely quantified theorem or conjecture without additional source information. No particular innovation distribution, moment condition, stationarity condition, dependence condition, parameter restriction, or bootstrap construction is imposed by this draft.

Short English statement

The question asks whether the proposed asymptotic and bootstrap theory for testing a $\operatorname{GARCH}(1,1)$ specification extends to higher-order and augmented GARCH processes. The baseline results and target model classes are insufficiently specified for a precise theorem. The supplied dated curation labels the problem partially solved; that assessment requires independent review.

Sources

• Supplied local problem-registry record, permanent UnsolvedMath ID $30002413$, title Specification Testing for Higher-Order GARCH Models, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record gives proposed year 2013 and names no individual proposer.
• Supplied report citation: Statistical Inference for Complex Time Series Data (2014), Oberwolfach Reports, pp. 2749--2823, DOI 10.4171/owr/2013/48. https://doi.org/10.4171/owr/2013/48. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labeled "Oberwolfach Report": https://ems.press/content/serial-article-files/46479.
• Supplied literature source, labeled "Leucht--Neumann--Kreiss": https://madoc.bib.uni-mannheim.de/35107/1/Leucht_%26_Neumann_%26_Kreiss_13-11.pdf.
• Supplied literature source, labeled "Duan": https://doi.org/10.1016/S0304-4076(97)00009-2.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and states that the $\operatorname{GARCH}(1,1)$ specification-test theory is established and related higher-order or augmented-GARCH results exist, but the exact conjectured generalization is not proved in full. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and describes its changes as normalization of notation, typography, and wording without changing mathematical content. These are supplied registry assertions, not independent verification of the formulation or mathematics in this draft.