Parameter Stability in Long-Horizon GARCH Change-Point Models

Definitions and mathematical statement

Let $I$ denote the observation index set of the economic time series under consideration, and let $\theta_t$ denote the parameter vector of its GARCH (generalized autoregressive conditional heteroskedasticity) component at index $t\in I$. This notation only expresses parameter constancy; the source does not specify $I$, the parameter space, the GARCH order, or a model equation. The constant-parameter assumption is \[ \exists\,\theta\quad\forall\,t\in I,\qquad \theta_t=\theta. \] The source describes this component as stationary, but does not specify the meaning of stationarity or conditions ensuring it.

Source question. For very long economic time series in a semiparametric change-point model, can the stationary GARCH component still consistently be assumed to have constant parameters?

The question asks whether the displayed assumption remains justified in that modelling setting. It does not specify whether "consistently" means statistical consistency of an estimator, compatibility with the model, or empirical adequacy. Neither "very long" nor an asymptotic regime is defined. The semiparametric decomposition, change-point structure, and relationship between changing dynamics and the stationary component are also unspecified. Consequently, the supplied record supports this methodological question and the displayed definition of parameter constancy, but does not determine a precise theorem, statistical testing problem, or universally quantified assertion.

Short English statement

The question asks whether a stationary GARCH component in a semiparametric change-point model can reasonably retain one parameter vector throughout a very long economic time series. The model, meaning of consistency, and long-series regime are unspecified, so a precise mathematical target requires further source context.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001411$, exact source problem code catalogue entry, title "Parameter Stability in Long-Horizon GARCH Change-Point Models," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the recorded proposed year is 2010.
• Supplied source citation: Mini-Workshop: Semiparametric Modelling of Multivariate Economic Time Series With Changing Dynamics (2010), Oberwolfach Reports, pp. 179--216, DOI 10.4171/owr/2010/05. Source URL and supplied literature reference labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2010/05. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions; no exact location within this report is supplied for the question.
• The registry labels the statement "exact" and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied provenance, not an independent verification performed for this draft.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "open" and describes the extracted sentence as a methodological question about allowing nonconstant parameters, with no precise theorem or later resolution identifiable from the supplied context. This is supplied dated curation, not an independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.