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Truncated sum-of-squares estimation of fractional time series models with generalized power law trend

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Truncated sum-of-squares estimation of fractional time series models with generalized power law trend. / Hualde, Javier; Nielsen, Morten Ørregaard.

In: Electronic Journal of Statistics, Vol. 16, No. 1, 2022, p. 2884-2946.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

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Hualde J, Nielsen MØ. Truncated sum-of-squares estimation of fractional time series models with generalized power law trend. Electronic Journal of Statistics. 2022;16(1):2884-2946. doi: 10.1214/22-EJS2009

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Hualde, Javier ; Nielsen, Morten Ørregaard. / Truncated sum-of-squares estimation of fractional time series models with generalized power law trend. In: Electronic Journal of Statistics. 2022 ; Vol. 16, No. 1. pp. 2884-2946.

Bibtex

@article{21fa2c687e524399852115622d7e569e,
title = "Truncated sum-of-squares estimation of fractional time series models with generalized power law trend",
abstract = "We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.",
keywords = "Asymptotic normality, consistency, deterministic trend, fractional process, generalized polynomial trend, generalized power law trend, noninvertibility, nonstationarity, sum-of-squares estimation",
author = "Javier Hualde and Nielsen, {Morten {\O}rregaard}",
year = "2022",
doi = "10.1214/22-EJS2009",
language = "English",
volume = "16",
pages = "2884--2946",
journal = "Electronic Journal of Statistics",
issn = "1935-7524",
publisher = "nstitute of Mathematical Statistics",
number = "1",

}

RIS

TY - JOUR

T1 - Truncated sum-of-squares estimation of fractional time series models with generalized power law trend

AU - Hualde, Javier

AU - Nielsen, Morten Ørregaard

PY - 2022

Y1 - 2022

N2 - We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.

AB - We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.

KW - Asymptotic normality

KW - consistency

KW - deterministic trend

KW - fractional process

KW - generalized polynomial trend

KW - generalized power law trend

KW - noninvertibility

KW - nonstationarity

KW - sum-of-squares estimation

UR - http://www.scopus.com/inward/record.url?scp=85130329829&partnerID=8YFLogxK

U2 - 10.1214/22-EJS2009

DO - 10.1214/22-EJS2009

M3 - Journal article

AN - SCOPUS:85130329829

VL - 16

SP - 2884

EP - 2946

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

SN - 1935-7524

IS - 1

ER -