Uniform Gaussian Approximation for The Quasi-Likelihood Estimator for a Weakly Dependent Nonlinear Time Series Models
Abstract
We study estimation and inference for a semiparametric class of time series models that specify only the conditional expectation, which is a known link function applied to a linear combination of past observations and covariates. The class covers count, binary, bounded and conditionally heteroskedastic responses within a single formulation, and the parameter is estimated by a quasi-likelihood estimating equation based on the first conditional moment. Under stationarity and a weak-dependence condition expressed through the functional dependence measure, we establish two results. First, using a Fuk--Nagaev inequality for weakly dependent sequences, we show that the estimator is localized in a shrinking neighbourhood of the true value with probability 1−o(n−1/2). Second, combining a Berry--Esseen bound for weakly dependent sequences with a Gaussian anti-concentration argument to control the remainder of the linear expansion, we obtain a Berry--Esseen bound for linear projections of the estimator, uniform over projection directions. From the projected bound we derive studentized confidence intervals with explicit coverage error and a conservative Bonferroni test for linear hypotheses on the parameters. For real data analysis, we extend the Beta autoregression for double-bounded data to an arbitrary link given by the inverse of a distribution function, and apply it to ten pairwise realized correlations of large-cap technology-stock returns, using Nasdaq and Dow~Jones index returns as covariates.
Related
- Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator
- Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design
- Existence and consistency of weighted maximum likelihood estimator for extreme quantile regression
- Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in r-Largest Generalized Extreme-Value Models
- A Proxy-likelihood Estimator for Multivariate Extremes Models with Intractable Likelihoods
BibTeX
@misc{debaly2026uniform,
title = {{Uniform Gaussian Approximation for The Quasi-Likelihood Estimator for a Weakly Dependent Nonlinear Time Series Models}},
author = {Zinsou Max Debaly and Arsene Brice Zotsa-Ngoufack},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04613},
eprint = {2609.04613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04613},
}
RIS
TY - UNPB
TI - Uniform Gaussian Approximation for The Quasi-Likelihood Estimator for a Weakly Dependent Nonlinear Time Series Models
AU - Zinsou Max Debaly
AU - Arsene Brice Zotsa-Ngoufack
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04613
UR - https://arxiv.org/abs/2609.04613
AB - We study estimation and inference for a semiparametric class of time series models that specify only the conditional expectation, which is a known link function applied to a linear combination of past observations and covariates. The class covers count, binary, bounded and conditionally heteroskedastic responses within a single formulation, and the parameter is estimated by a quasi-likelihood estimating equation based on the first conditional moment. Under stationarity and a weak-dependence condition expressed through the functional dependence measure, we establish two results. First, using a Fuk--Nagaev inequality for weakly dependent sequences, we show that the estimator is localized in a shrinking neighbourhood of the true value with probability 1−o(n^−1/2). Second, combining a Berry--Esseen bound for weakly dependent sequences with a Gaussian anti-concentration argument to control the remainder of the linear expansion, we obtain a Berry--Esseen bound for linear projections of the estimator, uniform over projection directions. From the projected bound we derive studentized confidence intervals with explicit coverage error and a conservative Bonferroni test for linear hypotheses on the parameters. For real data analysis, we extend the Beta autoregression for double-bounded data to an arbitrary link given by the inverse of a distribution function, and apply it to ten pairwise realized correlations of large-cap technology-stock returns, using Nasdaq and Dow~Jones index returns as covariates.
ER -