Semiparametric segment M-estimation for locally stationary diffusions

Abstract

Summary We develop and implement a novel M-estimation method for locally stationary diffusions observed at discrete time-points. We give sufficient conditions for the local stationarity of general time-inhomogeneous diffusions. Then we focus on locally stationary diffusions with time-varying parameters, for which we define our M-estimators and derive their limit theory.

Some novel aspects of quantile regression: local stationarity, random forests and optimal transportation

“…Importantly, by rescaling the observation period to the unit interval, estimators for the time-varying parameters can be obtained using windowed (in time) estimating equations. This concept has been successfully applied to various types of processes, including AR processes [Dahlhaus, 1997, Dahlhaus et al, 1999, ARCH processes [Dahlhaus andSubba Rao, 2006, Fryzlewicz et al, 2008], nonlinear AR processes [Vogt, 2012], scalar diffusion processes [Koo and Linton, 2012], Markov processes [Truquet, 2019] and [Deléamont and La Vecchia, 2019] (multivariate diffusions). Some recent developments of local stationarity for quantile spectral analysis are available in [Birr et al, 2017], while [Xu et al, 2022] (and in some sense also [Zhou and Wu, 2009]) study conditional quantile estimation.…”

“…The approach that we adopt in this paper has a spirit similar to the one of [Deléamont and La Vecchia, 2019], who explain how to conduct inference on the time-varying parameters of Markov processes and how to derive the asymptotics of the corresponding estimators. However, [Deléamont and La Vecchia, 2019] focus on the first two (infinitesimal) moments of the process, whilst here we study the problem of quantile regression and we look at the entire conditional distribution. To this end, we consider time-varying autoregression quantile estimation for a process that we observe over a time span [0, 𝑛].…”

“…This entails that notions like quantiles, check function, distribution function, signs, and ranks do not clearly extend to higher dimensions. Some solutions to this problem are already available, like the directional and direct approaches; see [Hallin and Šiman, 2017] p.187-193. Making use of statistical concepts related to optimal transportation theory (see and [La Vecchia et al, 2023] for a discussion on the use of Monge-Kantorovich results in statistics), [del Barrio et al, 2022] define nested conditional center-outward quantile regression contours and regions, with given conditional probability content irrespective of the underlying distribution. Their graphs constitute nested center-outward quantile regression tubes.…”

“…Their graphs constitute nested center-outward quantile regression tubes. [del Barrio et al, 2022] illustrate how to construct empirical counterparts of these population concepts, yielding interpretable empirical regions and contours. Their construction is based on two steps: in Step 1, one specifies a set of weights and uses them to construct an empirical distribution of the multivariate response variable conditional on some values of the multivariate covariates; in Step 2, one computes the corresponding empirical center-outward quantile map, resorting on Monge-Kantorovich's results.…”

“…At a high level, our procedure makes use of random forests as an adaptive neighbourhood classification tool, which we combine with the multiple-output centeroutward quantile regression of [del Barrio et al, 2022]. We grow the trees mimicking the logic of standard random forests, obtaining, for every regressors value, a set of weights for the original multivariate response.…”