In celebration of Masanobu Taniguchi’s 70th birthday, this book brings together advancements in statistical inference for time series and related models. It encompasses a wide range of models, including long-range dependence models, nonlinear conditionally heteroscedastic time series, locally stationary processes, integer-valued time series, Lévy Processes, complex-valued time series, categorical time series, exclusive topic models, and copula models. The book introduces cutting-edge methodologies, such as empirical likelihood methods, quantile regression, portmanteau tests, rank-based inference, change-point detection, goodness-of-fit testing, higher-order asymptotic expansion, minimum contrast estimation, optimal transportation, and topological methods, which are utilized, considered, or proposed for analyzing complex data through statistical inference for stochastic processes.
Prof. Tommaso Proietti, Prof. Alessandra Luati and I are thrilled to present our contribution titled “Generalized Linear Spectral Models for Locally Stationary Processes”. Our Essay introduces a class of parametric models designed specifically for locally stationary processes. These models incorporate a power parameter that affects the time-varying spectrum, allowing it to be represented locally using a finite low-dimensional Fourier polynomial. The coefficients of this polynomial hold significance as time-varying autocovariances, with their dynamics determined by a linear combination of smooth transition functions based on static parameters.
The estimation in the frequency domain relies on the generalized Whittle likelihood and the pre-periodogram, while model selection is carried out using information criteria. Change points are detected using a sequence of score tests. The paper establishes the consistency and asymptotic normality of the parametric estimators under mild assumptions on the time-varying parameters.