Maximum likelihood Estimation of locally stationary processes

Abstract

Time series analysis has attracted growing interest in the literature over recent years, driven by the need to model nonstationary phenomena whose statistical characteristics evolve over time. Among the most relevant frameworks developed to address this challenge are locally stationary processes, which provide a powerful compromise between flexibility and analytical tractability. These processes assume that, although a time series may be globally nonstationary, it can be well approximated locally by a stationary process in the neighborhood of each time point.The primary objective of this thesis is to study the estimation of parameters for locally stationary processes, with a particular focus on models featuring time-varying volatility. Since the parameters are allowed to change smoothly over time, classical estimation methods designed for stationary settings become inadequate. To overcome this limitation, we adopt a modeling approach based on time-varying parameter models and employ a nonparametric quasi-maximum likelihood estimation framework that relies on local likelihood approximation. The core idea consists in approximating the locally stationary process by a stationary one in the vicinity of a fixed rescaled time point. This local approximation justifies the construction of kernel-weighted estimators, where observations closer to the time point of interest receive higher weights. Under suitable regularity conditions, the proposed estimator is shown to be consistent and asymptotically normal, allowing for valid statistical inference on time-varying parameters. Special attention is devoted to time-varying GARCH models, which play a central role in modeling conditional volatility in financial time series. Classical GARCH models are extended by allowing their parameters to vary over time within the locally stationary framework. In addition to the standard Gaussian assumption, alternative distributions are considered to better capture heavy-tailed behavior commonly observed in financial data. In particular, Student’s t and Laplace distributions are incorporated into the quasi-likelihood framework.Theoretical results show that when the assumed distribution is correctly specified, the estimator converges to the true parameter values. In contrast, under likelihood misspecification, the estimator converges to pseudo-true parameters rather than the true ones. Extensive numerical experiments confirm the theoretical findings and demonstrate the robustness of the proposed methodology. Applications to real financial data further validate the approach, showing a strong agreement between estimated and observed volatility