Equilibrium in Multivariate Nonstationary Time Series
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Nonstationary time series systems appear routinely in economics, seismology, neuroscience, and physics, where stationarity is usually synonymous with equilibrium. Such systems are usually multidimensional, and their modeling, prediction, and control have tremendous social and scientific impacts. Isolating and identifying equilibrium or stationary features are of fundamental importance in prediction and control of such systems. This research project aims to develop methodologies for extracting aspects of multivariate nonstationary processes that display a sense of equilibrium or stationarity. It is interdisciplinary in nature and has immediate applications to the analysis of economics, seismology, and neuroscience data. A graduate student will be involved in the research.This research project aims to elevate the concept and theory of cointegration from multivariate integrated time series rooted in economics theory to the more general multivariate nonstationary time series setup in probability and statistics. In spite of its central role in econometrics in the last four decades and well-founded motivations in economics, the cointegration theory suffers from the requirements that the series be integrated (unit-root nonstationary) and satisfy a vector autoregressive and moving average model. The goal of this project is to avoid such restrictions and focus on general multivariate nonstationary time series. Three distinct methods for computing analogues of cointegrating vectors and the cointegrating rank will be developed. The first is a time-domain method in line with the classical (Johansen''s) approach that relies on the reduced rank regression and likelihood ratio tests. The second method is in the spectral domain and relies on the idea of projection pursuit. It searches for coefficients of candidate linear combinations by minimizing a projection index measuring the discrepancy between time-varying and constant spectral density functions. The third method is concerned with a time-varying cointegration setup where the coefficients are piecewise constant over time. Its successful implementation rests on a good solution of the problem of change-point detection for nonstationary processes, and a novel solution is explored in this research. The results will have immediate impact in settings where multivariate time series data are collected, such as in financial markets, epidemiology, environmental monitoring, and global change.