PERIODIC AUTOREGRESSIVE AND PERIODICALLY INTEGRATED MODELS

Abstract

There has been an enormous interest in analysing and modelling periodic time series. This thesis researches on the periodic time series models, including periodic autoregressive (PAR) models and periodically integrated autoregressive (PIAR) models. The thesis extends conventional PAR models to Fourier-PAR models which allow the PAR parameters to be represented by their Fourier coefficients. Selecting only the significant Fourier coefficients leads to a class of parsimonious Fourier-PAR models. The thesis researches on identification, estimation, diagnostic checking and forecasting for the full and parsimonious Fourier-PAR models. Most existing literatures study the quarterly PIAR models, however, their method would be inefficient when extending the quarterly period to general cases. This thesis develops a multi-companion method which uses the eigenvalues and eigenvectors of the multi-companion matrices to analyse PIAR models with general period. With the multi-companion method, the thesis gives a general definition for periodic integration, parametrizes the PIAR parameters, and proposes a two-step method to estimate PIAR models. The thesis fully discusses PIAR models with periodic integration order of one, PI(1). For PI(1) cases, the thesis derives limiting distributions of some important processes, asymptotic properties of OLS estimators, and two original theorems which are applied to unit root test. The two original theorems are significantly important to check the number of unit roots in PI(1) cases, and the corresponding Monte Carlo simulations are provided. At last, the thesis introduces a state space representation for PIAR models. The state space method is applied to deal with parameter estimation, and also to capture the stochastic trends driving the series to be periodically integrated. A Monte Carlo study with applying the state space method to estimate PIAR models is provided.

Date of Award	1 Aug 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Jingsong Yuan (Supervisor) & Georgi Boshnakov (Supervisor)