Estimating and forecasting univariate and multivariate time series using dynamic linear models

Abstract

This thesis investigates the place of dynamic linear models in time series analysis. These models, which are based on the state space representation of the time series where the dynamic parameters are the state variables, are particularly useful because they are applicable to non-stationary (as well as stationary) time series. In this thesis some forecasting algorithms using dynamic linear models are developed for univariate and multivariate time series, and the effectiveness of these algorithms is investigated. The algorithms are applied to some examples in biometrics and economics. The thesis is based on original research papers published hy the author, which are included as appendices 1-5. Chapter one surveys the field of time series analysis in order to get a broad overview of the subject. There are two main goals in time series analysis: the search for the underlying pattern in the time series, and the prediction of future values of the series. The first leads to the study of the time series in the frequency domain where a harmonic analysis of the time series decomposes it into a sum of sine waves at various frequencies. The second leads to the study of the time series in the time domain. Here the relationship between values a given distance apart is studied for each possible distance, and these relationships are used to predict future values. Chapter two develops the class of second order stationary stochastic processes in the frequency and time domains. The main tool of the frequency domain analysis, the spectral distribution of the process, and the main tool of the time domain analysis, the autocovariance function of the process, are seen to be Fourier transforms of each other. The effect of a time invariant linear filter on a second order stationary stochastic process is determined. It establishes the important results that if a stochastic process is Markovian, it must be the output of a first order linear system excited by Gaussian white noise. The Wiener-Hopf integral equation for the optimal (in terms of minimizing expected squared error) linear filter is developed using the principle of orthogonality. Chapter three introduces the state space representation of a linear dynamic system, develops the Kalman filter by using orthogonal projections, and establishes the recursive equations. Section 3 investigates the efficiency of the dynamic linear model applied to a linearly aggregated time series, and determines the efficiency of estimation using the aggregated time series relative to using the weighted sum of the component model estimators. While it is not surprising that the latter method is more efficient, the exact relative efficiency is developed here for the first time. Chapter four surveys the field of filtering in some non-Gaussian models. While the Kalman filter is still the optimal linear filter in the non-Gaussian case, it may be far from the optimal filter. Chapter five introduces multiprocess dynamic (regression) models to give a filter that is effective for short term forecasting in time series that are subject to abrupt, as well as evolutionary, changes in pattern, and may contain transients (outliers). In Section 3, the multivariate extension of the multiprocess dynamic linear model developed by the author is presented. The forecasting algorithm is used to forecast the quarterly inflation rate in the consumer price index, by using the inflation rates for the six main consumer price subindices, food, housing, household operations, apparel, transportation, and miscellaneous. This method was found to give an improvement over the univariate dynamic linear model forecasts, which in turn are improvements over the naive model forecasts. In Section 4, the biased perturbation multiprocess dynamic linear model developed by the author is presented and used to analyze growth hormone level data in young steers that are subject to dietary stress. The growth hormone level data exhibited the characteristics of pulsatile data, that is exponential decay towards a base level with occasional jumps in level when a pulse of growth hormone is released. The model was found to give an effective method of analyzing pulsatile data that can be implemented in real time. In Section 5, the multiprocess dynamic generalized linear model developed by the author is presented. It is shown to be an effective method for estimating a time varying population proportion. This thesis investigates dynamic linear time series models and has shown their place in the time series analysis framework. The relative efficiency of a dynamic linear model applied to a linearly aggregated time series has been determined. Estimation and forecasting algorithms based on dynamic linear time series models for univariate and multivariate time series have been developed and analyzed. They are effective for time series that contain outliers and exhibit occasional abrupt as well as evolutionary changes in pattern. These algorithms have been applied successfully to non-stationary time series in biology and economics.