Measuring dependence

Abstract

The degree of dependence inherent in a dataset, either in the time series domain or in multivariate analysis, commonly gives rise to two distinct types of processes: stationary and non-stationary (unit root). This thesis focuses on detecting the underlying degree of dependence in a certain dataset of unit (1; T) or higher (N; T) dimension. The first part of the thesis aims at identifying the intrinsic strength of structure in a large dimensional setup. It is known that all information needed for this purpose is contained in the column-sum norm of the variance-covariance matrix of the dataset. This approaches in unity at rate N , 0 < 1. The strength of structure can then be determined by the value of . On this basis, a summary statistic is constructed as a means of quantifying this parameter . The resulting non-linear least squares estimator is consistent for relatively small N=T ratios. The accuracy of the statistic is further checked by use of a bootstrap procedure. An application to three major stock market indices is also incorporated. The second part of the thesis introduces alternative approaches to testing the hypothesis that a certain time series contains a unit root. Contrary to traditional unit root tests that assume specific forms of dependence in the errors of their stationary alternatives, it is argued that properties of partial sums can be used to provide limiting distributions for a wide range of stationary processes, with short or long memory. A rescaled variance type test making use of partial sums provides a simpler procedure for assessing the existence of a unit root in a time series with reasonable size and power. A further approach examined entails making use of the properties of the periodogram at low frequencies. Again a unit root/stationarity test based on this methodology is simple to implement and gives satisfactory results for size and power.