Donald Poskitt

A Functional Data—Analytic Approach to Signal Discrimination

Motivated by specific problems involving radar-range profiles, we suggest techniques for real-time discrimination in the context of signal analysis. The key to our approach is to regard the signals as curves in the continuum and employ a functional data-analytic (FDA) method for dimension reduction, based on the FDA technique for principal coordinates analysis. This has the advantage, relative to competing methods such as canonical variates analysis, of providing a signal approximation that is best possible, in an L2 sense, for a given dimension. As a result, it produces particularly good discrimination. We explore the use of both nonparametric and Gaussian-based discriminators applied to the dimensionreduced data.

Specification of Echelon-Form VARMA Models

The echelon form of a vector autoregressive moving average (VARMA) model is considered. Its advantages over other identified VARMA representations are discussed. Furthermore, a general strategy for specifying echelon form VARMA models from data is presented. Specifically, procedures for choosing the Kronecker indices that characterize an echelon form are reviewed. The feasibility of the method is demonstrated by analyzing a well-known set of flour price time series and the term structure of German interest rates.

Properties of the Sieve Bootstrap for Fractionally Integrated and Non-Invertible Processes

In this paper we will investigate the consequences of applying the sieve bootstrap under regularity conditions that are sufficiently general to encompass both fractionally integrated and non-invertible processes. The sieve bootstrap is obtained by approximating the data generating process by an autoregression whose order h increases with the sample size T. The sieve bootstrap may be particularly useful in the analysis of fractionally integrated processes since the statistics of interest can often be non-pivotal with distributions that depend on the fractional index d. The validity of the sieve bootstrap is established and it is shown that when the sieve bootstrap is used to approximate the distribution of a general class of statistics admitting an Edgeworth expansion then the error rate achieved is of order O ( T β+d-1 ), for any β > 0. Practical implementation of the sieve bootstrap is considered and the results are illustrated using a canonical example.

Testing for Causation Using Infinite Order Vector Autoregressive Processes

Tests for Granger-causality have been performed in numerous empirical studies. These tests are usually based on finite order vector autoregressive (VAR) processes, and the assumption is made that the model fitted to the available data corresponds to the true data generating mechanism. In the present study, the more general assumption is made that a finite order VAR model is fitted to a potentially infinite order process. The order is assumed to increase with the sample size. Asymptotic properties of tests for Granger-causality as well as other types of causality concepts are derived. Some limited small sample results are obtained using simulation methods.

Estimating Orthogonal Impulse Responses via Vector Autoregressive Models

Impulse response functions from time series models are standard tools for analyzing the relationship between economic variables. The asymptotic distribution of orthogonalized impulse responses is derived under the assumption that finite order vector autoregressive (VAR) models are fitted to time series generated by possibly infinite order processes. The resulting asymptotic distributions of forecast error variance decompositions are also given.

Markov chain models, time series analysis and extreme value theory

Markov chain processes are becoming increasingly popular as a means of modelling various phenomena in different disciplines. For example, a new approach to the investigation of the electrical activity of molecular structures known as ion channels is to analyse raw digitized current recordings using Markov chain models. An outstanding question which arises with the application of such models is how to determine the number of states required for the Markov chain to characterize the observed process. In this paper we derive a realization theorem showing that observations on a finite state Markov chain embedded in continuous noise can be synthesized as values obtained from an autoregressive moving-average data generating mechanism. We then use this realization result to motivate the construction of a procedure for identifying the state dimension of the hidden Markov chain. The identification technique is based on a new approach to the estimation of the order of an autoregressive moving-average process. Conditions for the method to produce strongly consistent estimates of the state dimension are given. The asymptotic distribution of the statistic underlying the identification process is also presented and shown to yield critical values commensurate with the requirements for strong consistency.

Strongly Consistent Determination of Cointegrating Rank Via Canonical Correlations

This article is concerned with the statistical analysis of nonstationary, cointegrated time series. The estimation of the cointegrating structure of such time series is considered, and the problem of identifying the cointegrating rank is addressed. A methodology is presented that leads to strongly consistent estimates of this quantity. The identification is based on a canonical correlation analysis of the original variables and presents an alternative approach to those currently in vogue. The procedures are easily implemented and the practical relevance of the results obtained, which are founded on asymptotic theory, is demonstrated by means of a small simulation study.

On the specification of cointegrated autoregressive moving-average forecasting systems

Abstract This paper discusses equilibrium correction, echelon canonical form vector autoregressive moving-average, EC-ARMAE, forecasting systems. The echelon canonical form of a vector ARMA model is expanded by the inclusion of an equilibrium correction term to accommodate the possibility of cointegrated variables. A coherent procedure is presented for consistently estimating the Kronecker indices, which characterize the echelon form, and the cointegration rank, which is essential in the specification of the equilibrium correction term. A method of estimation that is fully efficient under Gaussian assumptions is also discussed. The computational burden of these techniques is very moderate because they are based on least squares calculations. The methodology is illustrated by examining a six-equation model of the US economy. An improvement in forecasting performance of the selected EC-ARMAE model over non-equilibrium correction and previously preferred vector AR equilibrium correction models is observed.

The selection and use of linear and bilinear time series models

Abstract This paper is concerned with problems associated with employing linear and bilinear processes to represent time series phenomena. It contains some discussion of theoretical reasons why bilinear models may prove to be useful for modelling non-Gaussian time series. Attention is paid to the effects of adopting a Bayesian stance to the use of model selection criteria when approaching the question of determining suitable parameterizations. These ideas are illustrated using a familiar data set and their ramifications for forecasting are explored. It transpires in this instance that the one-step ahead forecasts with the smallest mean squared error are generated by a noval combination of the two classes of models entertained.

Bayesian adaptive bandwidth kernel density estimation of irregular multivariate distributions

In this paper, we propose a new methodology for multivariate kernel density estimation in which data are categorized into low- and high-density regions as an underlying mechanism for assigning adaptive bandwidths. We derive the posterior density of the bandwidth parameters via the Kullback-Leibler divergence criterion and use a Markov chain Monte Carlo (MCMC) sampling algorithm to estimate the adaptive bandwidths. The resulting estimator is referred to as the tail-adaptive density estimator. Monte Carlo simulation results show that the tail-adaptive density estimator outperforms the global-bandwidth density estimators implemented using different global bandwidth selection rules. The inferential potential of the tail-adaptive density estimator is demonstrated by employing the estimator to estimate the bivariate density of daily index returns observed from the USA and Australian stock markets.