Bootstrap for order identification in ARMA(P,Q) structures

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Anselmo Chaves Neto
Thais Mariane Biembengut Faria

Abstract

The identification of de order p,q, of ARMA models is a critical step in time-series modelling. In classic Box-Jenkins method of identification the autocorrelation function (ACF) and the partial autocorrelation (PACF) function should be estimated, but the classical expressions used to measure the variability of the respective estimators are obtained on the basis of asymptotic results. In addition, when having sets of few observations, the traditional confidence intervals to test the null hypotheses display low performance. The bootstrap method may be an alternative for identifying the order of ARMA models, since it allows to obtain an approximation of the distribution of the statistics involved in this step. Therefore it is possible to obtain more accurate confidence intervals than those obtained by the classical method of identification. In this paper we propose a bootstrap procedure to identify the order of ARMA models. The algorithm was tested on simulated time series from models of structures AR(1), AR(2), AR(3), MA(1), MA(2), MA(3), ARMA(1,1) and ARMA (2,2). This way we determined the sampling distributions of ACF and PACF, free from the Gaussian assumption. The examples show that the bootstrap has good performance in samples of all sizes and that it is superior to the asymptotic method for small samples.

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Author Biographies

Anselmo Chaves Neto, Universidade Federal do Paraná

Anselmo Chaves Neto is Professor at the Federal University of Parana (UFPR), Department of Statistics and acts as a permanent teacher in PostGraduate Program in Numerical Methods in Engineering(PPGMNE). Has a PhD in electrical engineering from PUC - Rio de Janeiro (1991). Research, mainly on the following topics: Multivariate Statistical Methods, Time Series Prediction, Quality Engineering, Computationally Intensive Methods (Bootstrap and Jacknife), Pattern Recognition, Product Reliability and Structural Reliability.

Thais Mariane Biembengut Faria, Universidade Regional de Blumenau

Escola Técnica do Vale do Itajaí. Thais Mariane Biembengut Faria has a PhD in Numerical Methods in Engineering with specialization in Statistics (2014). She is currently professor at the University of Blumenau (FURB) incourses of Engineering and Management. Research in the areas of statistical inference, time series forecasts and also genetic algorithms.

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