Bootstrap for order identification in ARMA(P,Q) structures
Article Sidebar
Main Article Content
صندلی اداریAbstract
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.
Downloads
Article Details
1. Proposal of Policy for Free Access Periodics
Authors whom publish in this magazine should agree to the following terms:
a. Authors should keep the copyrights and grant to the magazine the right of the first publication, with the work simultaneously permitted under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 that allows the sharing of the work with recognition of the authorship of the work and initial publication in this magazine.
b. Authors should have authorization for assuming additional contracts separately, for non-exclusive distribution of the version of the work published in this magazine (e.g.: to publish in an institutional repository or as book chapter), with recognition of authorship and initial publication in this magazine.
c. Authors should have permission and should be stimulated to publish and to distribute its work online (e.g.: in institutional repositories or its personal page) to any point before or during the publishing process, since this can generate productive alterations, as well as increasing the impact and the citation of the published work (See The Effect of Free Access).
Proposal of Policy for Periodic that offer Postponed Free Access
Authors whom publish in this magazine should agree to the following terms:
a. Authors should keep the copyrights and grant to the magazine the right of the first publication, with the work simultaneously permitted under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 [SPECIFY TIME HERE] after the publication, allowing the sharing of the work with recognition of the authorship of the work and initial publication in this magazine.
b. Authors should have authorization for assuming additional contracts separately, for non-exclusive distribution of the version of the work published in this magazine (e.g.: to publish in institutional repository or as book chapter), with recognition of authorship and initial publication in this magazine.
c. Authors should have permission and should be stimulated to publish and to distribute its work online (e.g.: in institutional repositories or its personal page) to any point before or during the publishing process, since this can generate productive alterations, as well as increasing the impact and the citation of the published work (See The Effect of Free Access).
d. They allow some kind of open dissemination. Authors can disseminate their articles in open access, but with specific conditions imposed by the editor that are related to:
Version of the article that can be deposited in the repository:
Pre-print: before being reviewed by pairs.
Post-print: once reviewed by pairs, which can be:
The version of the author that has been accepted for publication.
The editor's version, that is, the article published in the magazine.
At which point the article can be made accessible in an open manner: before it is published in the magazine, immediately afterwards or if a period of seizure is required, which can range from six months to several years.
Where to leave open: on the author's personal web page, only departmental websites, the repository of the institution, the file of the research funding agency, among others.
References
ANDERSON, R. (1942) Distribuition of serial correlation coefficient. Annals of Mathematics Statistics, n. 13, p. 1-13
BARTLETT, M. S. (1946) On the theoretical specification and sampling properties of autocorrelated time-series. Journal of the Royal Statistical Society, v. 8, n. 27, p. 27-41.
BOX, G. E. P.; JENKINS, G. (1976) Time Series Analysis Forecasting and Control. Holen Day: New Jersey.
BOX, G. E. P.; JENKINS, G.; REINSEL, G. C. (1994) Time Series Analysis. Prentice Hall: New Jersey.
CAVALIERE, G.; TAYLOR, R. (2008) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory, n. 24, p. 43-71.
CHOI, B. S. (1992) Identification of ARMA Models. Springer: New York.
COSKUN, A.; CEYHAN, E.; INAL, T. C.; SERTESER, M; UNSAL, I. (2013) The comparison of parametric and nonparametric bootstrap methods for reference interval computation in small sample size groups. Accred Qual Assur, n. 18, p. 51-60.
EFRON, B. (1979) Bootstrap methods: another look at jackknife. Annals of Statistics v. 7, n. 1, p. 1-26.
EFRON, B. (1986) Bootstrap methods for standard errors confidence intervals and other measures of statistics accuracy. Statistical Science, v. 1, n. 1, p. 54-77.
MACHADO, M. A. S; SOUZA, R. C. (2012) Box & Jenkins model identification: A comparison of methodologies. Independent Journal of Management & Production, v. 3, n. 2, p. 54-61.
MINERVA, T.; POLI, I. (2001) ARMA models with genetic algorithms, in: Applications of Evolutionary Computing. Springer: New York, p. 335-342.
MORETTIN, P.; TOLOI, C. M. C. (2006) Análise de séries temporais. Blucher: São Paulo.
MULLER, U.; SCHICK, A.; WEFELMEYER, W. (2005) Weighted residual-based density estimators for nonlinear autoregressive models. Statistc Sinica, n. 15, p. 177-195.
NETO CHAVES, A. (1991) Bootstrap em séries temporais. Thesi (PhD in Eletric Engineering), PUC: Rio de Janeiro.
ONG, C. S.; HUANG, J. J.; TZENG, G. H. (2005) Model identification of arima family using genetic algorithms. Applied Mathematics and Computation, v. 164, n. 3, p. 885-912.
PAPARODITIS, E.; STREITBERG, B. (1992) Order identification statistics in stationary autoregressive moving-average models: vector autocorrelations and the bootstrap. Journal of Time Series Analysis, v. 13, n. 5, p. 415-434.
QUENOUILLE, M. H. (1949) Approximate tests of correlation in time-series. Journal of Statistical Computation and Simulation, n. 8, p. 75-80.
ROLF, S.; SPRAVE, J. (1997) Model identification and parameter estimation of arma models by means evolutionary algorithms. Computational Intelligence for Financial Engineering (CIFEr), v. 1, n. 997, p. 237-243.
SAAVEDRA, A.; CAO, R. (1999) Rate of convergence of a convolution-type estimator of the marginal density of a ma(1) process. Stochastic Process, n. 80, p. 129-155.
SENSIER, M.; VAN DIJK, D. (2004) Testing for volatility changes in u.s. macroeconomic time series. Review of Economics and Statistics, n. 86, p. 833–839.
SILVA, D. (1995) O método bootstrap e aplicações a regressão múltipla. Dissertation (Master in Statistics), Unicamp: Campinas.