*Periodicity.:*

**January - March 2015***e-ISSN......:*

**2236-269X**### Bootstrap for order identification in ARMA(P,Q) structures

#### 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.

#### Keywords

#### 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.

DOI: http://dx.doi.org/10.14807/ijmp.v6i1.244

#### Article Metrics

_{Metrics powered by PLOS ALM}

### Refbacks

- There are currently no refbacks.

Copyright (c)

LIBRARIES BY | ||||