Electronic Journal of Statistics

The multiple testing problem for Box-Pierce statistics

Tucker McElroy and Brian Monsell

Full-text: Open access

Abstract

We derive the exact joint asymptotic distribution for multiple Box-Pierce statistics, and use these results to determine appropriate critical values in joint testing problems of time series goodness-of-fit. By sequentially testing at various lags, we can identify specific problems with a model, and identify superior models. A novel $\alpha$-rationing scheme, motivated by the sequence of conditional probabilities for the statistical tests, is developed and implemented. This method can be used to produce critical values and p-values both for each step of the sequential testing procedure, and for the procedure as a whole. Efficient computational algorithms are discussed. Simulation studies assess the impact of finite samples on the real Type I error. It is also demonstrated empirically that the conventional $\chi^{2}$ critical values for the Box-Pierce statistics are too small, with a Type I error rate greater than the nominal; the new method does not suffer from this defect, and allows for more rigorous model-building. We illustrate on several time series how model defects can be identified and ameliorated.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 497-522.

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1399901042

Digital Object Identifier
doi:10.1214/14-EJS892

Mathematical Reviews number (MathSciNet)
MR3205731

Zentralblatt MATH identifier
1349.62418

Keywords
ARIMA models Ljung-Box statistic time series residuals

Citation

McElroy, Tucker; Monsell, Brian. The multiple testing problem for Box-Pierce statistics. Electron. J. Statist. 8 (2014), no. 1, 497--522. doi:10.1214/14-EJS892. https://projecteuclid.org/euclid.ejs/1399901042


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References

  • [1] Ansley, C. and Newbold, P. (1979). On the finite sample distribution of residual autocorrelations in autoregressive-moving average models. Biometrika 66 547–553.
  • [2] Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217–226.
  • [3] Box, G. and Pierce, D. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association 65 1509–1526.
  • [4] Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • [5] Findley, D. F. and Hood, C. C. H. (1999). X-12-ARIMA and its application to some Italian indicator Series. In Seasonal Adjustment Procedures – Experiences and Perspectives, pp. 231–251. Rome: Istituto Nazionale di Statistica (ISTAT).
  • [6] Findley, D. F., Monsell, B. C., Bell, W. R., Otto, M. C., and Chen, B. C. (1998). New capabilities and methods of the X-12-ARIMA seasonal adjustment program. Journal of Business and Economic Statistics 16 127–177 (with discussion).
  • [7] Golub, G. and Van Loan, C. (1996). Matrix Computations. Johns Hopkins University Press, Baltimore.
  • [8] Hosoya, Y. and Taniguchi, M. (1982). A central limit theorem for stationary processes and the parameter estimation of linear processes. Ann. Statist. 10 132–153.
  • [9] Imhof, J. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika 48 419–426.
  • [10] Kan, R. and Wang, X. (2010). On the distribution of sample autocorrelation coefficients. Journal of Econometrics 154 101–121.
  • [11] Katayama, N. (2008). An improvement of the portmanteau statistic. Journal of Time Series Analysis 29 359–370.
  • [12] Katayama, N. (2009). On multiple portmanteau tests. Journal of Time Series Analysis 30 487–504.
  • [13] Koopman, S., Harvey, A., and Doornik, J. (2000). STAMP 6.0: Structural Time Series Analyser, Modeller, and Predictor. Timberlake Consultants, London.
  • [14] Kwan, A. and Sim, A. (1996). On the finite-sample distribution of modified portmanteau tests for randomness of a Gaussian time series. Biometrika 83 938–943.
  • [15] Kwan, A. and Wu, Y. (1997). Further results on the finite-sample distribution of Monti’s portmanteau test for the adequacy of an $ARMA(p,q)$ model. Biometrika 84 733–736.
  • [16] Ljung, G. (1986). Diagnostic testing of univariate time series models. Biometrika 73 725–730.
  • [17] Ljung, G. and Box, G. (1978). On a measure of lack of fit in time series models. Biometrika 65 297–303.
  • [18] Maravall, A. and Caporello, G. (2004). Program TSW: Revised Reference Manual. Working Paper 2004, Research Department, Bank of Spain. http://www.bde.es.
  • [19] McElroy, T. (2008). Statistical properties of model-based signal extraction diagnostic tests. Communications in Statistics, Theory and Methods 37 591–616.
  • [20] McElroy, T. and Holan, S. (2009). A local spectral approach for assessing time series model misspecification. Journal of Multivariate Analysis 100 604–621.
  • [21] McElroy, T. and Monsell, B. (2014). Supplement to “The multiple testing problem for Box-Pierce statistics”. DOI:10.1214/14-EJS892SUPP.
  • [22] McElroy, T. and Wildi, M. (2013). Multi-step ahead estimation of time series models. International Journal of Forecasting 29 378–394.
  • [23] McLeod, A.I. (1978). On the distribution of residual autocorrelations in Box-Jenkins models. Journal of the Royal Statistical Society, Series B 40 296–302.
  • [24] Monti, A. (1994). A proposal for residual autocorrelation test in linear models. Biometrika 81 776–780.
  • [25] Peña, D. and Rodriguez, J. (2002). A powerful portmanteau test of lack of fit for time series. Journal of the American Statistical Association 97 601–610.
  • [26] R Development Core Team (2009). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org.
  • [27] Pollock, D. (1999). A Handbook of Time-Series Analysis, Signal Processing and Dynamics. Academic Press, New York.
  • [28] Slud, E. and Wei, L. (1982). Two-sample repeated significance tests based on the modified Wilcoxon statistic. Journal of the American Statistical Association 77 862–868.
  • [29] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer-Verlag, New York.
  • [30] Tziritas, G. (1987). On the distribution of positive-definite Gaussian quadratic forms. IEEE Transactions on Information Theory 33 895–906.

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