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Testing the stability of the functional autoregressive process

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  • Horváth, Lajos
  • Husková, Marie
  • Kokoszka, Piotr

Abstract

The functional autoregressive process has become a useful tool in the analysis of functional time series data. It is defined by the equation , in which the observations Xn and errors [epsilon]n are curves, and is an operator. To ensure meaningful inference and prediction based on this model, it is important to verify that the operator does not change with time. We propose a method for testing the constancy of against a change-point alternative which uses the functional principal component analysis. The test statistic is constructed to have a well-known asymptotic distribution, but the asymptotic justification of the procedure is very delicate. We develop a new truncation approach which together with Mensov's inequality can be used in other problems of functional time series analysis. The estimation of the principal components introduces asymptotically non-negligible terms, which however cancel because of the special form of our test statistic (CUSUM type). The test is implemented using the R package fda, and its finite sample performance is examined by application to credit card transaction data.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 101 (2010)
Issue (Month): 2 (February)
Pages: 352-367

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Handle: RePEc:eee:jmvana:v:101:y:2010:i:2:p:352-367

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Related research

Keywords: 62M10 Change-point Functional autoregressive process;

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References

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  1. Andrews, Donald W K & Monahan, J Christopher, 1992. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, Econometric Society, vol. 60(4), pages 953-66, July.
  2. Horváth, Lajos & Kokoszka, Piotr & Steinebach, Josef, 1999. "Testing for Changes in Multivariate Dependent Observations with an Application to Temperature Changes," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 68(1), pages 96-119, January.
  3. Peter Hall & Mohammad Hosseini-Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126.
  4. Gabrys, Robertas & Kokoszka, Piotr, 2007. "Portmanteau Test of Independence for Functional Observations," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 102, pages 1338-1348, December.
  5. Kargin, V. & Onatski, A., 2008. "Curve forecasting by functional autoregression," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 99(10), pages 2508-2526, November.
  6. Hansen, Bruce E., 1995. "Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1148-1171, October.
  7. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, Econometric Society, vol. 59(3), pages 817-58, May.
  8. P. M. Robinson, 1998. "Inference-Without-Smoothing in the Presence of Nonparametric Autocorrelation," Econometrica, Econometric Society, Econometric Society, vol. 66(5), pages 1163-1182, September.
  9. Antoniadis, Anestis & Sapatinas, Theofanis, 2003. "Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 87(1), pages 133-158, October.
  10. Philippe C. Besse, 2000. "Autoregressive Forecasting of Some Functional Climatic Variations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 673-687.
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Cited by:
  1. Horváth, Lajos & Kokoszka, Piotr & Rice, Gregory, 2014. "Testing stationarity of functional time series," Journal of Econometrics, Elsevier, Elsevier, vol. 179(1), pages 66-82.
  2. Zhou, Jie, 2011. "Maximum likelihood ratio test for the stability of sequence of Gaussian random processes," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 55(6), pages 2114-2127, June.
  3. A. Soltani & M. Hashemi, 2011. "Periodically correlated autoregressive Hilbertian processes," Statistical Inference for Stochastic Processes, Springer, Springer, vol. 14(2), pages 177-188, May.
  4. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, Springer, vol. 23(2), pages 219-255, June.
  5. Devin Didericksen & Piotr Kokoszka & Xi Zhang, 2012. "Empirical properties of forecasts with the functional autoregressive model," Computational Statistics, Springer, Springer, vol. 27(2), pages 285-298, June.

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