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TESTING FOR WHITE NOISE UNDER UNKNOWNDEPENDENCE AND ITS APPLICATIONS TO DIAGNOSTICCHECKING FOR TIME SERIES MODELS

Published online by Cambridge University Press:  27 August 2010

Xiaofeng Shao
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Abstract

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Testing for white noise has been well studied in theliterature of econometrics and statistics. For mostof the proposed test statistics, such as thewell-known Box–Pierce test statistic with fixed lagtruncation number, the asymptotic null distributionsare obtained under independent and identicallydistributed assumptions and may not be valid fordependent white noise. Because of recent popularityof conditional heteroskedastic models (e.g.,generalized autoregressive conditionalheteroskedastic [GARCH] models), which implynonlinear dependence with zero autocorrelation,there is a need to understand the asymptoticproperties of the existing test statistics underunknown dependence. In this paper, we show that theasymptotic null distribution of the Box–Pierce teststatistic with general weights still holds underunknown weak dependence as long as the lagtruncation number grows at an appropriate rate withincreasing sample size. Further applications todiagnostic checking of the autoregressive movingaverage (ARMA) and fractional autoregressiveintegrated moving average (FARIMA) models withdependent white noise errors are also addressed. Ourresults go beyond earlier ones by allowingnon-Gaussian and conditional heteroskedastic errorsin the ARMA and FARIMA models and providetheoretical support for some empirical findingsreported in the literature.

Information

Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 2 , April 2011, pp. 312 - 343
Copyright
Copyright © Cambridge University Press 2010

Footnotes

I thank Professor Pentti Saikkonen and tworeferees for constructive comments that led toimprovement of the paper. The work was supportedin part by NSF grant DMS-0804937.

References

REFERENCES

Bartlett, M.S. (1955) An Introduction to Stochastic Processes with Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Beran, J. (1992) A goodness-of-fit test for time series with long range dependence. Journal of the Royal Statistical Society, Series B 54, 749–760.Google Scholar
Beran, J. (1995) Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. Journal of the Royal Statistical Society, Series B 57, 659–672.Google Scholar
Box, G. & Pierce, D. (1970) Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association 65, 1509–1526.Google Scholar
Breidt, F.J., Davis, R.A., & Trindade, A.A. (2001) Least absolute deviation estimation for all-pass time series models. Annals of Statistics 29, 919–946.CrossRefGoogle Scholar
Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holden-Day.Google Scholar
Chen, W. & Deo, R.S. (2004a). A generalized portmanteau goodness-of-fit test for time series models. Econometric Theory 20, 382–416.Google Scholar
Chen, W. & Deo, R.S. (2004b) Power transformations to induce normality and their applications. Journal of the Royal Statistical Society, Series B 66, 117–130.Google Scholar
Chen, W. & Deo, R.S. (2006) The variance ratio statistic at large horizons. Econometric Theory 22, 206–234.Google Scholar
Chen, H. & Romano, J.P. (1999) Bootstrap-assisted goodness-of-fit tests in the frequency domain. Journal of Time Series Analysis 20, 619–654.Google Scholar
Delgado, M.A., Hidalgo, J., & Velasco, C. (2005) Distribution free goodness-of-fit tests for linear processes. Annals of Statistics 33, 2568–2609.CrossRefGoogle Scholar
Deo, R.S. (2000) Spectral tests of the martingale hypothesis under conditional heteroscedasticity. Journal of Econometrics 99, 291–315.Google Scholar
Durlauf, S. (1991) Spectral based testing for the martingale hypothesis. Journal of Econometrics 50, 1–19.Google Scholar
Elek, P. & Márkus, L. (2004) A long range dependent model with nonlinear innovations for simulating daily river flows. Natural Hazards and Earth System Sciences 4, 277–283.Google Scholar
Francq, C., Roy, R., & Zakoïan, J.M. (2005) Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100, 532–544.Google Scholar
Francq, C. & Zakoïan, J.M. (2000) Covariance matrix estimation for estimators of mixing weak ARMA-models. Journal of Statistical Planning and Inference 83, 369–394.Google Scholar
Francq, C. & Zakoïan, J.M. (2005) Recent results for linear time series models with non independent innovations. In Duchesne, P. & Rémillard, B. (eds.), Statistical Modeling and Analysis for Complex Data Problems, pp. 241–267. Springer-Verlag.Google Scholar
Granger, C.W.J. & Anderson, A.P. (1978) An Introduction to Bilinear Time Series Models. Vandenhoek and Ruprecht.Google Scholar
Granger, C.W.J. & Teräsvirta, T. (1993) Modelling Nonlinear Economic Relationships. Oxford University Press.Google Scholar
Grenander, U. & Rosenblatt, M. (1957) Statistical Analysis of Stationary Time Series. Wiley.Google Scholar
Guerre, E. & Lavergne, P. (2005) Data-driven rate-optimal specification testing in regression models. Annals of Statistics 33, 840–870.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. Academic Press.Google Scholar
Hidalgo, J. & Kreiss, J.P. (2006) Bootstrap specification tests for linear covariance stationary processes. Journal of Econometrics 133, 807–839.Google Scholar
Hong, Y. (1996) Consistent testing for serial correlation of unknown form. Econometrica 64, 837–864.Google Scholar
Hong, Y. & Lee, Y.J. (2003) Consistent Testing for Serial Uncorrelation of Unknown Form under General Conditional Heteroscedasticity. Preprint, Cornell University, Department of Economics.Google Scholar
Horowitz, J.L., Lobato, I.N., Nankervis, J.C., & Savin, N.E. (2006) Bootstrapping the Box-Pierce Q test: A robust test of uncorrelatedness. Journal of Econometrics 133, 841–862.Google Scholar
Horowitz, J.L. & Spokoiny, V.G. (2001) An adaptive rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599–631.Google Scholar
Hosoya, Y. (1997) A limit theory for long-range dependence and statistical inference on related models. Annals of Statistics 25, 105–137.CrossRefGoogle Scholar
Hsing, T. & Wu, W.B. (2004) On weighted U-statistics for stationary processes. Annals of Probability 32, 1600–1631.Google Scholar
Koopman, S.J., Oohs, M., & Carnero, M.A. (2007) Periodic seasonal Reg-ARFIMA-GARCH models for daily electricity spot prices. Journal of the American Statistical Association 102, 16–27.Google Scholar
Li, G. & Li, W.K. (2008) Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity. Biometrika 95, 399–414.Google Scholar
Lien, D. & Tse, Y.K. (1999) Forecasting the Nikkei spot index with fractional cointegration. Journal of Forecasting 18, 259–273.Google Scholar
Ling, S. & Li, W.K. (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184–1194.Google Scholar
Lobato, I.N., Nankervis, J.C., & Savin, N.E. (2002) Testing for zero autocorrelation in the presence of statistical dependence. Econometric Theory 18, 730–743.Google Scholar
Lumsdaine, R.L. & Ng, S. (1999) Testing for ARCH in the presence of a possibly misspecified conditional mean. Journal of Econometrics 93, 257–279.Google Scholar
Paparoditis, E. (2000) Spectral density based goodness-of-fit tests for time series models, Scandinavian Journal of Statistics 27, 143–176.Google Scholar
Priestley, M.B. (1981) Spectral Analysis and Time Series, vol. 1. Academic Press.Google Scholar
Robinson, P.M. (2005) Efficiency improvements in inference on stationary and nonstationary fractional time series. Annals of Statistics 33, 1800–1842.Google Scholar
Romano, J.L. & Thombs, L.A. (1996) Inference for autocorrelations under weak assumptions. Journal of the American Statistical Association 91, 590–600.Google Scholar
Shao, X. (2010) Nonstationarity-extended Whittle estimation. Econometric Theory 26, 1060–1087.Google Scholar
Shao, X. & Wu, W.B. (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 4, 1773–1801.Google Scholar
Wu, W.B. (2005) Nonlinear system theory: another look at dependence. Proceedings of the National Academy of Science 102, 14150–14154.Google Scholar
Wu, W.B. (2007) Strong invariance principles for dependent random variables. Annals of Probability 35, 2294–2320.Google Scholar
Wu, W.B. & Min, W. (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939–958.Google Scholar
Wu, W.B. & Shao, X. (2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436.Google Scholar
Wu, W.B. & Shao, X. (2007) A limit theorem for quadratic forms and its applications. Econometric Theory 23, 930–951.Google Scholar
Wu, W.B. & Woodroofe, M. (2004) Martingale approximations for sums of stationary processes. Annals of Probability 32, 1674–1690.Google Scholar