Bootstrapping the portmanteau tests in weak auto-regressive moving average models

Zhu, Ke (2015): Bootstrapping the portmanteau tests in weak auto-regressive moving average models.

Abstract

This paper uses a random weighting (RW) method to bootstrap the critical values for the Ljung-Box/Monti portmanteau tests and weighted Ljung-Box/Monti portmanteau tests in weak ARMA models. Unlike the existing methods, no user-chosen parameter is needed to implement the RW method. As an application, these four tests are used to check the model adequacy in power GARCH models. Simulation evidence indicates that the weighted portmanteau tests have the power advantage over other existing tests. A real example on S&P 500 index illustrates the merits of our testing procedure. As one extension work, the block-wise RW method is also studied.

• Item Type: MPRA Paper
• Language: English
• Keywords: Bootstrap method; Portmanteau test; Power GARCH models; Random weighting approach; Weak ARMA models; Weighted portmanteau test.
• Subjects: C - Mathematical and Quantitative Methods > C0 - General; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General

References

[1] Beltrao, K. and Bloomfield, P. (1987) Determining the bandwidth of a kernel spectrum estimate. Journal of Time Series Analysis 8, 21-38.

[2] Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489-502.

[3] Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307-327.

[4] Bollerslev, T., Chou, R.Y., and Kroner, K.F. (1992) ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics 52, 5-59.

[5] Box, G.E.P. and Pierce, D.A. (1970) Distribution of the residual autocorrelations in autoregressive integrated moving average time series models. Journal of the American Statistical Association 65, 1509-1526.

[6] Carbon, M. and Francq, C. (2011) Portmanteau goodness-of-fit test for asymmetric power GARCH models. Austrian Journal of Statistics 40, 55-64.

[7] Carrasco, M. and Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17-39.

[8] Chen, M. and Zhu, K. (2015) Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations. Forthcoming in Journal of Econometrics.

[9] Chen, K., Ying, Z., Zhang, H., and Zhao, L. (2008) Analysis of least absolute deviation. Biometrika 95, 107-122.

[10] Chen, K., Guo, S., Lin, Y., and Ying, Z. (2010) Least absolute relative error estimation. Journal of the American Statistical Association 105, 1104-1112.

[11] Deo, R.S. (2000) Spectral tests of the martingale hypothesis under conditional heteroskedasticity. Journal of Econometrics 99, 291-315.

[12] Ding, Z., Granger, C.W.J. and Engle, R.F. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83-106.

[13] Duchesne, P. and Francq, C. (2008) On diagnostic checking time series models with portmanteau test statistics based on generalized inverses and {2}-inverses. COMPSTAT 2008, Proceedings in Computational Statistics, (ed.) P. Brito, Heidelberg: Physica-Verlag, pp. 143-154.

[14] Durlauf, S.N. (1991) Spectral-based testing of the martingale hypothesis. Journal of Econometrics 50, 355-376.

[15] Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50, 987-1008.

[16] Escanciano, J.C. (2006) Goodness-of-fit tests for linear and non-linear time series models. Journal of the American Statistical Association 101, 531-541.

[17] Escanciano, J.C. (2007) Model checks using residual marked emprirical processes. Statistica Sinica 17, 115-138.

[18] Escanciano, J.C. and Velasco, C. (2006) Generalized spectral tests for the martingale difference hypothesis. Journal of Econometrics 134, 151-185.

[19] Escanciano, J.C. and Lobato, I.N. (2009) An automatic portmanteau test for serial correlation. Journal of Econometrics 151, 140-149.

[20] Fisher, T.J. and Gallagher, C.M. (2012) New weighted portmanteau statistics for time series goodness of fit testing. Journal of the American Statistical Association 107, 777-787.

[21] Francq, C., Roy, R., and Zakoian, J.M. (2005) Diagostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100, 532-544.

[22] Francq, C. and Zakoian, J.M. (1998) Estimating linear representations of nonlinear processes. Journal of Statistical Planning and Inference 68, 145-165.

[23] Francq, C. and Zakoian, J.M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley, Chichester, UK.

[24] Franses, P.H. and Van Dijk, R. (1996) Forecasting stock market volatility using (non-linear) Garch models. Journal of Forecasting 15, 229-235.

[25] Giot, P. and Laurent, S. (2004) Modelling daily value-at-risk using realized volatility and Arch type models. Journal of Empirical Finance 11, 379-398.

[26] Hall, P., Horowitz, J.L., and Jing, B.-Y. (1995) On blocking rules for the bootstrap with dependent data. Biometrica 82, 561-574.

[27] Higgins, M.L. and Bera, A.K. (1992) A class of nonlinear ARCH models. International Economic Review 33, 137-158.

[28] Hong, Y. (1996) Consistent testing for serial correlation of unknown form. Econometrica 64, 837-864.

[29] Hong, Y. (1999) Hypothesis testing in time series via the empirical characteristic function: a generalized spectral density approach. Journal of the American Statistical Association 94, 1201-1220.

[30] Hong, Y. and Lee, T.H. (2003) Diagnostic checking for adequacy of nonlinear time series models. Econometric Theory 19, 1065-1121.

[31] Hong, Y. and Lee, Y.-J. (2005) Generalized spectral testing for conditional mean models in time series with conditional heteroskedasticity of unknown form. Review of Economic Studies 72, 499-541.

[32] Horowitz, J.L., Lobato, I.N., Nankervis, J.C., and Savin, N.E. (2006) Bootstrapping the Box-Pierce Q test: A robust test of uncorrelatedness. Journal of Econometrics 133, 841-862.

[33] Jin, Z., Ying, Z. and Wei, L.J. (2001) A simple resampling method by perturbing the minimand. Biometrika 88, 381-390.

[34] Kuan, C.-M. and Lee, W.-M. (2006) Robust M tests without consistent estimation of the asymptotic covariance matrix. Journal of the American Statistical Association 101, 1264-1275.

[35] Lee, J. and Hong, Y. (2001) Testing for serial correlation of unknown form using wavelet methods. Econometric Theory 17, 386-423.

[36] Li, W.K. (1992) On the asymptotic standard errors of residual autocorrelations in nonlinear time series modelling. Biometrika 79, 435-437.

[37] Li, W.K. (2004) Diagnostic checks in time series. Chapman& Hall/CRC.

[38] Li, G., Leng, C. and Tsai, C.-L. (2014) A hybrid bootstrap approach to unit root tests. Journal of Time Series Analysis 35, 299-321.

[39] Li, G. and Li, W.K. (2005) Diagnostic checking for time series models with conditional heteroscedasticity estimated by the least absolute deviation approach. Biometrika 92, 691-701.

[40] Li, G. and Li, W.K. (2008). Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity. Biometrika 95, 399-414.

[41] Li, G., Li, Y. and Tsai, C.-L. (2014) Quantile correlations and quantile autoregressive modeling. Forthcoming in Journal of the American Statistical Association.

[42] Li, W.K. and Mak, T.K. (1994) On the squared residual autocorrelations in non-linear time series with conditional heteroscedasticity. Journal of Time Series Analysis 15, 627-636.

[43] Ljung, G.M. and Box, G.E.P. (1978) On a measure of lack of fit in time series models. Biometrika 65, 297-303.

[44] Lobato, I.N. (2001) Testing that a dependent process is uncorrelated. Journal of the American Statistical Association 96, 1066-1076.

[45] McKenzie, M. and Mitchell, H. (2002) Generalised asymmetric power arch modeling of exchange rate volatility. Applied Financial Economics 12, 555-564.

[46] McLeod, A.I. and Li, W.K. (1983) Diagnostic checking ARMA time series models using square-dresidual autocorrelations. Journal of Time Series Analysis 4, 269-273.

[47] McLeod, A.I. (1978) On the distribution of residual autocorrelations in Box-Jenkins method. Journal of the Royal Statistical Society B 40, 296-302.

[48] Monti, A.C. (1994) A proposal for a residual autocorrelation test in linear models. Biometrika 81, 776-780.

[49] Newey, W.K. and West, K.D. (1987) A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703-708.

[50] Pena, 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.

[51] Pena, D. and Rodriguez, J. (2006), The Log of the determinant of the autocorrelation matrix for testing goodness of fit in time series. Journal of Statistical Planning and Inference 136, 2706-2718.

[52] Romano, J.L. and Thombs, L.A. (1996) Inference for autocorrelations under weak assumptions. Journal of the American Statistical Association 91, 590-600.

[53] Politis, D.N., Romano, J., and Wolf, M. (1999) Subsampling Springer-Verla, New York.

[54] Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 45, 1129-1155.

[55] Shao, X. (2011a). Testing for white noise under unknown dependence and its applications to diagnostic checking for time series models. Econometric Theory 27, 312-343.

[56] Shao, X. (2011b) A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics 162, 213-224.

[57] Stinchcombe, M. and White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295-325.

[58] Sun, Y. (2014) Let's fix it: Fixed-b asymptotics versus small-b asymptotics in heteroskedasticity and autocorrelation robust inference. Journal of Econometrics 178, 659-677.

[59] Taylor, S. (1986) Modelling Financial Time Series. Wiley, New York.

[60] Tsay, R.S. (2005) Analysis of Financial Time Series (2nd ed.). New York: John Wiley&Sons, Incorporated.

[61] Wu, C.F.J. (1986) Jacknife, bootstrap and other resampling methods in regression analysis (with discussion). Annals of Statistics 14, 1261-1350.

[62] Zhu, K. and Ling, S. (2011) Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models. Annals of Statistics 39, 2131-2163.

[63] Zhu, K. and Ling, S. (2015) LADE-based inference for ARMA models with unspecified and heavy-tailed heteroscedastic noises. Forthcoming in Journal of the American Statistical Association.