Data-driven score test of fit for conditional distribution in the GARCH (1, 1) model

• Autorzy: Inglot T. , Stawiarski B.
• Języki publikacji: EN

Abstrakty

EN

A data-driven score test for a conditional distribution in the GARCH (1, 1) model is proposed. Conditional distribution assumption is verified by a score test, obtained from nesting the null density into an exponential family and then choosing the dimension of this exponential family by a score-based selection rule. A simulation study, which is provided, shows good empirical behaviour of the proposed test, outperforming in most cases the behaviour of competitive tests.

Słowa kluczowe

EN

GARCH (1, 1) model; noise distribution; efficient score vector; score test; BIC Schwarz selection rule; martingale difference array; central limit theorem; ergodic theorem; Monte Carlo simulation

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2005
• Tom: Vol. 25, Fasc. 2
• Strony: 331--362
• Opis fizyczny: Bibliogr. 40 poz., tab.

Twórcy

autor

Inglot T.

• Institute of Mathematics and Informatics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland
• Polish Academy of Sciences

autor

Stawiarski B.

• Institute of Mathematics and Informatics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland

Bibliografia

• [1] J. Bai, Testing parametric conditional distributions of dynamic models, Rev. Econom. Statist. 85 (2003), pp. 531-549.
• [2] R. T. Baillie and T. Bollerslev, The message in daily exchange rates. a conditional-variance tale, J. Bus. Econom. Statist. 7 (1989), pp. 297-305.
• [3] E. K. Berndt, B. H. Hall, R. E. Hall and J. A. Hausman, Estimation inference in nonlinear structural model, Ann. Econom. Social Measurement 4 (1974) pp. 653-665.
• [4] T. Bollerslev, Generalized autoregressive conditional heteroskedusticity, J. Econometrics 31 (1986), pp. 307-327.
• [5] T. Bollerslev, A conditionally heteroskedastic time-series model for speculative prices and rates of return data, Rev. Econometrics Statist. 69 (1987), pp. 542-547.
• [6] P. Bougerol and N. M. Picard, Stationarity of GARCH processes and of some nonnegative time series, J. Econometrics 52 (1992), pp. 115-127.
• [7] C. Brooks, S. P. Burke and G. Persand, Benchmarks and the accuracy of GARCH model estimation, Internat. J. Forecasting 17 (2001), pp. 45-56.
• [8] W. J. Bühler and P. S. Puri, On optional asymptotic tests of composite hypotheses with several constraints, Z. Wahrsch. Verw. Gebiete 5 (1966), pp. 71-88.
• [9] Y.-T. Chen, A new class of characteristic-function-based distribution tests and its application to GARCH model, Working paper, ISSP, Academia Sinica, 2002.
• [10] Y.-T. Chen and C.-M. Kuan, Generalized Jarque-Bera test of conditional normality, Working paper, Academia Sinica, 2002.
• [11] D. R. Cox and D. V. Hinkley, Theoretical Statistics, Chapman and Hall, London 1974.
• [12] F. X. Diebold, Empirical Modeling of Exchange Rate Dynamics, Springer, New York 1988.
• [13] F. C. Drost and C. A. Klaassen, Efficient estimation in semiparametric GARCH models, J. Econometrics 81 (1997), pp. 183-221.
• [14] G. R. Ducharme and P. Lafaye de Micheaux, Goodness-of-fit tests of normality for the innovation in ARMA models, J. Time Ser. Anal. 25 (2004), pp. 375-395.
• [15] R. F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50 (1982), pp. 987-1007.
• [16] G. Fiorentini, E. Sentana and G. Calzolari, On the validity of the Jarque-Bera normality test in conditionally heteroskedastic dynamic regression models, Econom. Lett. 83 (2004), pp. 307-312.
• [17] N. Friedman, Introduction to Ergodic Theory, Princeton, New Jersey, 1970.
• [18] C. Gouriéroux, ARCH Models and Financial Applications, Springer, New York 1997.
• [19] R. Ignacco1o, Goodness-of-fit tests for dependent data, Nonparametric Statistics 16 (2004), pp. 19-38.
• [20] T. Inglot, W. C. M. Kallenberg and T. Ledwina, Power approximations to and power comparison of smooth goodness-of-fit tests, Scand. J. Statist. 21 (1994), pp. 131-145.
• [21] T. Inglot, W. C. M. Kallenberg and T. Ledwina, Data driven smooth tests for composite hypotheses, Ann. Statist. 25 (1997), pp. 1222-1250.
• [22] T. Inglot and T. Ledwina, Intermediate approach to comparison of some goodness-of-fit tests, Ann. Inst. Statist. Math. 53 (2001), pp. 810-834.
• [23] W. C. M. Kallenberg and T. Ledwina, Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests, Ann. Statist. 23 (1995), pp. 1594-1608.
• [24] W. C. M. Kallenberg and T. Ledwina, Data driven tests for composite hypotheses: Comparison of powers, J. Stat. Comput. Simul. 59 (1997a), pp. 101-121.
• [25] W. C. M. Kallenberg and T. Ledwina, Data driven smooth tests when the hypothesis is composite, J. Amer. Statist. Assoc. 92 (1997b), pp. 1094-1104.
• [26] S. Kundu, S. Majumdar and K. Mukherjee, Central limit theorems revisited, Statist. Probab. Lett. 47 (2000), pp. 265-275.
• [27] T. Ledwina, Data driven version of Neyman's smooth test of fit, J. Amer. Statist. Assoc. 89 (1994), pp. 1000-1005.
• [28] S.-W. Lee and B. E. Hansen, Asymptotic theory for the GARCH (1, 1) quasi-maximum likelihood estimator, Econometric Theory 10 (1994), pp. 29-52.
• [29] W. K. Li and S. Ling, On fractionally integrated autoregressive moving average time series models with conditional heteroskedasticity, J. Amer. Statist. Assoc. 92 (1997), pp. 1184-1194.
• [30] W. K. Li, S. Ling and M. McAleer, A survey of recent theoretical results for time series models with GARCH errors, J. Econom. Survey 16 (2002), pp. 245-269.
• [31] S. Ling and M. McAleer, Asymptotic theory for a vector ARMA-GARCH model, Econometric Theory 19 (2003), pp. 278-308.
• [32] R. L. Lumsdaine, Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH (1, 1) and covariance stationary GARCH (1, 1) models, Econometrica 64 (1996), pp. 575-596.
• [33] T. K. Mak, H. Wong and W. K. Li, Estimation of nonlinear time series with conditional heteroscedastic variances by iteratively weighted least squares, Comput. Statist. Data Analysis 24 (1997), pp. 169-178.
• [34] H. Milbrodt and H. Strasser, On the asymptotic power of the two-sided Kolmogorov-Smirnov test, J. Statist. Plann. Inference 26 (1990), pp. 1-23.
• [35] A. Milhøj, The moment structure of ARCH processes, Scand. J. Statist. 12 (1985), pp. 281-292.
• [36] S. Mittnik, M. S. Paolella and S. T. Rachev, Unconditional and conditional distributional models for the Nikkei index, Asia-Pacific Financial Markets 5 (1998), pp. 99-128.
• [37] D. B. Nelson, Stationarity and persistence in the GARCH (1, 1) model, Econometric Theory 6 (1990), pp. 318-334.
• [38] J. Neyman, Smooth tests for goodness of fit, Skand. Aktuarietidskr. 20 (1937), pp. 149-199.
• [39] D. R. Thomas and D. A. Pierce, Neyman's smooth goodness-of-fit test when the hypothesis is composite, J. Amer. Statist. Assoc. 74 (1979), pp. 441-445.
• [40] A. A. Weiss, Asymptotic theory for ARCH models: Estimation and testing, Econometric Theory 2 (1986), pp. 107-131.