Nonlinearity of arch and stochastic volatility models and Bartlett’s formula

• Autorzy: Kokoszka P. S. , Politis D. N.
• Języki publikacji: EN

Abstrakty

EN

We review some notions of linearity of time series and show that ARCH or stochastic volatility (SV) processes are not only non-linear: they are not even weakly linear, i.e., they do not even have a martingale representation. Consequently, the use of Bartlett’s formula is unwarranted in the context of data typically modeled as ARCH or SV processes such as financial returns. More surprisingly, we show that even the squares of an ARCH or SV process are not weakly linear. Finally, we discuss an alternative estimator for the variance of sample autocorrelations that is applicable (and consistent) in the context of financial returns data.

Słowa kluczowe

EN

ARCH processes; GARCH processes; linear time series; stochastic volatility

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 1
• Strony: 47--59
• Opis fizyczny: Bibliogr. 26 poz., tab.

Twórcy

autor

Kokoszka P. S.

• Utah State University, Department of Mathematics and Statistics, Logan, UT 84322-3900, USA

autor

Politis D. N.

• University of California, Department of Mathematics, La Jolla, CA 92093-0112, USA

Bibliografia

• [1] M. S. Bartlett, On the theoretical specification and sampling properties of autocorrelated time series, J. Roy. Statist. Soc. Supplement 8 (1946), pp. 27-41. [Correction: 10 (1948), p. 200.]
• [2] A. Berlinet and C. Francq, On Bartlett’s formula for non-linear processes, J. Time Ser. Anal. 18 (1997), pp. 535-552.
• [3] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31 (1986), pp. 307-327.
• [4] R. C. Bradley, Basic properties of strong mixing conditions, in: Dependence in Probability and Statistics, E. Eberlein and M. S. Taqqu (Eds.), Birkhäuser, 1986, pp. 165-192.
• [5] P. Brockwell and R. Davis, Time Series: Theory and Methods, second edition, Springer, New York 1991. Nonlinearity of ARCH 59
• [6] M. Carasco and X. Chen, Mixing and moment properties of various GARCH and stochastic volatility models, Econometric Theory 18 (2002), pp. 17-39.
• [7] R. A. Davis and T. Mikosch, The sample autocorrelations of heavy-tailed processes with applications to ARCH, Ann. Statist. 26 (1998), pp. 2049-2080.
• [8] R. A. Davis and T. Mikosch, Point process convergence of stochastic volatility processes with application to sample autocorrelation, J. Appl. Probab. 38A (2001), pp. 93-104.
• [9] F. Diebold, Testing for serial correlation in the presence of ARCH, in: American Statistical Association: 1986 Proceedings of the Business and Economic Statistics Section, 1986, pp. 323-328.
• [10] R. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation, Econometrica 50 (1982), pp. 987-1008.
• [11] C. Francq and J.-M. Zakoian, Bartlett’s formula for a general class of non-linear processes, J. Time Ser. Anal. 30 (2009), pp. 449-465.
• [12] W. A. Fuller, Introduction to Statistical Time Series, second edition,Wiley, New York 1996.
• [13] L. Giraitis, P. Kokoszka and R. Leipus, Stationary ARCH models: dependence structure and Central Limit Theorem, Econometric Theory 16 (2000), pp. 3-22.
• [14] L. Giraitis, P. Robinson and D. Surgailis, A model for long memory conditional heteroskedasticity, Ann. Appl. Probab. 10 (2000), pp. 1002-1024.
• [15] C. Gouriéroux, ARCH Models and Financial Applications, Springer, New York 1997.
• [16] C. Granger and A. Andersen, An Introduction to Bilinear Time Series Models, Vandenhoeck and Ruprecht, Göttingen 1978.
• [17] E. J. Hannan and M. Deistler, The Statistical Theory of Linear Systems, Wiley, New York 1988.
• [18] O. Kallenberg, Foundations of Modern Probability, Springer, New York 1997.
• [19] I. N. Lobato, J. C. Nankervis and N. E. Savin, Testing for zero autocorrelation in the presence of statistical dependence, Econometric Theory 18 (2002), pp. 730-743.
• [20] D. N. Politis, The impact of bootstrap methods on time series analysis, Statist. Sci. 18 (2) (2003), pp. 219-230.
• [21] D. N. Politis, J. P. Romano and M. Wolf, Subsampling, Springer, New York 1999.
• [22] J. P. Romano and L. Thombs, Inference for autocorrelations under weak assumptions, J. Amer. Statist. Assoc. 91 (1996), pp. 590-600.
• [23] X. Shao and W. B. Wu, Asymptotic spectral theory for nonlinear time series, Ann. Statist. 35 (2007), pp. 1773-1801.
• [24] S. J. Taylor, Financial returns modelled by the product of two stochastic processes, a study of daily sugar prices 1961-1979, in: Time Series Analysis: Theory and Practice I, O. D. Anderson (Ed.), North Holland, Amsterdam 1982.
• [25] S. J. Taylor, Estimating the variances of autocorrelations calculated from financial time series, Applied Statistics 33 (3) (1984), pp. 300-308.
• [26] W. B. Wu, An asymptotic theory for sample covariances of Bernoulli shifts, Stochastic Process. Appl. 119 (2009), pp. 453-467.