Bernoulli

• Bernoulli
• Volume 13, Number 2 (2007), 389-422.

A recursive online algorithm for the estimation of time-varying ARCH parameters

Abstract

In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t−1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context – in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from Hölder classes of order between 1 and 2.

Article information

Source
Bernoulli, Volume 13, Number 2 (2007), 389-422.

Dates
First available in Project Euclid: 18 May 2007

https://projecteuclid.org/euclid.bj/1179498754

Digital Object Identifier
doi:10.3150/07-BEJ5009

Mathematical Reviews number (MathSciNet)
MR2331257

Zentralblatt MATH identifier
1127.62078

Citation

Dahlhaus, Rainer; Subba Rao, Suhasini. A recursive online algorithm for the estimation of time-varying ARCH parameters. Bernoulli 13 (2007), no. 2, 389--422. doi:10.3150/07-BEJ5009. https://projecteuclid.org/euclid.bj/1179498754

References

• Aguech, R., Moulines, E. and Priouret, P. (2000). On a perturbation approach for the analysis of stochastic tracking algorithms., SIAM J. Control Optim. 39 872–899.
• Chen, X. and White, H. (1998). Nonparametric learning with feedback., J. Economic Theory 82 190–222.
• Dahlhaus, R. and Subba Rao, S. (2006). A recursive online algorithm for the estimation of time-varying ARCH parameters. Technical, report.
• Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes., Ann. Statist. 34 1075–1114.
• Guo, L. (1994). Stability of recursive stochastic tracking algorithms., SIAM J. Control Optim. 32 1195–1225.
• Hall, P. and Heyde, C. (1980)., Martingale Limit Theory and Its Application. New York: Academic Press.
• Kushner, H. and Yin, G. (2003)., Stochastic Approximation and Recursive Algorithms and Applications. Berlin: Springer-Verlag.
• Ljung, L. and Söderström, T. (1983)., Theory and Practice of Recursive Identification. Cambridge, MA: MIT Press.
• Mikosch, T. and Stărică, C. (2000). Is it really long memory we see in financial returns? In P. Embrechts (ed.), Extremes and Integrated Risk Management, pp. 439–459. London: Risk Books.
• Mikosch, T. and Stărică, C. (2004). Non-stationarities in financial time series, the long-range dependence, and the IGARCH effects., Rev. Econometrics Statist. 86 378–390.
• Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for locally stationary time varying autoregressive processes., Ann. Statist. 33 2610–2654.
• Solo, V. (1981). The second order properties of a time series recursion., Ann. Statist. 9 307–317.
• Solo, V. (1989). The limiting behavior of the LMS., IEEE Trans. Acoust. Speech Signal Proc. 37 1909–1922.
• Subba Rao, S. (2004). On some nonstationary, nonlinear random processes and their stationary approximations., Adv. in Appl. Probab. 38 1155–1172.
• White, H. (1996). Parametric statistical estimation using artifical neural networks. In P. Smolensky, M. Mozer and D. Rumelhart (eds), Mathematical Perspectives on Neural Networks, pp. 719–775. Hillsdale, NJ: Lawrence Erlbaum Associates.