Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

Abstract

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.

Keywords:

• Autoregression
• Bootstrap
• GARCH

JEL Classification:

• C14
• C15
• C22
• C52

ACKNOWLEDGMENTS

Part of this research was conducted while the second author was serving as an advisor to the European Central Bank. We thank the editor, two anonymous referees, Peter R. Hansen, Atsushi Inoue, Guido Kuersteiner, Simone Manganelli, Nour Meddahi, Roch Roy, and Victoria Zinde-Walsh for helpful discussions. We also thank seminar participants at the June 2003 North American and the July 2003 Australasian Meeting of the Econometric Society, as well as the NBER-NSF Time Series Conference in Chicago in September of 2003. Gonçalves acknowledges financial support from the Fonds québécois pour la recherche sur la société et la culture (FQRSC), and the Social Science and Humanities Research Council of Canada (SSHRCC).

Notes

Source: Based on 20,000 Monte Carlo draws with 1,000 bootstrap replications each.

Source: See Table 1.

Source: See Table 1.