Near-integrated GARCH sequences

• 成果类型: Article
• 署名作者: Berkes, I; Horváth, L; Kokoszka, P
• 署名单位: Graz University of Technology; Hungarian Academy of Sciences; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Utah System of Higher Education; University of Utah; Utah System of Higher Education; Utah State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000783
• 发表日期: 2005
• 页码: 890-913
• 关键词: arch models; stationarity

摘要

Motivated by regularities observed in time series of returns on speculative assets, we develop an asymptotic theory of GARCH(1, 1) processes {y(k)} defined by the equations y(k) = sigmakepsilonk, sigma(k)(2) = omega + alphay(k-1)(2) + betasigma(k-1)(2) for which the sum a +,B approaches unity as the number of available observations tends to infinity. We call such sequences near-integrated. We show that the asymptotic behavior of near-integrated GARCH(1, 1) processes critically depends on the sign of y := alpha + beta-1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if gamma less than or equal to 0 and exponentially if gamma greater than or equal to 0. We establish an additive representation for the near-integrated GARCH(1, 1) processes which is more convenient to use than the traditional multiplicative Volterra series expansion.