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Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

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  • Chan, Ngai Hang
  • Zhang, Rong-Mao

Abstract

Let Xt=∑j=0∞cjεt−j be a moving average process with GARCH (1, 1) innovations {εt}. In this paper, the asymptotic behavior of the quadratic form Qn=∑j=1n∑s=1nb(t−s)XtXs is derived when the innovation {εt} is a long-memory and heavy-tailed process with tail index α, where {b(i)} is a sequence of constants. In particular, it is shown that when 1<α<4 and under certain regularity conditions, the limit distribution of Qn converges to a stable random variable with index α/2. However, when α≥4, Qn has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations.

Suggested Citation

  • Chan, Ngai Hang & Zhang, Rong-Mao, 2013. "Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 18-33.
    • Handle: RePEc:eee:jmvana:v:120:y:2013:i:c:p:18-33
    DOI: 10.1016/j.jmva.2013.04.007
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    References listed on IDEAS

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    1. Hwang, Eunju & Hong, Won-Tak, 2021. "A multivariate HAR-RV model with heteroscedastic errors and its WLS estimation," Economics Letters, Elsevier, vol. 203(C).

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