A test of conditional heteroscedasticity in time series
type: Item
instance of: scholarly article
title: A test of conditional heteroscedasticity in time series
zbMATH Open document ID: 0953.62092
DOI: 10.1007/BF02872047
author: Min Chen
author: Hongzhi An
published in: Science in China. Series A
publication date: 2000-08-29T00:00:00Z
review text: Let \((X_{t})_{t\in Z}\) be an autoregressive time series with finite second moments and conditional heteroscedasticity, denoted as AR-ARCH. That is \[ X_{t}=\phi_{1}X_{t-1}+...+\phi_{p}X_{t-p}+\varepsilon_{t},\;E\{\varepsilon _{t}|X_{t-1},...,X_{t-p}\}=0\;\text{a.s.}, \] \[ E\{\varepsilon_{t}^{2} |X_{t-1} ,...,X_{t-p}\}=a_{0}+a_{1}\varepsilon_{t-1}^{2}+...+a_{p}\varepsilon_{t-p} ^{2} \text{a.s.}, \] where \(1-\phi_{1}z-...-\phi_{p}z^{p}\neq 0,\) for all \(\left|z\right|\leq 1,\) \(a_{0}>0,\) \(a_{i}\geq 0,\) \(i=1,...,p\). This work deals with a test of conditional heteroscedasticity whether the conditional variance of the one-step forecast error is constant or not. More precisely, the null hypothesis is \(H_{0}:a_{i}=0,\) \(i=1,...,p\), vs. \(H_{1}:\) there exists \(i=1,...,p\) such that \(a_{i}>0\). The authors propose a new test based on a Cramér - von Mises type statistic and a goodness of fit statistic. The asymptotic consistency of the new test statistic is shown. The work is mathematically demanding, and hard to read for professionals not possessing good working knowledge in measure theory, stochastic processes, etc. Future work providing computational implementations of the proposed test will allow its assessment with real data and/or with Monte Carlo experiences.
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62G10
zbMATH DE Number: 1274854
zbMATH Keywords: nonlinear time series models
zbMATH Keywords: conditional heteroscedasticity
reviewed by: Oscar Bustos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1274854