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Estimation and testing for the parameters of ARCH(q) under ordered restriction

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  • Dehui Wang
  • Lixin Song
  • Ningzhong Shi

Abstract

. In this paper, we study a stationary ARCH(q) model with parameters α0,α1,α2,…,αq. It is known that the model requires all parameters αi to be non‐negative, but sometimes the usual algorithm based on Newton–Raphson's method leads us to obtain some negative solutions. So this study proposes a method of computing the maximum likelihood estimator (MLE) of parameters under the non‐negative restriction. A similar method is also proposed for the case where the parameters are restricted by a simple order: α1≥α2≥⋯≥αp. The strong consistency of the above two estimators is discussed. Furthermore, we consider the problem of testing homogeneity of parameters against the simple order restriction. We give the likelihood ratio (LR) test statistic for the testing problem and derive its asymptotic null distribution.

Suggested Citation

  • Dehui Wang & Lixin Song & Ningzhong Shi, 2004. "Estimation and testing for the parameters of ARCH(q) under ordered restriction," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 483-499, July.
    • Handle: RePEc:bla:jtsera:v:25:y:2004:i:4:p:483-499
    DOI: 10.1111/j.1467-9892.2004.01763.x
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    References listed on IDEAS

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    1. Lee, John H H & King, Maxwell L, 1994. "Correction [A Locally Most Mean Powerful Based Score Test for ARCH and GARCH Regression Disturbances]," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(1), pages 139-139, January.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    3. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(1), pages 107-131, April.
    4. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    1. Aknouche, Abdelhakim & Bentarzi, Mohamed, 2025. "Efficient two-stage estimation of cyclical ARCH models," MPRA Paper 127417, University Library of Munich, Germany.
    2. Fukang Zhu & Dehui Wang, 2011. "Estimation and testing for a Poisson autoregressive model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 73(2), pages 211-230, March.

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