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

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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.