Maximum Likelihood Estimate Of β-ARCH Model Under Restriction

of the paperIn this paper , we study a stationary B-ARCH model and proposed a method of computing the maximum likelihood estimator(MLE)of parameters under the nonnegative restriction . Namely , considr the model as follows :where for a known positive number 5 and a positive number M; is independent of {Xs,s < t) for each t; the restriction of the parameters is :let a be the real value of the parameters .In the second section of this paper , we discuss the strong consistency of 2 .THEORY Assume that the series Xt satisfies model (1) , EXt4 < + , , thenUse the same method ,we can get the same conclusion of a* which is under the restriction,namelyIn the third section of this paper , abstract expression of S* was given ,The lemma and theory are as follows :LEMMA Assume that the series Xt satisfies model (1) , and when THEORY Assume that the series Xt satisfies model (1) , , then Use the same method ,we can get the same conclusion of a* which is under the restriction,namelyIn section four , we assume that {Xt} follows model(1) , and , and cosider the following test problemWe can get some main conclusions as folows :THEORY1 Assume for the empty set and H1 is true for model(2.1) . Then for almost w , there exists an n0(w) such that for each for some J , where J depends on nand w .THEORY2 If the assumptions in Theorem 1 hold , then from the large sample point of view if and olny if satisfies the following conditions :ii). For unless iii). If and for some J' ,then According to the above two theories and the definition of AaJ , when n is large enough , we haveNext , we get the limiting behaviour of a* under H0 : THEORY Under H0 , we haveFor the hypothesis testing problem , likelihood ratio test statistics isand thelogarthm of likelihood ratio test statistics isLet T = -2 In A , then from the theory below we have the limiting property of T :THEORY Under H0 , for t > 0In the last section , there is the simulation of the parameters a1 and a2 Forthemore , a B-ARCH(1,2) model is made for the Dollar/Pound exchange of Jan 1985~ Dec 1994.