The Least Square Estimate Of ARCH Model Under Ordered Restriction

Autoregressive Conditional Heteroskedasticity (ARCH) model is very important in the study of nonlinear time series. In fact, the wave of stock price sequence is one focus of the money market and heteroskedasticity is impersonally existing phenomenon. In order to study the phemomenon actually, this model was first mentioned by Engle(1982) and developed by Engle and Kraft(1983). In this paper, we discuss a special case of ARCH model, namely ARCH(0, q).If a time series {Xt} satisfywhere and is independent with . The {Xt} is said to be ARCH(p, q).In this paper, we only discuss the stationary and ergodic ARCH(0, q) time series, namely {Xt} satisfyingwhere , , and t is independent with {Xs, s < t}. Let denote the o-field generated by {Xt-1,Xt-2...}.Prom thatwe study the functionwhereThen we will study the properties of the Least Square Estimate that minimizes Ln(a) in 0.Note a is the LSE of parameter a. a can be solved by = 0. And we can compute the LSE through the Quadratic Programming method.Matlab, Mathematica, Maple are all suitly solve the minimum.Firstly, we study the strong consistency of LSE d of the parameter a. Assume is the unknown true parameter, we should proov Lemma 1.1 Assuem that the time series {Xt} satisfies (1.1),EX4t < +, then we can use the parameter space , where M is a fixed large positive number.Lemma 1.2 Assume the time series {Xt} satisfies (1.1),EX4t < +, then for is the the unique minimum of ELn(a).Lemma 1.3 Assume the time series {Xt} satisfies (1.1),EX4t < + ,thenTheorem 1.1 Assume the time series {Xt} satisfies (1.1), EXf < +, a 0, thenTheorem 1.2 If the conditions of Theorem 1.1, thenwhen n , where the inner point set of .Theorem 1.3 Assume the time series {Xt} satisfies (1.1), EXf < +, a 10, then I (a) >0.Lemma 1.4 Assume the time senes {Xt} satisfies (1.1),a 02, and EX8t < +, thenFollowing , we study the asymptotic normality of a.Theorem 1.4 Assume the time series {Xt} satisfies (1.1),a 01, then n(a - Next, we consider the testing problem with order of ARCH(0,q). In many eco-nomical and financial studies, we expect that the effect of recent data is greater than that of remote data. This can be expressed asnamely parameters are restricted by a simple order.Suppose the time series {Xt} follows (1.1),a 0, consider the following testwhere Write the LSE of a in the restricted set C The set C can be rewrited aswhere Assume J is the subset of If write aJ as the minimum of Ln(a) in AJ, we can find the relationship of a* anda, aJ.Theorem 1.5 For (1.1), if and H\ hold, then from the large sample of view, there exists a J(J depends on w and n), we have a* = aJ. Namely, for almost Prom the large sample of view, we suppose : if , then Based on this supposition, we have the following theorem.Theorem 1.6 the conditions of Theorem 1.5 hold, then a {1,2, ... ,q -1} if and only if aJ satisfy the following conditions: Let satisfy i),ii),iii),}, when n is large enough,Thus, we obtain the exact expression of a*.Under H0, rewrite C0:Subsequently, we derive a representation of aJ by score function under H0. DefineBased on the above supposition, we obtainLemma 1.5 Under Ho, for each J {1,2, ... , q - 1},whereis the LSE of and ami is No. mi part of a. Andwhere denotes the limited distributions of nXn and Yn are same"From Lemma 1.5, for each JFurthermore,Let is the project matrix, and it is continuous. Sowhere Theorem 1.6 Under Ho,where k = q + 1 - rank(PJ).Lei AUJ is the set which composed with Uj satisfying 1,2,3?LetTheorem 1.7 Under H0,Theorem 1.8 Under H0, for t > 0,The results that we obtain for ARCH can be easily extended to B-ARCH with the similar score function , because ARCH is a special case of B-ARCH. And we omit the proofs.