YAN Hui,HU Hong-chang

(School of Mathematics and Statistics,Hubei Normal University,Huangshi 435002,China)

Abstract:In this paper,we study the hypothesis testing for the homogeneity of the Markov chain of the errors in linear models.By using the quasi-maximum likelihood estimates(QMLEs)of some unknown parameter and the methods of martingale-difference,the limiting distribution for likelihood ratio test statistics is obtained.

Keywords:linear model;Markov chain;homogeneity;hypothesis testing;martingale

1 Introduction

The theory and application of linear models with Markov type dependent errors recently attracted increasing research attention.In the case that the errors form a homogeneous Markov chain,one can see Maller[1],Pere[2],Fuller[3]and form a non-homogeneous Markov chain,see Azrak and Mélard[4],Carsoule and Franses[5],Dahlhaus[6],Kwoun and Yajima[7].It is well-known that compared with a homogeneous Markov chain,the limit behavior of a non-homogeneous Markov chain is much more complicated to handle.To simplify the models,we consider the hypothesis testing for the homogeneity of the process of errors in the following linear model

where xt∈ Rdare deterministic regressor vectors,β is a d-dimensional unknown parameter,and{εt}is a Markov chain with recursive formula as follows

where θ∈ R is an unknown parameter,φt(θ)is a real valued function on a compact set Θ which contains the true value θ0as an inner point,and the ηtare i.i.d.mean zero random variables(rvs)with finite variance σ2(also to be estimated).

It is obvious that the errors{εt}is a non-homogeneous Markov chain when the coefficient φt(θ)depends on t.This paper discusses the hypothesis testing for the homogeneity of Markov chain{εt}based on the quasi-maximum likelihood estimates(QMLEs)of the unknown parameters.Limiting distribution for likelihood ratio test statistics of hypotheses is obtained by the techniques of martingale-difference.

2 Preliminaries and Statement of Result

The log-likelihood of y2,y3,···,ynconditional on y1is defined by[1]

We maximize(2.1)to obtain QML estimators denoted by?βn,?θnand?σ2n(when they exsit).Then the corresponding estimators,satisfy[1]

Write the“true”model as

By(2.5)

We need the following conditions

(A2)There is a constant α>0 such that

for any t∈ {1,2,···,n}and θ∈ Θ.

Remark 2.1 Condition(A1)is often imposed in order to obtain the existence of the estimators in some linear models with Markov type errors,see e.g.Muller[1],Hu[8],Xu and Hu[9].

And[8,9]used condition(A2),Kwound and Yajima[7]used the first condition in(A2).Silvapulle[10],Tong et.al.[11]used the condition similar to(A3),when they discussed the asymptotic properties of the estimators in some linear and partial linear models.

Define(d+1)-vectorG=(β,θ),and

where

From eq.(5.29)in Hu[8],we have

where

In this paper,we consider the hypothesis

where the function ρ(θ)<1,θ∈ Θ and ρ(θ0) ≠0,ρ′(θ)is bounded on Θ.

The main result in this paper is the following theorem.

Theorem2.1 Assume(A1)–(A3).Suppose H0:φt(θ)= ρ(θ)holds.Then as n → ∞,whereis chi-square rv with m degrees of freedom.

3 Lemmas

Lemma 3.1Assume(A1)–(A3).Thenand,the QML estimators of β,θ and σ2in model(1.1)–(1.2)exist.And as n → ∞,

Proof See Theorem 3.1 and Theorem 3.2 in Hu[8].

Lemma 3.2 Assume(A2)and(A3).Then

Proof

From Lemma 4.1 in Xu and Hu[9],we haveThen

By recursive method,

Similarly,

Therefore,from(3.2),(3.4)and(3.5),

where c0is the bound of

4 Proof of Theorem

Using(2.2),(2.8)and(2.4),

By(2.8),

Then,from(2.4),

By(5.23)and(5.24)in Hu[8],as n→∞,|T2|=op(1),|T3|=op(1).Thus

From(2.7),

Using(4.1),(4.3),(4.4)and Taylor expansion,

Thus,

Now we give an approximation for.In fact,from eq.(5.28)in Hu[8],

Φn,Dn,Snsame as in(2.14)and(2.11).Then

which means

In view of Lemma 3.2,the law of large numbers holds for the sequenceNote thatthen

From(4.11)and(2.14),

Thus,from(4.8),(4.13),(4.14)and Lemma 3.2,straightforward calculus yields

We now finish the proof of Theorem 2.1.From(2.4)and(2.8),

By(2.2),(2.8)and(4.16),

In view of eq.(4.14)and Lemma 3.1,Lemma 3.2,

and

Then to prove that

we need only to show that

then,to obtain(4.21),it will suffice to verify the Lindeberg condition for the sequence

In fact,since

?n(θ0,σ0)=O(n)as n → ∞,then for every ε>0,we have

Now,we obtain(4.21)due to the central limit theorem for martingale difference array(Theorem 8.1 in Pollard[12]).Then we prove(4.18)from(4.20).