# Hypothesis Testing in Generalized Linear Models with Functional Coefficient Autoregressive Processes

Authors: Lei Song; Hongchang Hu; Xiaosheng Cheng
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The paper studies the hypothesis testing in generalized linear models with functional coefficient autoregressive (FCA) processes. The quasi-maximum likelihood (QML) estimators are given, which extend those estimators of Hu (2010) and Maller (2003). Asymptotic chi-squares distributions of pseudo likelihood ratio (LR) statistics are investigated.


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## Body

## 1. Introduction

Consider the following generalized linear model:(1.1)yt=g(xtTβ)+εt,t=1,2,…,n,
whereβ is d-dimensional unknown parameter,{εt,t=1,2,…,n} are functional coefficient autoregressive processes given by(1.2)ε1=η1,εt=ft(θ)εt-1+ηt,t=2,3,…,n,
where {ηt,t=1,2,…,n} are independent and identically distributed random variable errors with zero mean and finite variance σ2, θ is a one-dimensional unknown parameter, and ft(θ) is a real valued function defined on a compact set Θ which contains the true value θ0as an inner point and is a subset of R1. The values of θ0and σ2 are unknown. g(·) is a known continuous differentiable function.Model (1.1) includes many special cases, such as an ordinary regression model (whenft(θ)≡0,g(τ)=τ; see [1–7]), an ordinary generalized regression model (when ft(θ)≡0; see [8–13]), a linear regression model with constant coefficient autoregressive processes (when ft(θ)=θ, g(τ)=τ; see [14–16]), time-dependent and function coefficient autoregressive processes (when g(τ)=0; see [17]), constant coefficient autoregressive processes (when ft(θ)=θ, g(τ)=0; see [18–20]), time-dependent or time-varying autoregressive processes (when ft(θ)=at,g(τ)=0; see [21–23]), and a linear regression model with functional coefficient autoregressive processes (wheng(τ)=τ; see [24]). Many authors have discussed some special cases of models (1.1) and (1.2) (see [1–24]). However, few people investigate the model (1.1) with (1.2). This paper studies the model (1.1) with (1.2). The organization of this paper is as follows. In Section 2, some estimators are given by the quasi-maximum likelihood method. In Section 3, the main results are investigated. The proofs of the main results are presented in Section 4, with the conclusions and some open problems in Section 5.


## 2. The Quasi-Maximum Likelihood Estimate

Write the “true” model as(2.1)yt=g(xtTβ0)+et,t=1,2,…,n,(2.2)e1=η1,et=ft(θ0)et-1+ηt,t=2,3,…,n,
where g′(τ)=(dg(τ)/dτ)≠0,ft′(θ)=(dft(θ)/dθ)≠0. Define∏i=0-1ft-i(θ0)=1, and by (2.2), we have
(2.3)et=∑j=0t-1(∏i=0j-1ft-i(θ0))ηt-j.
Thus et is measurable with respect to the σ-field H generated byη1,η2,…,ηt, and(2.4)Eet=0,Var⁡(et)=σ02∑j=0t-1(∏i=0j-1ft-i2(θ0)).Assume at first that theηt are i.i.d. N(0,σ2), we get the log-likelihood of y2,…,yn conditional on y1 given by(2.5)Φn=ln⁡⁡Ln=-(n-1)ln⁡⁡σ22-∑t=2n(εt-ft(θ)εt-1)22σ2-(n-1)ln⁡⁡2π2.
At this stage we drop the normality assumption, but still maximize (2.5) to obtain QML estimators, denoted byσ̂n2,β̂n,θ̂n. The estimating equations for unknown parameters in (2.5) may be written as(2.6)∂Φn∂σ2=-n-12σ2+12σ4∑t=2n(εt-ft(θ)εt-1)2,(2.7)∂Φn∂θ=1σ2∑t=2nft′(θ)(εt-ft(θ)εt-1)εt-1,∂Φn∂βd×1=1σ2∑t=2n(εt-ft(θ)εt-1)⋅(g′(xtTβ)xt-ft(θ)g′(xt-1Tβ)xt-1).
Thus, σ̂n2,β̂n,θ̂n satisfy the following estimation equations
(2.8)σ̂n2=1n-1∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2,(2.9)∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)ft′(θ̂n)ε̂t-1=0,(2.10)∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)(g′(xtTβ̂n)xt-ft(θ̂n)g′(xt-1Tβ̂n)xt-1)=0,
where(2.11)ε̂t=yt-g(xtTβ̂n).Remark 2.1.
Ifg(xtTβ)=xtTβ, then the above equations become the same as Hu’s (see [24]). If ft(θ)=θ,g(xtTβ)=xtTβ, then the above equations become the same as Maller’s (see [15]). Thus we extend those QML estimators of Hu [24] and Maller [15].
For ease of exposition, we will introduce the following notations, which will be used later in the paper. Let(d+1)×1-vector φ=(βT,θ)T. Define(2.12)Sn(φ)=σ2∂Φn∂φ=σ2(∂Φn∂β,∂Φn∂θ),Fn(φ)=-σ2∂2Φn∂φ∂φT.
By (2.7), we have
(2.13)Fn(φ)=(Xn(φ,ω)U*∑t=2n((ft′2(θ)+ft(θ)ft′′(θ))εt-12-ft′′(θ)εtεt-1)),
where the * indicates that the elements are filled in by symmetry,
(2.14)Xn(φ,ω)=-σ2(∂2Φn∂β∂βT),U=∑t=2n(ft′(θ)εt-1g′(xtTβ)xt+ft′(θ)εtg′(xt-1Tβ)xt-1-2ft(θ)ft′(θ)εt-1g′(xt-1Tβ)xt-1),∂2Φn∂β∂βT=-1σ2∑t=2n(g′(xtTβ)xt-ft(θ)g′(xt-1Tβ)xt-1)(g′(xtTβ)xt-ft(θ)g′(xt-1Tβ)xt-1)T+1σ2∑t=2n(εt-ft(θ)εt-1)(g′′(xtTβ)xtxtT-ft(θ)g′′(xt-1Tβ)xt-1xt-1T).
Because {et-1} and {ηt} are mutually independent, we have
(2.15)Dn=E(Fn(φ0))=(Xn(φ0)00∑t=2nft′2(θ0)Eet-12)=(Xn(φ0)00Δ(θ0,σ0)),
where
(2.16)Xn(φ0)=∑t=2n(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1)(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1)T,Δ(θ0,σ0)=∑t=2nft′2(θ0)Eet-12=σ02∑t=2nft′2(θ0)∑j=0t-2(∏i=0j-1ft-i2(θ))=O(n).
By (2.8) (2.7) andEηt=0, we have
(2.17)σ02E(∂Φn∂β|β=β0)=∑t=2nEηt(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1)=0,σ02E(∂Φn∂θ|θ=θ0)=∑t=2nft′(θ0)E(ηtet-1)=0.


## 3. Statement of Main Results

In the section pseudo likelihood ratio (LR) statistics for various hypothesis tests of interest are derived. We consider the following hypothesis:(3.1)H1:g(⋅),f(⋅)are continuous functions, andf′(⋅)≠0,σ02>0.
When the parameter space is restricted by a hypothesis H0j, j=1,2,…,letβ̂jn,θ̂jn,σ̂jn2be the corresponding QML estimators of β,θ,σ2, and let(3.2)L̂jn=-2Φn(β̂jn,θ̂jn,σ̂jn2)
be minus twice the log-likelihood, evaluated at the fitted parameters. Also let
(3.3)L̂n=-2Φn(β̂n,θ̂n,σ̂n2),djn=L̂jn-L̂n
be the “deviance” statistic for testing H0jagainstH1. From (2.5) and (2.8),(3.4)L̂n=(n-1)ln⁡σ̂n2+(n-1)(1+ln⁡2π)
and similarly(3.5)L̂jn=(n-1)ln⁡σ̂jn2+(n-1)(1+ln⁡2π).In order to obtain our results, we give some sufficient conditions as follows.(A1)
Xn=∑t=2nxtxtT is positive definite for sufficiently large n and(3.6)lim⁡n→∞max⁡1≤t≤nxtTXn-1xt=O(n-α),∀α∈(12,1],lim⁡n→∞sup⁡|λ|max⁡(Xn-1/2ZnXn-T/2)<1,
whereZn=(1/2)∑t=2n(xtxt-1T+xt-1xtT) and |λ|max(·)denotes the maximum in absolute value of the eigenvalues of a symmetric matrix.(A2)
There is a constantα>0such that(3.7)∑j=1t(∏i=0j-1ft-i2(θ))≤α,max⁡1≤j≤n|∑t=j+1n(∏i=0t-j-1ft-i(θ0))|≤γ.(A3)
ft′(θ)=dft(θ)/dθ≠0andft′′(θ)=dft′(θ)/dθexist and are bounded, andg(·)is twice continuously differentiable, 0<m≤maxu|g′(u)|≤M<∞, 0<m̃≤maxu|g′′(u)|≤M̃<∞.Theorem 3.1.
Assume (2.1), (2.2) and (A1)–(A3).(1)
SupposeH01:ft(θ)=θ and g(u) is a continuous function,σ02>0 holds. Then
(3.8)d1n→Dχ12,n⟶∞.(2)
SupposeH02:ft(θ)=θ,g(u)=u,σ02>0 holds. Then
(3.9)d2n→Dχ12,n⟶∞.(3)
SupposeH03:ft(θ)=θ,g(u)=eu/(1+eu),σ02>0 holds. Then
(3.10)d3n→Dχ12,n⟶∞.


## 4. Proof of Theorem

To prove Theorem3.1, we first introduce the following lemmas.Lemma 4.1.
Suppose that (A1)–(A3) hold. Then, for allA>0,
(4.1)sup⁡φ∈Nn(A)‖Dn-1/2Fn(φ)Dn-T/2-Φn‖→P0,n→∞,
where
(4.2)Φn=diag⁡(Id,∑t=2nft′2(θ0)et-12Δn(θ0,σ0)),(4.3)Nn(A)={φ∈Rd+1:(φ-φ0)TDn(φ-φ0)≤A2}.Proof.
Similar to proof of Lemma4.1 in Hu [24], here we omit.Lemma 4.2.
Suppose that (A1)–(A3) hold. Thenφ̂n→φ0, σ̂n2→σ02 and
(4.4)Xn(β*,β**,θ̂n)⟶Xn(φ0),
where β*,β** are on the line ofβ0andβ̂n.Proof.
Similar to proof of Theorem3.1 in Hu [24], we easily prove thatφ̂n→φ0, and σ̂n2→σ02. Since (4.4) is easily proved, here we omit the proof (4.4).Proof of Theorem3.1.
Note thatSn(φ̂n)=0andFn(φ̂n) are nonsingular. By Taylor’s expansion, we have
(4.5)0=Sn(φ̂n)=Sn(φ0)-Fn(φ̃n)(φ̂n-φ0),
where φ̃n=aφ̂n+(1-a)φ0 for some 0≤a≤1. Since φ̂n∈Nn(A), also φ̃n∈Nn(A). By (4.1), we have
(4.6)Fn(φ̃n)=Dn1/2(Φn+Ãn)DnT/2.
Thus Ãn is a symmetric matrix withÃn→P0. By (4.5) and (4.6), we have
(4.7)DnT/2(φ̂n-φ0)=DnT/2Fn-1(φ̃n)Sn(φ0)=(Φn+Ãn)-1Dn-1/2Sn(φ0).
LetSn(φ),Fn(φ)denoteSn(β)(φ),Sn(θ)(φ),andFn(β)(φ),Fn(θ)(φ), respectively. By (4.7), we have
(4.8)ΦnDnT/2(β̂n-β0,θ̂n-θ0)=Dn-1/2(Sn(β)(φ0),Sn(θ)(φ0))+oP(1).
Note that
(4.9)ΦnDnT/2=(XnT/2(φ0)00(∑t=2nft′2(θ0)et-12)Δn(θ0,σ0)),Dn-1/2=(Xn-1/2(φ0)001Δn(θ0,σ0)).
By (2.15), (4.2) and (4.8), we get
(4.10)XnT/2(φ0)(β̂n-β0)=Xn-1/2(φ0)Sn(β)(φ0)+oP(1)=Xn-1/2(φ0)∑t=2nηt(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1)+oP(1),(4.11)∑t=2nft′2(θ0)et-12(θ̂n-θ0)=Sn(θ)(φ0)+oP(Δn(θ0,σ0))=∑t=2nft′(θ0)ηtet-1+oP(Δn(θ0,σ0)).
Note that
(4.12)εt=yt-g(xtTβ)=g′(xtTβ*)xtT(β0-β)+et
By (2.1), (2.11) and (4.12), we have
(4.13)ε̂t-ft(θ̂n)ε̂t-1=(g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n)+(et-ft(θ̂n)et-1).
By (4.13) and (2.10), we have
(4.14)∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2=∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)×(β0-β̂n)+(et-ft(θ̂n)et-1))=∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)(g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n)+∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)(et-ft(θ̂n)et-1)=∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)(et-ft(θ̂n)et-1).
By (4.13), we have
(4.15)(g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n)=(ε̂t-ft(θ̂n)ε̂t-1)-(et-ft(θ̂n)et-1).
By (4.15), we have
(4.16)∑t=2n((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))2=∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2+∑t=2n(et-ft(θ̂n)et-1)2-2∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)(et-ft(θ̂n)et-1)=∑t=2n(et-ft(θ̂n)et-1)2-∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2.
By (4.14) and (4.16), we have
(4.17)∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2=∑t=2n(et-ft(θ̂n)et-1)2-∑t=2n((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))2.
By (4.15), we have
(4.18)∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2=∑t=2n(et-ft(θ̂n)et-1)2+∑t=2n((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))2+2∑t=2n(et-ft(θ̂n)et-1)((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n)).
Thus, by (4.17) and (4.18), we have
(4.19)∑t=2n((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))2+∑t=2n(et-ft(θ̂n)et-1)((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))=0.
Since ηt=et-ft(θ0)et-1, we have
(4.20)∑t=2n(et-ft(θ̂n)et-1)2=∑t=2n(ηt+ft(θ0)et-1-ft(θ̂n)et-1)2=∑t=1nηt2+∑t=2n(ft(θ0)-ft(θ̂n))2et-12+2(ft(θ0)-ft(θ̂n))ηtet-1.
Thus, by (4.17), (4.20) and mean value theorem, we have
(4.21)(n-1)σ̂n2=∑t=2n(ε̂t-ft(θ̂n)ε̂t-1)2=∑t=1nηt2+∑t=2n(ft(θ0)-ft(θ̂n))2et-12+2(ft(θ0)-ft(θ̂n))ηtet-1-∑t=2n((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))2=∑t=1nηt2+(θ0-θ̂n)2∑t=2nft′2(θ̃)et-12+2(θ0-θ̂n)∑t=2nft′(θ̃)et-1ηt-∑t=2n((g′(xtTβ*)xtT-ft(θ̂n)g′(xt-1Tβ**)xt-1T)(β0-β̂n))2,
where θ̃=aθ0+(1-a)θ̂n for some 0≤a≤1.
It is easy to know that(4.22)(β̂n-β0)TXn(φ0)(β̂n-β0)=(∑t=2nηtXn-1/2(φ0)(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1))2+op(1).
By Lemma 4.2 and (4.22), we have
(4.23)(n-1)σ̂n2=∑t=1nηt2+(θ0-θ̂n)2∑t=2nft′2(θ̃)et-12+2(θ0-θ̂n)∑t=2nft′(θ̃)et-1ηt-(∑t=2nηtXn-1/2(β*,β**,θ̂n)(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1))2+oP(1)=∑t=1nηt2+(θ0-θ̂n)2∑t=2nft′2(θ̃)et-12+2(θ0-θ̂n)∑t=2nft′(θ̃)et-1ηt-(∑t=2nηtXn-1/2(φ0)(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1))2+oP(1).
Hence, by (4.11), we have
(4.24)θ̂n-θ0=∑t=2nft′(θ0)ηtet-1∑t=2nft′2(θ0)et-12+oP(Δn(θ0,σ0)∑t=2nft′2(θ0)et-12)=∑t=2nft′(θ0)ηtet-1∑t=2nft′2(θ0)et-12+oP(1∑t=2nft′2(θ0)et-12).
By (4.24), we have
(4.25)(θ0-θ̂n)2∑t=2nft′2(θ̃)et-12+2(θ0-θ̂n)∑t=2nft′(θ̃)et-1ηt=(∑t=2nft′(θ0)ηtet-1∑t=2nft′2(θ0)et-12+oP(1∑t=2nft′2(θ0)et-12))2∑t=2nft′2(θ̃)et-12+2(∑t=2nft′(θ0)ηtet-1∑t=2nft′2(θ0)et-12+oP(1∑t=2nft′2(θ0)et-12))∑t=2nft′(θ̃)et-1ηt+oP(1)=(∑t=2nft′(θ0)ηtet-1∑t=2nft′2(θ0)et-12+oP(1∑t=2nft′2(θ0)et-12))2∑t=2n(ft′(θ0)+o(1))2et-12+2(∑t=2nft′(θ0)ηtet-1∑t=2nft′2(θ0)et-12+oP(1∑t=2nft′2(θ0)et-12))⋅∑t=2n(ft′′(θ0)+o(1))et-1ηt+oP(1)=(∑t=2nft′(θ0)ηtet-1)2∑t=2nft′2(θ0)et-12-2(∑t=2nft′(θ0)ηtet-1)2∑t=2nft′2(θ0)et-12+oP(1)=-(∑t=2nft′(θ0)ηtet-1)2∑t=2nft′2(θ0)et-12+oP(1).
By Lemma 4.2, we have
(4.26)(n-1)σ̂n2=∑t=1nηt2-(∑t=2nft′(θ0)ηtet-1)2∑t=2nft′2(θ0)et-12-(∑t=2nηtXn-1/2(β*,β**,θ̂n)(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1))2+oP(1)=∑t=1nηt2-(∑t=2nft′(θ0)ηtet-1)2∑t=2nft′2(θ0)et-12-(∑t=2nηtXn-1/2(φ0)(g′(xtTβ0)xt-ft(θ0)g′(xt-1Tβ0)xt-1))2+oP(1).
Now, we prove (3.8). By (4.12), we have
(4.27)ε̂t(1)=yt-g(xtTβ̂1n)=g′(xtTβ̂1n*)xtT(β0-β̂1n)+et.
Note that
(4.28)εt-ft(θ0)εt-1=(g′(xtTβ*)xtT-ft(θ0)g′(xt-1Tβ**)xt-1T)(β0-β)+ηt.
From (4.28), we have
(4.29)ε̂t(1)-θ̂1nε̂t-1(1)=(g′(xtTβ̂1n*)xtT-θ̂1ng′(xt-1Tβ̂1n**)xt-1T)(β0-β̂1n)+ηt.
By (2.8) and (2.10), we have
(4.30)0=∑t=2n(ε̂t(1)-θ̂1nε̂t-1(1))(g′(xtTβ̂1n*)xt-θ̂1ng′(xt-1Tβ̂1n**)xt-1)=∑t=2n(g′(xtTβ̂1n*)xtT-θ̂1ng′(xt-1Tβ̂1n**)xt-1T)(β0-β̂1n)(g′(xtTβ̂1n*)xt-θ̂1ng′(xt-1Tβ̂1n**)xt-1)+∑t=2nηt(g′(xtTβ̂1n*)xt-θ̂1ng′(xt-1Tβ̂1n**)xt-1)=(β0-β̂1n)TX1n(β̂1n*,β̂1n**,θ̂1n)+∑t=2nηt(g′(xtTβ̂1n*)xt-θ̂1ng′(xt-1Tβ̂1n**)xt-1).
From (4.30), we obtain that
(4.31)β̂1n-β0=X1n-1(β̂1n*,β̂1n**,θ̂1n)∑t=2nηt(g′(xtTβ̂1n)xt-θ̂1ng′(xt-1Tβ̂1n)xt-1).
By (4.29), (4.31) and Lemma 4.2, we have
(4.32)(n-1)σ̂1n2=∑t=2n(ε̂t(1)-θ̂1nε̂t-1(1))2=∑t=1nηt2+(β0-β̂1n)TX1n(β̂1n*,β̂1n**,θ̂1n)(β0-β̂1n)+2(β0-β̂1n)T∑t=2nηt(g′(xtTβ̂1n*)xt-θ̂1ng′(xt-1Tβ̂1n**)xt-1)=∑t=1nηt2-(∑t=2nηtX1n-1/2(β̂1n*,β̂1n**,θ̂1n)(g′(xtTβ̂1n*)xt-θ̂1ng′(xt-1Tβ̂1n**)xt-1))2+op(1)=∑t=1nηt2-(∑t=2nηtX1n-1/2(φ0)(g′(xtTβ0)xt-θ0g′(xt-1Tβ0)xt-1))2+op.
By (3.3)–(3.5), we have
(4.33)d1n=L̂1n-L̂n=(n-1)ln⁡(σ̂1n2σ̂n2)=(n-1)((σ̂1n2σ̂n2)-1)+oP(1).
Under the H01, and by (4.26), (4.32) and (4.33), we have
(4.34)(n-1)(σ̂1n2-σ̂n2)σ̂n2=(∑t=2nηtet-1)2σ̂n2∑t=2net-12+oP(1)=(∑t=2nηtet-1)2σ02∑t=2net-12+oP(1).
It is easily proven that
(4.35)∑t=2nηtet-1σ0∑t=2net-12⟶N(0,1).
Thus, by (4.33)–(4.35), we finish the proof of (3.8).
Next we prove (3.9). UnderH02:ft(θ)=θ, g(u)=u, and yt=xtTβ0+et, we have(4.36)ε̂t(2)=yt-xtTβ̂2n=xtTβ0-xtTβ̂2n+et=xtT(β0-β̂2n)+et.
Hence
(4.37)ε̂t(2)-θ̂2nε̂t-1(2)=xtT(β0-β̂2n)+et-θ̂2n(xt-1T(β0-β̂2n)+et-1)=(xtT-θ̂2nxt-1T)(β0-β̂2n)+ηt.
By (2.8), (2.10), we have
(4.38)0=∑t=2n(ε̂t(2)-θ̂2nε̂t-1(2))(xt-θ̂2nxt-1)=∑t=2n(xtT-θ̂2nxt-1T)(β0-β̂2n)(xt-θ̂2nxt-1)+∑t=2nηt(xt-θ̂2nxt-1).
From (4.38), we obtain,
(4.39)β̂2n-β0=X2n-1(θ̂2n)∑t=2nηt(xt-θ̂2nxt-1).
Thus, by (4.37), (4.39) and Lemma 4.2, we have
(4.40)(n-1)σ̂2n2=∑t=2n(ε̂t(2)-θ̂2nε̂t-1(2))2=∑t=1nηt2+(β0-β̂2n)TX2n(θ̂2n)(β0-β̂2n)+2(β0-β̂2n)T∑t=2nηt(xt-θ̂2nxt-1)=∑t=1nηt2-(∑t=2nηt(xt-θ̂2nxt-1)T)X2n-1(θ̂2n)(∑t=2nηt(xt-θ̂2nxt-1))=∑t=1nηt2-(∑t=2nηtX2n-1/2(θ0)(xt-θ0xt-1))2+op(1).
By (3.3)–(3.5), we have
(4.41)d2n=L̂2n-L̂n=(n-1)ln⁡(σ̂2n2σ̂n2)=(n-1)((σ̂2n2σ̂n2)-1)+oP(1).
Under theH02, by (4.26), (4.40), and (4.41), we obtain
(4.42)(n-1)(σ̂2n2-σ̂n2)σ̂n2=(∑t=2nηtet-1)2σ̂n2∑t=2net-12+oP(1)=(∑t=2nηtet-1)2σ02∑t=2net-12+oP(1).
Thus, by (4.35), (4.42), (3.9) holds.
Finally, we prove (3.10). UnderH03, we have(4.43)ε̂t(3)=yt-extTβ̂3n*1+extTβ̂3n*=extTβ̂3n*(1+extTβ̂3n*)2xtT(β0-β̂3n)+et.
Thus
(4.44)ε̂t(3)-θ̂3nε̂t-1(3)=extTβ̂3n*(1+extTβ̂3n*)2xtT(β0-β̂3n)+et-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1T(β0-β̂3n)-θ̂3net-1=(extTβ̂3n*(1+extTβ̂3n*)2xtT-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1T)(β0-β̂3n)+ηt.
By (2.8) and (2.10), we have
(4.45)0=∑t=2n(ε̂t(3)-θ̂3nε̂t-1(3))(extTβ̂3n*(1+extTβ̂3n*)2xt-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1)=∑t=2n(extTβ̂3n*(1+extTβ̂3n*)2xtT-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1T)(β0-β̂3n)×(extTβ̂3n*(1+extTβ̂3n*)2xt-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1)+∑t=2nηt(extTβ̂3n*(1+extTβ̂3n*)2xt-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1)=(β0-β̂3n)TX3n(β̂3n*,β̂3n**,θ̂3n)+∑t=2nηt(extTβ̂3n*(1+extTβ̂3n*)2xt-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1).
From (4.45), we obtain
(4.46)β̂3n-β0=X3n-1(β̂3n*,β̂3n**,θ̂3n)∑t=2nηt(extTβ̂3n*(1+extTβ̂3n*)2xt-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1).
By (4.44), (4.46) and Lemma 4.2, we have
(4.47)(n-1)σ̂3n2=∑t=2n(ε̂t(3)-θ̂3nε̂t-1(3))2=∑t=1nηt2+(β0-β̂3n)TX3n(β̂3n*,β̂3n**,θ̂3n)(β0-β̂3n)+2(β0-β̂3n)T∑t=2nηt(extTβ̂3n*(1+extTβ̂3n*)2xt-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1)=∑t=1nηt2-((extTβ̂3n*(1+extTβ̂3n*)2xtT-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1T)∑t=2nηtX3n-1/2(β̂3n*,β̂3n**,θ̂3n)×(extTβ̂3n*(1+extTβ̂3n*)2xtT-θ̂3next-1Tβ̂3n**(1+ext-1Tβ̂3n**)2xt-1T))2=∑t=1nηt2-(∑t=2nηtX3n-1/2(φ0)(extTβ0(1+extTβ0)2xt-θ0ext-1Tβ0(1+ext-1Tβ0)2xt-1))2+op(1).
By (3.3)–(3.5), we know that
(4.48)d3n=L̂3n-L̂n=(n-1)ln⁡(σ̂3n2σ̂n2)=(n-1)((σ̂3n2σ̂n2)-1)+oP(1).
Under the H03, by (4.26), (4.47) and (4.48), we have
(4.49)(n-1)(σ̂3n2-σ̂n2)σ̂n2=(∑t=2nηtet-1)2σ02∑t=2net-12+oP(1).
Thus, (3.10) follows from (4.48), (4.49), and (4.35). Therefore, we complete the proof of Theorem 3.1.


## 5. Conclusions and Open Problems

In the paper, we consider the generalized linear mode with FCA processes, which includes many special cases, such as an ordinary regression model, an ordinary generalized regression model, a linear regression model with constant coefficient autoregressive processes, time-dependent and function coefficient autoregressive processes, constant coefficient autoregressive processes, time-dependent or time-varying autoregressive processes, and a linear regression model with functional coefficient autoregressive processes. And then we obtain the QML estimators for some unknown parameters in the generalized linear mode model and extend some estimators. At last, we use pseudo LR method to investigate three hypothesis tests of interest and obtain the asymptotic chi-squares distributions of statistics.However, several lines of future work remain open.(1) It is well known that a conventional time series can be regarded as the solution to a differential equation of integer order with the excitation of white noise in mathematics, and a fractal time series can be regarded as the solution to a differential equation of fractional order with a white noise in the domain of stochastic processes (see [25]). In the paper, {εt}is a conventional nonlinear time series. We may investigate some hypothesis tests by pseudo LR method when the{εt}is a fractal time series (the idea is given by an anonymous reviewer). In particular, we assume that(5.1)∑i=0pap-iDviεt=ηt,
where vp,vp-1,…,v0 is strictly decreasing sequence of nonnegative numbers, ai is a constant sequence, and Dv is the Riemann-Liouville integral operator of order v>0 given by(5.2)Dvh(t)=1Γ(v)∫0t(t-u)v-1h(u)du,
where Γ is the Gamma function, and h(t) is a piecewise continuous on(0,∞) and integrable on any finite subinterval of [0,∞)(See [25, 26]). Fractal time series may have a heavy-tailed probability distribution function and has been applied various fields of sciences and technologies (see [25, 27–32]). Thus it is very significant to investigate various regression models with fractal time series errors, including regression model (1.1) with (5.1).(2) We maybe investigate the others hypothesis tests, for example:H04:
ft(θ)=0,g(u)=u,σ02>0;H05:
ft(θ)=θ,g(u)=0,σ02>0;H06:
ft(θ)=0,g(u)=eu/(1+eu),σ02>0;H07:
ft(θ)=at and g(u) is a continuous function,σ02>0;H08:
ft(θ)=at,g(u)=u, σ02>0;H09:
ft(θ)=at,g(u)=eu/(1+eu), σ02>0.



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*Source: 862398-2012-05-23.xml*