# Statistical Portfolio Estimation under the Utility Function Depending on Exogenous Variables

Authors: Kenta Hamada; Dong Wei Ye; Masanobu Taniguchi
Journal: Advances in Decision Sciences
 (2012)

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## Abstract

In the estimation of portfolios, it is natural to assume that the utility function depends on exogenous variable. From this point of view, in this paper, we develop the estimation under the utility function depending on exogenous variable. To estimate the optimal portfolio, we introduce a function of moments of the return process and cumulant between the return processes and exogenous variable, where the function means a generalized version of portfolio weight function. First, assuming that exogenous variable is a random process, we derive the asymptotic distribution of the sample version of portfolio weight function. Then, an influence of exogenous variable on the return process is illuminated when exogenous variable has a shot noise in the frequency domain. Second, assuming that exogenous variable is nonstochastic, we derive the asymptotic distribution of the sample version of portfolio weight function. Then, an influence of exogenous variable on the return process is illuminated when exogenous variable has a harmonic trend. We also evaluate the influence of exogenous variable on the return process numerically.


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## Body

## 1. Introduction

In the usual theory of portfolio analysis, optimal portfolios are determined by the meanμ and the variance Σ of the portfolio return X={X(t)}. Several authors proposed estimators of optimal portfolios as functions of the sample mean μ̂ and the sample variance Σ̂ for independent returns of assets. However, empirical studies show that financial return processes are often dependent and non-Gaussian. Shiraishi and Taniguchi [1] showed that the above estimators are not asymptotically efficient generally if the returns are dependent. Under the non-Gaussianity, if we consider a general utility function U(·), the expected utility should depend on higher-order moments of the return. From this point of view, Shiraishi and Taniguchi [1] proposed the portfolios including higher-order moments of the return.However, empirical studies show that the utility function often depends on exogenous variableZ={Z(t)}. From this point of view, in this paper, we develop the estimation under the utility function depending on exogenous variable. Denote the optimal portfolio estimator by a function ĝ=g(θ̂)=g(𝔼̂(X),cov̂(X,X),cov̂(X,Z),cum̂(X,X,Z)) where hat (·̂) means the sample version of (·). Although Shiraishi and Taniguchi’s [1] setting does not include the exogenous variable Z(t) in ĝ, we can develop the asymptotic theory in the light of their work.First, assuming thatZ={Z(t)} is a random process, we derive the asymptotic distribution of ĝ. Then, an influence of Z on the return process is illuminated when Z has a shot noise in the frequency domain. Second, assuming that Z is a nonrandom sequence of variables which satisfy Grenander’s conditions, we also derive the asymptotic distribution of ĝ. Then an influence of Z on X is evaluated when Z is a sequence of harmonic functions. Numerical studies will be given, and they show some interesting features.The paper is organized as follows. Section2 introduces the optimal portfolio of the form ĝ and provides the asymptotic distribution of ĝ. Assuming that Z is a stochastic process, we derive the asymptotics of ĝ when Z has a shot noise in the frequency domain. The influence of Z on X is numerically evaluated in Section 2.2. Assuming that Z is a nonrandom sequence satisfying Grenander’s conditions, we derive the asymptotic distribution of ĝ. Section 3 provides numerical studies for the influence of Z on X when Z is a sequence of harmonic functions. The appendix gives the proofs of all the theorems.


## 2. Optimal Portfolio with the Exogenous Variables

Suppose the existence of a finite number of assets indexed byi,(i=1,…,p). Let X(t)=(X1(t),…,Xp(t))′ denote the random returns on p assets at time t, and let Z(t)=(Z1(t),…,Zq(t))′ denote the exogenous variables influencing on the utility function at time t. We write Y(t)=(X(t)′,Z(t)′)′=(X1(t),…,Xp(t),Z1(t),…,Zq(t))′.Since it is empirically observed that{X(t)} is non-Gaussian and dependent, we will assume that it is a non-Gaussian stationary process with the 3rd-order cumulants. Also, suppose that there exists a risk-free asset whose return is denoted by Y0(t). Let α0 and α=(α1,…,αp︷p,0,…,0︷q)′ be the portfolio weights at time t, and the portfolio is M(t)=Y(t)′α+Y0(t)α0 whose higher-order cumulants are written as(2.1)c1M(t)=cum{M(t)}=ca1αa1+Y0(t)α0,c2M(t)=cum{M(t),M(t)}=ca2a3αa2αa3,c3M(t)=cum{M(t),M(t),M(t)}=ca4a5a6αa4αa5αa6.We use Einstein’s summation convention here and throughout the paper. For a utility functionU(·), the expected utility can be approximated as(2.2)E[U(M(t))]≈U(c1M(t))+12!D2U(c1M(t))c2M(t)+13!D3U(c1M(t))c3M(t),
by Taylor expansion of order 3. The approximate optimal portfolio may be described as(2.3)max⁡α0,α{therighthandsideof(2.2)},subjecttoα0+∑i=1pαi=1.Solving (2.4), Shiraishi and Taniguchi [1] introduced the optimal portfolio depending on the mean, variance, and the third-order cumulants, and then derived the asymptotic distribution of a sample version estimator. Although our problem is different from that of Shiraishi and Taniguchi [1], we develop the discussion with the methods inspired by them.Introduce a portfolio estimator function based on observed higher-order cumulants,(2.4)ĝ≡g(θ̂)=g(Ê(X),cov⁡̂(X,X),cov⁡̂(X,Z),cum̂{X,X,Z}),
and assume that the function g(·) is p-dimensional and measurable, that is,(2.5)g(θ)∶g(E(X),cov⁡(X,X),cov⁡(X,Z),cum{X,X,Z})⟶Rp.Let the random process{Y(t)=(Y1(t),…,Yp+q(t))′} be a (p+q)-vector linear process generated by(2.6)Y(t)=∑j=0∞G(j)ε(t-j)+μ,
where {ε(t)} is a (p+q)-dimensional stationary process such that 𝔼{ε(t)}=0 and 𝔼{ε(s)ε(t)′}=δ(s,t)K, with K a nonsingular (p+q)×(p+q)-matrix, G(j)’s are (p+q)×(p+q)-matrices, and μ=(μ1,…,μp+q) is the mean vector of {Y(t)}. All the components ofY,ε,G,μ are real. Assuming that {ε(t)} has all order cumulants, let Qa1,…,aje(t1,…,tj-1) be the joint jth-order cumulant of εa1(t),εa2(t+t1),…,εaj(t+tj-1). In what follows we assume that, for each j=1,2,3,…,(2.7)∑t1,…,tj=-∞∞∑a1,…,aj=1p+q|Qa1,…,aje(t1,…,tj-1)|<∞,∑t=0∞‖G(t)‖<∞.LettingQa1,…,ajY(t1,…,tj-1) be the joint jth-order cumulant of Ya1(t),Ya2(t+t1),…,Yaj(t+tj-1), we define the jth-order cumulant spectral density by(2.8)fa1,…,aj(λ1,…,λj-1)=(12π)j-1∑t1,…,tj-1=-∞∞exp⁡{-i(λ1t1+⋯+λj-1tj-1)}×Qa1,…,ajY(t1,…,tj-1),which is expressed as(2.9)fa1,…,aj(λ1,…,λj-1)=∑b1,…,bj=1p+qka1b1(λ1+⋯+λj-1)ka2b2(-λ1)⋯kaj-1bj-1(-λj-2)kajbj(-λj-1)Q̃a1,…,aje(λ1,…,λj-1),
where Q̃a1,…,aje is the jth-order cumulant spectral density of εa1(t),…,εaj(t), kab(λ)=∑l=0∞Gab(l)eiλl and Gab(l) is the (a,b)th element of G(l). We introduce the following quantities:(2.10)ĉa1=1n∑s=1nYa1(s),ĉa2a3=1n∑s=1n(Ya2(s)-ĉa2)(Ya3(s)-ĉa3),ĉa4a5=1n∑s=1n(Ya4(s)-ĉa4)(Ya5(s)-ĉa5),ĉa6a7a8=1n∑s=1n(Ya6(s)-ĉa6)(Ya7(s)-ĉa7)(Ya8(s)-ĉa8),
where 1≤a1,a2,a3,a4,a6,a7≤pandp+1≤a5,a8≤p+q. Write the quantities that appeared in (2.4) by(2.11)θ̂=(ĉa1,ĉa2a3,ĉa4a5,ĉa6a7a8),θ=(ca1,ca2a3,ca4a5,ca6a7a8),
where ca1,…,aj≡Qa1,…,ajY(0,…,0). Then dimθ=dimθ̂=a+b+c+d, where a=p, b=p(p+1)/2, c=pq, d=p(p+1)q/2.First, we derive the asymptotics of the fundamental quantityθ̂.Theorem 2.1.
Under the assumptions,(2.12)n(θ̂-θ)⟶DN(0,Ω),(n⟶∞),
where
(2.13)Ω=(Ω11Ω12Ω13Ω14Ω21Ω22Ω23Ω24Ω31Ω32Ω33Ω34Ω41Ω42Ω43Ω44),
and the typical element of Ωij corresponding to the covariance between ĉΔ and ĉ∇ is denoted by V{(Δ)(∇)}, and
(2.14)V{(a1)(a1′)}=Ω11=2πfa1a1′(0),V{(a2a3)(a1′)}=Ω12=2π∫-ππfa2a3a1′(λ,0)dλ(=Ω21),V{(a4a5)(a1′)}=Ω13=2π∫-ππfa4a5a1′(λ,0)dλ(=Ω31),V{(a6a7a8)(a1′)}=Ω14=2π∬-ππfa6a7a8a1′(λ1,λ2,0)dλ1dλ2(=Ω41),V{(a2a3)(a2′a3′)}=Ω22=2π∬-ππfa2a3a2′a3′(λ1,λ2,-λ2)dλ1dλ2+2π∫-ππ{fa2a2′(λ)fa3a3′(-λ)+fa2a3′(λ)fa3a2′(-λ)}dλ,V{(a2a3)(a4′a5′)}=Ω23=2π∬-ππfa2a3a4′a5′(λ1,λ2,-λ2)dλ1dλ2+2π∫-ππ{fa2a4′(λ)fa3a5′(-λ)+fa2a5′(λ)fa3a4′(-λ)}dλ(=Ω32),V{(a2a3)(a6′a7′a8′)}=Ω24=2π∭-ππfa2a3a6′a7′a8′(λ1,λ2,λ3,-λ3)dλ1dλ2dλ3(=Ω42),V{(a4a5)(a4′a5′)}=Ω33=2π∬-ππfa4a5a4′a5′(λ1,λ2,-λ2)dλ1dλ2+2π∫-ππ{fa4a4′(λ)fa5a5′(-λ)+fa4a5′(λ)fa5a4′(-λ)}dλ,V{(a2a3)(a6′a7′a8′)}=Ω34=2π∭-ππfa4a5a6′a7′a8′(λ1,λ2,λ3,-λ3)dλ1dλ2dλ3(=Ω43),V{(a6a7a8)(a6′a7′a8′)}=Ω44=2π∫∭-ππfa6a7a8a6′a7′a8′(λ1,λ2,λ3,-λ3-λ4)dλ1⋯dλ4+2π∭-ππ∑ν1fai1ai2ai3ai4(λ1,λ2,λ3)fai5ai6×(-λ2-λ3)dλ1dλ2dλ3+2π∭-ππ∑ν2fai1ai2ai3(λ1,λ2)fai4ai5ai6×(λ3,-λ2-λ3)dλ1dλ2dλ3+2π∭-ππ∑ν3fai1ai2(λ1)fai3ai4(λ2)fai5ai6(-λ1-λ2)dλ1dλ2.In what follows we place all the proofs of theorems in the appendix.Next we discuss the estimation of portfoliog(θ). For this we assume that the portfolio function g(θ) is continuously differentiable. Henceforth, we use a unified estimator g(θ̂) for g(θ). The δ-method and Slutsky’s lemma imply the following.Theorem 2.2.
Under the assumptions(2.15)n(g(θ̂)-g(θ))⟶DN(0,(Dg)Ω(Dg)′),(n⟶∞),
where, Dg={∂igj;i=1,…,dimθ,j=1,…,p+q}.The quantitiesĉa6a7a8’s are the 3rd-order cumulants of the process, which show the non-Gaussianity. For the returns of five financial stocks IBM, Ford, Merck, HP, and EXXON, we calculated the standardized 3rd-order cumulants ŝ=ĉa6a7a8/(v̂2)3/2 where v̂2 is the sample variance of the stock. Table 1 below shows their values.Table 1
Standardized 3rd-order cumulants of the returns of five stocks from 2005/11/08 to 2011/11/08.
IBMFordMerckHPEXXONŝ0.056082.508790.795219.847770.34485From Table1 we observe that the five returns are non-Gaussian. In view of Theorem 2.1, it is possible to construct the (1-α) confidence interval for c=ca6a7a8 in the following form:(2.16)[ĉ-zαnΩ̂441/2,ĉ+zαnΩ̂441/2],
where zα is the upper level-α point of N(0,1) and Ω̂44 is a consistent estimator of Ω44 calculated by the method of Keenan [2] and Taniguchi [3].


### 2.1. Influence of Exogenous Variable

In this subsection we investigate an influence of the exogenous variablesZ(t) on the asymptotics of the portfolio estimator g(θ̂).Assume that the exogenous variables have “shot noise” in the frequency domain, that is,(2.17)Zaj(λ)=δ(λaj-λ),
where δ(·) is the Dirac delta function with period 2π, and λaj≠0, hence Zaj(λ) has one peak at λ+λaj≡0(mod2π).Theorem 2.3.
For (2.17), denote Ωij and V{(Δ)(∇)} in Theorem 2.1 by Ωij′ and V′{(Δ)(∇)}, respectively. That is, Ωij′ and V′{(Δ)(∇)} represent the asymptotic variance when the exogenous variables are shot noise. Then,
(2.18)V′{(a4a5)(a1′)}=Ω13′=0(=Ω31′),V′{(a6a7a8)(a1′)}=Ω14′=2πfa6a7a1′(λa8,0)(=Ω41′),V′{(a2a3)(a4′a5′)}=Ω23′=2πfa2a3a4′(λa5′,0)+2πfa2a4′(λa5)fa3a5′(-λa5′)+2πfa2a5′(-λa5′)fa3a4′(λa5)(=Ω32′),V′{(a2a3)(a6′a7′a8′)}=Ω24′=2π∬-ππfa2a3a6′a7′a8′(-λa8′,λ1,λ2,-λ2)dλ1dλ2(=Ω42′),V′{(a4a5)(a4′a5′)}=Ω33′=2πfa4a5a4′a5′(λa5′,λa5,-λa5)+2πfa4a4′(λa5)fa5a5′(-λa5)+2πfa4a5′(λa5)fa5a4′(-λa5),V′{(a4a5)(a1′a1′)}=Ω34′=2π∫-ππfa4a5a6′a7′a8′(-λa8′,λa5,λ,-λ)dλ(=Ω43′).


### 2.2. Numerical Studies for Stochastic Exogenous Variables

This subsection provides some numerical examples which show the influence ofZ(t) on Ωij.Example 2.4.
For a risk-free assetX0(t) and risky asset X(t), we consider construction of optimal portfolios αX(t)+α0X0(t). Here {X(t)} is the return process of the risky asset, which is generated by
(2.19)X(t)=θX(t-1)+ε(t)+μ1,
where 𝔼{ε(t)}=0, Var{ε(t)}=σ2. We assume that X0(t)=μ, and that the exogenous variable in the frequency domain is given by Z(λ)=δ(λ). Write,
(2.20)Y(t)=(X(t)X0(t)Z(t)′),
then
(2.21)Ω13′=Ω31′=V′{(a4a5)(a1′)}=0,Ω23′=Ω32′=V′{(a2a3)(a4′a5′)}=0,Ω33′=V{(a4a5)(a4′a5′)}=cum(a1,a1′)cum(a3,a3′)=σ21(1-θeiλa3),
which are covariances between Z(t) and X(t), and show an influence Z(t) on X(t). From Figure 1, it is seen that as θ tends to 1, and λa3 tends to 0, then Ω33′ increases. If θ tends to −1 and λa3 tends to -π, π, then Ω33′ also increases, which entails that the exogenous variables have big influence on the asymptotics of estimators when θ is close to the unit root of AR(2.2).Figure 1
Values ofΩ33′ for θ=-0.9(0.018)0.9, λa3=-π(0.06)π.Remark 2.5.
Ω13′ is robust for the shot noise in Z(t) at λ=λa3.


## 2.1. Influence of Exogenous Variable

In this subsection we investigate an influence of the exogenous variablesZ(t) on the asymptotics of the portfolio estimator g(θ̂).Assume that the exogenous variables have “shot noise” in the frequency domain, that is,(2.17)Zaj(λ)=δ(λaj-λ),
where δ(·) is the Dirac delta function with period 2π, and λaj≠0, hence Zaj(λ) has one peak at λ+λaj≡0(mod2π).Theorem 2.3.
For (2.17), denote Ωij and V{(Δ)(∇)} in Theorem 2.1 by Ωij′ and V′{(Δ)(∇)}, respectively. That is, Ωij′ and V′{(Δ)(∇)} represent the asymptotic variance when the exogenous variables are shot noise. Then,
(2.18)V′{(a4a5)(a1′)}=Ω13′=0(=Ω31′),V′{(a6a7a8)(a1′)}=Ω14′=2πfa6a7a1′(λa8,0)(=Ω41′),V′{(a2a3)(a4′a5′)}=Ω23′=2πfa2a3a4′(λa5′,0)+2πfa2a4′(λa5)fa3a5′(-λa5′)+2πfa2a5′(-λa5′)fa3a4′(λa5)(=Ω32′),V′{(a2a3)(a6′a7′a8′)}=Ω24′=2π∬-ππfa2a3a6′a7′a8′(-λa8′,λ1,λ2,-λ2)dλ1dλ2(=Ω42′),V′{(a4a5)(a4′a5′)}=Ω33′=2πfa4a5a4′a5′(λa5′,λa5,-λa5)+2πfa4a4′(λa5)fa5a5′(-λa5)+2πfa4a5′(λa5)fa5a4′(-λa5),V′{(a4a5)(a1′a1′)}=Ω34′=2π∫-ππfa4a5a6′a7′a8′(-λa8′,λa5,λ,-λ)dλ(=Ω43′).


## 2.2. Numerical Studies for Stochastic Exogenous Variables

This subsection provides some numerical examples which show the influence ofZ(t) on Ωij.Example 2.4.
For a risk-free assetX0(t) and risky asset X(t), we consider construction of optimal portfolios αX(t)+α0X0(t). Here {X(t)} is the return process of the risky asset, which is generated by
(2.19)X(t)=θX(t-1)+ε(t)+μ1,
where 𝔼{ε(t)}=0, Var{ε(t)}=σ2. We assume that X0(t)=μ, and that the exogenous variable in the frequency domain is given by Z(λ)=δ(λ). Write,
(2.20)Y(t)=(X(t)X0(t)Z(t)′),
then
(2.21)Ω13′=Ω31′=V′{(a4a5)(a1′)}=0,Ω23′=Ω32′=V′{(a2a3)(a4′a5′)}=0,Ω33′=V{(a4a5)(a4′a5′)}=cum(a1,a1′)cum(a3,a3′)=σ21(1-θeiλa3),
which are covariances between Z(t) and X(t), and show an influence Z(t) on X(t). From Figure 1, it is seen that as θ tends to 1, and λa3 tends to 0, then Ω33′ increases. If θ tends to −1 and λa3 tends to -π, π, then Ω33′ also increases, which entails that the exogenous variables have big influence on the asymptotics of estimators when θ is close to the unit root of AR(2.2).Figure 1
Values ofΩ33′ for θ=-0.9(0.018)0.9, λa3=-π(0.06)π.Remark 2.5.
Ω13′ is robust for the shot noise in Z(t) at λ=λa3.


## 3. Portfolio Estimation for Nonstochastic Exogenous Variables

So far we assumed that the sequence of exogenous variables{Z(t)} is a random stochastic process. In this section, assuming that {Z(t)} is a nonrandom sequence, we will propose a portfolio estimator, and elucidate the asymptotics. We introduce the following quantities,(3.1)Âj,k=∑t=1nXj(t)Zk(t)n∑t=1nZk2(t),B̂j,m,k=∑t=1nXj(t)Xm(t)Zk(t)n∑t=1nZk2(t).We assume thatZ(t)’s satisfy Grenander’s conditions (G1)–(G4) with(3.2)aj,k(n)(h)=∑t=1n-hZj(t+h)Zk(t).(G1)
limn→∞aj,j(n)(0)=∞,(j=1,…,q).(G2)
limn→∞Zj(n+1)2/aj,j(n)(0)=0,(j=1,…,q).(G3)
aj,k(n)(h)/aj,j(n)(h)ak,k(n)(h)=ρj,k(h)+o(1/n) for j,k=1,…,q,h∈Z.(G4)
the(p,p)-matrix Φ(0)≡{ρj,k(0):j,k=1,…,q} is regular.Under Grenander’s conditions, there exists a Hermitian matrix functionM(λ)={Mj,k(λ):j,k=1,…,q} with positive semidefinite increments such that(3.3)Φ(h)=∫-ππeihλdM(λ).M(λ) is the regression spectral measure of {Z(t)}. Next we discuss the asymptotics for sample versions of cov(X,Z) and cov{XX,Z}. For this we need the following assumption, There exists constant b>0 such that(3.4)det⁡{fX(λ)}≥b,
where fX(λ) is the spectral density matrix of {X(t)}.Theorem 3.1.
Under Grenander’s conditions and the assumption(3.5)n{Âj,k-Aj,k}⟶DN(0,Ωj,k),
where the (j′,k′)-th element of Ωj,k is given by
(3.6)V(j,k:j′,k′)=2π∫-ππfjj′(λ)dMk,k′(λ).Theorem 3.2.
Under Grenander’s conditions and the assumption(3.7)n{B̂j,m,k-Bj,m,k}⟶DN(0,Ωj,m,k),
where Ωj,m,k={V(j,m,k:j′,m′,k′)} with
(3.8)V(j,m,k:j′,m′,k′)=2π∫-ππ[∫-ππ{fjm′(λ-λ1)fmj′(λ1)+fjj′(λ-λ1)fmm′(λ1)}dλ1+∬-ππfjmj′m′(λ1,λ2-λ,λ2)dλ1λ2]dMk,k′(λ).


### 3.1. Numerical Studies for Nonstochastic Exogenous Variables

Letting{X(t)} and {Z(t)} be scalar processes, we investigate an influence of non-stochastic process {Z(t)} on {X(t)}. The figures below show influence of harmonic trends {Z(t)} on V(j,m,k:j′,m′,k′) in Ωj,m,k. In these cases V(j,m,k:j′,m′,k′) measures the amount of covariance between XX and Z.Example 3.3.
Let the return process{X(t)} and the exogenous process {Z(t)} be generated by
(3.9)X(t)=ε(t)-ηε(t-1),Z(t)=cos⁡(μt)+cos⁡(0.25μt),
where ε(t)’s are i. i. d. N(0,1) variables. Next, suppose that {Z(t)} consists of harmonic trends with period μ and the quarter period. We plotted the graph of V(j,m,k:j′,m′,k′) in Figure 2.Example 3.4.
Assume that{X(t)} and {Z(t)} are generated by
(3.10)X(t)-ηX(t-1)=ε(t),Z(t)=cos⁡(μt)+cos⁡(0.25μt).
We observe that there exist two peaks in Figure 3. If μ≈0 and |η|≈1, V(j,m,k:j′,m′,k′) increases rapidly. For further study it may be noted that Cheng et al. [4] discussed statistical estimation of generalized multiparameter likelihood models. Although these models are for independent samples, there is some possibility to apply them to our portfolio problem.Figure 2
Values ofV(j,m,k:j′,m′,k′) in Theorem 3.2 for η=-0.9(0.1)0.9, μ=-3.14(0.33)3.14.Figure 3
Values ofV(j,m,k:j′,m′,k′) for η=-0.9(0.1)0.9, μ=-3.14(0.33)3.14.


## 3.1. Numerical Studies for Nonstochastic Exogenous Variables

Letting{X(t)} and {Z(t)} be scalar processes, we investigate an influence of non-stochastic process {Z(t)} on {X(t)}. The figures below show influence of harmonic trends {Z(t)} on V(j,m,k:j′,m′,k′) in Ωj,m,k. In these cases V(j,m,k:j′,m′,k′) measures the amount of covariance between XX and Z.Example 3.3.
Let the return process{X(t)} and the exogenous process {Z(t)} be generated by
(3.9)X(t)=ε(t)-ηε(t-1),Z(t)=cos⁡(μt)+cos⁡(0.25μt),
where ε(t)’s are i. i. d. N(0,1) variables. Next, suppose that {Z(t)} consists of harmonic trends with period μ and the quarter period. We plotted the graph of V(j,m,k:j′,m′,k′) in Figure 2.Example 3.4.
Assume that{X(t)} and {Z(t)} are generated by
(3.10)X(t)-ηX(t-1)=ε(t),Z(t)=cos⁡(μt)+cos⁡(0.25μt).
We observe that there exist two peaks in Figure 3. If μ≈0 and |η|≈1, V(j,m,k:j′,m′,k′) increases rapidly. For further study it may be noted that Cheng et al. [4] discussed statistical estimation of generalized multiparameter likelihood models. Although these models are for independent samples, there is some possibility to apply them to our portfolio problem.Figure 2
Values ofV(j,m,k:j′,m′,k′) in Theorem 3.2 for η=-0.9(0.1)0.9, μ=-3.14(0.33)3.14.Figure 3
Values ofV(j,m,k:j′,m′,k′) for η=-0.9(0.1)0.9, μ=-3.14(0.33)3.14.



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