Statistical Portfolio Estimation under the Utility Function Depending on Exogenous Variables

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.

1. Introduction

In the usual theory of portfolio analysis, optimal portfolios are determined by the mean 𝜇 and the variance Σ of the portfolio return 𝑋={𝑋(𝑡)}. Several authors proposed estimators of optimal portfolios as functions of the sample mean 𝜇 and the sample variance Σ 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 variable 𝑍={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 𝑔=𝑔(𝜃)=𝑔(𝔼(𝑋),cov(𝑋,𝑋),cov(𝑋,𝑍),cum(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 𝑔, we can develop the asymptotic theory in the light of their work.

First, assuming that 𝑍={𝑍(𝑡)} is a random process, we derive the asymptotic distribution of 𝑔. 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 𝑔. 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. Section 2 introduces the optimal portfolio of the form 𝑔 and provides the asymptotic distribution of 𝑔. Assuming that 𝑍 is a stochastic process, we derive the asymptotics of 𝑔 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 𝑔. 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 by 𝑖,(𝑖=1,…,𝑝). Let 𝑋(𝑡)=(𝑋₁(𝑡),…,𝑋_𝑝(𝑡))^′ denote the random returns on 𝑝 assets at time 𝑡, and let 𝑍(𝑡)=(𝑍₁(𝑡),…,𝑍_𝑞(𝑡))^′ denote the exogenous variables influencing on the utility function at time 𝑡. We write 𝑌(𝑡)=(𝑋(𝑡)^′,𝑍(𝑡)^′)^′=(𝑋₁(𝑡),…,𝑋_𝑝(𝑡),𝑍₁(𝑡),…,𝑍_𝑞(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 3^rd-order cumulants. Also, suppose that there exists a risk-free asset whose return is denoted by 𝑌₀(𝑡). Let 𝛼₀ and 𝛼=(𝛼₁,…,^{𝑝0𝑥0𝑓𝑒37}𝛼_𝑝,0,…,^{𝑞0𝑥0𝑓𝑒37}0)^′ 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 as𝑐^𝑀₁(𝑡)=cum{𝑀(𝑡)}=𝑐^𝑎1𝛼_𝑎1+𝑌₀(𝑡)𝛼₀,𝑐^𝑀₂(𝑡)=cum{𝑀(𝑡),𝑀(𝑡)}=c^𝑎2𝑎3𝛼_𝑎2𝛼_𝑎3,𝑐^𝑀₃(𝑡)=cum{𝑀(𝑡),𝑀(𝑡),𝑀(𝑡)}=𝑐^{𝑎4𝑎5𝑎6}𝛼_𝑎4𝛼_𝑎5𝛼_𝑎6. (2.1)

We use Einstein’s summation convention here and throughout the paper. For a utility function 𝑈(⋅), the expected utility can be approximated as𝔼[𝑈(𝑀(𝑡))]≈𝑈𝑐^𝑀₁(𝑡)+1𝐷2!²𝑈𝑐^𝑀₁(𝑡)𝑐^𝑀₂(𝑡)+1𝐷3!³𝑈𝑐^𝑀₁(𝑡)𝑐^𝑀₃(𝑡), (2.2) by Taylor expansion of order 3. The approximate optimal portfolio may be described asmax_𝛼0,𝛼therighthandsideof(2.2),subjectto𝛼₀+^𝑝_𝑖=1𝛼_𝑖=1. (2.3)

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,𝑔≡𝑔𝜃=𝑔𝔼(𝑋),cov(𝑋,𝑋),cov(𝑋,𝑍),cum{𝑋,𝑋,𝑍}, (2.4) and assume that the function 𝑔(⋅) is 𝑝-dimensional and measurable, that is,𝑔(𝜃)∶𝑔(𝔼(𝑋),cov(𝑋,𝑋),cov(𝑋,𝑍),cum{𝑋,𝑋,𝑍})⟶𝐑^𝑝. (2.5)

Let the random process {𝐘(𝑡)=(𝑌₁(𝑡),…,𝑌_𝑝+𝑞(𝑡))^′} be a (𝑝+𝑞)-vector linear process generated by𝐘(𝑡)=^∞_𝑗=0𝐺𝑗𝜀𝑡−𝑗+𝝁, (2.6) 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,…,𝜇_𝑝+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 𝑄^𝑒_{𝑎1,…,𝑎𝑗}(𝑡₁,…,𝑡_𝑗−1) be the joint 𝑗th-order cumulant of 𝜀_𝑎1(𝑡),𝜀_𝑎2(𝑡+𝑡₁),…,𝜀_𝑎𝑗(𝑡+𝑡_𝑗−1). In what follows we assume that, for each 𝑗=1,2,3,…,^∞_{𝑡1,…,𝑡𝑗=−∞}^𝑝+𝑞_{𝑎1,…,𝑎𝑗=1}𝑄^𝑒_{𝑎1,…,𝑎𝑗}𝑡₁,…,𝑡_𝑗−1<∞,_∞𝑡∑₌₀‖𝐺(𝑡)‖<∞. (2.7)

Letting 𝑄^𝑌_{𝑎1,…,𝑎𝑗}(𝑡₁,…,𝑡_𝑗−1) be the joint 𝑗th-order cumulant of 𝑌_𝑎1(𝑡),𝑌_𝑎2(𝑡+𝑡₁),…,𝑌_𝑎𝑗(𝑡+𝑡_𝑗−1), we define the 𝑗th-order cumulant spectral density by𝑓_{𝑎1,…,𝑎𝑗}𝜆₁,…,𝜆_𝑗−1=12𝜋^∞𝑗−1_{𝑡1,…,𝑡𝑗−1=−∞}exp−𝑖𝜆₁𝑡₁+⋯+𝜆_𝑗−1𝑡_𝑗−1×𝑄^𝑌_{𝑎1,…,𝑎𝑗}𝑡₁,…,𝑡_𝑗−1, (2.8)

which is expressed as𝑓_{𝑎1,…,𝑎𝑗}𝜆₁,…,𝜆_𝑗−1=^𝑝+𝑞_{𝑏1,…,𝑏𝑗=1}𝑘_𝑎1𝑏1𝜆₁+⋯+𝜆_𝑗−1𝑘_𝑎2𝑏2−𝜆₁⋯𝑘_{𝑎𝑗−1𝑏𝑗−1}−𝜆_𝑗−2𝑘_{𝑎𝑗𝑏𝑗}−𝜆_𝑗−1𝑄^𝑒_{𝑎1,…,𝑎𝑗}𝜆₁,…,𝜆_𝑗−1, (2.9) where 𝑄^𝑒_{𝑎1,…,𝑎j} is the 𝑗th-order cumulant spectral density of 𝜀_𝑎1(𝑡),…,𝜀_𝑎𝑗(𝑡), 𝑘_𝑎𝑏(𝜆)=∑^∞_𝑙=0𝐺_𝑎𝑏(𝑙)𝑒^𝑖𝜆𝑙 and 𝐺_𝑎𝑏(𝑙) is the (𝑎,𝑏)th element of 𝐺(𝑙). We introduce the following quantities:𝑐̂^𝑎1=1𝑛^𝑛_𝑠=1𝑌_𝑎1(𝑠),𝑐̂^𝑎2𝑎3=1𝑛^𝑛_𝑠=1𝑌_𝑎2(𝑠)−𝑐̂^𝑎2𝑌_𝑎3(𝑠)−𝑐̂^𝑎3,𝑐̂^𝑎4𝑎5=1𝑛^𝑛_𝑠=1𝑌_𝑎4(𝑠)−𝑐̂^𝑎4𝑌_𝑎5(𝑠)−𝑐̂^𝑎5,𝑐̂^{𝑎6𝑎7𝑎8}=1𝑛^𝑛_𝑠=1𝑌_𝑎6(𝑠)−𝑐̂^𝑎6𝑌_𝑎7(𝑠)−𝑐̂^𝑎7𝑌_𝑎8(𝑠)−𝑐̂^𝑎8, (2.10) where 1≤𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₆,𝑎₇≤𝑝and𝑝+1≤𝑎₅,𝑎₈≤𝑝+𝑞. Write the quantities that appeared in (2.4) by𝜃=𝑐̂^𝑎1,𝑐̂^𝑎2𝑎3,𝑐̂^𝑎4𝑎5,𝑐̂^{𝑎6𝑎7𝑎8},𝜃=𝑐^𝑎1,𝑐^𝑎2𝑎3,𝑐^𝑎4𝑎5,𝑐^{𝑎6𝑎7𝑎8}, (2.11) where 𝑐^{𝑎1,…,𝑎𝑗}≡𝑄^𝑌_{𝑎1,…,𝑎𝑗}(0,…,0). Then dim𝜃=dim𝜃=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 quantity 𝜃.

Theorem 2.1. Under the assumptions, √𝑛𝜃−𝜃⟶^𝒟𝑁(𝟎,Ω),(𝑛⟶∞), (2.12) where ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₄Ω₂₁Ω₂₂Ω₂₃Ω₂₄Ω₃₁Ω₃₂Ω₃₃Ω₃₄Ω₄₁Ω₄₂Ω₄₃Ω₄₄, (2.13) and the typical element of Ω_𝑖𝑗 corresponding to the covariance between 𝑐^Δ̂ and 𝑐^∇̂ is denoted by 𝑉{(Δ)(∇)}, and 𝑉𝑎₁𝑎₁′=Ω₁₁=2𝜋𝑓_{𝑎1𝑎1′}(0),𝑉𝑎₂𝑎₃𝑎₁′=Ω₁₂=2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎3𝑎1′}(𝜆,0)𝑑𝜆=Ω₂₁,𝑉𝑎₄𝑎₅𝑎₁′=Ω₁₃=2𝜋^𝜋_−𝜋𝑓_{𝑎4𝑎5𝑎1′}(𝜆,0)𝑑𝜆=Ω₃₁,𝑉𝑎₆𝑎₇𝑎₈𝑎₁′=Ω₁₄=2𝜋^𝜋_−𝜋𝑓_{𝑎6𝑎7𝑎8𝑎1′}𝜆₁,𝜆₂,0𝑑𝜆₁𝑑𝜆₂=Ω₄₁,𝑉𝑎₂𝑎₃𝑎₂′𝑎₃′=Ω₂₂=2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎3𝑎2′𝑎3′}𝜆₁,𝜆₂,−𝜆₂𝑑𝜆₁𝑑𝜆₂+2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎2′}(𝜆)𝑓_{𝑎3𝑎3′}(−𝜆)+𝑓_{𝑎2𝑎3′}(𝜆)𝑓_{𝑎3𝑎2′}(𝑉𝑎−𝜆)𝑑𝜆,₂𝑎₃𝑎₄′𝑎₅′=Ω₂₃=2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎3𝑎4′𝑎5′}𝜆₁,𝜆₂,−𝜆₂𝑑𝜆₁𝑑𝜆₂+2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎4′}(𝜆)𝑓_{𝑎3𝑎5′}(−𝜆)+𝑓_{𝑎2𝑎5′}(𝜆)𝑓_{𝑎3𝑎4′}(−𝜆)𝑑𝜆=Ω₃₂,𝑉𝑎₂𝑎₃𝑎₆′𝑎₇′𝑎₈′=Ω₂₄=2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎3𝑎6′𝑎7′𝑎8′}𝜆₁,𝜆₂,𝜆₃,−𝜆₃𝑑𝜆₁𝑑𝜆₂𝑑𝜆₃=Ω₄₂,𝑉𝑎₄𝑎₅𝑎₄′𝑎₅′=Ω₃₃=2𝜋^𝜋_−𝜋𝑓_{𝑎4𝑎5𝑎4′𝑎5′}𝜆₁,𝜆₂,−𝜆₂𝑑𝜆₁𝑑𝜆₂+2𝜋^𝜋_−𝜋𝑓_{𝑎4𝑎4′}(𝜆)𝑓_{𝑎5𝑎5′}(−𝜆)+𝑓_{𝑎4𝑎5′}(𝜆)𝑓_{𝑎5𝑎4′}(𝑉𝑎−𝜆)𝑑𝜆,₂𝑎₃𝑎₆′𝑎₇′𝑎₈′=Ω₃₄=2𝜋^𝜋_−𝜋𝑓_{𝑎4𝑎5𝑎6′𝑎7′𝑎8′}𝜆₁,𝜆₂,𝜆₃,−𝜆₃𝑑𝜆₁𝑑𝜆₂𝑑𝜆₃=Ω₄₃,𝑉𝑎₆𝑎₇𝑎₈𝑎₆′𝑎₇′𝑎₈′=Ω₄₄=2𝜋^𝜋_−𝜋𝑓_{𝑎6𝑎7𝑎8𝑎6′𝑎7′𝑎8′}𝜆₁,𝜆₂,𝜆₃,−𝜆₃−𝜆₄𝑑𝜆₁⋯𝑑𝜆₄+2𝜋^𝜋_−𝜋𝜈1𝑓_{𝑎𝑖1𝑎𝑖2𝑎𝑖3𝑎𝑖4}𝜆₁,𝜆₂,𝜆₃𝑓_{𝑎𝑖5𝑎𝑖6}×−𝜆₂−𝜆₃𝑑𝜆₁𝑑𝜆₂𝑑𝜆₃+2𝜋^𝜋_−𝜋𝜈2𝑓_{𝑎𝑖1𝑎𝑖2𝑎𝑖3}𝜆₁,𝜆₂𝑓_{𝑎𝑖4𝑎𝑖5𝑎𝑖6}×𝜆₃,−𝜆₂−𝜆₃𝑑𝜆₁𝑑𝜆₂𝑑𝜆₃+2𝜋^𝜋_−𝜋𝜈3𝑓_{𝑎𝑖1𝑎𝑖2}𝜆₁𝑓_{𝑎𝑖3𝑎𝑖4}𝜆₂𝑓_{𝑎𝑖5𝑎𝑖6}−𝜆₁−𝜆₂𝑑𝜆₁𝑑𝜆₂. (2.14)

In what follows we place all the proofs of theorems in the appendix.

Next we discuss the estimation of portfolio 𝑔(𝜃). For this we assume that the portfolio function 𝑔(𝜃) is continuously differentiable. Henceforth, we use a unified estimator 𝑔(𝜃) for 𝑔(𝜃). The 𝛿-method and Slutsky’s lemma imply the following.

Theorem 2.2. Under the assumptions √𝑛𝑔𝜃−𝑔(𝜃)⟶^𝒟𝑁𝟎,𝐷𝑔Ω𝐷𝑔^′,(𝑛⟶∞), (2.15) where, 𝐷𝑔={𝜕_𝑖𝑔^𝑗;𝑖=1,…,dim𝜃,𝑗=1,…,𝑝+𝑞}.

The quantities 𝑐̂^{𝑎6𝑎7𝑎8}’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=𝑐̂^{𝑎6𝑎7𝑎8}/(𝑣²)^3/2 where 𝑣² 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.

From Table 1 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=𝑐^{𝑎6𝑎7𝑎8} in the following form:𝑧𝑐̂−^𝛼√𝑛Ω^1/2₄₄𝑧,𝑐̂+^𝛼√𝑛Ω^1/2₄₄, (2.16) where 𝑧_𝛼 is the upper level-𝛼 point of 𝑁(0,1) and Ω₄₄ 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 variables 𝑍(𝑡) on the asymptotics of the portfolio estimator 𝑔(𝜃).

Assume that the exogenous variables have “shot noise” in the frequency domain, that is,𝑍_𝑎𝑗(𝜆)=𝛿𝜆_𝑎𝑗−𝜆, (2.17) where 𝛿(⋅) is the Dirac delta function with period 2𝜋, and 𝜆_𝑎𝑗≠0, hence 𝑍_𝑎𝑗(𝜆) has one peak at 𝜆+𝜆_𝑎𝑗≡0(mod2𝜋).

Theorem 2.3. For (2.17), denote Ω_𝑖𝑗 and 𝑉{(Δ)(∇)} in Theorem 2.1 by Ω_𝑖𝑗′ and 𝑉′{(Δ)(∇)}, respectively. That is, Ω_𝑖𝑗′ and 𝑉′{(Δ)(∇)} represent the asymptotic variance when the exogenous variables are shot noise. Then, 𝑉^′𝑎₄𝑎₅𝑎₁^′=Ω₁₃′=0=Ω₃₁′,𝑉^′𝑎₆𝑎₇𝑎₈𝑎₁^′=Ω₁₄′=2𝜋𝑓_{𝑎6𝑎7𝑎1′}𝜆_𝑎8,0=Ω₄₁′,𝑉^′𝑎₂𝑎₃𝑎₄^′𝑎₅^′=Ω₂₃′=2𝜋𝑓_{𝑎2𝑎3𝑎4′}𝜆_𝑎5′,0+2𝜋𝑓_{𝑎2𝑎4′}𝜆_𝑎5𝑓_{𝑎3𝑎5′}−𝜆_𝑎5′+2𝜋𝑓_{𝑎2𝑎5′}−𝜆_𝑎5′𝑓_{𝑎3𝑎4′}𝜆_𝑎5=Ω^′₃₂,𝑉^′𝑎₂𝑎₃𝑎₆^′𝑎₇^′𝑎₈^′=Ω₂₄′=2𝜋^𝜋_−𝜋𝑓_{𝑎2𝑎3𝑎6′𝑎7′𝑎8′}−𝜆_𝑎8′,𝜆₁,𝜆₂,−𝜆₂𝑑𝜆₁𝑑𝜆₂=Ω₄₂′,𝑉^′𝑎₄𝑎₅𝑎₄^′𝑎₅^′=Ω₃₃′=2𝜋𝑓_{𝑎4𝑎5𝑎4′𝑎5′}𝜆_𝑎5′,𝜆_𝑎5,−𝜆_𝑎5+2𝜋𝑓_{𝑎4𝑎4′}𝜆_𝑎5𝑓_{𝑎5𝑎5′}−𝜆_𝑎5+2𝜋𝑓_{𝑎4𝑎5′}𝜆_𝑎5𝑓_{𝑎5𝑎4′}−𝜆_𝑎5,𝑉^′𝑎₄𝑎₅𝑎₁^′𝑎₁^′=Ω₃₄′=2𝜋^𝜋_−𝜋𝑓_{𝑎4𝑎5𝑎6′𝑎7′𝑎8′}−𝜆_𝑎8′,𝜆_𝑎5,𝜆,−𝜆𝑑𝜆=Ω^′₄₃. (2.18)

2.2. Numerical Studies for Stochastic Exogenous Variables

This subsection provides some numerical examples which show the influence of 𝑍(𝑡) on Ω_𝑖𝑗.

Example 2.4. For a risk-free asset 𝑋₀(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 𝑋(𝑡)=𝜃𝑋(𝑡−1)+𝜀(𝑡)+𝜇₁, (2.19) 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, 𝑌(𝑡)=𝑋(𝑡)𝑋₀(𝑡)𝑍(𝑡)^′, (2.20) then Ω₁₃′=Ω₃₁′=𝑉^′𝑎₄𝑎₅𝑎^′₁Ω=0,₂₃′=Ω₃₂′=𝑉^′𝑎₂𝑎₃𝑎^′₄𝑎^′₅Ω=0,₃₃′=𝑉𝑎₄𝑎₅𝑎^′₄𝑎^′₅=cum𝑎₁,𝑎^′₁cum𝑎₃,𝑎^′₃=𝜎²11−𝜃𝑒^{𝑖𝜆𝑎3}, (2.21) 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 𝜆_𝑎3 tends to 0, then Ω₃₃′ increases. If 𝜃 tends to −1 and 𝜆_𝑎3 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 of Ω33′ 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=𝜆_𝑎3.

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,𝐴_𝑗,𝑘=∑^𝑛_𝑡=1𝑋_𝑗(𝑡)𝑍_𝑘(𝑡)𝑛∑^𝑛_𝑡=1𝑍²_𝑘(𝑡),𝐵_{𝑗,𝑚,𝑘}=∑^𝑛_𝑡=1𝑋_𝑗(𝑡)𝑋_𝑚(𝑡)𝑍_𝑘(𝑡)𝑛∑^𝑛_𝑡=1𝑍²_𝑘(𝑡). (3.1)

We assume that 𝑍(𝑡)’s satisfy Grenander’s conditions (G1)–(G4) with𝑎^(𝑛)_𝑗,𝑘(ℎ)=^𝑛−ℎ_𝑡=1𝑍_𝑗(𝑡+ℎ)𝑍_𝑘(𝑡). (3.2)(G1)lim_𝑛→∞𝑎^(𝑛)_𝑗,𝑗(0)=∞,(𝑗=1,…,𝑞).(G2)lim_𝑛→∞𝑍_𝑗(𝑛+1)²/𝑎^(𝑛)_𝑗,𝑗(0)=0,(𝑗=1,…,𝑞).(G3)𝑎^(𝑛)_𝑗,𝑘(ℎ)/𝑎^(𝑛)_𝑗,𝑗(ℎ)𝑎^(𝑛)_𝑘,𝑘(ℎ)=𝜌_𝑗,𝑘(ℎ)+𝑜(1/√𝑛) for 𝑗,𝑘=1,…,𝑞,ℎ∈𝐙.(G4) the (𝑝,𝑝)-matrix Φ(0)≡{𝜌_𝑗,𝑘(0)∶𝑗,𝑘=1,…,𝑞} is regular.

Under Grenander’s conditions, there exists a Hermitian matrix function 𝐌(𝜆)={𝑀_𝑗,𝑘(𝜆)∶𝑗,𝑘=1,…,𝑞} with positive semidefinite increments such thatΦ(ℎ)=^𝜋_−𝜋𝑒^𝑖ℎ𝜆𝑑𝐌(𝜆). (3.3)

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 thatdet𝑓_𝑋(𝜆)≥𝑏, (3.4) where 𝑓_𝑋(𝜆) is the spectral density matrix of {𝑋(𝑡)}.

Theorem 3.1. Under Grenander’s conditions and the assumption √𝑛𝐴_𝑗,𝑘−𝐴⟶_𝑗,𝑘^𝒟𝑁0,Ω_𝑗,𝑘, (3.5) where the (𝑗^′,𝑘^′)-th element of Ω_𝑗,𝑘 is given by 𝑉𝑗,𝑘∶𝑗^′,𝑘^′=2𝜋^𝜋_−𝜋𝑓_𝑗𝑗′(𝜆)𝑑𝑀_𝑘,𝑘′(𝜆). (3.6)

Theorem 3.2. Under Grenander’s conditions and the assumption √𝑛𝐵_{𝑗,𝑚,𝑘}−𝐵⟶_{𝑗,𝑚,𝑘}^𝒟𝑁0,Ω_{𝑗,𝑚,𝑘}, (3.7) where Ω_{𝑗,𝑚,𝑘}={𝑉(𝑗,𝑚,𝑘∶𝑗^′,𝑚^′,𝑘^′)} with 𝑉𝑗,𝑚,𝑘∶𝑗^′,𝑚^′,𝑘^′=2𝜋^𝜋_−𝜋^𝜋_−𝜋𝑓_𝑗𝑚′𝜆−𝜆₁𝑓_𝑚𝑗′𝜆₁+𝑓_𝑗𝑗′𝜆−𝜆₁𝑓_𝑚𝑚′𝜆₁𝑑𝜆₁+^𝜋_−𝜋𝑓_{𝑗𝑚𝑗′𝑚′}𝜆₁,𝜆₂−𝜆,𝜆₂𝑑𝜆₁𝜆₂𝑑𝑀_𝑘,𝑘′(𝜆). (3.8)

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 Ω_{𝑗,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 𝑋(𝑡)(𝑡)(),𝑍(𝑡),=𝜀−𝜂𝜀t−1=cos𝜇𝑡+cos0.25𝜇𝑡 (3.9) 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 𝑋(𝑡)()(𝑡),𝑍(𝑡).−𝜂𝑋𝑡−1=𝜀=cos𝜇𝑡+cos0.25𝜇𝑡 (3.10) 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 of 𝑉(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 of 𝑉(𝑗,𝑚,𝑘∶𝑗′,𝑚′,𝑘′) for 𝜂=−0.9(0.1)0.9, 𝜇=−3.14(0.33)3.14.

Appendix

This section provides the proofs of theorems.

Proof of Theorem 2.1. Our setting includes the exogenous variables. Although Shiraishi and Taniguchi’s [1] setting does not include them, we can prove the theorem in line with Shiraishi and Taniguchi [1].
Let 𝑐̃^𝑎2𝑎3=1𝑛^𝑛_𝑠=1𝑌_𝑎2−𝜇_𝑎2𝑌_𝑎3−𝜇_𝑎3,𝑐̃^{𝑎6𝑎7𝑎8}=1𝑛^𝑛_𝑠=1𝑌_𝑎6−𝜇_𝑎6𝑌_𝑎7−𝜇_𝑎7𝑌_𝑎8−𝜇_𝑎8. (A.1) From Fuller [5], it is easy to see that 𝑐̂^𝑎6−𝜇_𝑎6𝑐̂^𝑎7−𝜇_𝑎7𝑐̂^𝑎8−𝜇_𝑎8=𝑜_𝑝1√𝑛,𝑐̃^𝑎6𝑎7𝑐̂^𝑎8−𝜇_𝑎8=𝑐^𝑎6𝑎7𝑐̂^𝑎8−𝜇_𝑎8+𝑜_𝑝1√𝑛, (A.2) where 1≤𝑎₆,𝑎₇≤𝑝,𝑝+1≤𝑎₈+𝑝+𝑞. (A.3) Then we can see that 𝑐̂^{𝑎6𝑎7𝑎8}=𝑐̃^{𝑎6𝑎7𝑎8}−^8∗_𝑘=6𝑐^{𝑎𝑗𝑘𝑎𝑖𝑘}𝑐̂^𝑎𝑘−𝜇_𝑎𝑘+𝑜_𝑝1√𝑛, (A.4) where ∑^8∗_𝑘=6 is the sum over 𝑘=6,7,8 with 𝑖_𝑘 and 𝑗_𝑘∈{6,7,8} satisfying 𝑖_𝑘<𝑗_𝑘;𝑘≠𝑖_𝑘,𝑗_𝑘.
Hence it follows that 𝑛Cov𝑐̂^{𝑎6𝑎7𝑎8}−𝑐^{𝑎6𝑎7𝑎8}𝑐̂^{𝑎6′𝑎7′𝑎8′}−𝑐^{𝑎6′𝑎7′𝑎8′}=𝑛cum𝑐̂^{𝑎6𝑎7𝑎8}−𝑐̂^{𝑎6′𝑎7′𝑎8′}−𝑛^8∗_𝑘=6cum𝑐^{𝑎𝑖𝑘𝑎𝑗𝑘}𝑐̂^𝑎𝑘−𝜇_𝑎𝑘,𝑐̃^{𝑎6′𝑎7′𝑎8′}−𝑛^8∗_𝑘′=6cum𝑐̃^{𝑎6𝑎7𝑎8},𝑐^{𝑎𝑖𝑘′𝑎𝑗𝑘′}𝑐̂^𝑎𝑘′−𝜇_𝑎𝑘′+𝑛^8∗_𝑘=6^8∗_𝑘′=6cum𝑐^{𝑎𝑖𝑘𝑎𝑗𝑘}𝑐^𝑎𝑘−𝜇_𝑎𝑘𝑐^{𝑎𝑖𝑘′a𝑗𝑘′}𝑐^𝑎𝑘′−𝜇_𝑎𝑘′+𝑜(1). (A.5) In what follows we assume that (𝑎_𝑖1,…,𝑎_𝑖6) is an arbitrary permutation of (𝑎₆,𝑎₇,𝑎₈,𝑎_6′,𝑎_7′,𝑎_8′), 1≤𝑎₆,𝑎₇,𝑎_6′,𝑎_7′≤𝑝,𝑝+1≤𝑎₈,𝑎_8′≤𝑝+𝑞. (A.6) Then we get ⟶^∞_𝑡=−∞𝑄^𝑌_{𝑎6𝑎7𝑎8𝑎6′𝑎7′𝑎8′}(0,0,𝑡,𝑡,𝑡)+_𝜈1𝑄^𝑌_{𝑎𝑖1𝑎𝑖2𝑎𝑖3𝑎𝑖4}(0,𝑡,𝑡)𝑄^𝑌_{𝑎𝑖5𝑎𝑖6}(𝑡)+_𝜈2𝑄^𝑌_{𝑎𝑖1𝑎𝑖2𝑎𝑖3}(0,𝑡)𝑄^𝑌_{𝑎𝑖4𝑎𝑖5𝑎𝑖6}(𝑡,𝑡)+_𝜈3𝑄^𝑌_{𝑎𝑖1𝑎𝑖2}(𝑡)𝑄^𝑌_{𝑎𝑖5𝑎𝑖4}(𝑡)+𝑄^𝑌_{𝑎𝑖5𝑎𝑖6}(𝑡). (A.7) By use of Fourier transform, we see that (A1)=2𝜋^𝜋_−𝜋𝑓_{𝑎6𝑎7𝑎8𝑎6′𝑎7′𝑎8′}𝜆₁,𝜆₂,𝜆₃,𝜆₄,−𝜆₃,−𝜆₄𝑑𝜆₁⋯𝑑𝜆₄+2𝜋^𝜋_−𝜋𝜈1𝑓_{𝑎𝑖1𝑎𝑖2𝑎𝑖3𝑎𝑖4}𝜆₁,𝜆₂,𝜆₃𝑓_{𝑎𝑖5𝑎𝑖6}−𝜆₂−𝜆₃𝑑𝜆₁𝑑𝜆₂𝑑𝜆₃+2𝜋^𝜋_−𝜋𝜈2𝑓_{𝑎𝑖1𝑎𝑖2𝑎𝑖3}𝜆₁,𝜆₂𝑓_{𝑎𝑖4𝑎𝑖5𝑎𝑖6}𝜆₃,−𝜆₂−𝜆₃𝑑𝜆₁𝑑𝜆₂𝑑𝜆₃+2𝜋^𝜋_−𝜋𝜈3𝑓_{𝑎𝑖1𝑎𝑖2}𝜆₁𝑓_{𝑎𝑖3𝑎𝑖4}𝜆₂𝑓_{𝑎𝑖5𝑎𝑖6}−𝜆₁−𝜆₂𝑑𝜆₁𝑑𝜆₂. (A.8) The other asymptotic covariances are similarly evaluated. Finally, it suffices to prove the asymptotic normality of √𝑛(𝜃−𝜃). For this we prove cum√𝑛𝜃^𝑎1−𝜃^𝑎1,…,√𝑛𝜃^𝑎𝑗−𝜃^𝑎𝑗⟶0,𝑗≥3, (A.9) where 𝜃^𝑎𝑗 and 𝜃^𝑎𝑗 are the 𝑖th component of 𝜃 and 𝜃, respectively. Let 𝜃^𝑎𝑗−𝜃^𝑎𝑗=𝑐̂^𝑏1−𝑐^𝑏1if𝑖=1,…,𝑗₁𝑐̂^𝑏2𝑏3−𝑐^𝑏2𝑏3if𝑖=𝑗₁+1,…,𝑗₁+𝑗₂𝑐̂^𝑏4𝑏5−𝑐^𝑏4𝑏5if𝑖=𝑗₁+𝑗₂+1,…,𝑗₁+𝑗₂+𝑗₃𝑐̂^{𝑏6𝑏7𝑏8}−𝑐^{𝑏6𝑏7𝑏8}if𝑖=𝑗₁+𝑗₂+𝑗₃…,𝑗₁+𝑗₂+𝑗₃+𝑗₄=𝑗. (A.10) Then, similarly as in Shiraishi and Taniguchi [1] we can see that cum√𝑛𝜃^𝑎1−𝜃^𝑎1,…,√𝑛𝜃^𝑎𝑗−𝜃^𝑎𝑗=𝑛^𝑛/2cum𝜃^𝑎1,…,𝜃^𝑎1+1,…,𝜃^{𝑎𝑗1234+𝑗+𝑗+𝑗}=𝑂𝑛^1−𝑗/2for𝑗≥3, (A.11) which implies the asymptotic normality of √𝑛(𝜃−𝜃).

Proof of Theorem 3.1. We write ∑^𝑛_𝑡=1𝑍²_𝑘(𝑡) by 𝑑_𝑡,𝑘(𝑛). Then, 𝔼𝐴_𝑗,𝑘=𝜇_𝑗∑^𝑛_𝑡=11⋅𝑍_𝑘(𝑡)√𝑛𝑑_𝑡,𝑘(𝑛)by(G3)=𝜇_𝑗^𝜋_−𝜋𝑑𝑀_0,𝑘(𝜆)1+𝑜√𝑛^say=𝐴_𝑗,𝑘, (A.12) which leads to 𝔼𝐴_𝑗,𝑘=𝐴_𝑗,𝑘1+𝑜√𝑛. (A.13) Next we evaluate the covariance: Cov√𝑛𝐴_𝑗,𝑘−𝐴_𝑗,𝑘,√𝑛𝐴_{𝑗′,𝑘′}−𝐴_{𝑗′,𝑘′}=𝑛𝔼𝐴_𝑗,𝑘−𝐴_𝑗,𝑘−𝔼𝐴_𝑗,𝑘−𝐴_𝑗,𝑘𝐴_{𝑗′,𝑘′}−𝐴_{𝑗′,𝑘′}−𝔼𝐴_{𝑗′,𝑘′}−𝐴_{𝑗′,𝑘′}=𝑛𝔼𝐴_𝑗,𝑘−𝐴_𝑗,𝑘1+𝑜√𝑛𝐴_{𝑗′,𝑘′}−𝐴_{𝑗′,𝑘′}1+𝑜√𝑛=𝑛𝔼𝐴_𝑗,𝑘−𝐴_𝑗,𝑘𝐴_{𝑗′,𝑘′}−𝐴_{𝑗′,𝑘′}(1)=+𝑜_{𝑛𝑛𝑡=1𝑠=1}𝑅_𝑗,𝑗′()𝑍𝑠−𝑡^𝑘(𝑡)𝑍_𝑘′(𝑠)𝑑_𝑡,𝑘(𝑛)𝑑_{𝑠⋅𝑘′}(𝑛)=^𝑛−1_{𝑙=−𝑛+1}𝑅_𝑗,𝑗′(𝑙)^𝑛_{𝑠=11≤𝑠−𝑙≤𝑛}𝑍_𝑘(𝑠−𝑙)𝑍_𝑘′(𝑠)𝑑_{𝑠−𝑙,𝑘}(𝑛)𝑑_{𝑠⋅𝑘′}(𝑛)⟶2𝜋^𝜋_−𝜋𝑓_𝑗𝑗′(𝜆)𝑑𝑀_𝑘,𝑘′(𝜆)=𝑉𝑗,𝑘∶𝑗^′,𝑘^′, (A.14) where 𝑅_𝑗,𝑗′(𝑠−𝑡) is the covariance function of 𝑋_𝑗(𝑡) and 𝑋_𝑗′(𝑠).
The asymptotic normality of √𝑛(𝐴_𝑗,𝑘−𝐴_𝑗,𝑘) can be shown if we prove cum√𝑛𝐴_𝑗1,𝑘1−𝐴_𝑗1,𝑘1,…,√𝑛𝐴_{𝑗𝑙,𝑘𝑙}−𝐴_{𝑗𝑙,𝑘𝑙}⟶0,𝑙≥3. (A.15) Similarly as in Theorem  5.11.1 of Brillinger [6], we can see that cum√𝑛𝐴_𝑗1,𝑘1−𝐴_𝑗1,𝑘1,…,√𝑛𝐴_{𝑗𝑙,𝑘𝑙}−𝐴_{𝑗𝑙,𝑘𝑙}=𝑛^𝑙/2cum𝐴_𝑗1,𝑘1,…,𝐴_{𝑗𝑙,𝑘𝑙}=𝑂𝑛^1−𝑙/2,for𝑙≥3. (A.16)

Proof of Theorem 3.2. First, it is seen that lim_𝑛→∞𝔼𝐵_{𝑗,𝑚,𝑘}=lim_𝑛→∞𝑅_𝑗,𝑚(0)∑^𝑛_𝑡=1𝑍_𝑘(𝑡)√𝑛𝑑_𝑡,𝑘(𝑛)⟶𝑅_𝑗,𝑚(0)^𝜋_−𝜋𝑑𝑀_0,𝑘(𝜆)=^say𝐵_{𝑗,𝑚,𝑘}by(G3). (A.17) We can evaluate the covariance as follows: Cov√𝑛𝐵_{𝑗,𝑘,𝑚}−𝐵_{𝑗,𝑘,𝑚},√𝑛𝐵_{𝑗′,𝑘′,𝑚′}−𝐵_{𝑗′,𝑘′,𝑚′}=𝑛𝔼𝐵_{𝑗,𝑘,𝑚}−𝐵_{𝑗,𝑘,𝑚}𝐵_{𝑗′,𝑘′,𝑚′}−𝐵_{𝑗′,𝑘′,𝑚′}=𝑛𝔼𝐵_{𝑗,𝑘,𝑚}𝐵_{𝑗′,𝑘′,𝑚′}−𝐵_{𝑗,𝑘,𝑚}𝐵_{𝑗′,𝑘′,𝑚′}=^𝑛𝑛_𝑡=1𝑠=1Cov𝑋_𝑗(𝑡)𝑋_𝑚(𝑡),𝑋_𝑗′(𝑠)𝑋_𝑚′(𝑠)^𝜋_−𝜋𝑑𝑀_𝑘,𝑘′(𝜆)(1)=+𝑜_{𝜋∞−𝜋𝑙=−∞}cum_{𝑗,𝑚,𝑗′,𝑚′}(0,𝑙,𝑙)+𝑅_𝑗,𝑚′(𝑙)𝑅_𝑚,𝑗′(𝑙)+𝑅_𝑗,𝑗′(𝑙)𝑅_𝑚,𝑚′(𝑙)𝑒^{−𝑖𝑙𝜆}𝑑𝑀_𝑘,𝑘′(𝜆)⟶2𝜋^𝜋_−𝜋^𝜋_−𝜋𝑓_𝑗𝑚′𝜆−𝜆₁𝑓_𝑚𝑗′𝜆₁+𝑓_𝑗𝑗′𝜆−𝜆₁𝑓_𝑚𝑚′𝜆₁𝑑𝜆₁+2𝜋^𝜋_−𝜋𝑓_{𝑗𝑚𝑗′𝑚′}𝜆₁,𝜆₂+𝜆,−𝜆₂𝑑𝜆₁𝑑𝜆₂𝑑𝑀_𝑘,𝑘′(𝜆)=𝑉𝑗,𝑚,𝑘∶𝑗^′,𝑚^′,𝑘^′. (A.18) Next we derive the asymptotic normality of √𝑛(𝐵_{𝑗,𝑚,𝑘}−𝐵_{𝑗,𝑚,𝑘}). For this we prove cum√𝑛𝐵_{𝑗1,𝑘1,𝑚1}−𝐵_{𝑗1,𝑘1,𝑚1},…,√𝑛𝐵_{𝑗𝑙,𝑘𝑙,𝑚𝑙}−𝐵_{𝑗𝑙,𝑘𝑙,𝑚𝑙}⟶0,𝑙≥3. (A.19) Similarly as in Theorem  5.11.1 of Brillinger [6], it is shown that cum√𝑛𝐵_{𝑗1,𝑚1,𝑘1}−𝐵_{𝑗1,𝑚1,𝑘1},…,√𝑛𝐵_{𝑗𝑙,𝑚𝑙,𝑘𝑙}−𝐵_{𝑗𝑙,𝑚𝑙,𝑘𝑙}=𝑛^𝑙/2cum𝐵_{𝑗1,𝑚1,𝑘1},…,𝐵_{𝑗𝑙,𝑚𝑙,𝑘𝑙}=𝑂𝑛^1−𝑙/2, (A.20) which proves the asymptotic normality.

Acknowledgments

The authors thank Professor Cathy Chen and two referees for their kind comments.