# Optimal Portfolio Estimation for Dependent Financial Returns with Generalized Empirical Likelihood

Authors: Hiroaki Ogata
Journal: Advances in Decision Sciences
 (2012)

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## Abstract

This paper proposes to use the method of generalized empirical
likelihood to find the optimal portfolio weights. The log-returns of
assets are modeled by multivariate stationary processes rather than
i.i.d. sequences. The variance of the portfolio is written by the spectral
density matrix, and we seek the portfolio weights which minimize it.


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## Body

## 1.  Introduction

The modern portfolio theory has been developed since circa the 1950s. It is common knowledge that Markowitz [1, 2] is a pioneer in this field. He introduced the so-called mean-variance theory, in which we try to maximize the expected return (minimize the variance) under the constant variance (the constant expected return). After that, many researchers followed, and portfolio theory has been greatly improved. For a comprehensive survey of this field, refer to Elton et al. [3], for example.Despite its sophisticated paradigm, we admit there exists several criticisms against the early portfolio theory. One of them is that it blindly assumes that the asset returns are normally distributed. As Mandelbrot [4] pointed out, the price changes in the financial market do not seem to be normally distributed. Therefore, it is appropriate to use the nonparametric estimation method to find the optimal portfolio. Furthermore, it is empirically observed that financial returns are dependent. Therefore, it is unreasonable to fit the independent model to it.One of the nonparametric techniques which has been capturing the spotlight recently is the empirical likelihood method. It was originally proposed by Owen [5, 6] as a method of inference based on a data-driven likelihood ratio function. Smith [7] and Newey and Smith [8] extended it to the generalized empirical likelihood (GEL). GEL can be also considered as an alternative of generalized methods of moments (GMM), and it is known that its asymptotic bias does not grow with the number of moment restrictions, while the bias of GMM often does.From the above point of view, we consider to find the optimal portfolio weights by using the GEL method under the multivariate stationary processes. The optimal portfolio weights are defined as the weights which minimize the variance of the return process with constant mean. The analysis is done in the frequency domain.This paper is organized as follows. Section2 explains about a frequency domain estimating function. In Section 3, we review the GEL method and mention the related asymptotic theory. Monte Carlo simulations and a real-data example are given in Section 4. Throughout this paper, A′ and A* indicate the transposition and adjoint of a matrix A, respectively.


## 2. Frequency Domain Estimating Function

Here, we are concerned with them-dimensional stationary process {X(t)}t∈ℤ with mean vector 0, the autocovariance matrix(2.1)Γ(h)=Cov[X(t+h),X(t)]=E[X(t+h)X(t)′],and spectral density matrix(2.2)f(λ)=12π∑h=-∞∞Γ(h)e-ihλ,-π≤λ<π.Suppose that information of an interestedp-dimensional parameter θ∈Θ⊂ℝp exists through a system of general estimating equations in frequency domain as follows. Let ϕj(λ;θ), (j=1,…,q,q≥p) be m×m matrix-valued continuous functions on [-π,π) satisfying ϕj(λ;θ)=ϕj(λ;θ)* and ϕj(-λ;θ)=ϕj(λ;θ)′. We assume that each ϕj(λ;θ) satisfies the spectral moment condition(2.3)∫-ππtr⁡{ϕj(λ;θ0)f(λ)}dλ=0(j=1,…,q),
where θ0=(θ10,…,θp0)′ is the true value of the parameter. By taking an appropriate function for ϕj(λ;θ), (2.3) can express the best portfolio weights as shown in Example 2.1.Example 2.1 (portfolio selection).
In this example, we setm=p=q. Let xi(t) be the log-return of ith asset (i=1,…,m) at time t and suppose that the process {X(t)=(x1(t),…,xm(t))′} is stationary with zero mean. Consider the portfolio p(t)=∑i=1mθixi(t), where θ=(θ1,…,θm)′ is a vector of weights, satisfying ∑i=1mθi=1. The process {p(t)} is a linear combination of the stationary process, hence {p(t)} is still stationary, and, from Herglotz’s theorem, its variance is
(2.4)Var⁡{p(t)}=θ′Var⁡{X(t)}θ=θ′{∫-ππf(λ)dλ}θ.
Our aim is to find the weights θ0=(θ10,…,θm0)′ that minimize the variance (the risk) of the portfolio p(t) under the constrain of ∑i=1mθi=1. The Lagrange function is given by
(2.5)L(θ,λ)=θ′{∫-ππf(λ)dλ}θ+ξ(θ′e-1),where e=(1,1,…,1)′ and ξ is Lagrange multiplier. The first order condition leads to
(2.6)(I-eθ0′)[∫-ππ{f(λ)+f(λ)′}dλ]θ0=0,
where I is an identity matrix. Now, for fixed j=1,…,m, consider to take
(2.7)ϕj(λ;θ)={2θj(1-θj)(j,j)thcomponent1-2θjθl(j,l)thand(l,j)thcomponent withl=1,…,mandl≠j-2θkθl(k,l)thcomponent withk,l=1,…,mandk≠j,l≠j.
Then, (2.3) coincides with the first order condition (2.6), which implies that the best portfolio weights can be solved with the framework of the spectral moment condition.Besides, we can express other important indices for time series. In what follows, several examples are given.Example 2.2 (autocorrelation).
Denote the autocovariance and the autocorrelation of the process{xi(t)} (ith component of the process {X(t)}) with lag h by γi(h) and ρi(h), respectively. Suppose that we are interested in the joint estimation of ρi(h) and ρj(k). Take
(2.8)ϕ1(λ;θ)={cos⁡⁡(hλ)-θ1(i,i)thcomponent0otherwise,ϕ2(λ;θ)={cos⁡(kλ)-θ2(j,j)thcomponent0otherwise.
Then, (2.3) leads to
(2.9)θ10={∫-ππexp⁡(ihλ)fii(λ)dλ}{∫-ππfii(λ)dλ}-1,θ20={∫-ππexp⁡(ikλ)fjj(λ)dλ}{∫-ππfjj(λ)dλ}-1.
From Herglotz’s theorem, θ10=γi(h)/γi(0), and θ20=γj(k)/γj(0). Then,  θ0=(θ10,θ20)′ corresponds to the desired autocorrelations ρ=(ρi(h),ρj(k))′. The idea can be directly extended to more than two autocorrelations.Example 2.3 (Whittle estimation).
In this example, we setp=q. Consider fitting a parametric spectral density model fθ(λ) to the true spectral density f(λ). The disparity between fθ(λ) and f(λ) is measured by the following criterion:
(2.10)D(fθ,f)≡∫-ππ[log⁡det⁡fθ(λ)+tr⁡{fθ(λ)-1f(λ)}]dλ,
which is based on Whittle likelihood. The purpose here is to seek the quasi-true value θ̲ defined by
(2.11)θ̲=argmin⁡θ∈ΘD(fθ,f).
Assume that the spectral density model is expressed by the following form:
(2.12)fθ(λ)={∑j=0∞Bθ(j)exp⁡(ijλ)}K{∑j=0∞Bθ(j)exp⁡(ijλ)}*,
where each Bθ(j) is an m×m matrix (Bθ(0) is defined as identity matrix), and K is an m×m symmetric matrix. The general linear process has this spectral form, so this assumption is not so restrictive. The key of this assumption is that the elements of the parameter θ do not depend on K. We call such a parameter innovation-free. Imagine that you fit the VARMA process, for example. Innovation-free implies that the elements of the parameter θ depend on only AR or MA coefficients and not on the covariance matrix of the innovation process. Now, let us consider the equation:
(2.13)∂∂θD(fθ,f)|θ=θ̲=0,
to find quasi-true value. The Kolmogorov’s formula says
(2.14)det⁡K=exp⁡[12π∫-ππlog⁡det⁡{2πfθ(λ)}dλ].
This implies that if θ is innovation-free, the quantity ∫-ππlogdet{fθ(λ)}dλ is independent of θ and (2.13) leads to
(2.15)∂∂θ∫-ππtr⁡{fθ(λ)-1f(λ)dλ}|θ=θ̲=0.
This corresponds to (2.3), when we set
(2.16)ϕj(λ;θ)=∂fθ(λ)-1∂θj(j=1,…,p),
so the quasi-true value can be expressed by the spectral moment condition.Based on the form of (2.3), we set the estimating function for θ as(2.17)m(λt;θ)=(tr⁡{ϕ1(λt;θ)In(λt)},…,tr⁡{ϕq(λt;θ)In(λt)})′,whereIn(λ) is the periodogram, defined by(2.18)In(λ)=(2πn)-1{∑t=1nX(t)exp⁡(itλ)}{∑t=1nX(t)exp⁡(itλ)}*,whereλt=(2πt)/n, (t=-[(n-1)/2],…,[n/2]). Then, we have
(2.19)2πn∑-[(n-1)/2][n/2]E[m(λt;θ)]⟶[∫-ππtr⁡{ϕj(λ;θ0)f(λ)}dλ]j=1,…,q=0.


## 3. Generalized Empirical Likelihood

Once we construct the estimating function, we can make use of the method of GEL as in the work by Smith [7] and Newey and Smith [8]. GEL is introduced as an alternative to GMM and it is pointed out that its asymptotic bias does not grow with the number of moment restrictions, while the bias of GMM often does.To describe GEL letρ(v) be a function of a scalar v which is concave on its domain, an open interval 𝒱 containing zero. Let Λ̂n(θ)={λ:λ′m(λt;θ)∈𝒱,t=1,…,n}. The estimator is the solution to a saddle point problem(3.1)θ̂GEL=argmin⁡θ∈Θsup⁡λ∈Λ̂n(θ)∑t=1nρ(λ′m(λt;θ)).The empirical likelihood (EL) estimator (cf. [9]) the exponential tilting (ET) estimator (cf. [10]), and the continuous updating estimator (CUE) (cf. [11]) are special cases with ρ(v)=log(1-v), ρ(v)=-ev and ρ(v)=-(1+v)2/2, respectively. Let Ω=E[m(λt;θ0)m(λt;θ0)′], G=E[∂m(λt;θ0)/∂θ], and Σ=(G′Ω-1G)-1. The following assumptions and theorems are due to Newey and Smith [8].Assumption 3.1.
(i)θ0∈Θ is the unique solution to (2.3).
(ii)Θ is compact.
(iii)m(λt;θ) is continuous at each θ∈Θ with probability one.
(iv)E[supθ∈Θ||m(λt;θ)||α]<∞ for some α>2.
(v)Ω is nonsingular.
(vi)ρ(v) is twice continuously differentiable in a neighborhood of zero.Theorem 3.2.
Let Assumption3.1 hold. Then  θ̂GEL→pθ0.Assumption 3.3.
(i)θ0∈int(Θ).
(ii)m(λt;θ) is continuously differentiable in a neighborhood 𝒩 of θ0 and E[supθ∈𝒩||∂m(λt;θ)/∂θ′||]<∞.
(iii)rank(G)=p.Theorem 3.4.
Let Assumptions3.1 and 3.3 hold. Then n(θ̂GEL-θ0)→pN(0,Σ).


## 4. Monte Carlo Studies and Illustration

In the first part of this section, we summarize the estimation results of Monte Carlo studies of the GEL method. We generate 200 observations from the following two-dimensional-AR(1) model(4.1)[X1(t)X2(2)]=[0.2-0.5-0.50.3][X1(t-1)X2(t-2)]+[U1(t)U2(t)],where{U(t)=(U1(t),U2(t))′}t∈ℤ is an i.i.d. innovation process, distributed to two-dimensional t-distribution whose correlation matrix is identity, and degree of freedom is 5. The true autocorrelations with lag 1 of this process are ρ1(1)=0.3894 and ρ2(1)=0.4761, respectively. As described in Example 2.2, we estimate ρ1(1) and ρ2(1) by using three types of frequency domain GEL method (EL, ET, and CUE). Table 1 shows the mean and standard deviation of the estimation results whose repetition time is 1000. All types work properly.Table 1
Estimated autocorrelations for two-dimensional-AR(1) model.
ELETCUEMeans.d.Means.d.Means.d.ρ1 (1)0.37790.08900.37600.09000.37970.0911ρ2 (1)0.46800.08320.46600.08170.46500.0855Next, we apply the proposed method to the returns of market index data. The sample consists of 7 weekly indices (S&P 500, Bovespa, CAC 40, AEX, ATX, HKHSI, and Nikkei) having 800 observations each: the initial date is April 30, 1993, and the ending date is August 22, 2008. Refer to Table2 for the market of each index.Table 2
Market.
IndexMarketS&P 500NYSEBovespaSão Paulo Stock ExchangeCAC 40Bourse de ParisAEXAmsterdam Stock ExchangeATXWiener BörseHKHSIHong Kong Exchanges and ClearingNikkeiTokyo Stock ExchangeAs shown in Example2.1, we use frequency domain GEL method to estimate the optimal portfolio weights, and the results are shown in Table 3. Bovespa and ATX account for large part in the optimal portfolio.Table 3
Estimated portfolio weights.
ELETCUES&P 5000.07590.06170.0001Bovespa0.66480.64870.6827CAC 400.00000.00000.0000AEX0.00000.00000.0000ATX0.25930.25580.3168HKHSI0.00000.00000.0000Nikkei0.00000.03380.0004



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*Source: 973173-2012-07-01.xml*