Statistical inference for the EU portfolio in high dimensions
Taras Bodnar, Solomiia Dmytriv, Yarema Okhrin, Nestor Parolya, Wolfgang Schmid
SStatistical inference for the EU portfolio in highdimensions
Taras Bodnar a , Solomiia Dmytriv b , Yarema Okhrin c , Nestor Parolya ∗ d and Wolfgang Schmid ba Department of Mathematics, Stockholm University, Stockholm, Sweden b Department of Statistics, European University Viadrina, Frankfurt(Oder), Germany c Department of Statistics, University of Augsburg, Augsburg, Germany d Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands
Abstract
In this paper, using the shrinkage-based approach for portfolio weights and modernresults from random matrix theory we construct an effective procedure for testing theefficiency of the expected utility (EU) portfolio and discuss the asymptotic behavior ofthe proposed test statistic under the high-dimensional asymptotic regime, namely whenthe number of assets p increases at the same rate as the sample size n such that theirratio p/n approaches a positive constant c ∈ (0 ,
1) as n → ∞ . We provide an extensivesimulation study where the power function and receiver operating characteristic curves ofthe test are analyzed. In the empirical study, the methodology is applied to the returns ofS&P 500 constituents. Keywords:
Finance; Portfolio analysis; Mean-variance optimal portfolio; Statistical test;Shrinkage estimator; Random matrix theory.
Following the mean-variance approach of Markowitz (1952), which is considered to be one ofthe most popular portfolio choice strategies, the weights of an optimal portfolio are obtained byminimizing the portfolio variance for a predefined level of the portfolio expected return. This setof optimal portfolios determines the efficient frontier in the mean-variance space. The Markowitzapproach formalizes the advantages of portfolio diversification and has become a benchmark forboth researchers and practitioners in portfolio management. ∗ Corresponding Author: Nestor Parolya. E-Mail: [email protected] a r X i v : . [ q -f i n . P M ] M a y arkowitz optimal portfolios, also known as mean-variance optimal portfolios, can also beobtained as solutions of other optimization problems (e.g., Bodnar et al. (2013)), like by maxi-mizing the expected quadratic utility (EU) function (see, Ingersoll (1987)) expressed as w (cid:48) µ − γ w (cid:48) Σw → max subject to w (cid:48) p = 1 , (1)where w = ( w , . . . , w p ) (cid:48) is the vector of portfolio weights, p is the p -dimensional vectorsof ones, µ and Σ are the mean vector and the covariance matrix of the random vector ofasset returns x = ( x , . . . , x p ) (cid:48) . The quantity γ > γ = ∞ , then the investor is fully risk averse and determines the investment strategyby minimizing the portfolio variance without paying attention to the expected portfolio return,i.e., constructs the so-called global minimum variance (GMV) portfolio. Under the assumptionthat the asset returns are normally distributed, the problem of maximization the mean-varianceobjective function (1) is equivalent to the maximization of the expected exponential utility,which implies constant absolute risk aversion (CARA). In this case, γ is equal to the investorsabsolute risk aversion coefficient (see, e.g., Ingersoll (1987)).We denote the solution of (1) by w EU and it is given by w EU = Σ − p (cid:48) p Σ − p + γ − Q µ , (2)where Q = Σ − − Σ − p (cid:48) p Σ − (cid:48) p Σ − p . (3)The case of fully risk averse investor, i.e., γ = ∞ , leads to the weights of the GMV portfolioexpressed as w GMV = Σ − p (cid:48) p Σ − p . (4)The derived formulas of optimal portfolio weights (2) and (4) cannot directly be used inpractice, since they both depend on unknown parameters of the data generating process. Themean vector µ and the covariance matrix Σ are not observable in practice and have to beestimated by using historical data for asset returns. This, however, introduces further sources ofrisk into the investment process, namely the estimation risk which has been ignored for a longtime in finance.The most commonly used approach to estimate the weights of optimal portfolios is based onsimple replacing the unknown first two moments of the asset returns by their sample counter-parts. As a result, we obtain a ”plug-in” estimator for the optimal portfolio weights also knownas its sample estimator, which is a traditional way to construct a portfolio in practice. Assumingthat the asset returns are independent and normally distributed Okhrin and Schmid (2006) ob-tain the asymptotic distribution of the sample estimator of the EU portfolio weights, while thecorresponding exact distributional results can be found in Bodnar and Schmid (2011). Furthertheoretical and practically relevant findings related to the characterization of the distribution ofthe sample estimator of the optimal portfolio weights and their characteristics can be found in2ang et al. (2015), Woodgate and Siegel (2015), Simaan et al. (2018), Zhao et al. (2019), amongothers.The use of the ”plug-in” estimators in practice has been widely criticized in statistical andfinancial literature. One of the main drawbacks of the sample estimators is the investors overop-timism about the optimality of the constructed portfolio. Several studies (see, e.g., Siegel andWoodgate (2007), Kan and Smith (2008), Bodnar and Bodnar (2010)) show with theoreticaland empirical arguments that the plug-in estimator of the efficient frontier overestimates thelocation of the true efficient frontier in the mean-variance space. This leads to too optimistictrading strategies which perform in practice typically much worse than expected.In recent years, other types of estimators for the optimal portfolio weights have been intro-duced in the literature. Some estimators attempt to improve the estimators for the parametersof the asset returns. Relying on the idea of Stein (1956) we can use a shrinkage estimator forthe mean vector and for the covariance matrix or its inverse (see, e.g., Bodnar et al. (2014) andBodnar et al. (2016)). Alternatively, one can apply the shrinkage method directly to portfolioweights as suggested by Golosnoy and Okhrin (2007), Okhrin and Schmid (2008), Frahm andMemmel (2010), etc. The goal of the approach is to reduce the estimation uncertainty and todecrease the variance in the estimated portfolio weights.The problem of assessing the estimation risk, when an optimal portfolio is constructed,becomes very challenging from the high-dimensional perspectives, i.e., when both the numberof included assets p and the sample size n tend to infinity simultaneously such that p/n tends to the concentration ratio c > n → ∞ (Bai and Shi (2011)). Under the classicalasymptotic regime, when the number of assets p is fixed and substantially smaller then thesample size n , the traditional ”plug-in” estimator of optimal portfolio weights is consistent(see, Okhrin and Schmid (2006), Bodnar and Schmid (2011)). On the other hand, the sampleestimators of the mean vector and of the covariance matrix are not longer feasible under thehigh-dimensional asymptotics (Bai and Silverstein (2010), Bai and Shi (2011), Bodnar, Okhrinand Parolya (2019)), which has a negative impact on the performance of the asset allocationstrategy. Moreover, the inverse covariance matrix does not exist anymore for c > c > We consider a financial market consisting of p risky assets. Let x t denote the p -dimensionalvector of the returns on risky assets at time t . Suppose that E ( x t ) = µ and Cov ( x t ) = Σ where Σ is assumed to be positive definite. Let x , x , . . . , x n be a sample of asset return vectorsconsisting of their n independent realizations and let X n = ( x , x , . . . , x n ) stand for the p × n data matrix. Throughout of the paper we assume that the asset returns are independent and4dentically normally distributed, i.e. x i ∼ N p ( µ , Σ ) , i = 1 , . . . , n .The sample estimators of µ and Σ are given by ¯x n = 1 n n (cid:88) j =1 x j and ˆ Σ n = 1 n − n (cid:88) j =1 ( x j − ¯x n ) ( x j − ¯x n ) (cid:48) . (5)Replacing µ and Σ in (2) by their sample estimators from (5), we obtain the sample estimatorof the EU portfolio weights expressed asˆ w EU = ˆ Σ − n p (cid:48) p ˆ Σ − n p + γ − ˆ Q n ˆ µ n , where ˆ Q n = ˆ Σ − n − ˆ Σ − n p (cid:48) p ˆ Σ − n (cid:48) p ˆ Σ − n p . (6)Okhrin and Schmid (2006) derive the analytical expression for the expectation and the co-variance matrix of ˆ w EU and obtain its asymptotic distribution assuming that the portfolio sizeis considerably smaller than the sample size. These results are extended in Bodnar and Schmid(2011) who derive the finite-sample distribution of the estimated EU portfolio weights and usethese results in the derivation of an asymptotic tests on the weights which we present in thenext subsection. At each time point an investor has to decide whether the holding portfolio is efficient or it hasto be adjusted (see, Bodnar and Schmid (2008), Bodnar and Schmid (2011)). This problemcan be presented as a special case of the general linear hypotheses formulated for the portfolioweights. Let L denote the k × p dimensional matrix of constants with k < p − r be the k -dimensional vector of constants. Bodnar and Schmid (2011) consider the followinghypotheses for linear combinations of the EU portfolio weights H : Lw EU = r against H : Lw EU (cid:54) = r , (7)If one sets L = [ I k O k,p − k ] in (7) where I k is the k -dimensional identity matrix and O k,p − k is the k × ( p − k ) matrix with zeros, then the null hypothesis states that the first k weightsin w EU are equal to the corresponding components defined by r . It also has to be noted thatwhole structure of the EU portfolio cannot be tested by using (7) because of the restrictionimposed on the number of linear combinations which should be smaller than p − L in each of the null hypotheses such thatall elements in w EU are tested. This leads to a multiple testing problem also discussed below.In order to test (7) for a given matrix L and a vector r , Bodnar and Schmid (2011) suggest5he following test statistic: T L = ( n − p + 1) ( ˆ w L − r ) (cid:48) (cid:32) L ˆ Q n L (cid:48) (cid:48) p ˆ Σ − n p + γ − L ˆ Q n L (cid:48) ¯x (cid:48) n ˆ Q n ¯x n + γ − ( L ˆ Q n L (cid:48) ¯x (cid:48) n ˆ Q n ¯x n − L ˆ Q n ¯x n ¯x (cid:48) n ˆ Q n L (cid:48) ) (cid:33) − × ( ˆ w L − r ) , (8) where ˆ w L = L ˆ w EU = L ˆ Σ − n p (cid:48) p ˆ Σ − n p + γ − L ˆ Q n ¯x n . (9)Bodnar and Schmid (2011) show that the test statistic T L can be asymptotically well ap-proximated by a non-central χ -distribution with k degrees of freedom and the non-centralityparameter λ = n ( w L − r ) (cid:48) (cid:18) LQL (cid:48) (cid:48) p Σ − p + γ − LQ n L (cid:48) µ (cid:48) Q µ + γ − ( LQL (cid:48) µ (cid:48) Q µ − LQ µµ (cid:48) QL (cid:48) ) (cid:19) − ( w L − r ) (10)with w L = Lw EU = LΣ − p (cid:48) p Σ − p + γ − LQ µ , (11)when both p and k are relatively small with respect to the sample size n . As a special case,we obtain the asymptotic distribution of T L under the null hypothesis. This appears to be a χ -distribution, i.e. T L ∼ χ k under the null hypothesis in (7).Since the asymptotic distribution of the test statistic T L is obtained under classical asymp-totic regime, this test, in general, is not applicable when the portfolio size is comparable to thesample size. We illustrate this point in Figure 1. Here we plot the kernel density estimator(KDE) of the distribution of the test statistic T L under the null hypothesis together with theasymptotic χ -distribution (green and red curves, respectively). For this purpose we generatesamples from a multivariate normal distribution with mean vector and covariance matrix asspecified in the numerical study of Section 4.1. The vector r consists of the first k componentsof the true EU portfolio weights and we set L = [ I k O k,p − k ] . For each sample we compute thevalue of the test statistic T L and then plot the KDE. To robustify the conclusions we set γ = 5 , p = 300 , c n = p/n ∈ { . , . } and k ∈ { , , } . We observe that already for k = 10the difference between the KDE and the asymptotic distribution is very large and this evidencebecomes stronger if k increases. For k = 100 the KDE shifts strongly to the right and is notshown to retain the same scaling on the x -axis. Table 1 gives the realized sizes (type I errors)of the considered test based on the 5000 independent replications and with the nominal level α = 0 .
05 . For different values of k ∈ { , , } , it can be seen that T L is highly inconsistentand has a much higher size than the nominal value α . We conclude that the test is highlyunreliable if we wish to test many or all weights simultaneously.6 .2 Improvement of the test based on Mahalanobis distance for large-dimensional portfolios Bodnar, Dette, Parolya and Thors´en (2019) show that the sample estimator of the EU portfolioweights is not consistent under the high-dimensional asymptotic regime, i.e., when p/n → c ∈ [0 ,
1) as n → ∞ . Moreover, they derive a consistent estimator for the elements of w EU anduse these findings to construct a high-dimensional asymptotic test on the finite number of linearcombinations of the EU portfolio weights.Let L be a k × p matrix of constant as defined in Section 2.1 and letˆ w GMV ; L = L ˆ w GMV = L ˆ Σ − n p (cid:48) p ˆ Σ − n p , ˆ s = ¯x (cid:48) n ˆ Q n ¯x n and ˆ η L = L ˆ Q n ¯x n ¯x (cid:48) n ˆ Q n ¯x n . (12)Assuming that k is finite, i.e. considerably smaller than both p and n , Bodnar, Dette, Parolyaand Thors´en (2019) prove thatˆ w GMV ; L a.s. → Lw GMV , ˆ s c = (1 − c n )ˆ s − c n a.s. → s (13)and ˆ η L ; c = ˆ s c + c n ˆ s c ˆ η L a.s. → η L (14)for c n = p/n → c ∈ [0 ,
1) as n → ∞ with s = µ (cid:48) Q µ and η L = LQ µµ (cid:48) Q µ . (15)The symbol a.s. → denotes the almost surely convergence.Using (13), Bodnar, Dette, Parolya and Thors´en (2019) propose a high-dimensional asymp-totic test on the hypotheses (7) with the test statistic given by T L ; c = ( n − p ) ( ˆ w L ; c − r ) (cid:48) ˆ Ω − L ; c ( ˆ w L ; c − r ) , (16)where ˆ w L ; c = ˆ w GMV ; L + γ − ˆ s c ˆ η L ; c (17)andˆ Ω L ; c = (cid:18) (cid:18) − c n ˆ s c + c n + (ˆ s c + c n ) γ − (cid:19) γ − + ˆ V c (cid:19) (1 − c n ) L ˆ Q n L (cid:62) + γ − (cid:40) − c n ) c n (ˆ s c + c n ) + 4(1 − c n ) c n ˆ s c (ˆ s c + 2 c n )(ˆ s c + c n ) + 2(1 − c n ) c n (ˆ s c + c n ) ˆ s c − ˆ s c (cid:41) ˆ η L ; c ˆ η (cid:48) L ; c , (18)7here ˆ V c = ˆ V GMV − c n with ˆ V GMV = 1 (cid:48) p ˆ Σ − n p (19)are the consistent and the sample estimators of the variance of the GMV portfolio (4), that is(see, e.g., Bodnar et al. (2018, p.387))ˆ V c a.s. → V GMV = 1 (cid:48) p Σ − p for c n = p/n → c ∈ [0 ,
1) as n → ∞ .The application of the results of Theorem 4.4 in Bodnar, Dette, Parolya and Thors´en (2019)leads to the high-dimensional asymptotic distribution of T L ; c under both hypotheses in (7).Namely, it holds that the asymptotic distribution of T L ; c under H is well approximated by anon-central χ -distribution with k degrees of freedom and non-centrality parameter given by λ c = ( n − p )( w L − r ) (cid:48) Ω − L ; c ( w L − r ) , (20)where Ω L ; c = (cid:18) (cid:18) − cs + c + ( s + c ) γ − (cid:19) γ − + V GMV (cid:19) (1 − c ) LQL (cid:62) + γ − (cid:40) − c ) c ( s + c ) + 4(1 − c ) c s ( s + 2 c )( s + c ) + 2(1 − c ) c ( s + c ) s − s (cid:41) η L η (cid:48) L . Moreover, T L ; c d → χ k under H , where the symbol d → denotes the convergence in distribu-tion.In Figure 1 we present the KDE of the distribution of T L ; c (blue curve) and compare it toits high-dimensional asymptotic distribution (red curve). The kernel density estimator as wellas the sizes of the test are obtained under the same simulation setup as one used at the end ofSection 2.1. The approximation works well and much better than in the case of T L for smallervalues of k , but discrepancy becomes large if k increases. The same conclusion can be drawnfrom Table 1. Here the method proposed by Bodnar, Dette, Parolya and Thors´en (2019) has amuch better realized size which still increases dramatically with growing k . Both tests based on the Mahalanobis distance are designed to test a finite number of linearrestrictions imposed on the EU portfolio weights. Although the high-dimensional test shows aconsiderable improvement in terms of the size (see, Figures 1 and Table 1), this test, similarlyto the test based on the statistic T L , cannot be applied to test the structure of the whole EUportfolio. In practice, one has to fix the number k of the EU portfolio weights (or their linearrestrictions) and apply the test T L ; c several times in order to cover the whole vector w EU .This approach is a single-step multiple test (see, Dickhaus (2014)) with the number of marginalhypotheses to be tested equal to [ p/k ]+ 1 . Since the dependence structure between the marginal8 en s i t y c = 0.3, k = 10 . . . . . Asymptotic T L T L ;c D en s i t y c = 0.8, k = 10 . . . . . Asymptotic T L T L ;c D en s i t y c = 0.3, k = 30 . . . . . Asymptotic T L T L ;c D en s i t y c = 0.8, k = 30 . . . . . Asymptotic T L T L ;c D en s i t y c = 0.3, k = 100 . . . .
30 50 70 90 110 130 150 170 190 210 230
Asymptotic T L ;c D en s i t y c = 0.8, k = 100 . . . .
30 50 70 90 110 130 150 170 190 210 230
Asymptotic T L ;c Figure 1:
The high-dimensional asymptotic χ approximation of the densities of T L and T L ; c together with their kernel density estimators for γ = 5 , p = 300 , c n = p/n ∈ { . , . } and k ∈ { , , } . tests is very complicated, one has to monitor the overall type I error rate by using the so-calledBonferroni correction (see, Dickhaus (2014)). This would worse the power properties of eachindividual test, especially when the number of tests is relatively large.As a solution to this challenging problem, we suggest a new approach for testing the structureof the EU portfolio by a single test. The new procedure is based on the shrinkage estimator ofthe EU portfolio weights as suggested by Bodnar et al. (2020) and extend our previous results9 = . k = 10 k = 30 k = 100 T L T L ; c c = . k = 10 k = 30 k = 100 T L T L ; c Table 1:
Empirical sizes of the tests based on T L and T L ; c using 5 · independent replications. obtained for the GMV portfolio in Bodnar, Dmytriv, Parolya and Schmid (2019), which is avery special case of the EU portfolio. In contrast to the EU portfolio, the weights of the GMVportfolio do not depend on the mean vector. As a result, the derivation of the test for the EUportfolio becomes a very challenging task and completely new results in random matrix theoryhave to be derived to handle it. The shrinkage estimator for the EU portfolio weights is a convex combination of the sampleestimator and a fixed well behaved target portfolio b ∈ R p with bounded expected return andvariance, i.e., R b = b (cid:48) µ < ∞ and V b = b (cid:48) Σ − b < ∞ uniformly in p . Thus, the shrinkageestimator is expressed asˆ w GSE = α n ˆ w EU + (1 − α n ) b with b (cid:48) p = 1 , (21)where α n is the shrinkage intensity. One of the main ideas behind the shrinkage estimator (21)is to reduce the large variability present in the sample estimator ˆ w EU by shrinking it to a vectorof constants. This approach might introduce a bias in the estimator, but on the other side itreduces the variability of the sample estimator considerably.Bodnar et al. (2020) determine the optimal shrinkage intensity α ∗ n as the solution of themaximization problem based on the mean-variance objective function. It is given by α ∗ n = ( ˆ w EU − b ) (cid:48) ( µ − γ Σb )( ˆ w EU − b ) (cid:48) Σ ( ˆ w EU − b ) (22)Since the expression of α ∗ n depends on both the population mean vector and covariance matrixand on their sample counterparts, it cannot be directly applied in practice. As such, Bodnaret al. (2020) propose a two-stage procedure. First, the deterministic quantity α ∗ which isasymptotically equivalent to α ∗ n is found. Second, it is consistently estimated under the high-dimensional asymptotic regime. 10t holds that (see, Bodnar et al. (2020, Theorem 2.1)) α ∗ = γ − ( R GMV − R b ) (cid:0) − c (cid:1) + γ ( V b − V GMV ) + γ − − c s − c V GMV − (cid:16) V GMV + γ − − c ( R b − R GMV ) (cid:17) + γ − (cid:16) s (1 − c ) + c (1 − c ) (cid:17) + V b , (23)where R GMV = (cid:48) p Σ − µ (cid:48) p Σ − p is the expected return of the GMV portfolio. Following Bodnar et al.(2020) we assume throughout the paper that uniformly in p the quadratic form (cid:48) Σ − p isbounded away from zero and µ (cid:48) Σ − µ is bounded from above by some positive constant. Theseconditions guarantee among others the boundedness of R GMV , V GMV and s as p → ∞ ,thus, keeping the limiting expressions coming further well defined asymptotically. Consistentestimators for the variance of the GMV portfolio V GMV and for the slope parameter of theefficient frontier s are given in (19) and (13), respectively. Bodnar et al. (2020) show that thesample estimators of R GMV , R b , and V b are consistent, that isˆ R GMV = (cid:48) p ˆ Σ − n ¯x n (cid:48) p ˆ Σ − n p a.s → R GMV , ˆ R b = b (cid:48) ¯x n a.s → R b , ˆ V b = b (cid:48) ˆ Σ n b a.s → V b , (24)for p/n → c ∈ [0 ,
1) as n → ∞ .Hence, a consistent estimator for α ∗ is constructed asˆ α ∗ c = γ − ( ˆ R GMV − ˆ R b ) (cid:16) − c n (cid:17) + γ ( ˆ V b − ˆ V c ) + γ − − c n ˆ s c − c n ˆ V c − (cid:16) ˆ V c + γ − − c n ( ˆ R b − ˆ R GMV ) (cid:17) + γ − (cid:16) ˆ s c (1 − c n ) + c n (1 − c n ) (cid:17) + ˆ V b , (25)while the bona fide shrinkage estimator for the weights of the EU portfolio are expressed asˆ w BF GSE = ˆ α ∗ c ˆ w EU + (1 − ˆ α ∗ c ) b . (26)Next, we prove that ˆ α ∗ c is asymptotically normally distributed. This result will then be usedto derive a test for the structure of the EU portfolio in Section3.2. Let α ∗ = AB and ˆ α ∗ c = ˆ A n ˆ B n .Then, we get √ n ( ˆ α ∗ c − α ∗ ) = √ n (cid:32) ˆ A n − A ˆ B n − A ( ˆ B n − B ) B ˆ B n (cid:33) = 1ˆ B n (cid:18) √ n ( ˆ A n − A ) − AB √ n ( ˆ B n − B ) (cid:19) = d (cid:48) ˆ B n √ n t + o P (1) (27)for p/n → c + o ( n − / ) as n → ∞ with 11 = ˆ R GMV − R GMV ˆ V c − V GMV ˆ s c − s ˆ R b − R b ˆ V b − V b and d = − c n (cid:0) − AB (cid:1) − γ (cid:16) AB (cid:16) − c n − (cid:17)(cid:17) γ − − c n (cid:16) − − c n ) AB (cid:17) − − − c n (cid:0) − AB (cid:1) γ (cid:0) − AB (cid:1) , (28)where the symbol o P (1) denotes a sequence which tends almost surely to zero. In Theorem 1we derive the asymptotic distribution of t . Theorem 1
Let x , . . . , x n be independent and identically distributed with x i ∼ N p ( µ , Σ ) for i = 1 , . . . , n with Σ positive definite. Then it holds that √ n t d → N ( , Ω α ) (29) for p/n → c ∈ [0 , as n → ∞ where Ω α = V GMV ( s +1)1 − c V GMV − V GMV ( R b − R GMV )0 2 V GMV − c V GMV (( s +1) + c − − c R b − R GMV ) − R b − R GMV ) V GMV R b − R GMV ) V b − V GMV ( R b − R GMV ) 2 V GMV − R b − R GMV ) V b . (30)Since ˆ B n a.s → B for pn → c ∈ [0 ,
1) as n → ∞ , the application of Slutsky’s lemma (c.f., DasGupta (2008, Theorem 1.5)) leads to the asymptoticdistribution of ˆ α ∗ c as given in Theorem 2. Theorem 2
Under the assumptions of Theorem 1, it holds that √ n ( ˆ α ∗ c − α ∗ ) d → N (0 , C α ) , (31) for p/n → c ∈ [0 , as n → ∞ where C α = 1 B d (cid:48) Ω α d . (32)Finally, using (13), (19), and (24) a consistent estimator for C α is given byˆ C α = 1ˆ B n d (cid:48) ˆ Ω α ; c d , (33)where ˆ Ω α ; c is a consistent estimator for Ω α expressed as12 Ω α ; c = ˆ V c (ˆ s c +1)1 − c V c − V c ( ˆ R b − ˆ R GMV )0 2 ˆ V c − c V c ((ˆ s c +1) + c − − c
2( ˆ R b − ˆ R GMV ) −
2( ˆ R b − ˆ R GMV ) ˆ V c R b − ˆ R GMV ) ˆ V b − V c ( ˆ R b − ˆ R GMV ) 2 ˆ V c −
2( ˆ R b − ˆ R GMV ) V b . (34) Remark 1
In the case of the investor who invests into the GMV portfolio ( γ = ∞ ), theformulas (23) and (25) simplify to α ∗ = (1 − c )( V b − V GMV ) c + (1 − c )( V b − V GMV ) and ˆ α ∗ c = (1 − c )( ˆ V b − ˆ V c ) c + (1 − c )( ˆ V b − ˆ V c ) . Moreover, the application of Theorem 1 leads to √ n ( ˆ α ∗ c − α ∗ ) → N (cid:18) , − c ) c ( L b + 1)((1 − c ) R b + c ) ((2 − c ) L b + c ) (cid:19) (35) for p/n → c ∈ (0 , as n → ∞ with L b = V b /V GMV − , which coincides with the resultsobtained in Theorem 2 of Bodnar, Dmytriv, Parolya and Schmid (2019). We use the properties of the shrinkage intensity α ∗ and of its consistent estimator ˆ α ∗ c to derivean asymptotic test on the structure of the EU portfolio. The testing hypotheses are given by H : w EU = w against H : w EU (cid:54) = w , (36)which, in contrast to the hypotheses considered in Section 2, allow to test the structure ofthe whole vector of the EU portfolio weights by using a single test avoiding the problem ofmultiplicity.Following Bodnar, Dmytriv, Parolya and Schmid (2019), the idea behind a statistical testbased on the shrinkage approach is the usage w as a fixed target portfolio, i.e., to set b = w in (21). Since w is the EU optimal portfolio under the null hypothesis in (36), its expectedreturn and variance should satisfy R w = R GMV + γ − s and V w = V GMV + γ − s. (37)As a result, the numerator in (23) becomes A ( w ) = ( R GMV − R b ) (cid:18) − c (cid:19) + γ ( V b − V GMV ) + γ − − c s = − γ − s (cid:18) − c (cid:19) + γ − s + γ − − c s = 0 , α ∗ = 0 under H . (38)Hence, for testing (36), one can derive a test on the hypotheses H : α ∗ ( w ) = 0 against H : α ∗ ( w ) (cid:54) = 0 , (39)where the notation α ∗ ( w ) denotes the optimal shrinkage intensity as in (23) computed withtarget portfolio w . It has to be noted that the hypotheses (36) and (39) are not equiva-lent. Nevertheless, the rejection of the null hypothesis in (39) ensures the rejection of the nullhypothesis in (36) meaning that w is not the EU optimal portfolio.Let ˆ α ∗ c ( w ) be the consistent estimator of α ∗ ( w ) as constructed in (25) when the shrinkagetarget is b = w . Then the application of Theorem 2 shows thatˆ α ∗ c ( w ) a.s. → pn → c ∈ [0 ,
1) as n → ∞ , when the null hypothesis in (36) is true.Moreover, since the numerator in the expression of α ∗ ( w ) in (23) under the null hypothesisin (39) is equal to zero, i.e. A = 0 where A is defined before (27), we get the following stochasticrepresentation of √ n ˆ α ∗ c ( w ) expressed as √ n ˆ α ∗ c ( w ) = 1ˆ B n d (cid:48) √ n t with d = − c n − γ γ − − c n − − − c n γ (40)and t is defined in (28). The application of Theorem 1 then leads to the following result Theorem 3
Assume that the conditions of Theorem 1 are fulfilled. Then, under the null hy-pothesis in (39) , it holds that √ n ˆ α ∗ c ( w ) d → N (0 , C α ;0 ) , (41) for p/n → c ∈ [0 , as n → ∞ with C α ;0 = B d (cid:48) Ω α d where Ω α is given in (30) and B isdefined before (27) . Replacing B and Ω α by their consistent estimators ˆ B n and ˆ Ω α ; c , we get a consistentestimator for C α ;0 expressed as ˆ C α ;0 = 1ˆ B n d (cid:48) ˆ Ω α ; c d . (42)Then for testing hypotheses (39), we obtain the following test statistic T α = √ n ˆ α ∗ c ( w ) (cid:113) ˆ C α ;0 = √ n ˆ α ∗ c ( w ) ˆ B n (cid:113) d (cid:48) ˆ Ω α ; c d , (43)where ˆ α ∗ c ( w ) with b = w and ˆ Ω α ; c are given in (25) and (34), respectively. Under the null14ypothesis in (39) we get that T α d → N (0 , p/n → c ∈ [0 ,
1) as n → ∞ and, hence, the hypothesis that w are the weights of theEU portfolio is rejected as soon as | T α | > z − β/ where z − β/ is the (1 − β/
2) quantile ofthe standard normal distribution. Under the alternative hypothesis in (39), the distribution of √ n ˆ α ∗ c ( w ) can still be well approximated by the normal distribution under the high-dimensionalasymptotic regime and d (cid:48) ˆ Ω α ; c d provides a consistent estimator of its asymptotic variance. Onthe other side, it does not hold that ˆ α ∗ c ( w ) a.s. → T α candetect the deviation in the null hypotheses of both (36) and (39). Remark 2
Using that s = γ ( R w − R GMV ) (see (37) ) and ˆ R w and ˆ R GMV are consistentestimators of R w and R GMV , respectively (see (24) ), another consistent estimator of Ω α under H in (39) is given by (cid:101) Ω α ; c = ˆ V c ( γ ( ˆ R w − ˆ R GMV )+1)1 − c V c − V c ( ˆ R w − ˆ R GMV )0 2 ˆ V c − c V c (( γ ( ˆ R w − ˆ R GMV )+1) + c − − c
2( ˆ R w − ˆ R GMV ) −
2( ˆ R w − ˆ R GMV ) ˆ V c R w − ˆ R GMV ) ˆ V w − V c ( ˆ R w − ˆ R GMV ) 2 ˆ V c −
2( ˆ R w − ˆ R GMV ) V w . (44) Then, the hypotheses in (39) can also be tested by using the following test statistic (cid:101) T α = √ n ˆ α ∗ c ( w ) ˆ B n (cid:113) d (cid:48) (cid:101) Ω α ; c d (45) which is asymptotically standard normally distributed under H in (39) . Remark 3
Using the duality between the test theory and confidence interval (see, Aitchison(1964)), the null hypothesis in (39) and consequently in (36) are rejected at significance level β as soon as the (1 − β ) confidence interval constructed for α ∗ ( w ) does not include zero. Thisconfidence interval in the case of the test T α has the boundaries ˆ α ∗ c ( w ) ± z − β/ √ n (cid:113) d (cid:48) ˆ Ω α ; c d ˆ B n , (46) while for the test based on (cid:101) T α we get ˆ α ∗ c ( w ) ± z − β/ √ n (cid:113) d (cid:48) (cid:101) Ω α ; c d ˆ B n . (47)To assess the precision of the asymptotic distribution we use a similar setting as in thelast section. In Figure 2 we show the KDEs of the distribution of the test statistics T α and15 T α under the null hypothesis together with their high-dimensional asymptotic distribution. Thelatter approximates the simulated exact distributions very precisely, although the the fit appearsto be slightly better for T α . The empirical size on both cases is close to the nominal size of5% as it is shown in Table 2. Summarizing, we conclude that the high-dimensional asymptoticdistribution provide a good approximation for proposed test statistics for different values of c . D en s i t y c = 0.3 . . . . . −4 −3 −2 −1 0 1 2 3 4 AsymptoticT a T~ a D en s i t y c = 0.8 . . . . . −4 −3 −2 −1 0 1 2 3 4 AsymptoticT a T~ a Figure 2:
The high-dimensional asymptotic normal approximation of the densities of T α and (cid:101) T α together with their kernel density estimators for γ = 5 , p = 300 and c n = p/n ∈ { . , . } . c = . = . T α T α Table 2:
Empirical sizes of the two tests based on T α and ˜ T α using 5 · replications. The performance of the derived test is investigated throughout an extensive simulation study.In particular, we explore the behavior of the test with respect to its power characteristics andreceiver operative characteristic curves. Additionally, we apply the derived inference procedureto the real data in this section.
The sample of asset returns x , x , . . . , x n are generated independently from N p ( µ , Σ ) . Tomimic the bahavior of real data we generate the eigenvalues of population covariance matrix Σ according to the law λ i = 0 . e δc ( i − /p , i = 1 , . . . p (see, Bodnar et al. (2020)) and take itseigenvectors from the spectral decomposition of the standard Wishart random matrix. Then,the covariance matrix is given as follows Σ = Θ Λ Θ (cid:48) , (48)16here Λ is a diagonal matrix of the predefined eigenvalues and Θ is a p × p matrix of eigen-vectors. By changing the value of δ , we can control the conditional index of the covariancematrix for different values of c . We set condition index equals to 450. This setting reflectsthe parametrisation we observed in the empirical study in the next section. The mean vector israndomly generated from U ( − . , .
2) , which also corresponds to the natural behavior of dailyasset returns.We assume that the portfolio weights and thus the shrinkage intensity change due to a changein the mean of asset returns. Under the alternative hypothesis, there is an additive shift to themean vector of the asset returns defined as µ = µ + (cid:15), (49)where (cid:15) = − a · (1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) m , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) m ) , where a = 0 . κ , κ ∈ { , , , . . . , } , m = 0 . p . Thus we assume that the expected returnon the assets with high variance decreases.We conduct the test at the significance level α = 0 .
05 . We put p = 300 and c ∈ { . , . } .The number of repetitions is 10 and γ = 5 . For the ROC curves we fix a at 0 .
08 . The resultsare illustrated in Figure 3. It can be seen that both tests display an overall consistency anda good performance in terms of power functions and ROC curves. The behavior is better forsmaller values of c and not substantially worse in case of c = 0 . In this section, we apply the derived theoretical results to real data. The objective is to determinethe periods where the shrinkage intensity is significantly different from zero and thus the EUoptimal portfolio is significantly different from the target or the benchmark portfolio b . Thisstudy is based on daily return data of all companies listed in the S&P 500 index for the periodfrom April 1999 to March 2020. We assume that the investor allocates her wealth to portfoliosof size p ∈ { , } with daily reallocation. She selects the first p assets in alphabetic orderfrom the available data. The sample size n is chosen to attain c ∈ { . , . , . } , i.e. n = p/c .We put γ = 5 which is a common value for the risk aversion coefficient in financial literature.As the target portfolio we consider the equally weighted portfolio with all weights equal 1 /p .Despite of its simplicity this portfolio appears to show a superior long-run performance anddominates many more sophisticated trading strategies (see DeMiguel et al. (2009)).Figure 4 shows the time series of estimated shrinkage intensities together with 95% confidenceintervals as defined in (47). If c = 0 . c increases, the sample available for a portfolio of afixed size gets smaller and the shrinkage intensity shifts towards zero. The benchmark portfolio17 hange P o w e r c = 0.3 . . . . . . T a T~ a False positive rate T r ue po s i t i v e r a t e c = 0.3, a = 0.08 . . . . . . T a T~ a Change P o w e r c = 0.8 . . . . . . T a T~ a False positive rate T r ue po s i t i v e r a t e c = 0.8, a = 0.08 . . . . . . T a T~ a Figure 3:
Empirical power functions of the proposed tests as a function of the change a (left) andROC curves of two tests for a = 0 .
08 (right) for different values of c according to the scenario givenin (49) and p = 300 . gets higher weight and for c = 0 . p from 100 to 300. Fixed c and larger p increase the samplesize n and has a stabilizing impact on the shrinkage intensity.We cannot reject the null hypothesis of the test based on ˜ T α in (45) that the shrinkageintensity is zero if the confidence intervals cover the zero value (see Remark 3 above). Thefigures reveal that we never opt for H if c = 0 . . c = 0 . H in (39). Similar behavior is observed for p = 300 too, however, here theintensities and their variances are more stable leading to less periods with not rejected H .Recall that a non-rejection of H in (39) does not guarantee that the weights of the EUportfolio coincide with the weights of the target portfolio. To elaborate on the difference betweenthe two portfolios and to get more economic insight into the dynamics of the intensities weconsider Figure 5. Here we plot the difference between the means and variances of the GMVand the equally weighted benchmark. These quantities determine the behavior of the empiricalshrinkage intensity in (23). On the one hand, we observe in Figure 4 that the shrinkage intensityincreases during a crisis period, e.g. 2002-2003 and 2008-2010. This seems to be surprising18 .000.250.500.751.00 Date c=0. 3, p=100, g =5 Date c=0. 3, p=300, g =5 Date c=0. 5, p=100, g =5 Date c=0. 5, p=300, g =5 Date c=0. 8, p=100, g =5 Date c=0. 8, p=300, g =5 Figure 4:
Estimated shrinkage intensities for the equally weighted portfolio as the target portfolio( p = 100 on the right and p = 300 on the left) with 95% pointwise confidence intervals. The blackdots indicate the periods with rejected H (1-values) and not rejected H (0- values). since the volatility of returns is high in this period and the equally weighted portfolio is believedto reduce the risk. However, Figure 5 shows that the variance of the benchmark portfolio ismuch higher (i.e. ˆ V b > ˆ V c ) and its return is much lower (i.e. ˆ R b < ˆ R GMV ) compared to theGMV portfolio in the crisis period leading to a higher relative precision and efficiency of the EUportfolio. On another hand, the mean returns and the variances are almost indistinguishable incalm periods leading to shrinkage intensities closer to zero and even insignificant for larger c ’s.Thus we conclude that non-rejecting H is driven by high similarity between the mean and thevariance of the target and GMV portfolios. 19
005 2010 2015 2020
Date V ^ b - V ^ c − . − . . . . Date R ^ c - R ^ b R ^ b - R ^ G M V Date
Figure 5:
Components of the estimated shrinkage intensity given in (25) using equally weighted targetfor c = 0 . p = 300 and γ = 5 . This paper is dedicated to portfolio selection problems driven by high-dimensional financialdata sets. In particular, we deal with optimal asset allocation in a high-dimensional asymptoticregime, namely when the number of assets and the sample size tend to infinity at the samerate. Due to the curse of dimensionality in the parameter estimation process, asset allocationfor such portfolios becomes a challenging task. Using the techniques from the theory of randommatrices, new inferential procedures based on the optimal shrinkage intensity for testing theefficiency of the high-dimensional EU portfolio are developed and the asymptotic distributionsof the proposed test statistics are derived. In extensive simulations, we show that the suggestedtests have excellent performance characteristics for various values of c . The practical advantageof the proposed procedures are demonstrated in en empirical study based on stocks includedinto the S&P 500 index. References
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Lemma 1
Let z , ..., z n be an independent sample from the p -dimensional standard normaldistribution and let S n = 1 n − n (cid:88) j =1 ( z j − ¯z )( z j − ¯z ) (cid:48) (50) be the corresponding sample covariance matrix. Let m , m , and m be the p -dimensionalvector of constants with the Euclidean norms equal to one. Then √ n m (cid:48) S n m − m (cid:48) S − n m − − c n m (cid:48) S − n m − − c n m (cid:48) m m (cid:48) S − n m − − c n d → N (cid:18) , c Θ ( m , m , m ) ◦ Λ (cid:19) , (51) with Θ ( m , m , m ) = n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m )( m (cid:48) m ) lim n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m ) n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m )( m (cid:48) m ) lim n →∞ ( m (cid:48) m ) 0 . . n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m )lim n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m ) lim n →∞ ( m (cid:48) m ) 1 (52) and Λ = c − c − c − c − c − c − c − c − c c (1 − c ) c (1 − c ) c (1 − c ) − c − c c (1 − c ) c (1 − c ) c (1 − c ) − c − c c (1 − c ) c (1 − c ) c (1 − c ) , (53) where the symbol ◦ denotes the Hadamard (elementwise) product of matrices. roof of Lemma 1: Since ( n − S n has a p -dimensional Wishart distribution with the identitycovariance matrix, we get that there exists a p × ( n −
1) matrix ˜Z whose entries are independentand standard normally distributed such that ( n − S n = ˜Z˜Z (cid:48) . The application of Theorem 2in Bai et al. (2011) leads to (51) with Θ as in (52) and Λ given by Λ = λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ with λ = (cid:90) a + a − z d F c ( z ) − (cid:18)(cid:90) a + a − z d F c ( z ) (cid:19) ,λ = 1 − (cid:90) a + a − z d F c ( z ) (cid:90) a + a − z d F c ( z ) ,λ = (cid:90) a + a − z d F c ( z ) − (cid:18)(cid:90) a + a − z d F c ( z ) (cid:19) where the function F c ( z ) denotes the cumulative distribution function of the Marchenko-Pasturlaw (see, Bai and Silverstein (2010)) for c < F c ( z ) = 12 πzc (cid:112) ( a + − z )( z − a − ) [ a − ,a + ] ( z ) dz, where a ± = (1 ± √ c ) . The moments of F c ( z ) present in Λ can be found in Glombeck (2014,Lemma 14). This completes the proof of the lemma. (cid:50) Lemma 2
Under the conditions of Theorem 1 it holds that √ n h = √ n (cid:48) p ˆ Σ − n ¯x n − − c n (cid:48) p Σ − µ (cid:48) p ˆ Σ − n p − − c n (cid:48) p Σ − p ¯x (cid:48) n ˆ Σ − n ¯x n − − c n µ (cid:48) Σ − µ − c n − c n b (cid:48) ¯x n − b (cid:48) µ b (cid:48) ˆ Σ n b − b (cid:48) Σb d → N ( , Ξ ) (54) for c n = p/n → c ∈ [0 , as n → ∞ with Ξ = − c ) V GMV (cid:18) s ∗ + R GMV V GMV (cid:19) − c ) R GMV V GMV − c ) R GMV s ∗ V GMV − c − − c R b − c ) R GMV V GMV − c ) V GMV − c ) R GMV V GMV − − c − c ) R GMV s ∗ V GMV − c ) R GMV V GMV − c ) (( s ∗ ) + c − R b − c − − c R b − c R b − c V b − − c R b − − c − − c R b V b , (55)24 here s ∗ = s + R GMV V GMV + 1 .Proof of Lemma 2:
Let a (cid:48) = ( a , a , a , a , a ) be an arbitrary vector of constants. Next, weshow that √ n a (cid:48) h d → N (0 , a (cid:48) Ξa ) , which will prove the statement of the lemma.Since x , ..., x n are independent and identically distributed with x i ∼ N p ( µ , Σ ) , we get that x i = µ + Σz i where z , ..., z n are independent standard normally distributed and Σ / is thesymmetric square root of Σ . Moreover, it holds that ¯x n = µ + Σ¯z n and ˆ Σ n = Σ / S n Σ / , where ¯z n = 1 n n (cid:88) i =1 z i and S n = 1 n − n (cid:88) i =1 ( z i − ¯z n )( z i − ¯z n ) (cid:48) . To this end, we have that ¯z n and S n are independent with √ n ¯z n standard normally distributedand ( n − S n standard Wishart distributed.Let ν = Σ − / µ . We get √ n a (cid:48) h = H ( ¯z n , S n ) + H ( ¯z n ) , with H ( ¯z n , S n ) = a (cid:113) (cid:48) p Σ − p (cid:112) ¯x (cid:48) n Σ − ¯x n √ n (cid:48) p ˆ Σ − n ¯x n (cid:113) (cid:48) p Σ − p (cid:112) ¯x (cid:48) n Σ − ¯x n − − c n (cid:48) p Σ − ¯x n (cid:113) (cid:48) p Σ − p (cid:112) ¯x (cid:48) n Σ − ¯x n + a (cid:48) p Σ − p √ n (cid:32) (cid:48) p ˆ Σ − n p (cid:48) p Σ − p − − c n (cid:33) + a ¯x (cid:48) n Σ − ¯x n √ n (cid:32) ¯x (cid:48) n ˆ Σ − n ¯x n ¯x (cid:48) n Σ − ¯x n − − c n (cid:33) + a b (cid:48) Σb √ n (cid:32) b (cid:48) ˆ Σ n bb (cid:48) Σb − (cid:33) = d (cid:48) ( ¯z n ) √ n h ( ¯z n , S n )and H ( ¯z n ) = a − c n √ n (cid:0) (cid:48) p Σ − ¯x n − (cid:48) p Σ − µ (cid:1) + a − c n √ n (cid:0) ¯x (cid:48) n Σ − ¯x n − µ (cid:48) Σ − µ − c n (cid:1) + a √ n ( b (cid:48) ¯x n − b (cid:48) µ )= a − c n √ n (cid:0) ( ¯z n + d ) (cid:48) ( ¯z n + d ) − d (cid:48) d − c n (cid:1) with d ( ¯z n ) = a b (cid:48) Σb a (cid:48) p Σ − p a (cid:113) (cid:48) p Σ − p (cid:112) ( ν + ¯z n ) (cid:48) ( ν + ¯z n ) a ( ν + ¯z n ) (cid:48) ( ν + ¯z n ) , = 1 − c n a (cid:18) a − c n Σ − / µ + a − c n ) Σ − / p + a Σ / b (cid:19) , and h ( ¯z n , S n ) = b (cid:48) Σ / S n Σ / bb (cid:48) Σb − (cid:48) p Σ − / S − n Σ − / p (cid:48) p Σ − p − − c n (cid:48) p Σ − / S − n ( ν + ¯z n ) √ (cid:48) p Σ − p √ ( ν + ¯z n ) (cid:48) ( ν + ¯z n ) − − c n (cid:48) p Σ − / ( ν + ¯z n ) √ (cid:48) p Σ − p √ ( ν + ¯z n ) (cid:48) ( ν + ¯z n )( ν + ¯z n ) (cid:48) S − n ( ν + ¯z n )( ν + ¯z n ) (cid:48) ( ν + ¯z n ) − − c n . Since S n and ¯z n are independent the conditional distribution of H ( ¯z n , S n ) given ¯z n = v coincides with H ( v , S n ) . Furthermore, the application of Lemma 1 to √ n h ( v , S n ) proves thatit is asymptotically normally distributed and, thus, the asymptotic stochastic representation of H ( ¯z n , S n ) is given by H ( ¯z n v , S n ) d = (cid:114) c (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) d (cid:48) Θ Σ / b √ b (cid:48) Σb , Σ − / p (cid:113) (cid:48) p Σ − p , ( ν + ¯z n ) (cid:112) ( ν + ¯z n ) (cid:48) ( ν + ¯z n ) ◦ Λ d ω , (56)where ω d → N (0 ,
1) and is independent of ¯z n and hence of H ( ¯z n ) . Finally, we havethat n ( ¯z n + d ) (cid:48) ( ¯z n + d ) has a non-central χ distribution with p degrees of freedom andnoncentrality parameter n d (cid:48) d . The application of Bodnar and Reiß (2016) leads to √ p (cid:18) n ( ¯z n + d ) (cid:48) ( ¯z n + d ) p − n d (cid:48) d p − (cid:19) d → N (cid:18) , d (cid:48) d c (cid:19) and, consequently, H ( ¯z n ) d = √ c n − c n a (cid:115) d (cid:48) d c n ω . (57)where ω d → N (0 ,
1) .Using that ν (cid:48) ¯z n a.s. → ¯z (cid:48) n ¯z n a.s. → c , the application of Slutsky’s lemma (c.f., DasGupta(2008, Theorem 1.5)) leads to √ n a (cid:48) h d → N (0 , a (cid:48) Ξa )for p/n → c + o ( n − / ) as n → ∞ where Ξ is given in (55). Since a is an arbitrary vector,the statement of Lemma 2 is proved. (cid:50) Proof of Theorem 1:
It holds thatˆ R GMV − R GMV = ˆ V GMV (cid:32)(cid:18) (cid:48) p ˆ Σ − n ¯x n − − c n (cid:48) p Σ − µ (cid:19) − R GMV (cid:18) (cid:48) p ˆ Σ − n p − − c n (cid:48) p Σ − p (cid:19)(cid:33) , ˆ V c − V GMV = − V GMV ˆ V GMV (cid:18) (cid:48) p ˆ Σ − n p − − c n (cid:48) p Σ − p (cid:19) ˆ s c − s GMV = (1 − c n ) (cid:18) ¯x (cid:48) n ˆ Σ − n ¯x n − − c n µ (cid:48) Σ − µ − c n − c n (cid:19) − (1 − c n ) (cid:32) ( (cid:48) p ˆ Σ − n ¯x n ) (cid:48) p ˆ Σ − n p − − c n ( (cid:48) p Σ − µ ) (cid:48) p Σ − p (cid:33) = (1 − c n ) (cid:18) ¯x (cid:48) n ˆ Σ − n ¯x n − − c n µ (cid:48) Σ − µ − c n − c n (cid:19) − (1 − c n ) ˆ V GMV (cid:32) (cid:18) (cid:48) p ˆ Σ − n ¯x n + 11 − c n R GMV V GMV (cid:19) (cid:18) (cid:48) p ˆ Σ − n ¯x n − − c n (cid:48) p Σ − µ (cid:19) − − c n R GMV V GMV (cid:18) (cid:48) p ˆ Σ − n p − − c n (cid:48) p Σ − p (cid:19) (cid:33) . Hence, √ n ˆ R GMV − R GMV ˆ V c − V GMV ˆ s c − s ˆ R b − R b ˆ V b − V b = D √ n h , with h is defined in (54) and D = (1 − c n ) ˆ V c − (1 − c n ) ˆ V c R GMV − (1 − c n ) ˆ V c V GMV − c n ) ˆ V c (cid:16) R GMV V GMV − ˆ R GMV ˆ V c (cid:17) (1 − c n ) ˆ V c R GMV V GMV (1 − c n ) 0 00 0 0 1 00 0 0 0 1 The application of ˆ R GMV a.s. → R GMV and ˆ V c a.s. → V GMV for p/n → c ∈ [0 ,
1) as n → ∞ , theresults of Lemma 2, and Slutsky’s lemma (c.f., DasGupta (2008, Theorem 1.5)) completes theproof of the theorem. (cid:50)(cid:50)