Honey, I shrunk the sample covariance matrix

The central message of this paper is that nobody should be using the samplecovariance matrix for the purpose of portfolio optimization. It containsestimation error of the kind most likely to perturb a mean-varianceoptimizer. In its place, we suggest using the matrix obtained from thesample covariance matrix through a transformation called shrinkage. Thistends to pull the most extreme coefficients towards more central values,thereby systematically reducing estimation error where it matters most.Statistically, the challenge is to know the optimal shrinkage intensity,and we give the formula for that. Without changing any other step in theportfolio optimization process, we show on actual stock market data thatshrinkage reduces tracking error relative to a benchmark index, andsubstantially increases the realized information ratio of the activeportfolio manager.

Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

This paper analyzes whether standard covariance matrix tests work whendimensionality is large, and in particular larger than sample size. Inthe latter case, the singularity of the sample covariance matrix makeslikelihood ratio tests degenerate, but other tests based on quadraticforms of sample covariance matrix eigenvalues remain well-defined. Westudy the consistency property and limiting distribution of these testsas dimensionality and sample size go to infinity together, with theirratio converging to a finite non-zero limit. We find that the existingtest for sphericity is robust against high dimensionality, but not thetest for equality of the covariance matrix to a given matrix. For thelatter test, we develop a new correction to the existing test statisticthat makes it robust against high dimensionality.

Improved estimation of the covariance matrix of stock returns with an application to portofolio selection

This paper proposes to estimate the covariance matrix of stock returnsby an optimally weighted average of two existing estimators: the samplecovariance matrix and single-index covariance matrix. This method isgenerally known as shrinkage, and it is standard in decision theory andin empirical Bayesian statistics. Our shrinkage estimator can be seenas a way to account for extra-market covariance without having to specifyan arbitrary multi-factor structure. For NYSE and AMEX stock returns from1972 to 1995, it can be used to select portfolios with significantly lowerout-of-sample variance than a set of existing estimators, includingmulti-factor models.

The mean-variance model from the inverse of the variance-covariance matrix

En este artículo, a partir de la inversa de la matriz de varianzas y covarianzas se obtiene el modelo Esperanza-Varianza de Markowitz siguiendo un camino más corto y matemáticamente riguroso. También se obtiene la ecuación de equilibrio del CAPM de Sharpe.

The mean-variance model from the inverse of the variance-covariance matrix

En este artículo, a partir de la inversa de la matriz de varianzas y covarianzas se obtiene el modelo Esperanza-Varianza de Markowitz siguiendo un camino más corto y matemáticamente riguroso. También se obtiene la ecuación de equilibrio del CAPM de Sharpe.

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting anyapproximation about the observation statistics or the a prioridistribution of the parameters. The proposed solution is obtainedminimizing the estimator variance subject to some constraints onthe estimator bias. The resulting optimal estimator is found todepend on the observation fourth-order moments that can be calculatedanalytically from the known signal model. Unfortunately,in most cases, the performance of this estimator is severely limitedby the residual bias inherent to nonlinear estimation problems.To overcome this limitation, the second-order minimum varianceunbiased estimator is deduced from the general solution by assumingaccurate prior information on the vector of parameters.This small-error approximation is adopted to design iterativeestimators or trackers. It is shown that the associated varianceconstitutes the lower bound for the variance of any unbiasedestimator based on the sample covariance matrix.The paper formulation is then applied to track the angle-of-arrival(AoA) of multiple digitally-modulated sources by means ofa uniform linear array. The optimal second-order tracker is comparedwith the classical maximum likelihood (ML) blind methodsthat are shown to be quadratic in the observed data as well. Simulationshave confirmed that the discrete nature of the transmittedsymbols can be exploited to improve considerably the discriminationof near sources in medium-to-high SNR scenarios.

Asymptotic robustness in multi-sample analysis of multivariate linear relations

Standard methods for the analysis of linear latent variable models oftenrely on the assumption that the vector of observed variables is normallydistributed. This normality assumption (NA) plays a crucial role inassessingoptimality of estimates, in computing standard errors, and in designinganasymptotic chi-square goodness-of-fit test. The asymptotic validity of NAinferences when the data deviates from normality has been calledasymptoticrobustness. In the present paper we extend previous work on asymptoticrobustnessto a general context of multi-sample analysis of linear latent variablemodels,with a latent component of the model allowed to be fixed across(hypothetical)sample replications, and with the asymptotic covariance matrix of thesamplemoments not necessarily finite. We will show that, under certainconditions,the matrix $\Gamma$ of asymptotic variances of the analyzed samplemomentscan be substituted by a matrix $\Omega$ that is a function only of thecross-product moments of the observed variables. The main advantage of thisis thatinferences based on $\Omega$ are readily available in standard softwareforcovariance structure analysis, and do not require to compute samplefourth-order moments. An illustration with simulated data in the context ofregressionwith errors in variables will be presented.

Multi-sample analysis of moment-structures: Asymptotic validity of inferences based on second-order moments

In moment structure analysis with nonnormal data, asymptotic valid inferences require the computation of a consistent (under general distributional assumptions) estimate of the matrix $\Gamma$ of asymptotic variances of sample second--order moments. Such a consistent estimate involves the fourth--order sample moments of the data. In practice, the use of fourth--order moments leads to computational burden and lack of robustness against small samples. In this paper we show that, under certain assumptions, correct asymptotic inferences can be attained when $\Gamma$ is replaced by a matrix $\Omega$ that involves only the second--order moments of the data. The present paper extends to the context of multi--sample analysis of second--order moment structures, results derived in the context of (simple--sample) covariance structure analysis (Satorra and Bentler, 1990). The results apply to a variety of estimation methods and general type of statistics. An example involving a test of equality of means under covariance restrictions illustrates theoretical aspects of the paper.

The algebraic equality of two asymptotic tests for the hypothesis that a normal distribution has a specified correlation matrix

It is proved the algebraic equality between Jennrich's (1970) asymptotic$X^2$ test for equality of correlation matrices, and a Wald test statisticderived from Neudecker and Wesselman's (1990) expression of theasymptoticvariance matrix of the sample correlation matrix.

Impact of incorrect assumptions on the covariance structure of random effects and/or residuals in nonlinear mixed models for repeated measures data

In this paper we analyse, using Monte Carlo simulation, the possible consequences of incorrect assumptions on the true structure of the random effects covariance matrix and the true correlation pattern of residuals, over the performance of an estimation method for nonlinear mixed models. The procedure under study is the well known linearization method due to Lindstrom and Bates (1990), implemented in the nlme library of S-Plus and R. Its performance is studied in terms of bias, mean square error (MSE), and true coverage of the associated asymptotic confidence intervals. Ignoring other criteria like the convenience of avoiding over parameterised models, it seems worst to erroneously assume some structure than do not assume any structure when this would be adequate.

Multivariate contemporaneous-threshold autoregressive models

This paper proposes a contemporaneous-threshold multivariate smooth transition autoregressive (C-MSTAR) model in which the regime weights depend on the ex ante probabilities that latent regime-specific variables exceed certain threshold values. A key feature of the model is that the transition function depends on all the parameters of the model as well as on the data. Since the mixing weights are also a function of the regime-specific innovation covariance matrix, the model can account for contemporaneous regime-specific co-movements of the variables. The stability and distributional properties of the proposed model are discussed, as well as issues of estimation, testing and forecasting. The practical usefulness of the C-MSTAR model is illustrated by examining the relationship between US stock prices and interest rates.

Robust Factor Analysis for Compositional Data

Factor analysis as frequent technique for multivariate data inspection is widely used also for compositional data analysis. The usual way is to use a centered logratio (clr)transformation to obtain the random vector y of dimension D. The factor model istheny = Λf + e (1)with the factors f of dimension k & D, the error term e, and the loadings matrix Λ.Using the usual model assumptions (see, e.g., Basilevsky, 1994), the factor analysismodel (1) can be written asCov(y) = ΛΛT + ψ (2)where ψ = Cov(e) has a diagonal form. The diagonal elements of ψ as well as theloadings matrix Λ are estimated from an estimation of Cov(y).Given observed clr transformed data Y as realizations of the random vectory. Outliers or deviations from the idealized model assumptions of factor analysiscan severely effect the parameter estimation. As a way out, robust estimation ofthe covariance matrix of Y will lead to robust estimates of Λ and ψ in (2), seePison et al. (2003). Well known robust covariance estimators with good statisticalproperties, like the MCD or the S-estimators (see, e.g. Maronna et al., 2006), relyon a full-rank data matrix Y which is not the case for clr transformed data (see,e.g., Aitchison, 1986).The isometric logratio (ilr) transformation (Egozcue et al., 2003) solves thissingularity problem. The data matrix Y is transformed to a matrix Z by usingan orthonormal basis of lower dimension. Using the ilr transformed data, a robustcovariance matrix C(Z) can be estimated. The result can be back-transformed tothe clr space byC(Y ) = V C(Z)V Twhere the matrix V with orthonormal columns comes from the relation betweenthe clr and the ilr transformation. Now the parameters in the model (2) can beestimated (Basilevsky, 1994) and the results have a direct interpretation since thelinks to the original variables are still preserved.The above procedure will be applied to data from geochemistry. Our specialinterest is on comparing the results with those of Reimann et al. (2002) for the Kolaproject data

Analyzing shapes as compositions of distances

We propose to analyze shapes as “compositions” of distances in Aitchison geometry asan alternate and complementary tool to classical shape analysis, especially when sizeis non-informative.Shapes are typically described by the location of user-chosen landmarks. Howeverthe shape – considered as invariant under scaling, translation, mirroring and rotation– does not uniquely define the location of landmarks. A simple approach is to usedistances of landmarks instead of the locations of landmarks them self. Distances arepositive numbers defined up to joint scaling, a mathematical structure quite similar tocompositions. The shape fixes only ratios of distances. Perturbations correspond torelative changes of the size of subshapes and of aspect ratios. The power transformincreases the expression of the shape by increasing distance ratios. In analogy to thesubcompositional consistency, results should not depend too much on the choice ofdistances, because different subsets of the pairwise distances of landmarks uniquelydefine the shape.Various compositional analysis tools can be applied to sets of distances directly or afterminor modifications concerning the singularity of the covariance matrix and yield resultswith direct interpretations in terms of shape changes. The remaining problem isthat not all sets of distances correspond to a valid shape. Nevertheless interpolated orpredicted shapes can be backtransformated by multidimensional scaling (when all pairwisedistances are used) or free geodetic adjustment (when sufficiently many distancesare used)

New insights on river water chemistry by using non-centred simplicial principal component analysis: a case study

The use of perturbation and power transformation operations permits the investigation of linear processes in the simplex as in a vectorial space. When the investigated geochemical processes can be constrained by the use of well-known starting point, the eigenvectors of the covariance matrix of a non-centred principalcomponent analysis allow to model compositional changes compared with a reference point.The results obtained for the chemistry of water collected in River Arno (central-northern Italy) have open new perspectives for considering relative changes of the analysed variables and to hypothesise the relative effect of different acting physical-chemical processes, thus posing the basis for a quantitative modelling

Modelling world investment markets using threshold conditional correlation models

In this paper we propose a parsimonious regime-switching approach to model the correlations between assets, the threshold conditional correlation (TCC) model. This method allows the dynamics of the correlations to change from one state (or regime) to another as a function of observable transition variables. Our model is similar in spirit to Silvennoinen and Teräsvirta (2009) and Pelletier (2006) but with the appealing feature that it does not suffer from the course of dimensionality. In particular, estimation of the parameters of the TCC involves a simple grid search procedure. In addition, it is easy to guarantee a positive definite correlation matrix because the TCC estimator is given by the sample correlation matrix, which is positive definite by construction. The methodology is illustrated by evaluating the behaviour of international equities, govenrment bonds and major exchange rates, first separately and then jointly. We also test and allow for different parts in the correlation matrix to be governed by different transition variables. For this, we estimate a multi-threshold TCC specification. Further, we evaluate the economic performance of the TCC model against a constant conditional correlation (CCC) estimator using a Diebold-Mariano type test. We conclude that threshold correlation modelling gives rise to a significant reduction in portfolio´s variance.