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• 1. Bauder, David
Stockholm University, Faculty of Science, Department of Mathematics.
BAYESIAN INFERENCE FOR THE TANGENT PORTFOLIO2018In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 21, no 8, article id 1850054Article in journal (Refereed)

In this paper, we consider the estimation of the weights of tangent portfolios from the Bayesian point of view assuming normal conditional distributions of the logarithmic returns. For diffuse and conjugate priors for the mean vector and the covariance matrix, we derive stochastic representations for the posterior distributions of the weights of tangent portfolio and their linear combinations. Separately, we provide the mean and variance of the posterior distributions, which are of key importance for portfolio selection. The analytic results are evaluated within a simulation study, where the precision of coverage intervals is assessed.

• 2. Bodnar, Rostyslav
Stockholm University, Faculty of Science, Department of Mathematics.
Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices2016In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 45, no 12, p. 3421-3440Article in journal (Refereed)

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.

• 3.
Stockholm University, Faculty of Science, Department of Mathematics.
Spectral analysis of the Moore-Penrose inverse of a large dimensional sample covariance matrix2016In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 148, p. 160-172Article in journal (Refereed)

For a sample of $n$ independent identically distributed $p$-dimensional centered random vectorswith covariance matrix $\bSigma_n$ let $\tilde{\bS}_n$ denote the usual sample covariance(centered by the mean) and $\bS_n$ the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where $p> n$. In this paper, we provide the limiting spectral distribution andcentral limit theorem for linear spectralstatistics of the Moore-Penrose inverse of $\bS_n$ and $\tilde{\bS}_n$. We consider the large dimensional asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ such that $p/n\rightarrow c\in (1, +\infty)$. We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of $\tilde{\bS}_n$ differs in the mean from that of $\bS_n$.

• 4.
Stockholm University, Faculty of Science, Department of Mathematics.
On the Simes inequality in elliptical models2017In: Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, E-ISSN 1572-9052, Vol. 69, no 1, p. 215-230Article in journal (Refereed)

We provide some necessary and some sufficient conditions for the validity of the inequality ofSimes in models with elliptical dependencies. Necessary conditions are presented in terms of sufficient conditions for the reverse Simes inequality. One application of our main results concerns the problem of model misspecification, in particular the case that the assumption of Gaussianity of test statistics is violated. Since our sufficient conditions require non-negativity of correlation coefficients between test statistics, we also develop two exact tests for vectors of correlation coefficients and compare their powers in computer simulations.

• 5.
Bowling Green State University, USA.
Robustness of the Inference Procedures for the Global Minimum Variance Portfolio Weights in a Skew Normal Model2015In: European Journal of Finance, ISSN 1351-847X, E-ISSN 1466-4364, Vol. 21, no 13-14, p. 1176-1194Article in journal (Refereed)

In this paper, we study the influence of skewness on the distributional properties of the estimated weightsof optimal portfolios and on the corresponding inference procedures derived for the optimal portfolioweights assuming that the asset returns are normally distributed. It is shown that even a simple form ofskewness in the asset returns can dramatically influence the performance of the test on the structure of theglobal minimum variance portfolio. The results obtained can be applied in the small sample case as well.Moreover, we introduce an estimation procedure for the parameters of the skew-normal distribution that isbased on the modified method of moments.A goodness-of-fit test for the matrix variate closed skew-normaldistribution has also been derived. In the empirical study, we apply our results to real data of several stocksincluded in the Dow Jones index.

• 6.
Stockholm University, Faculty of Science, Department of Mathematics.
Direct shrinkage estimation of large dimensional precision matrix2016In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 146, p. 223-236Article in journal (Refereed)

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables p -> infinity and the sample size n -> infinity so that p/n -> c is an element of (0, +infinity). The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. Using this result, we construct a bona fide optimal linear shrinkage estimator for the precision matrix in case c < 1. At the end, a simulation is provided where the suggested estimator is compared with the estimators proposed in the literature. The optimal shrinkage estimator shows significant improvement even for non-normally distributed data.

• 7.
Stockholm University, Faculty of Science, Department of Mathematics.
Dynamic Conditional Correlation Multiplicative Error Processes2016In: Journal of Empirical Finance, ISSN 0927-5398, E-ISSN 1879-1727, Vol. 36, p. 41-67Article in journal (Refereed)

We introduce a dynamic model for multivariate processes of (non-negative) high-frequency tradingvariables revealing time-varying conditional variances and correlations. Modeling the variables' conditional mean processes using a multiplicative error model, we map the resulting residuals into aGaussian domain using a copula-type transformation. Based on high-frequency volatility, cumulativetrading volumes, trade counts and market depth of various stocks traded at the NYSE, we show thatthe proposed transformation is supported by the data and allows capturing (multivariate) dynamicsin higher order moments. The latter are modeled using a DCC-GARCH specification. We suggest estimating the model by composite maximum likelihood which is sufficientlyflexible to be applicablein high dimensions. Strong empirical evidence for time-varying conditional (co-)variances in tradingprocesses supports the usefulness of the approach. Taking these higher-order dynamics explicitlyinto account significantly improves the goodness-of-fit and out-of-sample forecasts of the multiplicative error model.

• 8.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
On the Product of a Singular Wishart Matrix and a Singular Gaussian Vector in High Dimension2018In: Theory of Probability and Mathematical Statistics, ISSN 0868-6904, Vol. 99Article in journal (Refereed)

In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation of this product is derived, using which its characteristic function and asymptotic distribution under the double asymptotic regime are established. We further document a good finite sample performance of the obtained high-dimensional asymptotic distribution via an extensive Monte Carlo study.

• 9.
Stockholm University, Faculty of Science, Department of Mathematics.
Bayesian estimation of the global minimum variance portfolio2017In: European Journal of Operational Research, ISSN 0377-2217, E-ISSN 1872-6860, Vol. 256, no 1, p. 292-307Article in journal (Refereed)

In this paper we consider the estimation of the weights of optimal portfolios from the Bayesian point of view under the assumption that the conditional distributions of the logarithmic returns are normal. Using the standard priors for the mean vector and the covariance matrix, we derive the posterior distributions for the weights of the global minimum variance portfolio. Moreover, we reparameterize the model to allow informative and non-informative priors directly for the weights of the global minimum variance portfolio. The posterior distributions of the portfolio weights are derived in explicit form for almost all models. The models are compared by using the coverage probabilities of credible intervals. In an empirical study we analyze the posterior densities of the weights of an international portfolio.

• 10.
Stockholm University, Faculty of Science, Department of Mathematics.
A test for the global minimum variance portfolio for small sample and singular covariance2017In: AStA Advances in Statistical Analysis, ISSN 1863-8171, E-ISSN 1863-818X, Vol. 101, no 3, p. 253-265Article in journal (Refereed)

Recently, a test dealing with the linear hypothesis for the global minimum variance portfolio weights was obtained under the assumption of non-singular covariance matrix. However, the problem of potential multicollinearity and correlations of assets constitutes a limitation of the classical portfolio theory. Therefore, there is an interest in developing theory in the presence of singularities in the covariance matrix. In this paper, we extend the test by analyzing the portfolio weights in the small sample case with a singular population covariance matrix. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented.

• 11.
Stockholm University, Faculty of Science, Department of Mathematics.
Singular inverse Wishart distribution and its application to portfolio theory2016In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 143, p. 314-326Article in journal (Refereed)

The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented.

• 12.
Stockholm University, Faculty of Science, Department of Mathematics.
Optimal shrinkage estimator for high-dimensional mean vector2019In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 170, p. 63-79Article in journal (Refereed)

In this paper we derive the optimal linear shrinkage estimator for the high-dimensional mean vector using random matrix theory. The results are obtained under the assumption that both the dimension $p$ and the sample size $n$ tend to infinity in such a way that $p∕n\to c\in(0,\infty)$. Under weak conditions imposed on the underlying data generating mechanism, we find the asymptotic equivalents to the optimal shrinkage intensities and estimate them consistently. The proposed nonparametric estimator for the high-dimensional mean vector has a simple structure and is proven to minimize asymptotically, with probability 1, the quadratic loss when $c\in(0,1)$. When $c\in(1,\infty)$ we modify the estimator by using a feasible estimator for the precision covariance matrix. To this end, an exhaustive simulation study and an application to real data are provided where the proposed estimator is compared with known benchmarks from the literature. It turns out that the existing estimators of the mean vector, including the new proposal, converge to the sample mean vector when the true mean vector has an unbounded Euclidean norm.

• 13.
Stockholm University, Faculty of Science, Department of Mathematics.
Determination and estimation of risk aversion coefficients2018In: Computational Management Science, ISSN 1619-697X, E-ISSN 1619-6988, Vol. 15, no 2, p. 297-317Article in journal (Refereed)

In the paper we consider two types of utility functions often used in portfolio allocation problems, i.e. the exponential utility and the quadratic utility. We link the resulting optimal portfolios obtained by maximizing these utility functions to the corresponding optimal portfolios based on the minimum value-at-risk (VaR) approach. This allows us to provide analytic expressions for the risk aversion coefficients as functions of the VaR level. The results are initially derived under the assumption that the vector of asset returns is multivariate normally distributed and they are generalized to the class of elliptically contoured distributions thereafter. We find that the choice of the coefficients of risk aversion depends on the stochastic model used for the data generating process. Finally, we take the parameter uncertainty into account and present confidence intervals for the risk aversion coefficients of the considered utility functions. The theoretical results are validated in an empirical study.

• 14.
Stockholm University, Faculty of Science, Department of Mathematics.
A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function2015In: Annals of Operations Research, ISSN 0254-5330, E-ISSN 1572-9338, Vol. 229, no 1, p. 121-158Article in journal (Refereed)

In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived under weak conditions on the asset returns. No assumption on the correlation structure between different time points is needed and no assumption on the distribution is imposed. All expressions are presented in terms of the conditional mean vectors and the conditional covariance matrices. If the multivariate process of the asset returns is independent, it is shown that in the case without a riskless asset the solution is presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. The process dynamics are included only in the shape parameter of the utility function. If a riskless asset is present, then the multi-period optimal portfolio weights are proportional to the single-period solutions multiplied by time-varying constants which are dependent on the process dynamics. Remarkably, in the case of a portfolio selection with the tangency portfolio the multi-period solution coincides with the sequence of the single-period solutions. Finally, we compare the suggested strategies with existing multi-period portfolio allocation methods on real data.

• 15.
Stockholm University, Faculty of Science, Department of Mathematics.
Estimation of the global minimum variance portfolio in high dimensions2018In: European Journal of Operational Research, ISSN 0377-2217, E-ISSN 1872-6860, Vol. 266, no 1, p. 371-390Article in journal (Refereed)

We estimate the global minimum variance (GMV) portfolio in the high-dimensional case using results from random matrix theory. This approach leads to a shrinkage-type estimator which is distribution-free and optimal in the sense of minimizing the out-of-sample variance. Its asymptotic properties are investigated assuming that the number of assets p depends on the sample size n such that p/n -> c is an element of (0, + infinity) as n tends to infinity. The results are obtained under weak assumptions imposed on the distribution of the asset returns: only the existence of the fourth moments is required. Furthermore, we make no assumption on the upper bound of the spectrum of the covariance matrix. As a result, the theoretical findings are also valid if the dependencies between the asset returns are described by a factor model which appears to be very popular in the financial literature nowadays. This is also documented in a numerical study where the small- and large-sample behavior of the derived estimator is compared with existing estimators of the GMV portfolio. The resulting estimator shows significant improvements and it turns out to be robust if the assumption of normality is violated.

• 16.
Stockholm University, Faculty of Science, Department of Mathematics.
On the exact solution of the multi-period portfolio choice problem for an exponential utility under return predictability2015In: European Journal of Operational Research, ISSN 0377-2217, E-ISSN 1872-6860, Vol. 246, no 2, p. 528-542Article in journal (Refereed)

In this paper we derive the exact solution of the multi-period portfolio choice problem for an exponential utility function under return predictability. It is assumed that the asset returns depend on predictable variables and that the joint random process of the asset returns and the predictable variables follow a vector autoregressive process. We prove that the optimal portfolio weights depend on the covariance matrices of the next two periods and the conditional mean vector of the next period. The case without predictable variables and the case of independent asset returns are partial cases of our solution. Furthermore, we provide an exhaustive empirical study where the cumulative empirical distribution function of the investor's wealth is calculated using the exact solution. It is compared with the investment strategy obtained under the additional assumption that the asset returns are independently distributed.

• 17.
Stockholm University, Faculty of Science, Department of Mathematics.
The Exact Solution of Multi-period Portfolio Choice Problem with Exponential Utility2016In: Operations Research Proceedings 2014: Selected Papers of the Annual International Conference of the German Operations Research Society (GOR), RWTH Aachen University, Germany, September 2-5, 2014 / [ed] Marco Lübbecke, Arie Koster, Peter Letmathe, Reinhard Madlener, Britta Peis, Grit Walther, Springer, 2016, p. 45-51Chapter in book (Refereed)

In the current paper we derive the exact analytical solution of the multiperiod portfolio choice problem for an exponential utility function. It is assumed that the asset returns depend on predictable variables and that the joint random process of the asset returns follows a vector autoregression. We prove that the optimal portfolio weights depend on the covariance matrices of the next two periods and the conditional mean vector of the next period. The case without predictable variables and the case of independent asset returns are partial cases of our solution.

• 18.
Stockholm University, Faculty of Science, Department of Mathematics.
Exact and asymptotic tests on a factor model in low and large dimensions with applications2016In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 150, p. 125-151Article in journal (Refereed)

In the paper, we suggest three tests on the validity of a factor model which can be applied for both, small-dimensional and large-dimensional data. The exact and asymptotic distributions of the resulting test statistics are derived under classical and high dimensional asymptotic regimes. It is shown that the critical values of the proposed tests can be calibrated empirically by generating a sample from the inverse Wishart distribution with identity parameter matrix. The powers of the suggested tests are investigated by means of simulations. The results of the simulation study are consistent with the theoretical findings and provide general recommendations about the application of each of the three tests. Finally, the theoretical results are applied to two real data sets, which consist of returns on stocks from the DAX index and on stocks from the S&P 500 index. Our empirical results do not support the hypothesis that all linear dependencies between the returns can be entirely captured by the factors considered in the paper.

• 19.
Stockholm University, Faculty of Science, Department of Mathematics.
How risky is the optimal portfolio which maximizes the Sharpe ratio?2017In: AStA Advances in Statistical Analysis, ISSN 1863-8171, E-ISSN 1863-818X, Vol. 101, no 1, p. 1-28Article in journal (Refereed)

In this paper, we investigate the properties of the optimal portfolio in the sense of maximizing the Sharpe ratio (SR) and develop a procedure for the calculation of the risk of this portfolio. This is achieved by constructing an optimal portfolio which minimizes the Value-at-Risk (VaR) and at the same time coincides with the tangent (market) portfolio on the efficient frontier which is related to the SR portfolio. The resulting significance level of the minimum VaR portfolio is then used to determine the risk of both the market portfolio and the corresponding SR portfolio. However, the expression of this significance level depends on the unknown parameters which have to be estimated in practice. It leads to an estimator of the significance level whose distributional properties are investigated in detail. Based on these results, a confidence interval for the suggested risk measure of the SR portfolio is constructed and applied to real data. Both theoretical and empirical findings document that the SR portfolio is very risky since the corresponding significance level is smaller than 90 % in most of the considered cases.

• 20. Neumann, André
Stockholm University, Faculty of Science, Department of Mathematics.
Multivariate multiple test procedures based on nonparametric copula estimation2019In: Biometrical Journal, ISSN 0323-3847, E-ISSN 1521-4036, Vol. 61, no 1, p. 40-61Article in journal (Refereed)

Multivariate multiple test procedures have received growing attention recently. This is due to the fact that data generated by modern applications typically are highdimensional, but possess pronounced dependencies due to the technical mechanisms involved in the experiments. Hence, it is possible and often necessary to exploit these dependencies in order to achieve reasonable power. In the present paper, we express dependency structures in the most general manner, namely, by means of copula functions. One class of nonparametric copula estimators is constituted by Bernstein copulae. We extend previous statistical results regarding bivariate Bernstein copulae to the multivariate case and study their impact on multiple tests. In particular, we utilize them to derive asymptotic confidence regions for the family-wise error rate (FWER) of multiple test procedures that are empirically calibrated by making use of Bernstein copulae approximations of the dependency structure among the test statistics. This extends a similar approach by Stange et al. (2015) in the parametric case. A simulation study quantifies the gain in FWER level exhaustion and, consequently, power that can be achieved by exploiting the dependencies, in comparison with common threshold calibrations like the Bonferroni or Šidák corrections. Finally, we demonstrate an application of the proposed methodology to real-life data from insurance.

• 21.
Weierstrass Institute for Applied Analysis and Stochastics, Germany.
Humboldt-University, Germany. Weierstrass Institute for Applied Analysis and Stochastics, Germany.
Uncertainty quantification for the family-wise error rate in multivariate copula models2015In: AStA Advances in Statistical Analysis, ISSN 1863-8171, E-ISSN 1863-818X, Vol. 99, no 3, p. 281-310Article in journal (Refereed)

We derive confidence regions for the realized family-wise error rate(FWER) of certain multiple tests which are empirically calibrated at a given (global)level of significance. To this end, we regard the FWER as a derived parameter of a multivariate parametric copula model. It turns out that the resulting confidence regions aretypically very much concentrated around the target FWER level, while generic multiple tests with fixed thresholds are in general not FWER-exhausting. Since FWERlevel exhaustion and optimization of power are equivalent for the classes of multipletest problems studied in this paper, the aforementioned findings militate strongly infavor of estimating the dependency structure (i.e., copula) and incorporating it in amultivariate multiple test procedure. We illustrate our theoretical results by considering two particular classes of multiple test problems of practical relevance in detail,namely multiple tests for components of a mean vector and multiple support tests.

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