| 1. |
- Bodnar, Olha, senior lecturer, 1979-, et al.
(författare)
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Recent advances in shrinkage-based high-dimensional inference
- 2022
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Ingår i: Journal of Multivariate Analysis. - : Elsevier. - 0047-259X .- 1095-7243. ; 188
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Tidskriftsartikel (refereegranskat)abstract
- Recently, the shrinkage approach has increased its popularity in theoretical and applied statistics, especially, when point estimators for high-dimensional quantities have to be constructed. A shrinkage estimator is usually obtained by shrinking the sample estimator towards a deterministic target. This allows to reduce the high volatility that is commonly present in the sample estimator by introducing a bias such that the mean-square error of the shrinkage estimator becomes smaller than the one of the corresponding sample estimator. The procedure has shown great advantages especially in the high-dimensional problems where, in general case, the sample estimators are not consistent without imposing structural assumptions on model parameters.In this paper, we review the mostly used shrinkage estimators for the mean vector, covariance and precision matrices. The application in portfolio theory is provided where the weights of optimal portfolios are usually determined as functions of the mean vector and covariance matrix. Furthermore, a test theory on the mean-variance optimality of a given portfolio based on the shrinkage approach is presented as well.
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| 2. |
- Bodnar, Taras, et al.
(författare)
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A test for the global minimum variance portfolio for small sample and singular covariance
- 2017
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Ingår i: AStA Advances in Statistical Analysis. - : Springer. - 1863-8171 .- 1863-818X. ; 101:3, s. 253-265
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Bodnar, Taras, et al.
(författare)
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Bayesian estimation of the global minimum variance portfolio
- 2017
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Ingår i: European Journal of Operational Research. - : Elsevier. - 0377-2217 .- 1872-6860. ; 256:1, s. 292-307
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Bodnar, Taras, et al.
(författare)
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Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions
- 2019
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Ingår i: Scandinavian Journal of Statistics. - : John Wiley & Sons. - 0303-6898 .- 1467-9469. ; 46:2, s. 636-660
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix-variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large-dimensional asymptotic regime where the dimension p and the sample size n approach to infinity such that p/n → c ∈ [0, +∞) when the sample covariance matrix does not need to be invertible and p/n → c ∈ [0, 1) otherwise.
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| 5. |
- Bodnar, Taras, et al.
(författare)
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Discriminant analysis in small and large dimensions
- 2020
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Ingår i: Theory of Probability and Mathematical Statistics. - Providence, Rhode Island : American Mathematical Society (AMS). - 1547-7363 .- 0094-9000. ; 100, s. 21-41
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Tidskriftsartikel (refereegranskat)abstract
- We study the distributional properties of the linear discriminant function under the assumption of normality by comparing two groups with the same covariance matrix but different mean vectors. A stochastic representation for the discriminant function coefficients is derived, which is then used to obtain their asymptotic distribution under the high-dimensional asymptotic regime. We investigate the performance of the classification analysis based on the discriminant function in both small and large dimensions. A stochastic representation is established, which allows to compute the error rate in an efficient way. We further compare the calculated error rate with the optimal one obtained under the assumption that the covariance matrix and the two mean vectors are known. Finally, we present an analytical expression of the error rate calculated in the high-dimensional asymptotic regime. The finite-sample properties of the derived theoretical results are assessed via an extensive Monte Carlo study.
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| 6. |
- Bodnar, Taras, et al.
(författare)
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Singular inverse Wishart distribution and its application to portfolio theory
- 2016
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Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X .- 1095-7243. ; 143, s. 314-326
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Bodnar, Taras, et al.
(författare)
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Tangency portfolio weights for singular covariance matrix in small and large dimensions : estimation and test theory
- 2019
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Ingår i: Journal of Statistical Planning and Inference. - : Elsevier. - 0378-3758 .- 1873-1171. ; 201, s. 40-57
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive the finite-sample distribution of the estimated weights of the tangency portfolio when both the population and the sample covariance matrices are singular. These results are used in the derivation of a statistical test on the weights of the tangency portfolio where the distribution of the test statistic is obtained under both the null and the alternative hypotheses. Moreover, we establish the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sample size increase to infinity. The theoretical findings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index.
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| 8. |
- Muhinyuza, Stanislas, et al.
(författare)
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A test on the location of the tangency portfolio on the set of feasible portfolios
- 2020
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Ingår i: Applied Mathematics and Computation. - : Elsevier BV. - 0096-3003 .- 1873-5649. ; 386
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Tidskriftsartikel (refereegranskat)abstract
- Due to the problem of parameter uncertainty, specifying the location of the tangency portfolio (TP) on the set of feasible portfolios becomes a challenging task. The set of feasible portfolios is a parabola in the mean-variance space with optimal portfolios lying on its upper part. Using statistical test theory, we want to decide if the tangency portfolio is mean-variance efficient, i.e. if it belongs to the upper limb of the efficient frontier. In the opposite case, the investor would prefer to invest into the risk-free asset or into the global minimum variance portfolio which lies in the vertex of the set of feasible portfolios. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we suggest a test on the location of the tangency portfolio on the set of feasible portfolios. The distribution of the test statistic is derived under both hypotheses, which we use to assess the power of the test and construct a confidence interval. Moreover, out-of-sample performance of the test is evaluated based on real data. The robustness to the assumption of normality is investigated via an extensive simulation study where we show that the new test is robust to the violation of the normality assumption and can also be used for heavy-tailed stochastic models. Moreover, in an empirical study we apply the developed theory to real data. We find that when the sample size is relatively large and a stable period is present on the market, then the mean-variance efficiency of the tangency portfolio can be statistically justified.
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| 9. |
- Alfelt, Gustav
(författare)
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Modeling Realized Covariance of Asset Returns
- 2019
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, which consists of two papers, we consider the modeling of positive definitive symmetric matrices, in particular covariance matrices of financial asset returns. The return covariance matrix describes the magnitude in which prices of financial assets tend to change over time, and how price changes between different assets are related. It is an instrumental quantity in many financial applications, and furthermore, an important component in understanding the dynamics present prior to and during times of financial turbulence, such as the 2008 financial crisis.In the first paper, we provide several goodness-of-fit tests applicable to models driven by a centralized Wishart process. To apply such a distributional assumption has become a popular way of modeling the stochastic properties of time-series of realized covariance matrices for asset returns. The paper includes a simulation study that aims to investigate how the tests perform under model uncertainty stemming from parameter estimation. In addition, the presented methods are used to evaluate the fit of a typical model of realized covariance adapted to real data on six stocks traded on the New York Stock Exchange.The second paper considers positive definite and symmetric random matrices of the exponential family. Under certain conditions for this class of distributions, we derive the Stein-Haff identity. Furthermore, we determine this identity in the case of the matrix-variate gamma distribution and apply it in order to present an estimator that outperforms the maximum likelihood estimator in terms of Stein's loss function. Finally, a small simulation study is conducted to support the theoretical results.
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| 10. |
- Alfelt, Gustav, 1985-
(författare)
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Modeling the covariance matrix of financial asset returns
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The covariance matrix of asset returns, which describes the fluctuation of asset prices, plays a crucial role in understanding and predicting financial markets and economic systems. In recent years, the concept of realized covariance measures has become a popular way to accurately estimate return covariance matrices using high-frequency data. This thesis contains five research papers that study time series of realized covariance matrices, estimators for related random matrix distributions, and cases where the sample size is smaller than the number of assets considered.Paper I provides several goodness-of-fit tests for discrete realized covariance matrix time series models that are driven by an underlying Wishart process. The test methodology is based on an extended version of Bartlett's decomposition, allowing to obtain independent and standard normally distributed random variables under the null hypothesis. The paper includes a simulation study that investigates the tests' performance under parameter uncertainty, as well as an empirical application of the popular conditional autoregressive Wishart model fitted to data on six stocks traded over eight and a half years.Paper II derives the Stein-Haff identity for exponential random matrix distributions, a class which for example contains the Wishart distribution. It furthermore applies the derived identity to the matrix-variate gamma distribution, providing an estimator that dominates the maximum likelihood estimator in terms of Stein's loss function. Finally, the theoretical results are supported by a simulation study.Paper III supplies a novel closed-form estimator for the parameters of the matrix-variate gamma distribution. The estimator appears to have several benefits over the typically applied maximum likelihood estimator, as revealed in a simulation study. Applying the proposed estimator as a start value for the numerical optimization procedure required to find the maximum likelihood estimate is also shown to reduce computation time drastically, when compared to applying arbitrary start values.Paper IV introduces a new model for discrete time series of realized covariance matrices that obtain as singular. This case occur when the matrix dimension is larger than the number of high frequency returns available for each trading day. As the model naturally appears when a large number of assets are considered, the paper also focuses on maintaining estimation feasibility in high dimensions. The model is fitted to 20 years of high frequency data on 50 stocks, and is evaluated by out-of-sample forecast accuracy, where it outperforms the typically considered GARCH model with high statistical significance.Paper V is concerned with estimation of the tangency portfolio vector in the case where the number of assets is larger than the available sample size. The estimator contains the Moore-Penrose inverse of a Wishart distributed matrix, an object for which the mean and dispersion matrix are yet to be derived. Although no exact results exist, the paper extends the knowledge of statistical properties in portfolio theory by providing bounds and approximations for the moments of this estimator as well as exact results in special cases. Finally, the properties of the bounds and approximations are investigated through simulations.
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