| 1. |
- Alfelt, Gustav, et al.
(författare)
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Goodness-of-fit tests for centralized Wishart processes
- 2020
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Ingår i: Communications in Statistics - Theory and Methods. - : Informa UK Limited. - 0361-0926 .- 1532-415X. ; 49:20, s. 5060-5090
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present several goodness-of-fit tests for the centralized Wishart process, a popular matrix-variate time series model used to capture the stochastic properties of realized covariance matrices. The new test procedures are based on the extended Bartlett decomposition derived from the properties of the Wishart distribution and allows to obtain sets of independently and standard normally distributed random variables under the null hypothesis. Several tests for normality and independence are then applied to these variables in order to support or to reject the underlying assumption of a centralized Wishart process. In order to investigate the influence of estimated parameters on the suggested testing procedures in the finite-sample case, a simulation study is conducted. Finally, the new test methods are applied to real data consisting of realized covariance matrices computed for the returns on six assets traded on the New York Stock Exchange.
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| 2. |
- 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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| 3. |
- Alfelt, Gustav, 1985-, et al.
(författare)
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Singular Conditional Autoregressive Wishart Model for Realized Covariance Matrices
- 2022
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Ingår i: Journal of business & economic statistics. - : Taylor & Francis Group. - 0735-0015 .- 1537-2707. ; 41:3, s. 833-845
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Tidskriftsartikel (refereegranskat)abstract
- Realized covariance matrices are often constructed under the assumption that richness of intra-day return data is greater than the portfolio size, resulting in nonsingular matrix measures. However, when for example the portfolio size is large, assets suffer from illiquidity issues, or market microstructure noise deters sampling on very high frequencies, this relation is not guaranteed. Under these common conditions, realized covariance matrices may obtain as singular by construction. Motivated by this situation, we introduce the Singular Conditional Autoregressive Wishart (SCAW) model to capture the temporal dynamics of time series of singular realized covariance matrices, extending the rich literature on econometric Wishart time series models to the singular case. This model is furthermore developed by covariance targeting adapted to matrices and a sector wise BEKK-specification, allowing excellent scalability to large and extremely large portfolio sizes. Finally, the model is estimated to a 20-year long time series containing 50 stocks and to a 10-year long time series containing 300 stocks, and evaluated using out-of-sample forecast accuracy. It outperforms the benchmark models with high statistical significance and the parsimonious specifications perform better than the baseline SCAW model, while using considerably less parameters.
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| 4. |
- Bodnar, Taras, et al.
(författare)
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Quantile-based optimal portfolio selection
- 2021
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Ingår i: Computational Management Science. - : Springer Science and Business Media LLC. - 1619-697X .- 1619-6988. ; :18, s. 299-324
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Tidskriftsartikel (refereegranskat)abstract
- In this paper the concept of quantile-based optimal portfolio selection is introduced and a specific portfolio connected to it, the conditional value-of-return (CVoR) portfolio, is proposed. The CVoR is defined as the mean excess return or the conditional value-at-risk (CVaR) of the return distribution. The portfolio selection consists solely of quantile-based risk and return measures. Financial institutions that work in the context of Basel 4 use CVaR as a risk measure. In this regulatory framework sufficient and necessary conditions for optimality of the CVoR portfolio are provided under a general distributional assumption. Moreover, it is shown that the CVoR portfolio is mean-variance efficient when the returns are assumed to follow an elliptically contoured distribution. Under this assumption the closed-form expression for the weights and characteristics of the CVoR portfolio are obtained. Finally, the introduced methods are illustrated in an empirical study based on monthly data of returns on stocks included in the S&P index. It is shown that the new portfolio selection strategy outperforms several alternatives in terms of the final investor wealth.
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| 5. |
- Thorsén, Erik, 1989-
(författare)
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Optimal portfolios in the high-dimensional setting : Estimation and assessment of uncertainty
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Financial portfolios and diversification go hand in hand. Diversification is one of, if not, the best risk mitigation strategy there is. If an investment performs poorly, then it will not impact the performance of the portfolio much due to diversification. Modern Portfolio Theory (MPT) is a framework for constructing diversified portfolios. However, MPT relies on unknown parameters that need to be estimated. By using estimates, estimation uncertainty is introduced to the allocation problem. This thesis contains five papers which provide results on how to deal with estimation uncertainty in very large sample portfolios from the MPT framework. These results provide tools to better understand the investment process and the empirical results that can be observed.Paper I explores all of the portfolios that can be placed in the framework of MPT. The paper provides the sampling distribution for all optimal portfolios and their characteristics. This is done by assuming that the returns follow a multivariate normal distribution. Furthermore, the high-dimensional asymptotic joint distribution for the quantities of interest is derived. A simulation study shows that the high-dimensional distribution can provide a good approximation to the finite sample one.Paper II continues on the idea of paper I. It considers the quadratic utility allocation problem from paper I with an additional risk-free asset in the portfolio. The portfolio is usually known as the Tangency Portfolio (TP). The distribution of the sample TP weights is derived under a skew-normal distribution. Results show that skewness implies a bias in the finite sample TP weights. The bias dissapears in the high-dimensional distribution.Paper III takes on a practical aspect of investing, namely how to transition from one portfolio to another. A reallocation scheme is developed, which minimizes the out-of-sample variance of the Global Minimum Variance (GMV) portfolio, given a holding portfolio. The holding portfolio is the portfolio which an investor currently owns. An extensive simulation study show that the reallocation scheme can provide accurate estimates of the portfolio variance. Furthermore, an empirical application shows that the scheme provides the smallest out-of-sample variance in comparison to a number of benchmarks. The theoretical results from this paper are implemented in the DOSPortfolio R-package.Paper IV derives properties of two different performance measures for three different high-dimensional GMV portfolio estimators. The measures are the out-of-sample variance and loss. The former is always used as an evaluation metric in empirical applications. The results show that the latter metric, the out-of-sample loss, does not need the same stringent assumptions as the out-of-sample variance in the high-dimensional setting. Using the out-of-sample loss, the performance of the three different portfolios can be ordered. This order is verified in a simulation study and an empirical application.Paper V extends the results of papers III and IV. It introduces Thikonov regularization to the GMV portfolio weights as well as linear shrinkage. A simulation study shows that the method is preferable to a number of benchmarks. Furthermore, an empirical application shows that it can provide the smallest out-of-sample variance and provide good characteristics for the portfolio weights.
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| 6. |
- Wängberg, Tobias, et al.
(författare)
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Shape-aware stochastic neighbor embedding for robust data visualisations
- 2022
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Ingår i: BMC Bioinformatics. - : Springer Science and Business Media LLC. - 1471-2105. ; 23:1
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Tidskriftsartikel (refereegranskat)abstract
- Background: The t-distributed Stochastic Neighbor Embedding (t-SNE) algorithm has emerged as one of the leading methods for visualising high-dimensional (HD) data in a wide variety of fields, especially for revealing cluster structure in HD single-cell transcriptomics data. However, t-SNE often fails to correctly represent hierarchical relationships between clusters and creates spurious patterns in the embedding. In this work we generalised t-SNE using shape-aware graph distances to mitigate some of the limitations of the t-SNE. Although many methods have been recently proposed to circumvent the shortcomings of t-SNE, notably Uniform manifold approximation (UMAP) and Potential of heat diffusion for affinity-based transition embedding (PHATE), we see a clear advantage of the proposed graph-based method.Results: The superior performance of the proposed method is first demonstrated on simulated data, where a significant improvement compared to t-SNE, UMAP and PHATE, based on quantitative validation indices, is observed when visualising imbalanced, nonlinear, continuous and hierarchically structured data. Thereafter the ability of the proposed method compared to the competing methods to create faithfully low-dimensional embeddings is shown on two real-world data sets, the single-cell transcriptomics data and the MNIST image data. In addition, the only hyper-parameter of the method can be automatically chosen in a data-driven way, which is consistently optimal across all test cases in this study.Conclusions: In this work we show that the proposed shape-aware stochastic neighbor embedding method creates low-dimensional visualisations that robustly and accurately reveal key structures of high-dimensional data.
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| 7. |
- Wängberg, Tobias
(författare)
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Statistical methods for molecular tissue-profiling
- 2023
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Through the advent of sequencing technologies, researchers have been able to map the entire human genome. The next major step is to classify the cells in order to obtain a cellular basis for health and disease. The previous ground for a taxonomy of the human cells have been mainly based on biological function and morphology. Now due to the advances in single-cell sequencing a molecular basis for classification is possible by grouping the cells based on their gene expression profiles. Development of statistical tools for the analysis of such data is the main focus of the two papers included in this thesis.In the first paper we propose a method that enables visualising the high dimensional sequencing data in a 2-dimensional scatter plot. It is demonstrated that the method is able to robustly reveal key structures of the data, such as clusters, hierarchical organisations of cell types and continuous developmental trajectories. The proposed method is shown to outperform existing state-of-the-art methods on simulated and real data sets. In the second paper we first propose a method for clustering single-cell sequencing data. The method is shown to more accurately group the data compared to the commonly used Louvain clustering method. The quality of the clustering is monitored by validation indices. Unlike the previous method, the proposed method outputs cluster membership probabilities that thus takes into account the uncertainty in assigning cells to cell types. The proposed method is automatic in the sense that their are no free parameters needing to be subjectively set by the user. In addition, a mathematical model is presented that the describes the stochastic evolution of the number of messenger RNA molecules present in a cell. This can serve as a basis for statistical inference and comparison of cell types. Finally, approximations of the stationary distribution are derived based on perturbation analysis.
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