This thesis studies the time-varying dependence between financial markets using a Gaussian Copula model. The study focuses on the dynamic Gaussian Copula dependence parameter 𝜌𝑖𝑗,𝑡, which characterizes the dependence relationship between two financial markets after filtering out the dynamic features of their marginal distributions. The thesis combines GARCH-type marginal modelling, probability integral transforms, and likelihood-based estimation of several static and dynamic Gaussian copula specifications.

The first part introduces the theoretical background on time series, copulas, maximum likelihood estimation, Generalized Autoregressive Score (GAS) updating, and GARCH marginal models. The second part uses simulation studies to explain why the observed behaviour of a time series can be affected by marginal mean and volatility dynamics and why copula parameters should be estimated using filtered pseudo-observations rather than the raw return series. The third part performs a Monte Carlo study to assess the finite sample properties of selected specifications of time-varying dependence. It concludes that estimation precision increases with sample size, while the complexity of recursive and score-driven models makes them computationally more demanding.

The empirical analysis uses data from five financial markets: stocks, silver, gold, Bitcoin, and an energy market. The results show that cross-market dependence is not constant over time. Different market pairs are characterized by different dependence specifications. This thesis uses Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to select the dependence specifications. Using AIC, GAS-type models are frequently selected, while BIC more often favours simpler constant or recursive specifications.

Finally, this thesis studies the minimum-variance portfolio allocation and CoVaR-based downside risk analysis by using the estimated dependence paths. The dynamic minimum-variance portfolio results show that time-varying dependence estimates can reduce portfolio volatility compared with an equal-weight portfolio. The ΔCoVaR analysis shows that downside risk transmission changes over time, differs by direction, and varies substantially across market pairs.

Overall, the thesis provides a framework for time-varying Gaussian copula models to analyse dynamic financial market dependence and its implications for portfolio and risk management.
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Copulas are useful for studying how several random variables depend on each other, because they allow the dependence structure to be separated from the individual marginal distributions. This makes them particularly useful when the main interest is in whether the dependence itself changes over time. In this thesis, I study this problem for a multivariate time series whose dependence is described by a copula from a fixed parametric family. The main goal is to test whether the copula parameter remains constant over the whole sample or changes once at an unknown point in time. For each possible change-point, a model with one common copula parameter is compared with a model that allows different parameters before and after the split. Since the change location is unknown, the final test statistic is obtained by taking the largest value over a trimmed set of candidate change-points.

I first consider the ideal case, where the marginal distributions are known. In this setting, the true copula observations are available directly. The main difficulty is that the test does not use only one fixed split point, but searches over many possible locations. Therefore, the likelihood approximations need to hold uniformly over all candidate change-points. To do this, I first control the score process and the observed information matrices uniformly over the candidate splits. These results are then used to describe the estimation errors of the maximum likelihood estimators. From there, the likelihood ratio statistic can be rewritten as a quadratic form involving a centered score process. This process converges to a standard multivariate Brownian bridge, which gives the limiting distribution of the maximized test statistic.

The feasible case is more difficult because the marginal distributions are unknown. The true copula observations are then replaced by rank-based pseudo-observations. For every possible change-point, the ranks are recomputed separately in the two resulting segments. Because of this, the score terms are no longer simple partial sums of independent observations. To handle this, I use the sequential localrank empirical copula process and combine it with a function-indexed integration-by-parts argument. This makes it possible to derive the weak limit of the local-rank score process and, from this, the limiting distribution of the feasible likelihood ratio statistic. The resulting limit still has a Brownian-bridge-type structure, but its covariance also reflects the additional effect of estimating the marginal distributions through ranks.

Finally, I study the finite-sample behaviour of the test through simulations using Gaussian and Student 𝑡 copulas. The results show a clear general pattern: the test becomes more powerful when the sample size increases or when the change in dependence becomes larger. Small changes are harder to detect, especially for the feasible procedure based on local ranks. At the same time, the difference between the ideal and feasible procedures becomes smaller for larger samples and stronger changes in dependence. Overall, the thesis develops the theoretical justification for a likelihood-ratio change-point test in copula models and studies how the procedure behaves in finite samples.

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Functional Principal Component Analysis (FPCA) is a powerful method for identifying the dominant modes of variation within a large collection of curves, thereby revealing their underlying low-dimensional structure. However, the existing FPCA framework assumes that the functional data are directly observed. This thesis develops a new theoretical framework for applying FPCA when the functional objects are estimated rather than observed, focusing specifically on estimated conditional Kendall's tau (CKT) curves. The analysis is supported by additional theoretical work on conditional U-statistics, which form the basis for establishing the asymptotic properties of the proposed method. In the theoretical setting where the CKT curves are estimated over the entire domain, we prove the consistency of the associated Hilbert-Schmidt integral operator and under suitable regularity and asymptotic conditions, we show that the spectral decomposition of the estimated operator converges to the true one with an error rate of  O(h² + (nh^d)^-½). We then extend the analysis to the practically relevant setting in which the curves are observed on a finite grid of N points and establish convergence of the spectral properties with a rate of  O(N^-1/d + h² + (nh^d)^-½). This illustrates the bias-variance trade-off with respect to the choice of the bandwidth h, as well as the numerical bias introduced by the discretization step, N^-1/d.
We support our theoretical results with an empirical study of global financial assets, applying ARMA-GARCH filtering and conditioning on different markets to recover the most dominant modes of variation within a large dataset of CKT curves.
We find that all CKT curves can be approximately summarized by only two degrees of freedom: a constant baseline level, representing the average conditional Kendall's tau, and a parabolic component that contrasts extreme market conditions with more typical periods, capturing the contagion effects.
Ultimately, this framework provides a foundation for future applications, including evaluating the simplifying assumption. ... Read more

When a population is divided into subgroups – such as men and women, or young and old individuals – researchers sometimes decide in advance how many people to sample from each group. If these fixed proportions do not reflect the true population proportions, the sample average becomes systematically biased, making any conclusions drawn from it unreliable. Surprisingly, this distortion cannot be corrected simply by choosing a stricter significance threshold: under the null hypothesis, the test continues to reject too often, regardless of how the rejection rule is adjusted.

This thesis investigates how such bias can be detected and corrected. When the magnitude of the bias is known, it can be adjusted for directly, restoring the reliability of the results. If the bias is not known exactly but is bounded within a certain range, two alternative strategies can still guarantee reliable conclusions, albeit at the cost of reduced sensitivity to small effects. However, if no information about the distortion is available, the situation becomes hopeless: no method can reliably distinguish genuine findings from random fluctuations.

The thesis therefore demonstrates that some knowledge about the degree of sampling imbalance is not merely beneficial, but essential for drawing meaningful statistical conclusions. ... Read more

Voluntary carbon markets are at an early stage of development, characterized by low and irregular trading frequency. Such limited activity results in extended periods of unchanged prices and a high incidence of zero returns, making voluntary carbon markets a typical example of illiquid financial markets. To model dependence in such settings, we propose a multivariate zero-inflated GARCH-X model. The model extends the existing zero-inflated GARCH model to a multivariate setup, incorporating both a GARCH-X component with binary trading indicators as exogenous covariates and a time-step specification that updates only when trades occur.

The multivariate extension incorporates two different types of dependence. First, cross-dependence in trading activity is modeled using Markov networks applied to binary trading indicators. Second, cross-dependence in returns is analyzed using a copula-GARCH framework applied to residuals, with residuals corresponding to zero returns treated as undefined values. To make copula methods applicable to zero-inflated data, we introduce a joint probability integral transform approach. In this construction, the univariate marginals are defined conditional on the simultaneous trading activity of each asset, rather than conditioning only on each asset’s own trading activity. We prove that this method yields a consistent copula estimator when applied to the subset of observations where all assets have simultaneous non-zero trading activity. Dependence is quantified using both unconditional and conditional Kendall’s tau, estimated via kernel-based methods.

Theoretical results include consistency and asymptotic normality of the quasi-maximum likelihood estimator for the multivariate model under stationary covariates. The empirical study covers seven voluntary carbon credits and six conventional financial assets. We find no significant dependence between carbon credits and conventional assets, but observe strong correlation within the carbon market, especially among nature-based credits. Results suggest that voluntary carbon markets may operate independently of more liquid assets and are influenced by peer pricing due to a lack of standardization. ... Read more

This report investigates the enumeration of labeled directed acyclic graphs (DAGs) under various structural constraints, extending an inclusion–exclusion recurrence introduced by R.W. Robinson. Starting from the enumeration of general DAGs via out-point partitioning, the recursive method is adapted to count more specialized classes such as DAGs with a fixed number of arcs or out-points. The Robinson method is also applied to create a formula for the enumeration of rooted directed trees, polytrees, and a special triangle structure. For each of these constrained graph classes, both the closed-form expressions and the derived recursive enumeration formulas are explained, showcasing the versatility and mathematical elegance of Robinson’s technique. A main focus of this report is the derivation and interpretation of the Robinson recurrence that adjusts the choose-attach model while using the local attachment rules. This report also presents visual illustrations of the original Robinson method and its adaptations. The results of these formulas are shown in graphs and tables to show the exponential growth of each class. The results bring together several enumeration problems in graph theory under a single recursive framework, offering both theoretical insight and practical enumeration formulas. The contributions of this thesis provide an analysis of DAG enumeration problems, bridging theoretical insights and practical relevance. These results have important implications for a wide range of applications, including Bayesian networks, causal inference modeling, scheduling, and network design. Finally, promising directions for future research are mentioned, emphasizing potential extensions to more complex graph structures and advanced enumeration methodologies. ... Read more

This thesis explores the growing complexity of contemporary financial markets, which is a consequence of a world that is increasingly interconnected and correlated. This evolution highlights the necessity of understanding and accurately modeling these underlying relationships, which translates into the need of incorporating more complex models into portfolio optimization, breaking away from Harry Markowitz’s foundational Portfolio Optimization Theory. While Markowitz’s model has been effective, the complexity of modern financial instruments demands more sophisticated approaches. This study focuses on the application of copulas and vine models to portfolio optimization, aiming to understand how these advanced models can enhance the optimization process by accurately capturing dependencies among financial assets. In particular, this thesis investigates the benefits of integrating copula-GARCH models, a combination of time series modelling where the residuals are modelled using copulas or vine models, into portfolio theory. Through this approach, the research aims to extend existing knowledge and highlight the specific advantages provided by these models in portfolio optimization. ... Read more

Statistical inference of low-frequency time series is a challenge present in various fields, such as financial risk management and weather forecasting. Practical difficulties arise due to the scarcity of non-overlapping observations. The “direct method”, which directly uses the available low-frequency data to construct estimators, often results in inaccurate estimations.

In this thesis, we propose a novel “simulation-based method” for statistical inference of low-frequency time series that result from the aggregation of a higher-frequency time series over a period of time. We start by estimating the distribution of this higher-frequency process. We then simulate a large number of paths from this estimated distribution. By independently aggregating each simulated path, we generate corresponding low-frequency data. This provides us with a large simulated dataset of the low-frequency process, which enables us to apply estimation procedures and bypass the limitations posed by the shortage of original low-frequency data.

We also provide a theoretical framework and propose three families of estimators constructed from the estimated higher-frequency distribution, analyzing their properties under additional assumptions. Through a comprehensive simulation study, we compare the simulation-based method with the traditional direct method across different scenarios and objectives. While our study focuses on the marginal distributions of low-frequency processes, the simulation-based method’s applicability extends to joint distributions across multiple time points. This research offers a robust method for parameter estimation when faced with limited low-frequency data. ... Read more

Dividing a population into subgroups and conducting research on this population including the subgroups comes with a challenge. This stratified sampling relies on information about the share of the subgroups in the population. Sometimes the proportions in the sample are not taken equal to the true proportions of the population. This can be corrected through the use of particular estimators taking these proportions into account.
In this thesis, different estimators for the true population mean and variance are defined and examined in terms of bias and variance in the case of two subgroups. Weighing the measurements according to the true proportions creates unbiased estimators for both the mean and the variance. These unbiased estimators are compared with other, biased, estimators, including naive ones in which the influence of different subgroups is not taken into account. The naive estimators are not only biased, they also have a variance of the same order as the unbiased ones. When the true proportions are not available, one can only take a guess. A guess lying close to the true proportions leads to a smaller bias and therefore a better estimator. This underlines the importance of obtaining sufficient knowledge about the population. ... Read more

Moving through immersive virtual reality (VR) is commonly achieved by physically walking in the real room or using other techniques like an omnidirectional treadmill or walk-in-place. Roomscale walking is most similar to normal walking but is limited by physical space. However, other techniques can cause user experience issues such as VR sickness, balance problems, and feeling unnatural. Newer locomotion techniques are available such as powered VR shoes, which are shoes with motorized treadmills underneath. While walking, the shoes drive the user backward and actively negate the forward velocity, reducing the needed physical space. Yet, there is little evidence of the effect of powered VR shoes on user experience, which part of this work addresses. Additionally, previous research shows mismatched VR motion (optical flow) can increase VR sickness, cognitive load, and break presence. However, full-gait locomotion studies often focus on the device, neglecting optical flow, and what is the best body part to control optical flow direction is still an open question. Therefore, we first developed a novel algorithm to convert leg-based walking to optical flow while walking on VR shoes, which may also be used for other full-gait locomotion techniques. We conducted a study with 20 participants to find which of four optical flow implementations, differing in VR motion direction, resulted in the best user experience. These direction conditions were based on body-mounted trackers: i) head orientation, ii) hip orientation, iii) standing foot velocity direction, and iv) average orientation of both feet. Head-oriented walking resulted in a significantly worse user experience compared to other conditions, with no significant differences among any other conditions. Additionally, we found no effect of optical flow on VR sickness, Mental Effort, and Presence, contrary to previous studies, but instead significant differences in Ease of Use, Input responsiveness, and Appropriateness, and indication that other user experience factors might be impacted more. Finally, we discovered that walking on VR shoes, although not completely comfortable and natural, was learnable within 10 minutes for all participants under 60 years old. ... Read more

Through the expansion of large-scale service systems and the exponential growth of data generated by complex IT infrastructure components, gaining a comprehensive overview of the different levels of service within an IT system has become increasingly challenging. In particular, this brought to the fore the question from a large commercial bank of how IT monitoring data streams generated by their complex IT infrastructure can be associated with one another.

In more detail, the data from the monitoring stream consists (among other things) of a message and a time stamp. Moreover, the monitoring data stream of this bank consists of two natures of information. These natures are either automatically generated warnings in the form of events or unplanned outages, referred to as incidents. The events and incidents are referred to as arrivals. As a first requirement to obtain better granularity, both event and incident messages with similar semantics should be grouped together. To this extent, the message component from each arrival is transformed into a numerical vector, the dimension of the obtained vector is reduced, and the collection of vectors is clustered. Once the individual arrival from the IT monitoring data stream is attached to a cluster based on their message component, the arrival is assigned a mark. This mark consists of a combination of the assigned cluster, the nature, and three different levels of service from the IT architecture on which the arrival occurred.

From a mathematical point of view, we can now view the monitoring data stream from different levels of service as a marked point process. Our primary focus centers on a specific category of marked point processes, known as marked Hawkes processes. Given the marked Hawkes process, we assume that each arrival from the IT monitoring data stream results in an instantaneous increase in the probability of some other arrivals in the near future. From here, we estimate the excitation matrix, representing the instantaneous increases among all assigned marks. Once the estimated excitation matrix is obtained, we decompose it into the different levels of service as defined within the mark. In particular, the decomposition has been performed through means of hierarchical linear models. Finally, the decomposition resulted in a comprehensive overview of the excitation behavior in large-scale service systems. This overview can directly be incorporated into the field of Software Architecture in order to uncover associations within complex IT infrastructures. ... Read more

The pair-copula Bayesian network (PCBN) is a Bayesian network (BN) where the conditional probability functions are modeled using pair-copula constructions. By assigning bivariate conditional copulas to the arcs of the BN, one finds a proper joint density which can flexibly model all kinds of dependence structures. It is a known problem that the PCBN may require numerical integration to perform computations such as sampling and likelihood-inference. To address this issue we propose novel restrictions on the graphical structure and assignment of copulas such that integration will not be required. The resulting restricted PCBN offers significant computational benefits. We establish how to estimate and conduct a structure search for the restricted PCBN. A simulation study shows that a restricted PCBN is able to model non-Gaussian dependence structures more accurately than the widely used Gaussian Bayesian network. ... Read more

In this thesis, we explore the structure of consistent bootstrap statistics in hypothesis testing. Bootstrap, as a very useful technique when theoretical distributions are not available or when the sample size is small, enjoys a lot of interest from applied statisticians. Historically, guidelines for performing Bootstrap have been proposed. One of the guidelines proposed is to center the bootstrap statistic around the true statistic, calculated from the original sample. The second, is to perform resampling in a way such that the new sample reflects the hypothesis tested. However, both of the guidelines are proposed based mostly on an empirical point of view. In this project, we show that the calculation of the bootstrap statistic is directly related to the way the new sample is generated. We describe the specific conditions under which the Bootstrap statistic should or should not be centered around the true. As mentioned the resampling scheme that is picked directly influences this choice. The motivation is derived from the independence test and the same arguments apply to the regression slope test. Finally, we provide a generalized setting where a consistent bootstrap statistic is provided, based on the resampling scheme that is picked. ... Read more

This paper presents a novel approach for the estimation of conditional multivariate cumulative distribution functions (CDFs) within a nonparametric framework. To achieve this, we introduce a binary random variable that indirectly represents conditional CDFs and construct a dataset by pairing input vectors with the binary variables. We developed a general approach compatible with various machine learning methods.
We have also developed an R package that facilitates the application of machine learning methods. This package leverages a range of machine learning models, including decision trees, neural networks, random forests, and bagging neural networks. Through systematic learning of the intricate relationships between the covariates and the binary variables, we effectively estimate conditional CDFs.
To enhance the accuracy and reliability of the estimated CDFs, we incorporate a rearrangement technique which transforms the estimated functions into monotonic representations, aligning them more closely with the target CDFs and mitigating potential inconsistencies [6].
Through simulations, we evaluate the performance of the estimation approach under various scenarios and assess the impact of sample size and correlation on estimation accuracy, using Mean Integrated Squared Error as a key performance metric. The results demonstrate the effectiveness and robustness of the methodology in estimating conditional CDFs, providing a valuable tool for capturing complex dependencies in multivariate data, with potential applications in risk assessment, finance, and environmental modeling.
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This thesis aims to improve Coolblue's direct demand estimation model for substitutable products. Their current model consists of three sub-models which all provide their direct demand estimations. For every product, the direct demand is taken from one of the sub-models based on their performance in estimating the sales. The sub-models are the mean, linear and expectation-maximisation (EM) model. The linear model gives the most accurate expectations, whereas the mean model scores the lowest. Therefore, we have improved the mean model's estimations by creating a new estimator. Furthermore, we have investigated if an out-of-stock (OOS) period influences the sales and, therefore, the direct demand estimations. From this investigation, we conclude that for a part of the products, the sales are affected by OOS periods. However, these results are dependent on how they are investigated. Moreover, the OOS periods' influences are as likely to be positive as negative on the sales. Therefore it is challenging to react to these influences. ... Read more

In this thesis, we present simulation studies of a non-parametric estimator, proposed by Liebscher (2005). This estimator uses a well-known non-parametric estimator called kernel density estimator. Non-parametric estimation is used when the parametric distribution of a given dataset is unknown. This technique is then applied with an assumption that the distribution in question has a density, so that its density can be estimated. The density that we are interested in, belongs to a class of elliptical densities which has a similar contour shape as the Gaussian distribution. One of an example of such density is the density of multivariate normal distribution. Liebscher’s estimator uses elliptical density to circumvent the ’curse of dimensionality’. The ’curse of dimensionality’ often appears in non-parametric estimation. When a high dimensional dataset is applied to a nonparametric estimator, the convergence rate of the estimator becomes slow. This is what we refer to as the ’curse of dimensionality’. Liebscher’s estimator circumvents this ’curse’ by assuming that the multi-dimensional dataset is sampled from an elliptical distribution. The estimator then transforms the dataset into one-dimensional dataset, so that we can use the univariate kernel density estimation, instead of the multivariate ones. We use Liebscher’s estimator to estimate the generator of elliptical distribution. The generator is a positive real-valued function. Liebscher’s estimator depends on two parameters: the bandwidth parameter and the tuning parameter around the boundary. In this thesis, we investigate how these two parameter influence the performance of the estimator. We start with the case when the simulated dataset is sampled from the standard multivariate normal distribution. Then, we apply the estimator on a different elliptical distribution with different generator. As it turns out, finding the parameter such that the estimator gives a minimal error is a difficult task. This is because the area where the error is small, depends on the generator. We also observe that as we increase the dimension of the simulated dataset, the computational time of the estimator increases as well. Lastly, the estimator is applied to Wisconsin breast cancer dataset. The estimator is used to study the accuracy of a Bayes’ classifier is based on estimated posterior probabilites. From the study, it appears that the role of the tuning parameter is smaller in comparison the bandwidth parameter, in changing the accuracy of the classifier. ... Read more

In this thesis, we have examined conditional dependence in a financial context using conditional Kendall’s tau (CKT). The conditional Kendall’s tau is a measure of concordance between two random variables given some covariates. This thesis covers topics related to conditional Kendall’s tau such as (conditional) copulas. We study non-parametric estimators of the conditional Kendall’s tau using kernel density estimation and kernel regression. An application of the non-parametric estimator to the returns of thirteen different financial assets is finally provided. The assets consist of stock indices, bonds, futures and exchange rates. Further, we apply Principal Component Analysis (PCA) on the conditional Kendall’s tau data matrix to increase the interpretability. In general, it seems that conditional dependence is slightly larger in the tails for all assets. Moreover, the conditional dependence for each group of assets is discussed. It seems that the degree
of the conditional dependence relates to characteristics of an asset such as geographical properties and type of asset. ... Read more

Kendall’s tau and conditional Kendall’s tau matrices are multivariate (conditional) dependence measures between the components of a random vector. For large dimensions, available estimators are computationally expensive and can be improved by averaging. Under structural assumptions on the underlying Kendall’s tau and conditional Kendall’s tau matrices, we introduce new estimators that have a significantly reduced computational cost while keeping a similar error level. In the unconditional setting we assume that, up to reordering, the underlying Kendall’s tau matrix is block-structured with constant values in each of the off-diagonal blocks. The estimators take advantage of this block structure by averaging over (part of) the pairwise estimates in each of the off-diagonal blocks. Derived explicit variance expressions show their improved efficiency. In the conditional setting, the conditional Kendall’s tau matrix is assumed to have a constant block structure, independently of the conditioning variable. Conditional Kendall’s tau matrix estimators are constructed similarly as in the unconditional case by averaging over (part of) the pairwise conditional Kendall’s tau estimators. We establish their joint asymptotic normality, and show that the asymptotic variance is reduced compared to the naive estimators. Then, we perform a simulation study which displays the improved performance of both the unconditional and conditional estimators. Finally, the estimators are used for estimating the value at risk of a large stock portfolio; backtesting illustrates the obtained improvements compared to the previous estimators.
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The COVID-19 crisis heavily affected financial stock markets. In March 2020 stock prices dropped immensely and markets became extremely volatile. In this report we model three European stock markets before and during the COVID-19 crisis to determine whether the dynamics of financial markets changed structurally compared to previous periods. Stock markets play a big role in our economy and can cause economic disruption when they crash. Therefore, it can be very useful to understand the dynamics of the stock market. Econometricians are nowadays often asked to model the non-constant volatility (conditional heteroscedasticity) of financial time series. This report uses Generalised AutoRegressive Conditionally Heteroscedastic (GARCH) models that are known for their precise modeling of conditional heteroscedasticity. They are also known for their ability to capture the key stylised facts, the common empirical properties applicable to all types of stock markets. This report includes general definitions and characteristics of the GARCH(p, q) process and covers topics such as autocorrelation of returns, kurtosis, leptokurticity and volatility clustering. For estimating the GARCH models this report uses the statistical program R which estimates the parameters by the Quasi-Maximum Likelihood Estimation (QMLE) method. We modeled different periods before and during COVID-19 and compared the estimated parameters with the corresponding 95% confidence intervals. To test the accuracy of the estimations we performed parametric bootstrapping. Throughout the report, models for the Dutch Amsterdam Exchange (AEX) index, the French Cotation Assistée Continu (CAC 40) index and the German Deutsche Aktien (DAX) index are analysed and compared. It seems that European markets may experience the impact of stock market crashes differently. The DAX index shows significant changes in the dynamics of the stock market due to the COVID-19 crisis whereas the AEX and CAC 40 index do not. ... Read more