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Time reversibility of random process
Paclík, Ondřej ; Hlubinka, Daniel (advisor) ; Hudecová, Šárka (referee)
Random processes can be used to describe the evolution of a real systems over time. Discrete-time Markov chains are random processes that meet special assumptions, but they still have a lot of practical applications. Some chains have the property that it is impossible to tell if they are being observed when the passage of time is reversed. We call such chains time reversible. In this paper, we define a time reversible Markov chain with discrete time, we show how it can be verified that a given chain is time reversible, and we introduce basic properties and examples of time reversible chains. At the same time, we apply the knowledge of time reversibility to the problem of finding the stationary distribution of specific Markov chains. 1
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Independence testing for series of Poisson variables
Jurčo, Tomáš ; Hudecová, Šárka (advisor) ; Hlávka, Zdeněk (referee)
This thesis deals with tests of independence for time series of identically distributed Poisson random variables. In the introductory part, important terms and definitions are defined, in particular the autocorrelation function, its estimates and INAR(1) model. Three types of tests of independence are described in the thesis - tests based on estimates of the autocorrelation function, simple runs test and tests based on contingency tables. These tests are compared in a simulation study under the null hypothesis of independence and under the alternative of INAR(1) model. 1
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Distributed lag models
Dian, Patrik ; Cipra, Tomáš (advisor) ; Hudecová, Šárka (referee)
The aim of this bachelor thesis is to unite the theory about distribu- ted lag models and autoregressive distributed lag model, which includes lagged dependent variables and application of these models on real data. The properties of these models are also presented. Dynamic models are highly used for financial and economic data because of their ability to capture lagged effect on dependent variable. As a similar topic there are mentioned models of intervention analysis which are used to examine the external effects on time series and to model the in- terventions using indicator variables. Finally, applications of mentioned models on two data sets are introduced and analysis of the effect of coronavirus pandemic on time series is demonstrated. 1
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Robust estimation of autocorrelation function
Lain, Michal ; Hudecová, Šárka (advisor)
The autocorrelation function is a basic tool for time series analysis. The clas- sical estimation is very sensitive to outliers and can lead to misleading results. This thesis deals with robust estimations of the autocorrelation function, which is more resistant to the outliers than the classical estimation. There are presen- ted following approaches: leaving out the outliers from the data, replacement the average with the median, data transformation, the estimation of another coeffici- ent, robust estimation of the partial autocorrelation function or linear regression. The thesis describes the applicability of the presented methods, their advantages and disadvantages and necessary assumptions. All the approaches are compared in simulation study and applied to real financial data. 1
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Combining multivariate volatility forecasts in portfolio optimization
Šípka, Stanislav ; Hendrych, Radek (advisor) ; Hudecová, Šárka (referee)
The selection of the best-performing model is always a challenge when solving financial-economic problems. The final model might prove to be suboptimal even after a short time if the economic climate changes suddenly. This thesis aims to construct a final model capable of estimating large-scale covariance matrices via the utilization of time-varying weights. A set of multivariate GARCH mod- els to be used as an input in the final combined estimate is used to introduce a weighting scheme based on the metrics of risk-adjusted return of the individ- ual model portfolios. As large-scale modeling often faces problems connected with the underlying dimensionality, the composite likelihood approach to model parameter estimation is proposed as a solution and compared to the standard maximum likelihood and its SVD modification. The resulting weighted covari- ance matrix prediction is used to construct optimal portfolios and their properties are compared in an empirical study. The thesis is concluded by noting the real-life limitation and possible improvements of the defined investing methodology. 1
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Beta regression
Štěpán, Marek ; Hudecová, Šárka (advisor) ; Omelka, Marek (referee)
The thesis deals with a beta regression model suitable for analysing data whose range of values is the interval (0, 1). The model assumes a conditional beta distribution for the response given covariates, and its structure is similar to generalised linear models. The model is defined and its basic properties are investigated. The asymptotic distribution of the maximum likelihood estimates is provided. A possible extension to situations where the response in the data attains one of the boundary values is considered and referred to as c-inflated beta regression model. For both models, statistical inference and model diagnostics are discussed. The practical part of the thesis involves two Monte Carlo studies and two real data analyses. The first simulation study compares the performance of the global goodness-of-fit measures for link selection, while the second study explores various approaches to the analysis of the inflated beta distribution response. Alternative initial values are proposed for the cases where the algorithm did not converge. The practical usage of the model is illustrated on a model of proportions of tertiary educated people in European countries, and the proportion of household income spent on education in the Philippines. 1
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ARFIMA time series models
Vdovičenko, Martin ; Hudecová, Šárka (advisor) ; Prášková, Zuzana (referee)
The thesis deal with long-memory processes which are defined by several ways. The main concern is dedicated to ARFIMA model, to its basic properties and its application. Next, graphical, semiparametric and parametric estimation methods of ARFIMA parameters are described in detail. Five selected R packages are introduced that are suitable for modeling long-memory processes. We discuss their basic functions with description of input arguments and output. Finally, the application of the packages on real data is discussed according to results of~each function. Data sample comes from the Nile River and represents its yearly minimal water levels. Powered by TCPDF (www.tcpdf.org)
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Selected methods of time series analysis with STATISTICA
Indrová, Magdalena ; Hudecová, Šárka (advisor) ; Zichová, Jitka (referee)
This work deals with the use of STATISTICA software for the basic analysis of time series. The thesis is focused on time series decomposition, mainly on the trend elimination. First, the basic methods of the analysis are described theoretically, namely, trend modeling using mathematical curves (polynomial, exponential, logistic and Gompertz) and adaptive approach (moving averages, simple exponential smoothing and Holt's method). These methods are then applied to three selected data sets (unnamed bank's balance sheet from 1998 to 1993, ship construction trends between 1820 and 1997, and CZK/EUR Exchange rate from 1998 to 2012). All analytical procedures are described in detail and individual program outputs are thoroughly explained and commented.
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