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This dissertation contains five chapters. An introduction and a summary of the research are given in Chapter 1. The other four chapters present theoretical results on the moment structure of GARCH processes. Some chapters also contain empirical examples in order to illustrate applications of the theory. The focus, however, is mainly on statistical theory. Chapter 2 considers the moments of a family of first-order GARCH processes. First, a general condition of the existence of any integer moment of the absolute values of the observations is given. Second, a general expression of this moments as a function of lower-order moments is derived. Third, the kurtosis and the autocorrelation function of the squared and absolute-valued observations are derived. The results apply to a host of different GARCH parameterizations. Finally, the existence, or the lack of it, of the theoretical counterpart to the so-called Taylor effect for some members of this GARCH family is discussed. The asymmetric power ARCH model is a recent addition to time series models that may be used for predicting volatility. Its performance is compared with that of standard models of conditional heteroskedasticity such as GARCH. This has previously been done empirically. In Chapter 3 the same issue is studied theoretically using unconditional fractional moments for the A-PARCH model that are derived for the purpose. The role of the heteroskedasticity parameter of the A-PARCH process is highlighted and compared with corresponding empirical results involving autocorrelation functions of power-transformed absolute-valued return series.In Chapter 4, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given as well as an expression for the moment itself. Furthermore, the autocorrelation function of the centred and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.Nonnegativity constraints on the parameters of the GARCH(p,q) model may be relaxed without giving up the requirement of the conditional variance remaining nonnegative with probability one. Chapter 5 looks into the consequences of adopting these less severe constraints in the GARCH(2,2) case and its two second-order special cases, GARCH(2,1) and GARCH(1,2). This is done by comparing the autocorrelation function of squared observations under these two sets of constraints. The less severe constraints allow more flexibility in the shape of the autocorrelation function than the constraints restricting the parameters to be nonnegative. The theory is illustrated by an empirical example. / Revised versions of chapters 2-5 have been published as:He, C. and T. Teräsvirta, "Properties of moments of a amily of GARCH processes" in Journal of Econometrics, Vol. 92, No. 1, 1999, pp173-192.He, C. and T. Teräsvirta, "Statistical Properties of the Asymmetric Power ARCH Process" in R.F. Engle and H. White (eds) Cointegration, causality, and forecasting. Festschrift in honour of Clive W.J. Granger, chapter 19, pp 462-474, Oxford University Press, 1999.He, C. and T. Teräsvirta, "Fourth moment structure of the GARCH(p,q) process" in Econometric Theory, Vol. 15, 1999, pp 824-846.He, C. and T. Teräsvirta, "Properties of the autocorrelation function of squared observations for second order GARCH processes under two sets of parameter constraints" in Journal of Time Series Analysis, Vol. 20, No. 1, January 1999, pp 23-30.

The statements that include sufficient detail to identify the probability distributions of future prices are asset price dynamics. In this research, using the empirical methods that could explain the historical prices and discuss about how prices change we investigate various important characteristics of the dynamics of asset pricing. The volatility changes can explain very important facts about the asset returns. Volatility could gauge the variability of prices over time. In order to do the volatility modelling we use the conditional heteroskedasticitc models. One of the most powerful tools to do so is using the idea of autoregressive conditional heteroskedastic process or ARCH models, which fill the gap in both academic and practical literature. In this work we detect the asymmetric volatility effect and investigate long memory properties in volatility in Canadian stock market index, using daily data from 1979 through 2015. On one hand, we show that there is an asymmetry in the equity market index. This is an important indication of how information impacts the market. On the other hand, we investigate for the long-range dependency in volatility and discuss how the shocks are persistence. By using the long memory-GARCH models, we not only take care of both short and long memory, but also we compute the d parameter that stands for the fractional decay of the series. By considering the breaks in our dataset, we compare our findings on different conditions to find the most suitable fit. We present the best fit for GARCH, EGARCH, APARCH, GJR-GARCH, FIGARCH, FIAPARCH, and FIEGARCH models.

The thesis concerns the topic of forecasting using Neural Networks, particu- larly the return and volatility forecasting in the volatile period of Covid-19. The thesis uses adjusted close daily data from Jan 1, 2000, until Jan 1, 2021, of the S&P index and Prague Exchange Stock index (PX). The comparison was between the vanilla econometrical model, a neural network model, and a hybrid neural network model. Hybrid neural networks were constructed with an additional feature column of the fitted econometrical model. Additionally to comparing the prediction, a risk-return trade-o analysis of the forecasted series was conducted. The test period for all models was from Jan 1, 2020, until Jan 1, 2021, where predictions were made. During the test period, MSE be- tween predicted and true values was extracted and compared. The results are that the hybrid model outperformed both econometrical as well as only neural networks models. Furthermore, the risk-return trade-o forecast provided by the hybrid model fares better than the other ones. JEL Classification C53, C81 Keywords Financial Time Series, Forecasting, Neural Net- works, ARIMA, GARCH Title Forecasting with Neural Network during Covid- 19 Crisis Author's e-mail tiep.luud@gmail.com Supervisor's e-mail barunik@fsv.cuni.cz

A necessidade de antecipar e identificar variações de acontecimentos apontam para uma nova direção nos mercados de bolsa de valores e vem de encontro às análises das oscilações de preços de ativos financeiros. Esta necessidade leva a argumentar sobre novas alternativas na predição de séries temporais financeiras utilizando métodos de aprendizado de máquinas e vários modelos têm sido desenvolvidos para efetuar a análise e a previsão de dados de ativos financeiros. Este trabalho tem por objetivo propor o desenvolvimento de um modelo de previsão adaptativo baseado em um sistema SVR-wavelet híbrido, que integra modelos de wavelets e Support Vector Regression (SVR) na previsão de séries financeiras. O método consiste na utilização da Transformada de Wavelet Discreta (DWT) a fim de decompor dados de séries de ativos financeiros que são utilizados como variáveis de entrada do SVR com o objetivo de prever dados futuros de ativos financeiros. O modelo proposto é aplicado a um conjunto de ativos financeiros do tipo Foreign Exchange Market (FOREX), Mercado Global de Câmbio, obtidos a partir de uma base de conhecimento público. As séries são ajustadas gerando-se novas predições das séries originais, que são comparadas com outros modelos tradicionais tais como o modelo Autorregressivo Integrado de Médias Móveis (ARIMA), o modelo Autorregressivo Fracionário Integrado de Médias Móveis (ARFIMA), o modelo Autorregressivo Condicional com Heterocedasticidade Generalizado (GARCH) e o modelo SVR tradicional com Kernel. Além disso, realizam-se testes de normalidade e de raiz unitária para distribuição não linear, tal como testes de correlação, para constatar que as séries temporais FOREX são adequadas para a comprovação do modelo híbrido SVR-wavelet e posterior comparação com modelos tradicionais. Verifica-se também a aderência ao Expoente de Hurst por meio da estatística de Reescalonamento (R/S). / The necessity to anticipate and identify changes in events points to a new direction in the stock exchange market and reaches the analysis of the oscillations of prices of financial assets. This necessity leads to an argument about new alternatives in the prediction of financial time series using machine learning methods. Several models have been developed to perform the analysis and prediction of financial asset data. This thesis aims to propose the development of SVR-wavelet model, an adaptive and hybrid prediction model, which integrates wavelet models and Support Vector Regression (SVR), for prediction of Financial Time Series, particularly Foreign Exchange Market (FOREX), obtained from a public knowledge base. The method consists of using the Discrete Wavelets Transform (DWT) to decompose data from FOREX time series, that are used as SVR input variables to predict new data. The series are adjusted by generating new predictions of the original series, which are compared with other traditional models such as the Autoregressive Integrated Moving Average model (ARIMA), the Autoregressive Fractionally Integrated Moving Average model (ARFIMA), the Generalized Autoregressive Conditional Heteroskedasticity model (GARCH) and the traditional SVR model with Kernel. In addition, normality and unit root tests for non-linear distribution, and correlation tests, are performed to verify that the FOREX time series are adequate for the verification of SVR-wavelet hybrid model and comparison with traditional models. There is also the adherence to the Hurst Exponent through the statistical Rescaled Range (R/S).

This work aims to describe the method of recursive estimation of time series with conditional volatility, used mainly in finance. First, there are described the basic types of models with conditional heteroskedasticity (GARCH) and princi- ples of state-space modeling demonstrated by means of linear models AR and ARMA. Subsequently, there are derived algorithms for recursive estimation of parameters of the GARCH model and its possible modifications including the ones for which recursive estimation formulas have not been yet derived in lit- erature. These algorithms are tested in a simulation study, where their appli- cability in practice is investigated. Finally, we apply these algorithms to real high-frequency data from the stock exchange. The practical part is done us- ing the software Mathematica 11.3. The work also serves as an overview of the current state of online modeling of financial time series. 1

<p>Forecasting the stock market is a complex task, partly because of the random walk behavior of the stock price series. The task is further complicated by the noise, outliers and missing values that are common in financial time series. Despite of this, the subject receives a fair amount of attention, which probably can be attributed to the potential rewards that follows from being able to forecast the stock market.</p><p>Since artificial neural networks are capable of exploiting non-linear relations in the data, they are suitable to use when forecasting the stock market. In addition to this, they are able to outperform the classic autoregressive linear models.</p><p>The objective of this thesis is to investigate if the stock market can be forecasted, using the so called error correction neural network. This is accomplished through the development of a method aimed at finding the optimum forecast model.</p><p>The results of this thesis indicates that the developed method can be applied successfully when forecasting the stock market. Of the five stocks that were forecasted in this thesis using forecast models based on the developed method, all generated positive returns. This suggests that the stock market can be forecasted using neural networks.</p>

Forecasting the stock market is a complex task, partly because of the random walk behavior of the stock price series. The task is further complicated by the noise, outliers and missing values that are common in financial time series. Despite of this, the subject receives a fair amount of attention, which probably can be attributed to the potential rewards that follows from being able to forecast the stock market. Since artificial neural networks are capable of exploiting non-linear relations in the data, they are suitable to use when forecasting the stock market. In addition to this, they are able to outperform the classic autoregressive linear models. The objective of this thesis is to investigate if the stock market can be forecasted, using the so called error correction neural network. This is accomplished through the development of a method aimed at finding the optimum forecast model. The results of this thesis indicates that the developed method can be applied successfully when forecasting the stock market. Of the five stocks that were forecasted in this thesis using forecast models based on the developed method, all generated positive returns. This suggests that the stock market can be forecasted using neural networks.

Financial time-series forecasting has long been discussed because of its vitality for making informed investment decisions. This kind of problem, however, is intrinsically challenging due to the data dynamics in nature. Most of the research works in the past focus on artificial neural network (ANN)-based approaches. It has been pointed out that such approaches suffer from explanatory power and generalized prediction ability though. The objective of this research is thus to propose a hybrid approach for stock price forecasting. Independent component analysis (ICA) is employed to reveal the latent structure of the observed time-series and remove noise and redundancy in the structure. It further assists clustering analysis. Support vector regression (SVR) models are then applied to enhance the generalization ability with separate models built based on the time-series data of companies in each individual cluster. Two experiments are conducted accordingly. The results show that SVR has robust accuracy performance. More importantly, SVR models with ICA-based clustered data perform better than the single SVR model with all data involved. Our proposed approach does enhance the generalization ability of the forecasting models, which justifies the feasibility of its applications.