First, we implement smooth transition **GARCH** **model** with the second-order logistic function on the return spread for the in-sample period using equations 4 and 5. For optimizing the parameters, we consider a sample size of 2,000 for simulating the data using MCMC **model**. In order to implement this **model**, we use Metropolis-Hastings method and consider the burn-in sample of 1,000 and a total sample of 50,000 iterations. We only take the second half of each iteration (Martin, A. D., Quinn, K. M., & Park, J. H., 2011).

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This paper extends the SEMIFAR **model** to a SEMIFAR-**GARCH** **model**, so that conditional heteroskedasticity in financial time series can also be modelled by the SEMIFAR **model**. A semiparametric estimation procedure is proposed. Asymptotic results on the SEMIFAR **model** are extended to the current proposal. It is shown in particular that the same asymptotic results obtained in Beran and Feng (2001) for the SEMIFAR **model** with i.i.d. normal innovations hold for the SEMIFAR-**GARCH** **model** under the much weaker condition that the **GARCH** innovation process has finite fourth moments. These theoretical results and the important property that the estimates of the FARIMA and **GARCH** parameter vectors are independent of each other, allow us to apply the data-driven SEMIFAR algorithms to estimate the trend and the FARIMA parameters in the SEMIFAR-**GARCH** **model**. It is proposed to estimate the **GARCH** parameter from the approximated **GARCH** innovations calculated by inverting the final residuals. Data examples show that the proposed algorithm works well. Further extensions of the SEMIFAR **model** are also possible. For instance, a seasonal component can also be introduced into the mean function to **model** daily periodicity in high-frequency financial data.

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In the next step, the heteroscedasticity test is implemented and it shows that our traffic data contains volatility periods. Thus, we can proceed to build **GARCH** **model** based on the multiplicative seasonal ARIMA **model** that we achieved. Following the steps mentioned above, **GARCH** (1, 1) assuming GED formulates which has the smallest measure of forecast error, i.e. MAE and RMSE, should be chosen as the one with the most accurate fit of the time series **model**. MAE indicates that the average difference between the forecast and the observed value of the **model** is 0.080042, while RMSE and MAPE are 0.131390 and 276.0843, respectively.

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Based on empirical evidences for some of FTSE100 companies, this paper examines two **GARCH** models with jumps to evaluate the impact of news flow intensity on stock volatility. First it will be considered the well-known **GARCH** **model** with jumps proposed in [1]. Then we will introduce the **GARCH**-Jumps **model** augmented with news intensity and obtain some empirical results. The main assumption of the **model** is that jump intensity might change over time and that jump intensity depends linearly on the number of news. It is not clear whether news adds any value to a jump-**GARCH** **model**. However, the comparison of the values of log likelihood shows that the **GARCH**-Jumps **model** augmented with news intensity performs slightly better than ”pure” **GARCH** or the **GARCH** **model** with Jumps. We

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We call our **model** GARCHX models since the constant in **GARCH** models is replaced by an extra term, i.e., the lagged cross-sectional market volatility, and thus the GARCHX **model** does not need additional parameters. Note that the cross-sectional market volatility is lagged to make the GARCHX mode conditional. The GARCHX **model** is simple, but includes information on some important factors, especially the market factor, via the cross-sectional volatility. Our **model** is a special case of the multivariate Factor-**GARCH** **model** of Engle, Ng, and Rothchild (1990) in the sense that only one factor, i.e., the market factor is included. Note that the main problem in the multivariate **GARCH** models is that the number of parameters to be estimated grows very fast and we need to impose some restrictions to make the conditional covariance matrix positive definite. Several methods have been suggested to solve these problems; see chapter 12, Campbell, Lo, and MacKinlay (1997).

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The research on asset volatility in financial market is the such as capital assets pricing, financial derivatives pricing, and financial risk measurement. The premise of quantitative financial analysis is to accurately measure and predict asset quality. Therefore, the measurement are a hotspot of research all the time. To measure and predict asset volatility accurately, Bollerslev built a generalised ARCH (**GARCH**) **model** based on the ARCH **model**. Then, **GARCH** **model** was extended. The **GARCH** process is often preferred by financial professionals because it provides a more real-world context than other forms when trying to predict the prices and general process for a **GARCH** **model** involves three steps. The first is to estimate a

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In this paper we propose a modification of the standard **GARCH** **model**, which allows time varying persistence in the volatility dynamics. Namely, a lower degree of persistence is assigned to extreme returns taking place in highly volatile periods rather than to shocks of lower magnitude occurring in tranquil periods. However, the **model** structure could be easily modified to account for more general situations in which variations in the volatility persistence originate from different sources such as, for example, leverage effects and intraday or intraweek seasonal effects in volatility. It is important to note that, on an observational ground, our **model** is able to reproduce most of the stylized features for which RS-**GARCH** **model** have been designed but, at the same time, it is still characterized by tractable inference procedures.

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term u t is i.i.d. with zero mean and unit variance. In order to ensure easily the positivity of the conditional variance we impose the restrictions ω > 0, α ≥ 0 and β ≥ 0. For simplicity, we assume that µ t is constant. The sum α + β measures the persistence of a shock to the conditional variance in equation (2). When a **GARCH** **model** is estimated using daily or higher frequency data, the estimate of this sum tends to be close to one, indicating that the volatility process is highly persistent and the second moment of the return process may not exist. However it was argued that the high persistence may artificially result from regime shifts in the **GARCH** parameters over time, see Diebold (1986), Lamoureux and Lastrapes (1990), and Mikosch and Starica (2004), among others.

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By now, the **GARCH** **model** of Bollerslev (1986) has been extended to several classes of multivariate **GARCH** models (see Bauwens, Laurent and Rombouts (2006)). **GARCH** itself has come a long way since Robert Engle’s (1982) pioneering paper on the ARCH. Multivariate **GARCH** (MGARCH) research focuses on ways of simplifying the variance- covariance matrix where the number of parameters to be estimated explodes for higher dimensions making estimation costly and computationally intractable. The approaches to simplify the estimation of the parameters of the variance-covariance matrix are now well- developed although suggestions have been made to come up with models that account for economic theory as a basis for simplifying this matrix, Diebold (2004).

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Abstract. We analyze the possibility of improving the prediction of stock market indicators by adding information about public mood ex- pressed in Twitter posts. To estimate public mood, we analysed frequen- cies of 175 emotional markers - words, emoticons, acronyms and ab- breviations - in more than two billion tweets collected via Twitter API over a period from 13.02.2013 to 22.04.2015. We explored the Granger causality relations between stock market returns of S&P500, DJIA, Ap- ple, Google, Facebook, Pfizer and Exxon Mobil and emotional markers frequencies. We found that 17 emotional markers out of 175 are Granger causes of changes in returns without reverse effect. These frequencies were tested by Bayes Information Criteria to determine whether they provide additional information to the baseline ARMAX-**GARCH** **model**. We found Twitter data can provide additional information and managed to improve prediction as compared to a **model** based solely on emotional markers.

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In the papers [11] and [12] authors analyze the impact of extraneous sources of information (viz. news and trade volume) on stock volatility by considering some augmented Generalized Autoregressive Conditional Heteroscedasticity (**GARCH**) models. Following the study of [13], it was supposed that trading volume can be considered as a pro- portional proxy for information arrivals to the market. Also it was considered the daily number of press releases on a stock (news intensity) as an alternative explanatory variable in the basic equation of **GARCH** **model**.

This paper seeks to uncover the non-linear characteristics of uncertainty underlying the US inflation rates over the period 1971-2015 within a regime-switching framework. Accordingly, we employ two variants of a Markov regime-switching **GARCH** **model**, one with normally distributed errors (MS-**GARCH**-N) and another with t-distributed errors (MS-**GARCH**-t), and compare their relative in-sample as well as out-of-sample performances with those of their standard single-regime counterparts. Consistent with the findings in existing studies, both of our regime-switching models are successful in identifying the year 1984 as the breakpoint in inflation volatility. Among other interesting results is a new finding that the process of switching to the low volatility regime started around April, 1979 and continued until mid 1983. This time frame is matched with the period of aggressive monetary policy changes implemented by the then Fed chairman Paul Volcker. As regards the performance in forecasting uncertainty, for shorter horizons spanning 1 to 5 months, MS-**GARCH**-N forecasts are found to outperform all other models whereas for 8 to 12-month ahead forecasts MS-**GARCH**-t appears superior.

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This study explores the potential of hybrid ARIMA-**GARCH** **model** in handling volatile data. The price of crude oil data will be used for this purpose. This report consists of five chapters. Chapter 1 presents the research framework. It starts with the introduction of time series and followed by the statement of problem, the objectives of the study, the scope of the study and the significance of the study.

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We propose an estimation method for the so–called power **GARCH** **model** with stable Paretian (SP) innovations. The method is based on the integrated weighted squared distance between the characteristic function of the SP distribution and an empirical counterpart computed from the **GARCH** residuals. Under fairly standard conditions the estimator was shown to be consistent. Its asymptotic distribution however proved non–standard and in fact splits into two parts: One regular Gaussian distribution cor- responding to the parameters of the SP law, while the other part of the distribution corresponding to the **GARCH** parameters is singular and in particular it is concen- trated on a hyperplane. For the regular Gaussian part it was possible to even optimize the choice of the weight function so that the estimators of the SP parameters attain minimum variance.

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discussed in the paper by M. Paolella and L. Taschini [1.32] and authors suggest two ways. The first way is to do only the unconditional analysis of the tails of the data, in other words, to avoid the zeros-problem, because the zeros are in the centre. The second one is a conditional analysis using mixed-normal and mixed- stable **GARCH** models. In this paper we don’t compare results of estimation using MS ARMA-**GARCH**-M and mixed **GARCH** **model**. We just consider results of regime switching **GARCH** **model** and a **model** from the previous paper [1.19] where we excluded zeros and estimated ARMA-t-**GARCH** **model** as O. Sabbaghi and N. Sabbaghi [1.35].

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The log-**GARCH** **model** provides a flexible framework for the modelling of economic uncertainty, financial volatility and other positively valued variables. Its exponen- tial specification ensures fitted volatilities are positive, allows for flexible dynamics, simplifies inference when parameters are equal to zero under the null, and the log- transform makes the **model** robust to jumps or outliers. An additional advantage is that the **model** admits ARMA-like representations. This means log-**GARCH** mod- els can readily be estimated by means of widely available software, and enables a vast range of well-known time-series results and methods. This chapter provides an overview of the log-**GARCH** **model** and its ARMA representation(s), and of how estimation can be implemented in practice. After the introduction, we delineate the univariate log-**GARCH** **model** with volatility asymmetry (“leverage”), and show how its (nonlinear) ARMA representation is obtained. Next, stationary covariates (“X”) are added, before a first-order specification with asymmetry is illustrated em- pirically. Then we turn our attention to multivariate log-**GARCH**-X models. We start by presenting the multivariate specification in its general form, but quickly turn our focus to specifications that can be estimated equation-by-equation – even in the presence of Dynamic Conditional Correlations (DCCs) of unknown form. Next, a multivariate non-stationary log-**GARCH**-X **model** is formulated, in which the X-covariates can be both stationary and/or nonstationary. A common critique directed towards the log-**GARCH** **model** is that its ARCH terms may not exist in the presence of inliers. An own Section is devoted to how this can be handled in prac- tice. Next, the generalisation of log-**GARCH** models to logarithmic Multiplicative Error Models (MEMs) is made explicit. Finally, the chapter concludes.

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consistency of the QMLE was obtained under only the second-order moment condi- tion. Unlike Weiss (1986) and Pantula (1989) for the univariate case, the asymptotic normality of the QMLE for the vector ARCH **model** requires only the second-order moment of the unconditional errors, and the finite fourth-order moment of the condi- tional errors. The asymptotic normality of the QMLE for the vector ARMA-ARCH **model** was proved using the fourth-order moment, which is an extension of Weiss (1986) and Pantula (1989). For the general vector ARMA-**GARCH** **model**, the asymptotic normality of the QMLE requires the assumption that the sixth-order moment exists. Whether this result will hold under some weaker moment condi- tions remains to be proved.

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term u t is i.i.d. with zero mean and unit variance. In order to ensure easily the positivity of the conditional variance we impose the restrictions ω > 0, α ≥ 0 and β ≥ 0. For simplicity, we assume that µ t is constant. The sum α + β measures the persistence of a shock to the conditional variance in equation (2). When a **GARCH** **model** is estimated using daily or higher frequency data, the estimate of this sum tends to be close to one, indicating that the volatility process is highly persistent and the second moment of the return process may not exist. However it was argued that the high persistence may artificially result from regime shifts in the **GARCH** parameters over time, see Diebold (1986), Lamoureux and Lastrapes (1990), and Mikosch and Starica (2004), among others.

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In this study, we focus on the class of BL-**GARCH** models, which is initially introduced by Storti & Vitale [1] in order to handle leverage effects and volatility clustering. First we illustrate some properties of BL-**GARCH** (1, 2) **model**, like the positivity, stationarity and marginal distribution; then we study the statistical inference, apply the composite likelihood on panel of BL-**GARCH** (1, 2) **model**, and study the asymptotic behavior of the estimators, like the consistency property and the asymptotic normality.