What investors often wish to insure is that the maximum possible loss of their portfolios falling below a certain value. Namely, the maximum possible loss that a portfolio will lose under normal market fluctuations, with a given confidence level, over a certain time horizon, it is known shortly as “value at risk (VaR).” However, when it comes to the hedging strategy taking in the derivative markets for the minimum VaR, many investors simply thinking it is a hedging ratio in one at beginning, then a lot of effective model came out from both academia and industry over the years.We pioneer deriving a combined and dynamic hedging model- exponentially weighted moving average-generalized autoregressive **conditional** **heteroskedasticity** (GARCH) (1,1)-M applicable to the real financial markets based on previous studies. The results in this paper turn out that the model we build is not only excellent for the pursuit for the minimum VaR but also practical for the actual situation where the variances of financial price data are time-varying.In this paper we calculate the optimal decay factor 0.93325 which is the best match to the Hu-Shen 300 stock index market, withdraw uniform 0.9400, and use the Cornish-Fisher function to correct the quantile of the normal distribution, get the final hedging ratios and the minimum VaR.

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Volatility and correlation are important metrics of risk evaluation for financial markets worldwide. The latter have shown that these tools are varying over time, thus, they require an appropriate estimation models to adequately capture their dynamics. Multivariate generalized autoregressive **conditional** **heteroskedasticity** (GARCH) models were developed for this purpose and have known a great success. The purpose of this article is to examine the performance of multivariate GARCH models to estimate variance covariance matrices in application to 10 years of daily stock prices in Moroccan stock markets. The estimation is done through the most widely used multivariate GARCH models, dynamic **conditional** correlation (DCC) and Baba, Engle, Kraft and Kroner (BEKK) models. A comparison of estimated results is done using multiple statistical tests and with application to volatility forecast and value at risk (VaR) calculation. The results show that BEKK model performs better than DCC in modeling variance covariance matrices and that both models failed to adequately estimate VaR.

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Various statistical models to forecast gold prices are available in the literatures such as Autoregressive Integrated Moving Average (ARIMA) (Pitigalaarachchi et al., 2016, Davis et al., 2014, Ali Khan, 2013), Generalized Autoregressive **Conditional** **Heteroskedasticity** (GARCH) family models (Mahalingam et al., 2015, Gencer and Musoglu, 2014, Trück and Liang, 2012, Ping et al., 2013, Sopipan et al., 2012, Kumari and Tan, 2014), Hybrid models of ARIMA and GARCH (Yaziz et al., 2013, Ahmad et al., 2015).

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Our models for US bonds approximate a nonlinear adjustment mecha- nism via a simple variable addition to an otherwise ordinary VAR model. Moreover, incorporating **conditional** **heteroskedasticity** can be done via stan- dard methods. Hence, they are much less complex to estimate, from a com- putational point of view, than multivariate threshold models and can also be used when the number of time series is greater than two.

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ARCH-type models, which originate from econometrics, give us an appropriate framework for studying this prob- lem. Volatility (i.e. time-varying variance) clustering, in which large changes tend to follow large changes, and small changes tend to follow small changes, has been well recognized in financial time series. This phenomenon is called **conditional** **heteroskedasticity**, and can be modeled by ARCH-type models, including the ARCH model proposed by Engle (1982) and the later extension GARCH (general- ized ARCH) model proposed by Bollerslev (1986), etc. Ac- cordingly, when a time series exhibits autoregressive condi- tionally **heteroskedasticity**, we say it has the ARCH effect or GARCH effect. ARCH-type models have been widely used to model the ARCH effect for economic and financial time series.

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In a recent contribution to the financial econometrics literature, Chu et al. (2017) provide the first examination of the time-series price behaviour of the most popular cryptocurrencies. However, insufficient attention was paid to correctly diagnosing the distribution of GARCH innovations. When these data issues are controlled for, their results lack robustness and may lead to either underestimation or overestimation of future risks. The main aim of this paper therefore is to provide an improved econometric specification. Particular attention is paid to correctly diagnosing the distribution of GARCH innovations by means of Kolmogorov type non-parametric tests and Khmaladze’s martingale transformation. Numerical computation is carried out by implementing a Gauss-Kronrod quadrature. Parameters of GARCH models are estimated using maximum likelihood. For calculating P-values, the parametric bootstrap method is used. Further reference is made to the merits and demerits of statistical techniques presented in the related and recently published literature. Keywords: Autoregressive **conditional** **heteroskedasticity** (ARCH), generalized autoregressive **conditional** **heteroskedasticity** (GARCH), market volatility, nonlinear time series, Khmaladze transform.

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A first step to introduce a functional heteroskedastic framework in the tradition of Engle (1982) was un- dertaken in H¨ormann et al. (2013). These authors found conditions for the existence of functional ARCH(1) processes, for which the **conditional** variance depends on the whole (intra-day) path of the previous obser- vation. In the spirit of Bollerslev (1986), this paper introduces a functional model in which the **conditional** volatility function of the present observation is given as a functional linear combination of the (intra-day) paths of both the past squared observation and its volatility function. The following definition is central. Definition 1.1. A sequence of random functions (y i : i ∈ Z ) is called a functional GARCH process of orders

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4.3 In- sample evaluation and parameters estimates of all GARCH models for Malaysian Sukuk return series, using the entire dataset and assuming three different distribut[r]

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The data set we will analyze is the General Index of Athens Stock Exchange 1 (hereafter GI). There are totally 2982 observations from 31 July 1987 to 30 July 1999. Define y t = log ( p t p t − 1 ) as the continuously compounded rate of return for GI at time t ( t = 1 ,..., 2981 ) , where p t is the daily closing price of GI. In the following lines, we estimate a model to examine several issues previously investigated in the economics and financial literature namely a) the relation between the level of market risk and required return, b) the asymmetry between positive and negative returns in their effects on **conditional** variance, c) fat tails in the **conditional** distribution of returns d) the contribution of non-trading days to volatility e) the inverse relation between volatility and serial correlation and f) the non-synchronous trading.

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N LM AC H is the repli ation of fat tails; the estimation results indi ate however that this pro ess is preferred to ARCH models using a student-t as onditional distribution only in one [r]

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Recently, volatility modeling has been a very active and extensive research area in empirical finance and time series econometrics for both academics and practitioners. GARCH models have been the most widely used in this regard. However, GARCH models have been found to have serious limitations empirically among which includes, but not limited to; failure to take into account leverage effect in financial asset returns. As such so many models have been proposed in trying to solve the limitations of the leverage effect in GARCH models two of which are the EGARCH and the TARCH models. The EGARCH model is the most highly used model. It however has its limitations which include, but not limited to; stability conditions in general and existence of unconditional moments in particular depend on the **conditional** density, failure to capture leverage effect when the parameters are of the same signs, assuming independence of the innovations, lack of asymptotic theory for its estimators et cetera. This paper therefore is geared at extending/improving on the EGARCH model by taking into account the said empirical limitations. The main objective of this paper therefore is to develop a volatility model that solves the problems faced by the exponential GARCH model. Using the Quasi-maximum likelihood estimation technique coupled with martingale techniques, while relaxing the independence assumption of the innovations; the paper has shown that the proposed asymmetric volatility model not only provides strongly consistent estimators but also provides asymptotically efficient estimators.

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This paper presents a new method for estimating linear triangular models where mea- surement error or endogeneity e¤ects one of the regressors. Examples of these types of models include (i) asset return factor models where one of the factors is either measured inaccurately or an imperfect proxy for the true, latent, factor or (ii) restricted VAR models from the empirical macro literature. The traditional approach to identifying these models is through the use of exclusionary restrictions on parameters a¤ecting the **conditional** mean or, equivalently, through the assumed existence of valid instruments. In contrast, this pa- per demonstrates how a certain parametric speci…cation of the **conditional** heteroskedasticty (CH) a¤ecting the structural errors to the triangular system allows for identi…cation in the absence of traditional instruments. As such, this paper contributes to the literature on iden- ti…cation through various forms of **heteroskedasticity**. Based on this identi…cation result, a continuous updating estimator (CUE) is proposed that is shown to be consistent and asymp- totically normal. It is also robust to many moments bias. This estimator performs well in Monte Carlo experiments under moment existence criteria that allow for varying fat-tailed processes. The estimator is also applied to estimating market betas from the familiar CAPM, o¤ering promising results for the ability of these estimates to price expected returns in the cross-section.

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In this paper, a number of univariate and multivariate ARCH models are presented and their estimation is discussed. The main features of what seem to be most widely used ARCH models are described with emphasis on their practical relevance. It is not an attempt to cover the whole of the literature on the technical details of the models, which is very extensive. (A comprehensive survey of the most important theoretical developments in ARCH type modeling covering the period up to 1993 was given by Bollerslev et al. (1994)). The aim is to give the broad framework of the most important models used today in the economic applications. A careful selection of references is provided so that more detailed examination of particular topics can be made by the interested reader. In particular, an anthology of representations of ARCH models that have been considered in the literature is provided (section 2), including representations that have been proposed for accounting for relationships between the **conditional** mean and the **conditional** variance (section 3) and methods of estimation of their parameters (section 4). Generalizations of these models suggested in the literature in multivariate contexts are also discussed (section 5). Section 6 gives a brief description of other methods of estimating volatility. Finally, section 7 is concerned with interpretation and implementation issues of ARCH models in financial applications.

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The autoregressive **conditional** **heteroskedasticity** (ARCH) model introduced by Fredrick Engel in 1982 is the first model that assumed that volatility is not constant. ARCH models are commonly employed in modelling financial time series that exhibit time-varying volatility clustering, that is, period of swings interspersed with periods of relative calm. (Grek, 2014; Wikipedia, 2017).

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Our analysis differs from the standard literature considering the consistency of the PML esti- mator of an index function interpretable as a **conditional** expectation, a **conditional** median and/or a **conditional** variance [see Gouriéroux et al. (1984), Bollerslev and Wooldridge (1992), Gouriéroux and Monfort (1995), Chapter 8]. It also differs from analyses where the shape of the true and/or pseudo error distribution is constrained. Due to the lack of ex-ante interpretation of the index in model (1.1), and to the lack of restrictions on the pseudo-distribution, we introduce a parameter λ allowing for consistency of PML estimators of θ(β 0 ) for any pseudo-distribution satisfying minimal regularity conditions. Our analysis is thus not limited to a class such as the linear exponential family [as in Gouriéroux et al. (1984)], the conditionally heteroskedastic models [as in Francq et al. (2011), Fan et al. (2014)], or to multivariate conditionally heteroskedastic dynamic regression mod- els [as in Fiorentini, Sentana (2016)]. Section 2 illustrates the usefulness of the consistency result by considering the examples of regression model with **conditional** **heteroskedasticity**, Cholesky ARCH model, and model with homogenous spatial interactions. Section 3 derives the main result of the paper, that is, the consistency of the PML estimator. In the rest of the paper, we focus on linear (affine) transformation models. These models are studied in Section 4. We discuss the choice of the appropriate parametrization for the examples of Section 2 and provide other applications to network models, to the multivariate regression model with **conditional** **heteroskedasticity**, and to models for observations of volatility matrices. Assumptions and identification issues are discussed in Section 5, with special focus on the Cholesky ARCH model and on an interaction model. Section 6 derives the asymptotic distribution of the PML estimator for a class of linear commutative transformation models. Section 7 concludes. Additional results and proofs are collected in an on-line Appendix.

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This paper investigates the trend in the monthly real price of oil between 1990 and 2008 with a generalized autoregressive **conditional** **heteroskedasticity** (GARCH) model. Trend and volatility are estimated jointly with the maximum likelihood estimation. There is long persistence in the variance of oil price shocks, and a GARCH unit root (GUR) test can potentially yield a significant power gain relative to the augmented Dickey-Fuller (ADF) test. After allowing for nonlinearity, the evidence supports a deterministic trend in the price of oil. The deterministic trend implies that influence of a price shock is transitory and policy efforts to restore a predictable price after a shock would be unwarranted in the long run.

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Keywords: Air Quality Surveillance, Atmospheric Pollution, Autoregressive Conditional Heteroskedasticity modelling, Control Charts, Diag-aVECH, Multivariate Statistical Process Monitorin[r]

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In this paper an asymmetric autoregressive **conditional** **heteroskedasticity** (ARCH) model and a Levy-stable distribution are applied to some well-known financial indices (DAX30, FTSE20, FTSE100 and SP500), using a rolling sample of constant size, in order to investigate whether the values of the estimated parameters of the models change over time. Although, there are changes in the estimated parameters reflecting that structural properties and trading behaviour alter over time, the ARCH model adequately forecasts the one-day-ahead volatility. A simulation study is run to investigate whether the time variant attitude holds in the case of a generated ARCH data process revealing that even in that case the rolling-sampled parameters are time-varying.

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Non-fundamentalness arises when observed variables do not contain enough infor- mation to recover structural shocks. This paper propose a new test to empirically detect non-fundamentalness, which is robust to the **conditional** **heteroskedasticity** of unknown form, does not need information outside of the specified model and could be accomplished with a standard F-test. A Monte Carlo study based on a DSGE model is conducted to examine the finite sample performance of the test. I apply the proposed test to the U.S. quarterly data to identify the dynamic effects of supply and demand disturbances on real GNP and unemployment.

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This research applies recursive Structural Vector Auto Regression (SVAR) model with short- run restriction by testing two kinds of shocks: monetary and volatility. The first SVAR estimates the shock of contractionary monetary policy on Taiwan’s key monthly macroeconomic variables including exports, CPI, exchange rate, money supply, and Taiwan Weighted Stock Exchange (TWSE) Index. The second SVAR estimates the shock of Generalized Autoregressive **Conditional** **Heteroskedasticity** (GARCH) volatility of TWSE on Taiwan’s daily exchange rate, overnight interbank loan rate, and Taiwan Government Bond Index (TGBI). The first SVAR model shows that prize puzzle has been evident in Taiwan. The second SVAR model found flight to safety into bond market after the volatility shock in equity market. Combining the results of both models and based on other literature reviews, we can also infer that effectiveness of contractionary and expansionary monetary policy exhibits non- linearity in Taiwan.

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