Univariate GARCH Research Articles

Estimation and asymptotic properties of a stationary univariate GARCH(p, q) process

Estimation and asymptotic properties of a stationary univariate GARCH(p, q) process

Discussion on the Effectiveness of the Copula-GARCH Method to Detect Risk of a Portfolio Containing Bitcoin

Since it was invented by Satoshi Nakamoto in 2008, Bitcoin has drawn considerable attention both from the financial industry and government supervisory departments, and there is no unanimity on Bitcoin’s nature in the academic field. Some people may think Bitcoin is more like an asset than a currency. And investors’ motivations for incorporating Bitcoin into their portfolios may vary. Might there be a better way to deal with the risk-detection issue associated with such a unique and ambiguous object? The copula-GARCH method has been proven in much of the literature to be a better way than the traditional ways to estimate the value at risk (VaR) of portfolios. When it comes to a portfolio containing Bitcoin, can it still maintain its superiority? In this study, gold and Ethereum were each used to construct a portfolio with Bitcoin. We collected a total of 2,246 daily adjusted closing prices from July 23, 2010, to March 12, 2019. As for the copula-GARCH model, we selected four constant and two time-varying copula models combined with GARCH Student-t residuals to fit the joint distribution of the two assets in the portfolios. The traditional methods refer to the historical simulation, the variance-covariance method, the EWMA method, and the univariate GARCH VaR method. We adopted each method to compute corresponding one-day VaRs. Our results indicated that for the portfolios containing Bitcoin and Ethereum, the copula-GARCH method performed better than traditional methods, while for the portfolio consisting of Bitcoin and gold, traditional methods performed better. Our results may suggest that the copula-GARCH method may not be suitable in the extremely low correlation case.

Modeling exchange rate volatility in the gambia using dynamic conditonal correlaton model

The relationship between different international stock markets is of importance for both financial practitioners and academicians in order to manage risks. Especially after the financial crisis, the pronounced financial contagion draws the public attention to look into such associations. However, measuring and modelling dependence structure becomes complicated when asset returns present non-linear, non-Gaussian and dynamic features. This paper examines the time-varying conditional correlations to the weekly exchange rate returns for the USD, EURO and GBP against the Gambian Dalasi (GMD) during the period 2000 to 2017. We use a dynamic conditional correlation (DCC) multivariate GARCH model. This model can be simplified by estimating univariate GARCH models for each return series, and then, using transformed residuals resulting from the first stage, estimating a conditional correlation estimator. DCC-GARCH model was implemented for two different assumptions of the error distribution; assuming Gaussian and Student t-distribution. Empirical results show substantial evidence of significant increase in conditional correlation. It is also clean that, the Student t-distributed errors better forecast the conditional correlation.

Hedge funds portfolio optimisation using a vine copula-GARCH-EVT-CVaR model

This paper investigates the conditional value-at-risk (CVaR) hedge funds portfolio optimisation approach using a univariate GARCH type model, extreme value theory (EVT) and the vine copula to determine the optimal allocation for hedge funds portfolio. First, we apply the generalised pareto distribution (GPD) to model the tails of the innovation of each hedge funds strategy return. Second, we capture the interdependence structure between hedge funds strategies and construct vine copula-GARCH-EVT model. Then, we combine it with Monte Carlo simulation and mean-CVaR model to optimise hedge funds portfolio, in order to estimate the risk more accurately. The empirical results of five Hedge funds indexes show that the C-vine copula can better characterise the interdependence structure between the different hedge funds strategies and the performance of C-vine copula-GARCH-EVT-CVaR model is better that of multivariate copulas-GARCH-EVT-CVaR models in portfolio optimisation.

Simulation analysis of financial stock market based on machine learning and GARCH model

With the rapid development of the world’s financial industry, the complexity and relevance of risks are gradually increasing. At present, there are still some deficiencies in the model for measuring financial risk. In view of this, this study analyzes the financial stock market and combines VAR model and GARCH model to conduct financial analysis. Moreover, this study uses the standard deviation in the statistical characteristics of the data to characterize the fluctuation of futures, and then uses the univariate GARCH model to measure the fluctuation. In addition, this study combines the examples to analyze the effectiveness of the model, and compares the predicted data with the actual data to verify the model performance. The results show that the algorithm proposed in this paper has certain effectiveness, and through this research algorithm, investors, speculators or macro decision makers in the futures market can obtain some inspiration.

Hedge funds portfolio optimisation using a vine copula-GARCH-EVT-CVaR model

This paper investigates the conditional value-at-risk (CVaR) hedge funds portfolio optimisation approach using a univariate GARCH type model, extreme value theory (EVT) and the vine copula to determine the optimal allocation for hedge funds portfolio. First, we apply the generalised pareto distribution (GPD) to model the tails of the innovation of each hedge funds strategy return. Second, we capture the interdependence structure between hedge funds strategies and construct vine copula-GARCH-EVT model. Then, we combine it with Monte Carlo simulation and mean-CVaR model to optimise hedge funds portfolio, in order to estimate the risk more accurately. The empirical results of five Hedge funds indexes show that the C-vine copula can better characterise the interdependence structure between the different hedge funds strategies and the performance of C-vine copula-GARCH-EVT-CVaR model is better that of multivariate copulas-GARCH-EVT-CVaR models in portfolio optimisation.

On variable ordination of modified Cholesky decomposition for estimating time‐varying covariance matrices

SummaryEstimating time‐varying covariance matrices of the vector of interest is challenging both computationally and statistically due to a large number of constrained parameters. In this work, we consider an order‐averaged Cholesky‐log‐GARCH (OA‐CLGARCH) model for estimating time‐varying covariance matrices through the orthogonal transformations of the vector based on the modified Cholesky decomposition. The proposed method is to transform the vector at each time as a linear transformation of uncorrelated latent variables and then to use simple univariate GARCH models to model them separately. But the modified Cholesky decomposition relies on a given order of variables, which is often not available, to sequentially orthogonalize the variables. The proposed method develops an order‐averaged strategy for the Cholesky‐GARCH method to alleviate the effect of order of variables. The merits of the proposed method are illustrated through simulations and real‐data studies.

Volatility Modelling and Dynamic Linkages between Pakistani and Leading Foreign Stock Markets: A Multivariate GARCH Analysis

It is essential for financial institutions and academicians to understand volatility spillover and financial market returns. However, previous studies examined the effects of direct spillover only and ignored those of the newly emerging stock markets. Therefore, this study attempts to estimate the time-varying volatility of Pakistani and leading foreign stock markets. It also tries to explore the direct and indirect volatility spillover effect between Pakistani and eight leading foreign stock markets. Daily data were used from nine international equity markets (KSE 100, NIKKEI 225, HIS, S&P 500, NASDAQ 100, DOW JONES, GADXI, FTSE 350 and DFMGI) for the period between 2005 and 2016. The univariate GARCH and GJR models were employed for analysing volatility, and the multivariate GARCH Diagonal BEKK model was used to explore direct and indirect volatility spillover effects. In order to analyse the volatility spillover effect during and after the global financial crisis period, the data were categorised into two periods: between 2005 and 2009 and between 2010 and 2016. The Chow break-point test was also employed to identify structural breaks in return series due to global financial crises. Direct and indirect spillover effects were found between KSE100, S&P 500, NASDAQ 100, DOW JONES and DFMGI. Keywords: Volatility, Spillover, Equity Market, Financial Crisis and GARCH

A Non-Elliptical Orthogonal GARCH Model for Portfolio Selection under Transaction Costs

Covariance matrix forecasts for portfolio optimization have to balance sensitivity to new data points with stability in order to avoid excessive rebalancing. To achieve this, a new robust orthogonal GARCH model for a multivariate set of non-Gaussian asset returns is proposed. The conditional return distribution is multivariate generalized hyperbolic and the dispersion matrix dynamics are driven by the leading factors in a principle component decomposition. Each of these leading factors is endowed with a univariate GARCH structure, while the remaining eigenvalues are kept constant over time. Joint maximum likelihood estimation of all model parameters is performed via an expectation maximization algorithm, and is applicable in high dimensions. The new model generates realistic correlation forecasts even for large asset universes and captures rising pairwise correlations in periods of market distress better than numerous competing models. Moreover, it leads to improved forecasts of an eigenvalue-based financial systemic risk indicator. Crucially, it generates portfolios with much lower turnover and superior risk-adjusted returns net of transaction costs, outperforming the equally weighted strategy even under high transaction fees.

Multivariate GARCH Model and Its Application to Bivariate Model

Multivariate GARCH model is a development of the univariate GARCH model. The multivariate GARCH model can be viewed as a conditional heteroskedasticity model in a multivariate time series. This paper discusses the parameterization of covariance matrices such as Vech model representation, BEEK model and Constant Correlation model. For parameter estimation the maximum likelihood method is used. Furthermore, multivariate GARCH model application is applied for bivariate model

Analysing Existence of Volatility Persistence in Sub-Sahara Africa Stock Markets

The aim of this paper was to analyse volatility persistence in Sub-Sahara stock markets. The study concentrated on selected markets including Ghana, Nigeria and South Africa by analysing univariate GARCH (1,1) model using monthly data from January 2000 to December 2017. Estimates from descriptive statistics show that the mean monthly returns are positive for the Sub-Sahara stock markets, but the South Africa stock market generates more returns than Nigeria and Ghana within the study period. Skewness coefficients show that the stock returns distributions of the Sub-Sahara Africa stock markets are negatively skewed. Excess kurtosis is positive for all the stock markets returns. The Jarque-Bera statistics indicate the stock markets’ series are not normally distributed. Unit roots tests results indicate that the Sub-Sahara Africa stock markets series are integrated of order one. The results of the GARCH (1,1) model provide evidence to show that the Sub-Sahara Africa stock markets exhibit volatility clustering and persistence. The study therefore concludes that there is volatility persistence in Sub-Sahara Africa Stock Markets. The study therefore recommends that Sub-Sahara Africa portfolio managers watch movements in stock market volatility as part of their portfolio management strategy and formulate cushion policies to mitigate effects of volatility shocks. DOI: https://doi.org/10.26905/afr.v2i1.3263

Asymmetric risk impacts of Chinese tourists to Taiwan

AbstractSince 2008, when Taiwan's President Ma Ying‐Jeou relaxed the Cross‐Strait policy, China has become Taiwan's largest source of international tourism. In order to understand the risk persistence of Chinese tourists, the paper investigates the short‐run and long‐run persistence of shocks to the change rate of Chinese tourists to Taiwan, which is a novel empirical analysis. The daily data used for the empirical analysis are from January 1, 2013, to February 28, 2018. McAleer's fundamental equation in tourism finance is used to link the change rate of tourist arrivals and the change in tourist revenues. Three widely used univariate conditional volatility models, namely, GARCH(1,1), GJR(1,1), and EGARCH(1,1), are used to measure the short‐run and long‐run persistence of shocks, as well as symmetric, asymmetric, and leverage effects. Three different heterogeneous autoregressive (HAR) models, HAR(1), HAR(1,7) and HAR(1,7,28), are considered as alternative mean equations for capturing a variety of long memory effects. The mean equations associated with GARCH(1,1), GJR(1,1), and EGARCH(1,1) are used to analyze the risk persistence of the change in Chinese tourists. The exponential smoothing process is used to adjust the seasonality around the trend in Chinese tourists. The empirical results show asymmetric impacts of positive and negative shocks on the volatility of the change in the number of group‐type and medical‐type tourists, whereas individual‐type tourists display a symmetric volatility pattern. Somewhat unusually, leverage effects are observed in EGARCH for medical‐type tourists, which shows a negative correlation between shocks in tourist numbers and the subsequent shocks to volatility. For both group‐type and medical‐type tourists, the asymmetric impacts on volatility show that negative shocks have larger effects than do positive shocks. The leverage effect in EGARCH for medical‐type tourists implies that larger shocks would decrease volatility in the change in the numbers of medical‐type tourists. These novel empirical results suggest that Taiwan's tourism authorities should act to prevent the negative shocks for the group‐type and medical‐type Chinese tourists to dampen the shocks that arise from having fewer Chinese tourists to Taiwan.

Modelling Change Point in GARCH Models

This research paper use PELT algorithm and GARCH models to conduct volatility change point analysis and to model and forecast change point in volatility of USD/KES data. This study employed simulated data and data from Central Bank of Kenya for the period between January 2005 to December 2018. The estimates and actual values of change points in volatility did not differ after analysis. The USD/KES data exhibited volatility clustering in some time periods. The volatility adjusted GARCH models outperformed plain models. The simulated estimates of GARCH models were almost converging to the parameters from USD/KES data using the same models. The GARCH models that incorporate change points registered better forecasting performance compared to the plain models. The PGARCH, TGARCH and GJRGARCH models had the same forecasting performance measures in absence and presence of change points. The study recognized TGARCH (1,1) as the best model for modelling and forecasting. Banks can use univariate GARCH models in conjunction with PELT algorithm to track loan defaulters. Hospitals can use the same technique to determine the most recurring diseases. Companies can apply the same to determine abnormal profits and losses. The technique can be applied in other sectors like in meteorology.

The variance implied conditional correlation

ABSTRACTWe apply univariate GARCH models to construct a computationally simple filter for estimating the conditional correlation matrix of asset returns. The proposed Variance Implied Conditional Correlation (VICC) exploits the polarization result that links the correlation between two standardized variables with the variances of linear combinations thereof. In a Monte Carlo study, we show that the VICC yields accurate correlation estimates for common choices of the correlation dynamics. We also provide an empirical application to cross hedging that confirms the effectiveness of the VICC.

Estimating volatility transmission between oil prices and the US Dollar exchange rate under structural breaks

This study examines volatility dynamics of oil prices and the US dollar exchange rate using univariate and bivariate GARCH models using data from January 2, 2000 to December 31, 2015. The modified iterative cumulative sum of square (ICSS) algorithm is employed to identify structural breaks in the variance of the two return series. I find no evidence of volatility transmission between oil prices and the US dollar exchange rate if structural breaks are ignored in the model. However, after accounting for structural breaks in variance in the bivariate GARCH model, I find significant volatility transmission between oil prices and the US dollar exchange rate. I also show that dynamic risk minimizing hedge ratios substantially change when breaks are incorporated in the bivariate GARCH model.

Asymmetry and leverage in GARCH models: a News Impact Curve perspective

ABSTRACTModels for conditional heteroskedasticity belonging to the GARCH class are now common tools in many economics and finance applications. Among the many possible competing univariate GARCH models, one of the most interesting groups allows for the presence of the so-called asymmetry or leverage effect. In our view, asymmetry and leverage are two distinct phenomena, both inspired by the seminal work of Black in 1976. We propose definitions of leverage and asymmetry that build on the News Impact Curve, allowing to easily and coherently verify if they are both present. We show that several GARCH models are asymmetric but none is allowing for a proper leverage effect. Finally, we extend the leverage definition to a local leverage effect and show that the AGARCH model is coherent with the presence of local leverage. An empirical analysis completes the paper.

Measuring portfolio risk of non-energy commodity using time-varying vine copula

In literature, many empirical studies have used the vine copula to measure and analyse the risk of stock indices, energetic products or crypto currencies portfolio (e.g., Zhang et al., 2014; Shahzad et al., 2018; Boako et al., 2019). These studies used mainly the VaR and ES with different versions of vine copula. However, few works have focused on a portfolio composed of non-energy commodities. Our aim in this work is to model the dependence between non-energy commodities returns by modelling standardised residuals obtained from univariate GARCH model by different versions of time varying vine copula, to quantify risk of N-dimensional non-energy commodity portfolio by VaR and ES and to compare the predictive performance of this method with traditional and competitive traditional univariate VaR methods. Empirical results suggest that risk quantifies generated by AR-GARCH vine copula methods with student-t distribution are sufficiently accurate at both low and high confidence levels. Given these results, we recommend the application of vine copula method to understanding the non-energy commodity behaviours which are very important to investors, producers, consumers, and policy makers.

CAPTURING THE ASYMMETRIC IMPACT OF NEGATIVE VERSUS POSITIVE SHOCKS TO VOLATILITY OF A MAJOR INDEX

ABSTRACT Volatility in stock indexes has affected investing decisions everywhere. This paper studies the volatility behavior of S&P 500 index over a time period of nine years. The time period is of crucial importance due to the most recent recession the US economy that had its effects on most European and Asian economies. The study utilizes a Univariate (asymmetric) GARCH model to estimate the conditional variance of the return series. Results show asymmetric behavior and high persistence of shocks to volatility. These results will help better understand the volatility behavior of stock indexes and provide major implications for investors and researchers. Keywords asymmetric, impact, volatility, index

Uncertainty and Exchange Rate Volatility: The Case of Mexico

This paper investigates the effect of domestic and global uncertainty on the volatility of the Mexican peso US dollar exchange rate during the period January 1999 - July 2018. The analysis is performed in two stages. In the first stage, we derive an exchange rate volatility measure by estimating a univariate GARCH(1,1) model. In the second stage, we regress the estimated exchange rate volatility on domestic and global uncertainty proxies, built with data from the survey of professional forecasters conducted monthly by Banco de Mexico since 1999. In order to capture the expectations of a broad set of economic agents, we also include as regressors both the Chicago Board Options Exchange Volatility Index (VIX) and Baker, Bloom and Davis global economic policy uncertainty index (EPU). A trade uncertainty index based on Google trends is incorporated in the model to analyze if uncertainty regarding NAFTA and Mexico’s trade liberalization policies has had an impact on the exchange rate volatility. The price of a barrel of Mexican oil mix is also included since the price volatility of this commodity has been shown to affect the peso US dollar exchange rate volatility. In order to control for the surprise component of macroeconomic data announcements the analysis controls for an inflation surprise and a GDP surprise. Finally, we assess whether the effect of domestic political uncertainty and domestic economic uncertainty on exchange rate volatility is amplified in election and recession periods. The main results show that the domestic political uncertainty measure, the VIX and the global EPU indices are the main drivers of the exchange rate volatility. We also find that in recession periods, higher domestic economic uncertainty is associated with higher exchange rate volatility. These findings are robust to alternative exchange rate volatility proxies and econometric techniques.

The Interrelationship of Return and Volatility between Exchange Rate and Financial Variables

The purpose of this research is to explore the relationship between exchange rate and 12 financial variables in order to find useful implications to predict the movement of short-term exchange rate. As a result of using various analytical techniques such as univariate GARCH, VAR, multivariate GARCH, cointegration, and VECM, we found some valuable information on exchange rate movement. First, the KOSPI is considered as the most useful financial variable affecting short-term exchange rate changes. Second, CDS premium, S&P500, and foreign net stock buying are also meaningful variables that affect the short-term exchange rate. Third, when the Korean won depreciates, factors of instability in the domestic financial market (i.e., falling stock price, rising stock market risk, and rising country default risk) also accompany such currency depreciation. When the won appreciates, the dollar value depreciates relative to major countries’ currencies. Fourth, the volatility of the dollar index is the only variable that affects the volatility of the exchange rate. Fifth, the CDS premium is the most useful financial variable affecting long-term exchange rate changes since the CDS premium and exchange rate are in long-term equilibrium relationship. The exchange rate, not the CDS premium, adjusts for deviation from the long-term equilibrium.