EMPIRICAL PERFORMANCE OF GARCH MODELS WITH HEAVY‐TAILED INNOVATIONS

We introduce a new type of heavy‐tailed distribution, the normal reciprocal inverse Gaussian distribution (NRIG), to the GARCH and Glosten‐Jagannathan‐Runkle (1993) GARCH models, and compare its empirical performance with two other popular types of heavy‐tailed distribution, the Student's t distribution and the normal inverse Gaussian distribution (NIG), using a variety of asset return series. Our results illustrate that there is no overwhelmingly dominant distribution in fitting the data under the GARCH framework, although the NRIG distribution performs slightly better than the other two types of distribution. For market indexes series, it is important to introduce both GJR‐terms and the NRIG distribution to improve the models’ performance, but it is ambiguous for individual stock prices series. Our results also show the GJR‐GARCH NRIG model has practical advantages in quantitative risk management. Finally, the convergence of numerical solutions in maximum‐likelihood estimation of GARCH and GJR‐GARCH models with the three types of heavy‐tailed distribution is investigated.     

QUANTILE FORECASTS OF FINANCIAL RETURNS USING REALIZED GARCH MODELS*

This article applies the realized generalized autoregressive conditional heteroskedasticity (GARCH) model, which incorporates the GARCH model with realized volatility, to quantile forecasts of financial returns, such as Value‐at‐Risk and expected shortfall. Student's t‐ and skewed Student's t‐distributions as well as normal distribution are used for the return distribution. The main results for the S&P 500 stock index are: (i) the realized GARCH model with the skewed Student's t‐distribution performs better than that with the normal and Student's t‐distributions and the exponential GARCH model using the daily returns only; and (ii) using the realized kernel to take account of microstructure noise does not improve the performance.     

Modelling the dynamics of Bitcoin and Litecoin: GARCH versus stochastic volatility models

We examine and compare a large number of generalized autoregressive conditional heteroskedastic (GARCH) and stochastic volatility (SV) models using series of Bitcoin and Litecoin price returns to assess the model fit for dynamics of these cryptocurrency price returns series. The various models examined include the standard GARCH(1,1) and SV with an AR(1) log-volatility process, as well as more flexible models with jumps, volatility in mean, leverage effects, t-distributed and moving average innovations. We report that the best model for Bitcoin is SV-t while it is GARCH-t for Litecoin. Overall, the t-class of models performs better than other classes for both cryptocurrencies. For Bitcoin, the SV models consistently outperform the GARCH models and the same holds true for Litecoin in most cases. Finally, the comparison of GARCH models with GARCH-GJR models reveals that the leverage effect is not significant for cryptocurrencies, suggesting that these do not behave like stock prices.     

Bayesian model comparison by Markov chain simulation: Illustration using stock market data

The paper illustrates the computation of marginal likelihoods and Bayes factors when Markov Chain Monte Carlo has been used to produce draws from a model’s posterior distribution. The method is based on Raftery (1996) and does not require that Gibbs sampling is used or conditional posterior distributions are available in closed form. Models used include a normal finite mixture, a GARCH and a Student t -model as alternative models for the Standard and Poor’s stock returns.     

On the asymmetric impact of macro–variables on volatility

We extend the GARCH–MIDAS model to take into account possible different impacts from positive and negative macroeconomic variations on financial market volatility: a Monte Carlo simulation which shows good properties of the estimator with realistic sample sizes. The empirical application is performed on the daily S&P500 volatility dynamics with the U.S. monthly industrial production and national activity index as additional (signed) determinants. We estimate the Relative Marginal Effect of macro variable movements on volatility at different lags. In the out-of-sample analysis, our proposed GARCH–MIDAS model not only statistically outperforms the competing specifications (GARCH, GJR-GARCH and GARCH–MIDAS models), but shows significant utility gains for a mean-variance investor under different risk aversion parameters. Attention to robustness is given by choosing different samples and estimating the model in an international context (six different stock markets).     

Early warning signals using AVaRs of infinitely divisible GARCH models — evidence from stock index markets

Classical time series models have failed to properly assess the risks that are associated with large adverse stock price behaviour. This article contributes to autoregressive moving average model–GARCH (ARMA–GARCH) models with standard infinitely divisible innovations and assesses the performance of these models by comparing them with other time series models that have normal innovation. We discuss the limitations of value at risk (VaR) and aim to develop early warning signal models using average value at risk (AVaRs) based on the ARMA–GARCH model with standard infinitely divisible innovations. Empirical results for the daily Dow Jones Industrial Average Index, the England Financial Times Stock Exchange 100 Index and the Japan Nikkei 225 Index reveal that estimating AVaRs for the ARMA–GARCH model with standard infinitely divisible innovations offers an improvement over prevailing models for evaluating stock market risk exposure during periods of distress in financial markets and provides a suitable early warning signal in both extreme events and highly volatile markets.     

Structural breaks and GARCH models of stock return volatility: The case of South Africa

This paper investigates the empirical relevance of structural breaks in forecasting stock return volatility using both in-sample and out-of-sample tests applied to daily returns of the Johannesburg Stock Exchange (JSE) All Share Index from 07/02/1995 to 08/25/2010. We find evidence of structural breaks in the unconditional variance of the stock returns series over the period, with high levels of persistence and variability in the parameter estimates of the GARCH(1,1) model across the sub-samples defined by the structural breaks. This indicates that structural breaks are empirically relevant to stock return volatility in South Africa. However, based on the out-of-sample forecasting exercise, we find that even though there structural breaks in the volatility, there are no statistical gains from using competing models that explicitly accounts for structural breaks, relative to a GARCH(1,1) model with expanding window. This could be because of the fact that the two identified structural breaks occurred in our out-of-sample, and recursive estimation of the GARCH(1,1) model is perhaps sufficient to account for the effect of the breaks on the parameter estimates. Finally, we highlight that, given the point of the breaks, perhaps what seems more important in South Africa, is accounting for leverage effects, especially in terms of long-horizon forecasting of stock return volatility.     

Using monthly returns to model conditional heteroscedasticity

This empirical study examines the extent of non–linearity in a multivariate model of monthly financial series. To capture the conditional heteroscedasticity in the series, both the GARCH(1,1) and GARCH(1,1)–in–mean models are employed. The conditional errors are assumed to follow the normal and Student– t distributions. The non–linearity in the residuals of a standard OLS regression are also assessed. It is found that the OLS residuals as well as conditional errors of the GARCH models exhibit strong non–linearity. Under the Student density, the extent of non–linearity in the GARCH conditional errors was generally similar to those of the standard OLS. The GARCH–in–mean regression generated the worse out–of–sample forecasts.     

Testing for nonlinearities in German bank stock returns

In this paper nonlinear structures in German bank stock returns are investigated in a stochastic modelling framework. In the first step we show the existence of a nonlinear return structure by means of the McLeod-Li and the BDS test. In the second step we focus our analysis on the kinds of nonlinearity actually present in bank stock data. On the basis of the Hsieh test it is possible to discriminate with high power additive from multiplicative dependencies to provide guidance for the choice of an adequate class of stochastic models. It is shown that the multiplicative dependencies predominating the bank stock returns can be captured by low order GARCH models.     

Tail dependence analysis of stock markets using extreme value theory

Financial risk modelling frequently uses the assumption of a normal distribution when considering the return series which is inefficient if the data is not normally distributed or if it exhibits extreme tails. Estimation of tail dependence between financial assets plays a vital role in various aspects of financial risk modelling including portfolio theory and hedging amongst applications. Extreme Value Theory (EVT) provides well established methods for considering univariate and multivariate tail distributions which are useful for forecasting financial risk or modelling the tail dependence of risky assets. The empirical analysis in this article uses nonparametric measures based on bivariate EVT to investigate asymptotic dependence and estimate the degree of tail dependence of the ASX-All Ordinaries daily returns with four other international markets, viz., the S&P-500, Nikkei-225, DAX-30 and Heng-Seng for both extreme right and left tails of the return distribution. It is investigated whether the asymptotic dependence between these markets is related to the heteroscedasticity present in the logarithmic return series using GARCH filters. The empirical evidence shows that the asymptotic extreme tail dependence between stock markets does not necessarily exist and rather can be associated with the heteroscedasticity present in the financial time series of the various stock markets.     

Long memory in the volatility of China stock returns

In this paper, we empirically examine the volatility process of China's stock market returns using daily and weekly Shanghai and Shenzhen stock indices during January 1990 to August 2008. To investigate the property of the process, we used the FIGARCH (fractionally integrated GARCH) model including GARCH and IGARCH processes as special cases. Since the FIGARCH model allows fractional integration order, it can detect hyperbolically decaying volatility processes which cannot be explained by previous models with integer integration order. Our results show that the Shanghai and Shenzhen stock indices exhibit long-term dependencies. The long memory properties of the Shanghai and Shenzhen stock markets do not seem to be spuriously induced without exception.     

A note on the estimated GARCH coefficients from the S&P1500 universe

ABSTRACT

We employ 1440 stocks listed in the S&P Composite 1500 Index of the NYSE. Three benchmark GARCH models are estimated for the returns of each individual stock under three alternative distributions (Normal, t and GED). We provide summary statistics for all the GARCH coefficients derived from 11,520 regressions. The EGARCH model with GED errors emerges as the preferred choice for the individual stocks in the S&P 1500 universe when non-negativity and stationarity constraints in the conditional variance are imposed. 57% of the constraint’s violations are taking place in the S&P small cap stocks.     

基于FIGARCH模型的中国房地产指数的实证分析

在T分布和正态分布假设下采用GARCH模型和FIGARCH模型对上证地产股指数日收益率序列进行建模分析,结果表明,上证地产股指数日收益率序列的波动具有显著的长记忆性,表明外部冲击对波动有着长期的影响。因此,采用FIGARCH模型建模的效果优于采用GARCH模型建模的效果,并且在T分布假设下拟合模型,其效果优于在正态分布假设下拟合的模型。     

Conditional volatility and correlations of weekly returns and the VaR analysis of 2008 stock market crash

Modelling of conditional volatilities and correlations across asset returns is an integral part of portfolio decision making and risk management. Over the past three decades there has been a trend towards increased asset return correlations across markets, a trend which has been accentuated during the recent financial crisis. We shall examine the nature of asset return correlations using weekly returns on futures markets and investigate the extent to which multivariate volatility models proposed in the literature can be used to formally characterize and quantify market risk. In particular, we ask how adequate these models are for modelling market risk at times of financial crisis. In doing so we consider a multivariate t version of the Gaussian dynamic conditional correlation (DCC) model proposed by Engle (2002), and show that the t-DCC model passes the usual diagnostic tests based on probability integral transforms, but fails the value at risk (VaR) based diagnostics when applied to the post 2007 period that includes the recent financial crisis.     

A new investor sentiment indicator (ISI) based on artificial intelligence: A powerful return predictor in China

This paper utilizes deep learning approach widely documented in artificial intelligence, and proposes an investor-sentiment indicator (ISI) that is consistent with the purpose of forecasting stock market returns. We find that ISI is positively correlated with future stock market returns at a monthly frequency, but negatively associated with subsequent returns over a longer horizon. Moreover, ISI outperforms other well-recognized predictors both in and out of sample, and can predict cross-sectional stock returns sorted by industry. We also show a positive association between monthly ISI and dividend growth rate, which indicates that investors’ expectations about future cash flows may contribute to the return predictability of ISI.     

Further empirical evidence on the forecasting of volatility with smooth transition exponential smoothing

Smooth transition exponential smoothing (STES) uses a logistic function of a user-specified transition variable as adaptive time varying smoothing parameter. This paper empirically addresses three aspects of the use of STES for volatility forecasting. Previous empirical results showed the method performing well in comparison with fixed parameter exponential smoothing and a variety of GARCH models. However, those results related only to forecasting weekly volatility. In this paper, we address the use of STES for forecasting daily volatility. A second issue that we evaluate is the robustness of STES in the presence of extreme outlying observations. The third aspect that we consider is the use of trading volume within a transition variable in the STES method. Our simulation results suggest that STES performs well in terms of robustness, when compared with standard methods and several alternative robust methods. Analysis using stock return data shows that STES has the potential to outperform standard and robust forms of fixed parameter exponential smoothing and GARCH models. The results suggest the use of the sign and size of past shocks as STES transition variables, and provide no clear support for the incorporation of trading volume in a transition variable.     

Conditional correlations in the returns on oil companies stock prices and their determinants

The identification of the forces that drive stock returns and the dynamics of their associated volatilities is a major concern in empirical economics and finance. This analysis is extremely important for determining optimal hedging strategies. This paper investigates the stock prices’ returns and their financial risk factors for several integrated oil companies, namely Bp (BP), Chevron-Texaco (CVX), Eni (ENI), Exxon-Mobil (XOM), Royal Dutch (RD) and Total-Fina Elf (TFE). We measure the actual co-risk in stock returns and their determinants “within” and “between” the different oil companies, using multivariate cointegration techniques in modelling the conditional mean, as well as multivariate GARCH models for the conditional variances. The distinguishing features of this paper are: (i) focus on the determinants of the market value of each company using the cointegrated VAR/VECM methodology; (ii) specification of the conditional variances of VECM residuals with the Constant Conditional Correlation (CCC) multivariate GARCH model of Bollerslev [(1990) Review of Economics and Statistics 72:498–505] and the Dynamic Conditional Correlation (DCC) multivariate GARCH model of Engle [(2002) Journal of Business and Economic Statistics 20:339–350]; (iii) discussion of the performance of optimal hedge ratios calculated with the DCC estimates. The “within” and “between” DCC indicate time-varying interdependence between stock return volatilities and their determinants. Moreover, DCC models are shown to produce more accurate hedging strategies.     

The persistence and asymmetric volatility in the Nigerian stock bull and bear markets

This paper considers the persistence and asymmetric volatility at each market phase of the Nigerian All Share Index (ASI). The estimate of the fractional difference parameter is used as a stability measure of the degree of persistence in the level of the series and in the absolute/squared returns, which are used as proxies for the volatility. Both semi-parametric and parametric methods are applied. Forms of Generalized Autoregressive Conditionally Heteroscedastic (GARCH) models, which include fractional integration and asymmetric variants are estimated at each market phase of the stock returns. The results show that the level of persistence differs between the two market phases in both level and squared/absolute return series. Apart from general asymmetry and persistence in Nigerian stocks, each market phase still presents significant persistence and asymmetry.     

人民币汇率与股市收益的动态关联性实证研究

利用基于广义误差分布（GED）的多因素TGARCH模型,实证分析了2005年7月21日汇改后人民币汇率与中国股市收益的动态关系。估计结果显示：人民币汇率对股市收益有显著的价格扩散效应,汇率上升会引起上证指数收益率较大幅度的上升;股市收益波动存在显著的ARCH效应和GARCH效应,并且等强度的正向或负向新息的冲击会引起股市波动的非对称反应,正向冲击比同强度的负向冲击能带来股市更大的未来波动。     

The impact of interest rate and exchange rate volatility on banks' stock returns and volatility: Evidence from Turkey

This paper investigates the effects of interest rate and foreign exchange rate changes on Turkish banks' stock returns using the OLS and GARCH estimation models. The results suggest that interest rate and exchange rate changes have a negative and significant impact on the conditional bank stock return. Also, bank stock return sensitivities are found to be stronger for market return than interest rates and exchange rates, implying that market return plays an important role in determining the dynamics of conditional return of bank stocks. The results further indicate that interest rate and exchange rate volatility are the major determinants of the conditional bank stock return volatility.