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.     

On the origin of high persistence in GARCH-models

We show that the (Baillie and Chung, 2001) minimum distance estimates of the GARCH (1,1) model induce spurious persistence in the volatility when there are structural changes in the mean of the process.     

ASYMMETRIC INFORMATION IMPACTS: EVIDENCE FROM THE AUSTRALIAN TREASURY-BOND FUTURES MARKET

Abstract.  The effect of information flows on the return volatility of Australian 3-year Treasury bond futures is examined using linear and non-linear GARCH models. Results show significant asymmetric information effects, where bad news has a greater impact on volatility than good news and a non-linear Threshold ARCH(1,1) in mean model provides the most accurate estimation of return volatility. Diagnostic tests confirm this finding and out of sample forecasting error statistics verify that the Threshold ARCH(1,1) in mean model yields the lowest forecasting error. The Threshold ARCH(1,1)-M model is best at capturing the asymmetric information impact on the Australian three-year T-Bond futures return volatility.     

The risk-return relation and VIX: evidence from the S&P 500

A significantly positive risk-return relation for the S&P 500 market index is detected if the squared implied volatility index (VIX) is allowed for as an exogenous variable in the conditional variance equation of the parsimonious GARCH(1,1) model. This result holds for both daily and weekly observations, for extended conditional mean and variance specifications, and is robust to sub-samples. We show that the conditional variance obtained from the GARCH model with VIX has better predictive ability for realized volatility than the conditional variance from GARCH without VIX and VIX itself, thereby documenting an important information content of VIX for conditional variance. The results are interpreted as evidence that adding VIX squared in the conditional variance equation yields a better measure of conditional variance which, subsequently, uncovers a strong risk-return relation.     

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.     

The linkages,persistence, asymmetry in the volatility,the price discovery and efficiency,and the effect of the US subprime mortgage financial crisis on the spot and the futures market’s returns: the case of India

This article examines the effects of persistence, asymmetry and the US subprime mortgage crisis on the volatility of the returns and also the price discovery, efficiency and the linkages and causality between the spot and futures volatility by using various classes of the ARCH and GARCH models, and through the Granger’s causality. We have used two indices: one for spot and the other for futures, for the daily data from 12 June 2000 to 30 September 2013 from Nifty stock indices. We have then tested for ARCH effects, and subsequently employed various models of the ARCH and GARCH conditional volatility. The GARCH(1,1) model is found to be significant, and it implies that the returns are not autocorrelated and have ‘short memory’. It supports the hypothesis of the efficiency of the markets. The negative ‘news’ has more significant effect on volatility, corroborating the ‘leverage impact’ in finance on market volatility. We have also tested the volatility spillover effects. The two methods we employed support the spillover effects and the causality is bidirectional. We also have used the dummy variable for the US subprime mortgage financial crisis and found that they are statistically significant. Indian stock market is thus integrated to the world stock markets.     

Modeling long memory volatility using realized measures of volatility: A realized HAR GARCH model

Long memory is an important feature of the volatility of financial returns. We document that the recently developed Realized GARCH model (Hansen et al., 2012) is insufficient for capturing the long memory of underlying volatility. We develop a parsimonious variant of the Realized GARCH model by introducing the HAR specification of Corsi (2009) into the volatility dynamics. A comparison of the theoretical and sample autocorrelation functions shows that the new model specification better captures the long memory dynamics of volatility. We calculate the multi-period out-of-sample volatility forecasts for several return series and find that the new model is a significant improvement over the classic Realized GARCH model.     

Is Beta-t-EGARCH(1,1) superior to GARCH(1,1)?

Statistical performance, in-sample point forecast precision and out-of-sample density forecast precision of GARCH(1,1) and Beta-t-EGARCH(1,1) models are compared. We study the volatility of nine global industry indices for period from April 2006 to July 2010. Competing models are estimated for periods before, during and after the United States (US) financial crisis of 2008. The results provide evidence of the superior out-of-sample predictive performance of Beta-t-EGARCH compared to GARCH after the US financial crisis.     

GARCH with omitted persistent covariate

This paper analyzes the effect of omitting a persistent covariate in the GARCH-X model. In particular, we show that if the relevant persistent covariate is omitted and the usual GARCH(1,1) model is fitted, the model will be estimated approximately as an IGARCH model. This may well explain the ubiquitous evidence of the IGARCH in empirical volatility analysis.     

Altered bank and exchange volatility

This paper examines the behaviour of the Czech crown's exchange rate when pegged to a currency basket. The peg is supposed to limit the overall instability of the currency. The GARCH(1,1) model with a dummy variable for the volatility response is used to account for a change in the width of the fluctuation band. The results of this paper show that volatility of the exchange rate decreased after a much wider fluctuation band was introduced to limit movements of the currency basket index.     

Improving GARCH volatility forecasts with regime-switching GARCH

Many researchers use GARCH models to generate volatility forecasts. Using data on three major U.S. dollar exchange rates we show that such forecasts are too high in volatile periods. We argue that this is due to the high persistence of shocks in GARCH forecasts. To obtain more flexibility regarding volatility persistence, this paper generalizes the GARCH model by distinguishing two regimes with different volatility levels; GARCH effects are allowed within each regime. The resulting Markov regime-switching GARCH model improves on existing variants, for instance by making multi-period-ahead volatility forecasting a convenient recursive procedure. The empirical analysis demonstrates that the model resolves the problem with the high single-regime GARCH forecasts and that it yields significantly better out-of-sample volatility forecasts. First Version Received: November 2000/Final Version Received: August 2001     

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).     

Forecasting stock market volatility and information content of implied volatility index

This study investigates the incremental information content of implied volatility index relative to the GARCH family models in forecasting volatility of the three Asia-Pacific stock markets, namely India, Australia and Hong Kong. To examine the in-sample information content, the conditional variance equations of GARCH family models are augmented by incorporating implied volatility index as an explanatory variable. The return-based realized variance and the range-based realized variance constructed from 5-min data are used as proxy for latent volatility. To assess the out-of-sample forecast performance, we generate one-day-ahead rolling forecasts and employ the Mincer–Zarnowitz regression and encompassing regression. We find that the inclusion of implied volatility index in the conditional variance equation of GARCH family model reduces volatility persistence and improves model fitness. The significant and positive coefficient of implied volatility index in the augmented GARCH family models suggests that it contains relevant information in describing the volatility process. The study finds that volatility index is a biased forecast but possesses relevant information in explaining future realized volatility. The results of encompassing regression suggest that implied volatility index contains additional information relevant for forecasting stock market volatility beyond the information contained in the GARCH family model forecasts.     

Exchange rate volatility and United Kingdom trade: evidence from Canada,Japan and New Zealand

This paper investigates the influence of exchange rate volatility on the real imports of the United Kingdom from Canada, Japan and New Zealand during the period 1980–2003. The Johansen multivariate cointegration method and the constrained error correction (general-to-specific) method are applied to study the relationship between real imports and its determinants (including exchange rate volatility). Conditional variance from the GARCH(1,1) model is applied as exchange rate volatility. Both nominal and real exchange rates are employed in the empirical study. Results indicate a significant effect of the exchange rate volatility on real imports. These exchange rate volatility effects are mostly positive. The author thanks an anonymous referee, the editor and Myles Wallace for several useful comments and suggestions. Any remaining errors and omissions are the author’s responsibility alone.     

货币冲击、房地产收益波动与最优货币政策选择

与传统资产定价模型中风险收益权衡关系相悖,我国房地产市场存在投资异象和波动长记忆性特征。文章利用泰勒规则(Taylor Rule)的利率缺口,在剔除市场预期之后测度了中国市场的货币政策冲击,并基于房地产投资回报的时序数据波动聚集性和时变性特征构建GARCH(1,1)-M模型,以此度量我国房地产市场投资收益的波动演变路径,解释了央行实施加息的货币政策后当期房价反而上涨的投资现象。文章还立足于房地产市场参与人的投资特征,从行为金融学的全新研究视角出发,建立包含行为资产定价的动态模型经济系统,研究资产价格波动与最优货币政策选择问题,求得相应闭型解,为实施关注资产价格波动的最优货币政策提供理论基础。     

Asymmetry, realised volatility and stock return risk estimates

In this paper we estimate minimum capital risk requirements for short and long positions with three investment horizons, using the traditional GARCH model and two other GARCH-type models that incorporate the possibility of asymmetric responses of volatility to price changes. We also address the problem of the extremely high estimated persistence of the GARCH model to generate observed volatility patterns by including realised volatility as an explanatory variable into the model??s variance equation. The results suggest that the inclusion of realised volatility improves the GARCH forecastability as well as its ability to calculate accurate minimum capital risk requirements and makes it quite competitive when compared with asymmetric conditional heteroscedastic models such as the GJR and the EGARCH.     

Does VIX or volume improve GARCH volatility forecasts?

This article considers whether the inclusion of two additional variables can improve volatility forecasts over a standard GARCH-based model. We consider three alternative ways of incorporating the volatility index (VIX) and trading volume as exogenous variables within a selection of GARCH models. We are particularly interested in whether these variables have additional incremental forecast power over and above the baseline GARCH specification. Our results suggest that both the VIX and volume do provide some additional forecast power, and this is generally improved when considering both of these series jointly in the model. However, while the results may be statistically significant the gain is marginal and the coefficient values small. Moreover, in a horse race exercise VIX does not outperform the GARCH approach. In answering the question of whether VIX produces better forecasts than the GARCH model, then the answer is no, but the informational content of VIX cannot be ignored and should be incorporated into forecast regressions.     

On the effects of rare disasters and uncertainty shocks for risk premia in non-linear DSGE models

This paper studies how rare disasters and uncertainty shocks affect risk premia in DSGE models approximated to second and third order. Based on an extension of the results in Schmitt-Grohé and Uribe (2004) to third order, we derive propositions for how rare disasters, stochastic volatility, and GARCH affect any type of risk premia in a wide class of DSGE models. To quantify the effects, we set up a standard New Keynesian DSGE model where total factor productivity includes rare disasters, stochastic volatility, and GARCH. We find that rare disasters increase the level of the 10-year nominal term premium, whereas a key effect of uncertainty shocks, i.e. stochastic volatility and GARCH, is an increase in the variability of this premium.     

Modelling conditional heteroskedasticity: Application to the “IBEX-35” stock-return index

This paper compares alternative time-varying volatility models for daily stock-returns using data from Spanish equity index IBEX-35. Specifically, we estimate a parametric family of models of generalized autoregressive heteroskedasticity (which nests the most popular symmetric and asymmetric GARCH models), a semiparametric GARCH model, the generalized quadratic ARCH model, the stochastic volatility model, the Poisson Jump Diffusion model and, finally, a nonparametric model. Those models which use conditional standard deviation (specifically, TGARCH and AGARCH models) produce better fits than all other GARCH models. We also compare the within sample predictive power of all models using a standard efficiency test. Our results show that the asymmetric behaviour of responses is a statistically significant characteristic of these data. Moreover, we observe that specifications with a distribution which allows for fatter tails than a normal distribution do not necessarily outperform specifications with a normal distribution.     

Stock Returns and Volatility: an empirical study of Chinese stock markets

This paper uses GARCH models to analyse the relationship between returns and volatility on the Shanghai and Shenzhen Stock Exchanges in China. Empirical estimates using the sample data from 21 May 1992 to 2 February 1996 suggest that the variances of the returns in the two markets are best modeled by the GARCH-M (1,1) specification. Volatility transmission between the two markets (the volatility spill-over effect) is also found to exist. The results of one month ahead ex ante forecasts show that the conditional variances of the returns of the two stock markets exhibit a similar pattern.