Simultaneously determined,time-varying hedge ratios in the soybean complex

Time-varying hedge ratios are derived which account for the dynamic characteristics of prices in the soybean complex. A multivariate generalized autogressive heteroskedastic (MGARCH) model, along with other conditional models, is used to specify the relevant covariance matrix. While the time-varying representations of the variance matrix are statistically appropriateex anteand ex posthedging effectiveness indicate that they provide minimal gain to hedging in terms of mean return and reduction in variance over a constant conditional procedure. Whether similar findings arise from other applications of GARCH models to optimal hedging is a question for further research.     

Robust estimation of the simplified multivariate GARCH model

In this paper, robust M-estimation of multivariate GARCH models are considered. The simplified GARCH model is chosen that involves the estimation of only univariate GARCH models, and hence easy to estimate, and does not put additional constraints on the model. The results of Monte Carlo simulations showed that accurate estimates of conditional correlations can be obtained using these robust estimators when the errors are heavy-tailed. We also investigate the forecasting performance of the class of robust estimators in predicting value-at-risk using various evaluation measures and collect empirical evidences of the better predictive potential of estimators such as LAD and B-estimator over the widely-used quasi-maximum likelihood estimator for the estimation and prediction of multivariate GARCH models. Applications to real data sets are also presented.     

Warrant pricing under GARCH diffusion model

The GARCH diffusion model has attracted a great deal of attention in recent years, as it is able to describe financial time series better, when comparing to many other models. This paper considers the problem of warrant pricing when the underlying asset follows the GARCH diffusion model. An analytical approximate solution for European option prices is derived by means of Fourier transform. The approximate solution can be quickly computed by the fast Fourier transform (FFT) algorithm. Monte Carlo simulations show that this approximate solution is correct and the FFT is accurate and efficient, and hence it enables us to investigate the volatility smile implied by the GARCH diffusion model. Then a method is developed to provide the maximum likelihood (ML) estimation of the GARCH diffusion model based on the efficient importance sampling (EIS) procedure. Furthermore, the empirical performance of the GARCH diffusion model applied to the valuation of Hang Seng Index (HSI) warrants traded on the Hong Kong Stock Exchange (HKEx) is investigated. Empirical results show that the GARCH diffusion model outperforms the Black–Scholes (B–S) model in terms of the pricing accuracy, indicating that the pricing model incorporated with stochastic volatility can improve the pricing of warrants.     

Asymmetric Conditional Volatility and Firm Size: Evidence from Australian Equity Portfolios

This paper examines the relationship between firm size and equity volatility for two portfolios of Australian equities. Univariate and Multivariate GARCH models are used to demonstrate that conditional variance is related to firm size. There is strong evidence to suggest that the variance-covariance matrix of returns is time varying and asymmetric. A negative innovation to the return of the large firm portfolio results in higher levels of conditional volatility in the small firm portfolio than would be the case for a positive innovation of equal magnitude. News about own returns appears to determine the conditional variance of the portfolio of large firms. The conditional covariance between the two portfolios also displays evidence of asymmetry.     

Finite-sample size distortion of the AESTAR unit root test: GARCH,corrected variance–covariance matrix estimators and adjusted critical values

The behaviour of the asymmetric exponential smooth transition autoregressive (AESTAR) unit root test, which allows for asymmetric and nonlinear reversion to equilibrium, is examined in the presence of generalized autoregressive conditional heteroscedasticity (GARCH). It is found that while the test is relatively robust in the presence of ‘low volatility’ GARCH processes, it exhibits substantial size distortion when large values of the volatility parameter are considered. Attempted resolution via the routine application of heteroscedasticity consistent (or ‘corrected’) covariance matrix estimators (HCCMEs) is shown to result in overwhelming size distortion due to their impact upon the finite-sample distribution of the underlying test statistic. However, application of a corrected HCCME, in combination with critical values derived specifically under its use, results in the control of test size. Analogous results for the Dickey–Fuller (1979) test are presented to permit comparison with a test considering linear, symmetric adjustment. It is found that the AESTAR test is subject to far greater distortion than its linear, symmetric alternative. In summary, the results indicate that caution must be exercised when applying the AESTAR test to macroeconomic and financial time series, particularly if routine application of corrected covariance matrix estimators occurs.     

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.     

A latent dynamic factor approach to forecasting multivariate stock market volatility

This paper proposes a latent dynamic factor model for high-dimensional realized covariance matrices of stock returns. The approach is based on the matrix logarithm and combines common latent factors driven by HAR processes and idiosyncratic autoregressive dynamics. The model accounts for positive definiteness of covariance matrices without imposing parametric restrictions. Simulated Bayesian parameter estimates are obtained using basic Markov chain Monte Carlo methods. An empirical application to 5-dimensional and 30-dimensional realized covariance matrices shows remarkably good forecasting results, in-sample and out-of-sample.     

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.     

Dynamic model averaging in large model spaces using dynamic Occam׳s window

Bayesian model averaging has become a widely used approach to accounting for uncertainty about the structural form of the model generating the data. When data arrive sequentially and the generating model can change over time, Dynamic Model Averaging (DMA) extends model averaging to deal with this situation. Often in macroeconomics, however, many candidate explanatory variables are available and the number of possible models becomes too large for DMA to be applied in its original form. We propose a new method for this situation which allows us to perform DMA without considering the whole model space, but using a subset of models and dynamically optimizing the choice of models at each point in time. This yields a dynamic form of Occam׳s window. We evaluate the method in the context of the problem of nowcasting GDP in the Euro area. We find that its forecasting performance compares well with that of other methods.     

基于ARIMA与GARCH模型的国际油价预测比较分析

在分析影响油价波动因素的基础上,利用1986年1月至2010年12月的WTI国际原油价格月度数据,分别建立ARIMA和GARCH模型对油价进行预测。并通过对2011年1月至2012年4月WTI原油价格进行外推预测,检验模型的预测效果。比较分析发现,在短期预测中,ARIMA和GARCH模型对油价的预测均比较准确,但当油价由于受到重大事件的影响而有较大波动时,模型的预测精度下降;在长期预测中,GARCH模型的预测效果优于ARIMA模型;整体来看,GARCH模型预测的精度高于ARIMA模型。因此,在国际油价预测中,用GARCH模型是比较合适的。     

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.     

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 volatility of Australian traded goods’ prices

It is generally accepted that the Australian economy is continually subject to unanticipated shocks, particularly, unexpected swings in the prices of Australia's internationally- traded goods. This article empirically investigates the nature and extent of volatility in import and export prices faced by the Australian production sector. It estimates multivariate GARCH models of the stochastic processes generating the prices of imports and exports, and of important components of exports and imports. This article proposes an index of volatility, which is used to provide a summary measure of the extent of volatility in a multivariate context. The overall conclusion is that the price growth rates for Australia's traded goods exhibit considerable time variation in volatility and that these price growth rates are highly and positively correlated with each other.     

Generalized Method of Moment estimation of multivariate multifractal models

Multifractal processes have recently been introduced as a new tool for modeling the stylized facts of financial markets and have been found to consistently provide certain gains in performance over basic volatility models for a broad range of assets and for various risk management purposes. Due to computational constraints, multivariate extensions of the baseline univariate multifractal framework are, however, still very sparse so far. In this paper, we introduce a parsimoniously designed multivariate multifractal model, and we implement its estimation via a Generalized Methods of Moments (GMM) algorithm. Monte Carlo studies show that the performance of this GMM estimator for bivariate and trivariate models is similar to GMM estimation for univariate multifractal models. An empirical application shows that the multivariate multifractal model improves upon the volatility forecasts of multivariate GARCH over medium to long forecast horizons.     

Maximum-likelihood estimation of misspecified models

Misspecified models occur frequently in econometric practice. It is therefore important to study the sampling distribution of maximum-likelihood estimators of parameters of misspecified models. This note exhibits the asymptotic covariance matrix of the ML estimator of a misspecified model. It points out that the expression for this matrix given by White is incorrect except for the very special case, rarely occuring in econometrics, that each observation is independent and identically distributed. An illustration using the standard linear regression model is provided.     

GARCH inadequacy for modelling exchange rates: empirical evidence from Latin America

This article checks for the adequacy of using GARCH models in exchange rate series. Using the Hinich portmanteau bicorrelation test, we find that a GARCH formulation or any of its variants fails to capture the data generating process of the main Latin American exchange rates. Our results highlight the potential of having misleading public policy when estimates are based in GARCH types of models. This article also complements recent similar findings encountered in European and Asian economies.     

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.     

Maximum likelihood estimation of sum-constrained linear models with insufficient observations

A maximum likelihood procedure for estimating sum-constrained linear models is presented, which seems to provide a good balance between excessive observational requirements on the one hand and an unduly restrictive specification of the contemporaneous covariance matrix on the other.     

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

Models 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.     

Volatility in productivity and the impact on unemployment

Volatility, and the uncertainty it creates, has long been recognized as a factor in economic decision making. Since hiring occurs before shocks to productivity are realized, firms’ investment in new labour is inherently risky. How large a role uncertainty in productivity has on aggregate unemployment is an empirical question that we attempt to answer. In this paper we measure the impact of higher volatility in labour productivity on the unemployment rate in the U.S. economy using a SVAR-GARCH-M model. Using the conditional standard deviation of productivity innovations from a multivariate GARCH model to measure uncertainty, we provide compelling evidence that unemployment increases with volatility. This estimated relative effect is actually larger for positive productivity shocks leading to unemployment declines only 60% as large as would have occurred using models that exclude uncertainty.