Standard GARCH Model Research Articles

Multiscale local change point detection with applications to value-at-risk

This paper offers a new approach to modeling and forecasting of nonstationary time series with applications to volatility modeling for financial data. The approach is based on the assumption of local homogeneity: for every time point, there exists a historical interval of homogeneity, in which the volatility parameter can be well approximated by a constant. The proposed procedure recovers this interval from the data using the local change point (LCP) analysis. Afterward, the estimate of the volatility can be simply obtained by local averaging. The approach carefully addresses the question of choosing the tuning parameters of the procedure using the so-called “propagation” condition. The main result claims a new “oracle” inequality in terms of the modeling bias which measures the quality of the local constant approximation. This result yields the optimal rate of estimation for smooth and piecewise constant volatility functions. Then, the new procedure is applied to some data sets and a comparison with a standard GARCH model is also provided. Finally, we discuss applications of the new method to the Value at Risk problem. The numerical results demonstrate a very reasonable performance of the new method.

Bivariate mixed normal GARCH models and out-of-sample hedge performances

This study compares bivariate mixed normal GARCH models with standard bivariate GARCH models in terms of the percentage variance reduction of the out-of-sample hedged portfolio and also statistical significance tests of performance improvements using Superior Predictive Ability statistics. All competing models are applied to corn and wheat futures and empirical results demonstrate that the standard BEKK-GARCH model significantly outperforms the other competing GARCH models at shorter horizons. However, as the hedge horizon is extended to longer than 10 days, it is evident that the mixed normal BEKK-GARCH model is the best at the usual significance level of 5%.

Modeling Pork Supply Response and Price Volatility: The Case of Greece

This paper examines the supply response of the Greek pork market. A GARCH process is used to estimate expected price and price volatility, while price and supply equations are estimated jointly. In addition to the standard GARCH model, several different symmetric, asymmetric, and nonlinear GARCH models are estimated. The empirical results indicate that among the estimated GARCH models, the quadratic NAGARCH model seems to better describe producers' price volatility, which was found to be an important risk factor of the supply response function of the Greek pork market. Furthermore, the empirical findings show that feed price is an important cost factor of the supply response function and that high uncertainty restricts the expansion of the Greek pork sector. Finally, the model provides forecasts for quantity supplied, producers' price, and price volatility.

Time-varying joint distribution through copulas

The analysis of temporal dependence in multivariate time series is considered. The dependence structure between the marginal series is modelled through the use of copulas which, unlike the correlation matrix, give a complete description of the joint distribution. The parameters of the copula function vary through time, following certain evolution equations depending on their previous values and the historical data. The marginal time series follow standard univariate GARCH models. Full Bayesian inference is developed where the whole set of model parameters is estimated simultaneously. This represents an essential difference from previous approaches in the literature where the marginal and the copula parameters are estimated separately in two consecutive steps. Moreover, a Bayesian procedure is proposed for the estimation of several measures of risk, such as the variance, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of a portfolio of assets, providing point estimates and predictive intervals. The proposed copula model enables to capture the dependence structure between the individual assets which strongly influences these risk measures. Finally, the problem of optimal portfolio selection based on the estimation of mean–variance, mean–VaR and mean–CVaR efficient frontiers is also addressed. The proposed approach is illustrated with simulated and real financial time series.

Seasonality, returns and volatility on the Stock Exchange of Mauritius

This article investigates the effects of any seasonality on stock market returns and volatility on the Stock Exchange of Mauritius. A standard GARCH model was used on daily SEMDEX returns from 1998 to 2006. The results obtained indicate that the return series are leptokurtic, indicating a higher peak and a thicker tail than a normal distribution. Also, the mean returns on Fridays seem to be the highest while average returns on Mondays turn out to be insignificant. Finally, significant effects of weekdays were found on the conditional variance on the stock returns.

Asymptotic variance–covariance matrix of sample autocorrelations for threshold-asymmetric GARCH processes

In the field of financial time series, threshold-asymmetric conditional variance models can be used to explain asymmetric volatilities [C.W. Li and W.K. Li, On a double-threshold autoregressive heteroscedastic time series model, J. Appl. Econometrics 11 (1996), pp. 253–274]. In this paper, we consider a broad class of threshold-asymmetric GARCH processes (TAGARCH, hereafter) including standard ARCH and GARCH models as special cases. Since sample autocorrelation function provides a useful information to identify an appropriate time-series model for the data, we derive asymptotic distributions of sample autocorrelations both for original process and for squared process. It is verified that standard errors of sample autocorrelations for TAGARCH models are significantly different from unity for lower lags and they are exponentially converging to unity for higher lags. Furthermore they are shown to be asymptotically dependent while being independent of standard GARCH models. These results will be interesting in the light of the fact that TAGARCH processes are serially uncorrelated. A simulation study is reported to illustrate our results.

Economic Liberalization and Conditional Volatility of Exchange Rate in Sub-Saharan Africa: Asymmetric GARCH Analysis

Abstract: For small open economies, an understanding of movements in the exchange rate is imperative in analyzing trade and capital flows. In addition, reliable forecasting of exchange rate volatility is important in risk-taking assessment and investment decision-making, both of which are critical to long-term growth. Using an asymmetric GARCH-type approach, this paper examines the implications of economic liberalization on the stochastic behavior of the exchange rate series in a sample of sub-Sahara African (SSA) countries over the 1970–2004 period. The results indicate that exchange rate volatility is variable, and is less volatile under fixed exchange rate regime (pre-economic liberalization) and higher under flexible regime (post-economic liberalization), that is, it is asymmetric. For most of the countries, the EGARCH and TGARCH models are robust to parameter stability and gives better forecasting performance compared to the standard GARCH model.

The Out-of-Sample Forecasting of Hedged Portfolio Variances using Bivariate Mixed Normal GARCH Models

In this study we compare two standard bivariate GARCH models with new bivariate mixed normal GARCH models in terms of the percentage variance reduction of the out-of-sample hedged portfolio and the statistical significance test of the performance improvements using Hansen’s (2001) Superior Predictive Ability statistics. All competing models are applied to estimate time-varying hedge ratios for corn and wheat, traded on the Chicago Board of Trade. The empirical results demonstrate that the standard BEKK-GARCH model significantly outperforms the other competing GARCH models at shorter horizons. However, as the hedge horizon is extended to longer than 10 days, it is evident that the mixed normal BEKK-GARCH model is the best at the usual significance level of 5%.

Forecasting daily volatility with intraday data

The aim of this paper is to assess to what extent intraday data can explain and predict end-of-the-day volatility. Using a realized volatility measure as proposed by Andersen, T., T. Bollerslev, F. Diebold, and P. Labys. 2001. The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96: 42–55, we hypothesize that volatility generated at the start of the day is an important predictor of daily volatility either on its own accord or in conjunction with information about the seasonal pattern characterizing intraday volatility. We address the question of how much information needs to arrive to the market before a good predictor can be formed. Using data from a specialist market (NYSE), a dealer market (Nasdaq) and a continuous auction market (Paris Bourse), we investigate how different trading structures may affect intraday volatility formation. As a preview to our results, we find that the explanatory power of first-hour volatility for daily volatility is as high as 68%, whereas the average volatility generated during this first hour is <30%. Comparison to a standard GARCH model shows that the forecasts based on the intraday data are generally highly informative both on their own accord and in combination with the GARCH forecasts.

The Limiting Properties of the QMLE in a General Class of Asymmetric Volatility Models

In this paper we analyze the limiting properties of the estimated parameters in a general class of asymmetric volatility models which are closely related to the traditional exponential GARCH model. The new representation has three main advantages over the traditional EGARCH: (1) It allows a much more flexible representation of the conditional variance function. (2) It is possible to provide a complete characterization of the asymptotic distribution of the QML estimator based on the new class of nonlinear volatility models, something which has proven very difficult even for the traditional EGARCH. (3) It can produce asymmetric news impact curves where, contrary to the traditional EGARCH, the resulting variances do not excessively exceed the ones associated with the standard GARCH model, irrespectively of the sign of an impact of moderate size. Furthermore, the new class of models considered can create a wide array of news impact curves which provide the researcher with a richer choice set relative to the traditional. We also show in a Monte Carlo experiment the good finite sample performance of our asymptotic theoretical results and we compare them with those obtained from a parametric and the residual based bootstrap. Finally, we provide an empirical illustration.

Supply Response and Price Volatility in the Greek Pork Industry

This paper examines the supply response of the Greek pork market. A GARCH process is used to estimate expected price and price volatility, while price and supply equations are estimated jointly. In addition to the standard GARCH model, several different symmetric, asymmetric and nonlinear GARCH models are estimated. The empirical results indicate that among the estimated GARCH models the quadratic NAGARCH model seems to describe better producers' price volatility, which was found to be an important risk factor of the supply repose function of the Greek pork market. Furthermore, the empirical findings show that feed price is an important risk factor of the supply repose function and that high uncertainty restricts the expansion of the Greek pork sector. Finally, the model provides forecasts for quantity supplied, producers' price and price volatility.

Volatility Components, Affine Restrictions and Non-Normal Innovations

Recent work by Engle and Lee (1999) shows that allowing for long-run and short-run components greatly enhances a GARCH model's ability to fit daily equity return dynamics. Using the risk-neutralization in Duan (1995), we assess the option valuation performance of the Engle-Lee model and compare it to the standard one-component GARCH(1,1) model. We also compare these non-affine GARCH models to one- and two- component models from the class of affine GARCH models developed in Heston and Nandi (2000). Using the option pricing methodology in Duan (1999), we then compare the four conditionally normal GARCH models to four conditionally non-normal versions. As in Hsieh and Ritchken (2005), we find that non-affine models dominate affine models both in terms of fitting return and in terms of option valuation. For the affine models, we find strong evidence in favor of the component structure for both returns and options; for the non-affine models, the evidence is somewhat less convincing in option valuation. The evidence in favor of the non-normal GED models is strong when fitting daily returns, but the non-normal models do not provide much improvement when valuing options.

How useful is intraday data for evaluating daily Value-at-Risk?: Evidence from three Euro rates

Previous research concerned with the investigation of intraday data has typically sought to model that data using techniques to control for intraday periodicity, has applied models of short-horizon and long-horizon dependencies, or has utilised intraday data in the construction of realised variance. Using Euro exchange rate data, we apply these different modelling strategies in forecasting daily volatility and calculating Value-at-Risk measures, benchmarked against a standard GARCH model for daily and raw intraday returns. Our results suggest that the use of intraday data provides improved daily volatility and VaR forecasts relative to daily data and daily realised volatility. Further, use of the raw intraday data, or intraday data subjected to a simple standardisation procedure, provides better forecasts and VaR measures than more complicated models for intraday periodicity. These results also hold in a multi-asset portfolio setting.

Adaptive pointwise estimation in time-inhomogeneous time-series models

This paper offers a new method for estimation and forecasting of the linear and nonlinear time series when the stationarity assumption is violated. Our general local parametric approach particularly applies to general varying-coefficient parametric models, such as AR or GARCH, whose coefficients may arbitrarily vary with time. Global parametric, smooth transition, and changepoint models are special cases. The method is based on an adaptive pointwise selection of the largest interval of homogeneity with a given right-end point by a local change-point analysis. We construct locally adaptive estimates that can perform this task and investigate them both from the theoretical point of view and by Monte Carlo simulations. In the particular case of GARCH estimation, the proposed method is applied to stock-index series and is shown to outperform the standard parametric GARCH model.

Volatility Components, Affine Restrictions and Non-Normal Innovations

Recent work by Engle and Lee (1999) shows that allowing for long-run and short-run components greatly enhances a GARCH model’s ability fit daily equity return dynamics. Using the risk-neutralization in Duan (1995), we assess the option valuation performance of the Engle-Lee model and compare it to the standard one-component GARCH(1,1) model. We also compare these non-affine GARCH models to one- and two- component models from the class of affine GARCH models developed in Heston and Nandi (2000). Using the option pricing methodology in Duan (1999), we then compare the four conditionally normal GARCH models to four conditionally non-normal versions. As in Hsieh and Ritchken (2005), we find that non-affine models dominate affine models both in terms of fitting return and in terms of option valuation. For the affine models we find strong evidence in favor of the component structure for both returns and options, but for the non-affine models the evidence is much less strong in option valuation. The evidence in favor of the non-normal models is strong when fitting daily returns, but the non-normal models do not provide much improvement when valuing options.

Modelling Conditional and Unconditional Heteroskedasticity with Smoothly Time-Varying Structure

In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the conditional variance to have a smooth time-varying structure of either additive or multiplicative type. The suggested parameterizations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. A modelling strategy for these new time-varying parameter GARCH models is developed. It relies on a sequence of Lagrange multiplier tests, and the adequacy of the estimated models is investigated by Lagrange multiplier type misspecification tests. Finite-sample properties of these procedures and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance.

제곱수익률 그래프와 TGARCH 모형을 이용한 비대칭 변동성 분석

일반적인 ARCH 형태의 모형들은 자산수익률의 급첨 (leptokurtic; heavy-tail) 성질과 변동성 집중 (volatility clustering) 현상 등의 특징을 잘 포착해내는 반면, 수익률의 부호에 따른 비대칭 레버리지 효과 (leverage effect)는 반영 할 수 없다는 단점을 가진다. 따라서 최근 금융 시계열 분야에서는 비대칭-조건부-이분산 시계열 모형에 대한 관심이 높아지고 있다. 본 연구에서는 국내 금융 시계열자료 (KOSPI, KOSDAQ, 환율, 채권, 주요종목의 주가)의 수익률 제곱을 그래프화 하여 비대칭 이분산성을 시각적으로 탐지하고 이를 바탕으로 비대 칭 TGARCH(1,1) 모형을 적합한 후 기존의 대칭 GARCH(1,1) 모형과 비교분석하고자 한다. As is pointed out by Gourieroux (1997), the volatility effects in financial time series vary according to the signs of the return rates and therefore asymmetric Threshold-GARCH (TGARCH, henceforth) processes are natural extensions of the standard GARCH toward asymmetric volatility modeling. For preliminary detection of asymmetry in volatility, we suggest graphs of squared-log-returns for various financial time series including KOSPI, KOSDAQ and won-Euro exchange rate. Next, asymmetric TGARCH(1,1) model fits are provided in comparisons with standard GARCH(1.1) models.

Metal volatility in presence of oil and interest rate shocks

This study uses three “two factor” volatility models of the GARCH family to examine the volatility behavior of three strategic commodities: gold, silver and copper, in the presence of crude oil and interest rate shocks. The results of the standard GARCH models suggest that gold and silver have almost the same volatility persistence which is greater than that of copper. The CGARCH estimates indicate that the (short-run) transitory component of volatility converges to zero much faster for copper than for gold and silver in this sequence. However, the permanent volatility component demonstrates equally strong persistence in the long-run for all three metals. The EGARCH results suggest that the leverage effect is present and significant for copper only, implying that gold and silver can be good investment in anticipation of bad times. Past oil shock does not impact all three metals similarly. Monetary policy and to leaser extent the oil shocks have calming effects on precious metals but not on copper if the T bill rate is used. Crises such as the 2003 Iraq war heighten metal volatility. These results have implications for derivatives valuations, using gold as a reserve asset, risk analysis, and for the commodity-exporting countries and commodity-producing firms.

Foreign Exchange Volatility Shifts and Futures Hedging: An ICSS-GARCH Approach

In this study, the impact of volatility regime shifts on volatility persistence and hedge ratio estimation is determined for four major currencies using an iterated cumulative sums of squares (ICSS)-GARCH model. Employing a standard GARCH (1,1) model as the benchmark, within-sample results demonstrate that the inclusion of volatility shifts substantially reduces volatility persistence and the significance of the ARCH and GARCH coefficients. In terms of hedging effectiveness, the ICSS-GARCH model outperforms the standard GARCH model for all four currencies. In comparison to two constant volatility models, the standard GARCH model yields the lowest performance, whereas the ICSS-GARCH model performs at least as well as these models. In out-of-sample analysis, the GARCH model provides substantial variance reductions relative to the constant volatility models. Moreover, the ICSS-GARCH model yields positive variance reductions relative to all competing models, including the standard GARCH model. The results suggest that in cases where dynamic hedging is important, sudden shifts in volatility should not be ignored.

Estimation and tests for power-transformed and threshold GARCH models.

Consider a class of power-transformed and threshold GARCH (PTTGRACH) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa [2004. Stationarity and moment structure for Box–Cox transformed threshold GARCH(1,1) processes. Statistics & Probability Letters 68, 209–220.] and includes the standard GARCH model and many other models as special cases. We first establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has finite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy-tailed errors. Furthermore, the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the financial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH model, we give in the appendix a necessary and sufficient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.