A conditionally heteroskedastic independent factor model with an application to financial stock returns

We propose a new conditionally heteroskedastic factor model, the GICA-GARCH model, which combines independent component analysis (ICA) and multivariate GARCH (MGARCH) models. This model assumes that the data are generated by a set of underlying independent components (ICs) that capture the co-movements among the observations, which are assumed to be conditionally heteroskedastic. The GICA-GARCH model separates the estimation of the ICs from their fitting with a univariate ARMA-GARCH model. Here, we will use two ICA approaches to find the ICs: the first estimates the components, maximizing their non-Gaussianity, while the second exploits the temporal structure of the data. After estimating and identifying the common ICs, we fit a univariate GARCH model to each of them in order to estimate their univariate conditional variances. The GICA-GARCH model then provides a new framework for modelling the multivariate conditional heteroskedasticity in which we can explain and forecast the conditional covariances of the observations by modelling the univariate conditional variances of a few common ICs. We report some simulation experiments to show the ability of ICA to discover leading factors in a multivariate vector of financial data. Finally, we present an empirical application to the Madrid stock market, where we evaluate the forecasting performances of the GICA-GARCH and two additional factor GARCH models: the orthogonal GARCH and the conditionally uncorrelated components GARCH.     

MULTIVARIATE VOLATILITY MODELING OF ELECTRICITY FUTURES

We model the dynamic volatility and correlation structure of electricity futures of the European Energy Exchange index. We use a new multiplicative dynamic conditional correlation (mDCC) model to separate long‐run from short‐run components. We allow for smooth changes in the unconditional volatilities and correlations through a multiplicative component that we estimate nonparametrically. For the short‐run dynamics, we use a GJR‐GARCH model for the conditional variances and augmented DCC models for the conditional correlations. We also introduce exogenous variables to account for congestion and delivery date effects in short‐term conditional variances. We find different correlation dynamics for long‐ and short‐term contracts and the new model achieves higher forecasting performance compared \to a standard DCC model. Copyright © 2012 John Wiley & Sons, Ltd.     

Non-stationarity in garch models: A bayesian analysis

First, the non-stationarity properties of the conditional variances in the GARCH(1, 1) model are analysed using the concept of infinite persistence of shocks. Given a time sequence of probabilities for increasing/decreasing conditional variances, a theoretical formula for quasi-strict non-stationarity is defined. The resulting conditions for the GARCH(1,1) model are shown to differ from the weak stationarity conditions mainly used in the literature. Bayesian statistical analysis using Monte Carlo integration is applied to analyse both stationarity concepts for the conditional variances of the US 3-month treasury bill rate. Interest rates are known for their weakly non-stationary conditional variances but, using a quasi-strict stationarity measure, it is shown that the conditional variances are likely to be stationary. Second, the level of the treasury bill rate is analysed for non-stationarity using Bayesian unit root methods. The disturbances of the GARCH model for the treasury bill rate are t-distributed. It is shown that the unit root parameter is negatively correlated with the degrees-of-freedom parameter. Imposing normally distributed disturbances leads therefore to underestimation of the non-stationarity in the level of the treasury bill rate.     

REALIZED BETA GARCH: A MULTIVARIATE GARCH MODEL WITH REALIZED MEASURES OF VOLATILITY

We introduce a multivariate generalized autoregressive conditional heteroskedasticity (GARCH) model that incorporates realized measures of variances and covariances. Realized measures extract information about the current levels of volatilities and correlations from high‐frequency data, which is particularly useful for modeling financial returns during periods of rapid changes in the underlying covariance structure. When applied to market returns in conjunction with returns on an individual asset, the model yields a dynamic model specification of the conditional regression coefficient that is known as the beta. We apply the model to a large set of assets and find the conditional betas to be far more variable than usually found with rolling‐window regressions based exclusively on daily returns. In the empirical part of the paper, we examine the cross‐sectional as well as the time variation of the conditional beta series during the financial crises. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating and predicting multivariate volatility thresholds in global stock markets

We propose a general double tree structured AR‐GARCH model for the analysis of global equity index returns. The model extends previous approaches by incorporating (i) several multivariate thresholds in conditional means and volatilities of index returns and (ii) a richer specification for the impact of lagged foreign (US) index returns in each threshold. We evaluate the out‐of‐sample forecasting power of our model for eight major equity indices in comparison to some existing volatility models in the literature. We find strong evidence for more than one multivariate threshold (more than two regimes) in conditional means and variances of global equity index returns. Such multivariate thresholds are affected by foreign (US) lagged index returns and yield a higher out‐of‐sample predictive power for our tree structured model setting. Copyright © 2006 John Wiley & Sons, Ltd.     

Copula-based multivariate GARCH model with uncorrelated dependent errors

Multivariate GARCH (MGARCH) models are usually estimated under multivariate normality. In this paper, for non-elliptically distributed financial returns, we propose copula-based multivariate GARCH (C-MGARCH) model with uncorrelated dependent errors, which are generated through a linear combination of dependent random variables. The dependence structure is controlled by a copula function. Our new C-MGARCH model nests a conventional MGARCH model as a special case. The aim of this paper is to model MGARCH for non-normal multivariate distributions using copulas. We model the conditional correlation (by MGARCH) and the remaining dependence (by a copula) separately and simultaneously. We apply this idea to three MGARCH models, namely, the dynamic conditional correlation (DCC) model of Engle [Engle, R.F., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350], the varying correlation (VC) model of Tse and Tsui [Tse, Y.K., Tsui, A.K., 2002. A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. Journal of Business and Economic Statistics 20, 351–362], and the BEKK model of Engle and Kroner [Engle, R.F., Kroner, K.F., 1995. Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150]. Empirical analysis with three foreign exchange rates indicates that the C-MGARCH models outperform DCC, VC, and BEKK in terms of in-sample model selection and out-of-sample multivariate density forecast, and in terms of these criteria the choice of copula functions is more important than the choice of the volatility models.     

DCC- and DECO-HEAVY: Multivariate GARCH models based on realized variances and correlations

This paper introduces the scalar DCC-HEAVY and DECO-HEAVY models for conditional variances and correlations of daily returns based on measures of realized variances and correlations built from intraday data. Formulas for multi-step forecasts of conditional variances and correlations are provided. Asymmetric versions of the models are developed. An empirical study shows that in terms of forecasts the scalar HEAVY models outperform the scalar BEKK-HEAVY model based on realized covariances and the scalar BEKK, DCC, and DECO multivariate GARCH models based exclusively on daily data.     

Modelling Regime‐Specific Stock Price Volatility*

Single‐state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state‐dependent behaviour of volatility nor explain why the equity index skew persists into long‐dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state‐dependent leverage effect to NM‐GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time‐varying volatility risk premia. An empirical study on European equity indices identifies two‐state asymmetric NM‐GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean‐reversion and leverage effects during crash markets are quite different from those in the stable regime.     

Information flows and stock market volatility

This study examines how news is distributed across stocks. A model is developed that categorizes a stock's latent news into normal and nonnormal news, and allows both types of news to be filtered through to other stocks. This is achieved by formulating a model that jointly incorporates a multivariate lognormal‐Poisson jump process (for nonnormal news) and a multivariate GARCH process (for normal news), in addition to a news (or shock) transmission mechanism that allows the shocks from both processes to impact intertemporally on all stocks in the system. The relationship between news and the expected volatility surface is explored and a unique news impact surface is derived that depends on time, news magnitude, and news type. We find that the effect of nonnormal news on volatility expectations typically builds up before dissipating, with the news transmission mechanism effectively crowding‐out normal news and crowding‐in nonnormal news. Moreover, in contrast to the standard approach for measuring leverage effects using asymmetric generalized autoregressive conditional heteroskedasticity models, we find that leverage effects stem predominantly from nonnormal news. Finally, we find that the capacity to identify positively or negatively correlated stock returns is ambiguous in the short term, and depends heavily on the behavior of the nonnormal news component.     

A Portfolio Index GARCH model

This paper develops the structure of a parsimonious Portfolio Index (PI) GARCH model. Unlike the conventional approach to Portfolio Index returns, which employs the univariate ARCH class, the PI-GARCH approach incorporates the effects on individual assets, leading to a better understanding of portfolio risk management, and achieves greater accuracy in forecasting Value-at-Risk (VaR) thresholds. For various asymmetric GARCH models, a Portfolio Index Composite News Impact Surface (PI-CNIS) is developed to measure the effects of news on the conditional variances. The paper also investigates the finite sample properties of the PI-GARCH model. The empirical example shows that the asymmetric PI-GARCH-t model outperforms the GJR-t model and the filtered historical simulation with a t distribution in forecasting VaR thresholds.     

Optimal Control of Heteroscedastic Macroeconomic Models

This paper analyses the implications of heteroscedasticity for optimal macroeconomic policy and welfare. We find that changes in the variance structure driven by exogenous processes like generalized autoregressive conditional heteroscedasticity (GARCH) affect welfare but not the optimal feedback rule. However, changes in the variance structure driven by state‐dependent processes affect both. We also derive certainty‐equivalent transformations of state‐dependent volatility models that allow standard quadratic dynamic programming algorithms to be employed to study optimal policy. These results are illustrated numerically using a reduced‐form model of the US economy in which changes in volatility are driven by a GARCH process and the rate of inflation. Copyright © 2015 John Wiley & Sons, Ltd.     

Sequential conditional correlations: Inference and evaluation

This paper presents a new approach to the modeling of conditional correlation matrices within the multivariate GARCH framework. The procedure, which consists of breaking the matrix into the product of a sequence of matrices with desirable characteristics, in effect converts a highly dimensional and intractable optimization problem into a series of simple and feasible estimations. This in turn allows for richer parameterizations and complex functional forms for the single components. An empirical application involving the conditional second moments of 69 selected stocks from the NASDAQ100 shows how the new procedure results in strikingly accurate measures of the conditional correlations.     

The dynamic connectedness and hedging opportunities of implied and realized volatility: Evidence from clean energy ETFs

This paper aims to examine dynamic connectedness and hedging opportunities between the realized volatilities of clean energy ETFs and energy implied volatilities through Time-Varying Parameter Vector Autoregression Model (TVP-VAR) and Asymmetric Dynamic Conditional Correlation (ADCC) GARCH models. TVP-VAR analysis results show that dynamic connectedness increases during turbulence periods. We also determine that clean energy ETFs such as PBW, QCLN, SMOG, and TAN are net volatility transmitters. Surprisingly, OVX is a net volatility receiver, especially with the developments after the Paris Agreement in 2016.As a result of the ADCC GARCH analysis, we determine that the conditional correlation between clean energy ETFs and implied volatility ETFs is asymmetric, and negative information shocks increase the conditional correlation. Although OVX is a cheap alternative for hedging long position risks in clean energy ETFs, VXXLE is more effective than OVX in terms of hedging effectiveness. These findings provide insight for individual and institutional investors, and portfolio managers on how negative and positive shocks change the conditional correlation between assets at different levels.     

Normal mixture GARCH(1,1): applications to exchange rate modelling

Some recent specifications for GARCH error processes explicitly assume a conditional variance that is generated by a mixture of normal components, albeit with some parameter restrictions. This paper analyses the general normal mixture GARCH(1,1) model which can capture time variation in both conditional skewness and kurtosis. A main focus of the paper is to provide evidence that, for modelling exchange rates, generalized two‐component normal mixture GARCH(1,1) models perform better than those with three or more components, and better than symmetric and skewed Student's t‐GARCH models. In addition to the extensive empirical results based on simulation and on historical data on three US dollar foreign exchange rates (British pound, euro and Japanese yen), we derive: expressions for the conditional and unconditional moments of all models; parameter conditions to ensure that the second and fourth conditional and unconditional moments are positive and finite; and analytic derivatives for the maximum likelihood estimation of the model parameters and standard errors of the estimates. Copyright © 2006 John Wiley & Sons, Ltd.     

Conditional heteroscedasticity of exchange rates: further results based on the fractionally integrated approach

A recent article (Tse, 1998 ) published in this journal analysed the conditional heteroscedasticity of the yen–dollar exchange rate based on the fractionally integrated asymmetric power ARCH model. In this paper, we present replication results using Tse's ( 1998 ) yen–dollar series. We also examine the robustness of Tse's ( 1998 ) findings across different currencies, sample periods and non‐nested GARCH‐type models. Unlike Tse ( 1998 ), we find some evidence of asymmetric conditional volatility for daily returns of currencies measured against the dollar or the yen. Copyright © 2004 John Wiley & Sons, Ltd.     

Stock returns and volatility: Evidence from the Athens Stock market index

This paper investigates the volatility of the Athens Stock excess stock returns over the period 1990–1999 through the comparison of various conditional hetero-skedasticity models. The empirical results indicate that there is significant evidence for asymmetry in stock returns, which is captured by a quadratic GARCH specification model, while there is strong persistence of shocks into volatility.     

Modelling the conditional volatility of commodity index futures as a regime switching process

Commodity index futures offer a versatile tool for gaining different forms of exposure to commodity markets. Volatility is a critical input in many of these applications. This paper examines issues in modelling the conditional variance of futures returns based on the Goldman Sachs Commodity Index (GSCI). Given that commodity markets tend to be ‘choppy’ (Webb, 1987 ), a general econometric model is proposed that allows for abrupt changes or regime shifts in volatility, transition probabilities which vary explicitly with observable fundamentals such as the basis, GARCH dynamics, seasonal variations and conditional leptokurtosis. The model is applied to daily futures returns on the GSCI over 1992–1997. The results show clear evidence of regime shifts in conditional mean and volatility. Once regime shifts are accounted for, GARCH effects are minimal. Consistent with the theory of storage, returns are more likely to switch to the high‐variance state when the basis is negative than when the basis is positive. The regime switching model also performs well in forecasting the daily volatility compared to standard GARCH models without regime switches. The model should be of interest to sophisticated traders who base their trading strategies on short‐term volatility movements, managed commodity funds interested in hedging an underlying diversified portfolio of commodities and investors of options and other derivatives tied to GSCI futures contracts. Copyright © 2001 John Wiley & Sons, Ltd.     

Constructing Multi‐Country Rational Expectations Models

This article considers some of the technical issues involved in using the global vector autoregression (GVAR) approach to construct a multi‐country rational expectations (RE) model and illustrates them with a new Keynesian model for 33 countries estimated with quarterly data over the period 1980–2011. The issues considered are: the measurement of steady states; the determination of exchange rates and the specification of the short‐run country‐specific models; the identification and estimation of the model subject to the theoretical constraints required for a determinate rational expectations solution; the solution of a large RE model; the structure and estimation of the covariance matrix and the simulation of shocks. The model used as an illustration shows that global demand and supply shocks are the most important drivers of output, inflation and interest rates in the long run. By contrast, monetary or exchange rate shocks have only a short‐run impact in the evolution of the world economy. The article also shows the importance of international connections, directly as well as indirectly through spillover effects. Overall, ignoring global inter‐connections as country‐specific models do, could give rise to misleading conclusions.     

Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations

ARCH and GARCH models are widely used to model financial market volatilities in risk management applications. Considering a GARCH model with heavy-tailed innovations, we characterize the limiting distribution of an estimator of the conditional value-at-risk (VaR), which corresponds to the extremal quantile of the conditional distribution of the GARCH process. We propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. Finally, we apply the proposed approach to an energy market data set.     

Analytical quasi maximum likelihood inference in multivariate volatility models

Quasi maximum likelihood estimation and inference in multivariate volatility models remains a challenging computational task if, for example, the dimension of the parameter space is high. One of the reasons is that typically numerical procedures are used to compute the score and the Hessian, and often they are numerically unstable. We provide analytical formulae for the score and the Hessian for a variety of multivariate GARCH models including the Vec and BEKK specifications as well as the recent dynamic conditional correlation model. By means of a Monte Carlo investigation of the BEKK–GARCH model we illustrate that employing analytical derivatives for inference is clearly preferable to numerical methods.