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.     

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.     

Temporal aggregation of volatility models

In this paper, we consider temporal aggregation of volatility models. We introduce semiparametric volatility models, termed square-root stochastic autoregressive volatility (SR-SARV), which are characterized by autoregressive dynamics of the stochastic variance. Our class encompasses the usual GARCH models and various asymmetric GARCH models. Moreover, our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables. The SR-SARV class is a natural extension of the class of weak GARCH models. This extension has four advantages: (i) we do not assume that fourth moments are finite; (ii) we allow for asymmetries (skewness, leverage effect) that are excluded from weak GARCH models; (iii) we derive conditional moment restrictions and (iv) our framework allows us to study temporal aggregation of IGARCH models.     

Recent Theoretical Results for Time Series Models with GARCH Errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA–GARCH are summarized. Various new ARCH–type models, including double threshold ARCH and GARCH, ARFIMA–GARCH, CHARMA and vector ARMA–GARCH, are also reviewed.     

Measuring Spot Variance Spillovers when (Co)variances are Time‐varying – The Case of Multivariate GARCH Models

We propose global and disaggregated spillover indices that allow us to assess variance and covariance spillovers, locally in time and conditionally on time‐t information. Key to our approach is the vector moving average representation of the half‐vectorized ‘squared’ multivariate GARCH process of the popular BEKK model. In an empirical application to a four‐dimensional system of broad asset classes (equity, fixed income, foreign exchange and commodities), we illustrate the new spillover indices at various levels of (dis)aggregation. Moreover, we demonstrate that they are informative of the value‐at‐risk violations of portfolios composed of the considered asset classes.     

金融时间序列中加性异常值的鉴别与校正

借鉴Franses and Ghijsel[1](1999)和Charles and Darne[2](2005)提出的鉴别和校正金融序列加性异常值的方法,以GARCH模型为例,对我国的上证综合指数和深圳成分指数进行了加性异常值的鉴定与校正,并对校正后的残差进行了正态检验。结果表明该方法效果显著,进行异常值校正后的GARCH(1,1),更好地拟合金融时间序列中的尖峰厚尾和波动丛聚性的特性,纠正了正态分布的GARCH(1,1)对时间序列拟合的偏误。     

Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

Evaluating interval forecasts of high‐frequency financial data

A number of methods of evaluating the validity of interval forecasts of financial data are analysed, and illustrated using intraday FTSE100 index futures returns. Some existing interval forecast evaluation techniques, such as the Markov chain approach of Christoffersen ( 1998 ), are shown to be inappropriate in the presence of periodic heteroscedasticity. Instead, we consider a regression‐based test, and a modified version of Christoffersen's Markov chain test for independence, and analyse their properties when the financial time series exhibit periodic volatility. These approaches lead to different conclusions when interval forecasts of FTSE100 index futures returns generated by various GARCH(1,1) and periodic GARCH(1,1) models are evaluated. Copyright © 2003 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.     

Structural breaks and GARCH models of exchange rate volatility

We investigate the empirical relevance of structural breaks for GARCH models of exchange rate volatility using both in‐sample and out‐of‐sample tests. We find significant evidence of structural breaks in the unconditional variance of seven of eight US dollar exchange rate return series over the 1980–2005 period—implying unstable GARCH processes for these exchange rates—and GARCH(1,1) parameter estimates often vary substantially across the subsamples defined by the structural breaks. We also find that it almost always pays to allow for structural breaks when forecasting exchange rate return volatility in real time. Combining forecasts from different models that accommodate structural breaks in volatility in various ways appears to offer a reliable method for improving volatility forecast accuracy given the uncertainty surrounding the timing and size of the structural breaks. Copyright © 2008 John Wiley & Sons, Ltd.     

An econometric analysis of asymmetric volatility: Theory and application to patents

The purpose in registering patents is to protect the intellectual property of the rightful owners. Deterministic and stochastic trends in registered patents can be used to describe a country's technological capabilities and act as a proxy for innovation. This paper presents an econometric analysis of the symmetric and asymmetric volatility of the patent share, which is based on the number of registered patents for the top 12 foreign patenting countries in the USA. International rankings based on the number of foreign US patents, patent intensity (or patents per capita), patent share, the rate of assigned patents for commercial exploitation, and average rank scores, are given for the top 12 foreign countries. Monthly time series data from the United States Patent and Trademark Office for January 1975 to December 1998 are used to estimate symmetric and asymmetric models of the time-varying volatility of the patent share, namely US patents registered by each of the top 12 foreign countries relative to total US patents. A weak sufficient condition for the consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the univariate GJR(1,1) model is established under non-normality of the conditional shocks. The empirical results provide a diagnostic validation of the regularity conditions underlying the GJR(1,1) model, specifically the log-moment condition for consistency and asymptotic normality of the QMLE, and the computationally more straightforward but stronger second and fourth moment conditions. Of the symmetric and asymmetric models estimated, AR(1)–EGARCH(1,1) is found to be suitable for most countries, while AR(1)–GARCH(1,1) and AR(1)–GJR(1,1) also provide useful insights. Non-nested procedures are developed to test AR(1)–GARCH(1,1) versus AR(1)–EGARCH(1,1), and AR(1)–GJR(1,1) versus AR(1)–EGARCH(1,1).     

A forecast comparison of volatility models: does anything beat a GARCH(1,1)?

We compare 330 ARCH‐type models in terms of their ability to describe the conditional variance. The models are compared out‐of‐sample using DM–$ exchange rate data and IBM return data, where the latter is based on a new data set of realized variance. We find no evidence that a GARCH(1,1) is outperformed by more sophisticated models in our analysis of exchange rates, whereas the GARCH(1,1) is clearly inferior to models that can accommodate a leverage effect in our analysis of IBM returns. The models are compared with the test for superior predictive ability (SPA) and the reality check for data snooping (RC). Our empirical results show that the RC lacks power to an extent that makes it unable to distinguish ‘good’ and ‘bad’ models in our analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

Testing multivariate distributions in GARCH models

In this paper, we consider testing distributional assumptions in multivariate GARCH models based on empirical processes. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions. We then apply the Khmaladze's martingale transformation (K-transformation) to the empirical process in the presence of estimated parameters. The K-transformation eliminates the effect of parameter estimation, allowing a distribution-free test statistic to be constructed. We show that the K-transformation takes a very simple form for testing multivariate normal and multivariate t-distributions. The procedure is applied to a multivariate financial time series data set.     

Does investor sentiment impact the returns and volatility of Islamic equities?

In this paper, we estimate generalized autoregressive conditional heteroskedasticity (GARCH) and vector autoregressive (VAR) models to examine whether investor sentiment impacts the returns and volatility of various U.S. Dow Jones Islamic equity indices. The results from GARCH estimations show that changes in investor sentiment are positively correlated with the returns of the Shari’ah-compliant market portfolio. In addition, we find similar results for the three Shari’ah-compliant firm-size portfolios (i.e., large-, medium-, and small-cap). However, this relationship is stronger for harder to arbitrage Shari’ah-compliant stocks; that is, investor sentiment has a greater influence on small-cap equities. Additionally, estimations from the vector autoregressive model confirm the aforementioned results. In terms of volatility, GARCH estimations suggest that bullish shifts in investor sentiment in the current period are accompanied by lower conditional volatility in the ensuing period. In general, our findings suggest that as noise traders create more risk the market seems to reward them with higher expected returns.     

Shock and volatility spillovers among equity sectors of the Gulf Arab stock markets

The major objectives of this study are twofold. The first objective is to examine the dynamic volatility and volatility transmission in a multivariate setting using the VAR(1)–GARCH(1,1) model for three major sectors, namely, Service, Banking and Industrial/or Insurance, in four Gulf Cooperation Council (GCC)’s economies (Kuwait, Qatar, Saudi Arabia and UAE). The second is to use the models’ results to compute and analyze the optimal weights and hedge ratios for two-sector portfolio holdings, comprised of the three sectors for each country. The results suggest that past own volatilities matter more than past shocks and there are moderate volatility spillovers between the sectors within the individual countries, with the exception of Qatar. Moreover, the values for ratios of hedging long positions with short positions in the GCC sectors are smaller than those for the US equity sectors. The optimal portfolio weights favor the Banking/financial sector for Qatar, Saudi Arabia and UAE and the Industrial sector for Kuwait.     

Inflation Uncertainty, Output Growth Uncertainty and Macroeconomic Performance

We use a bivariate generalized autoregressive conditionally heteroskedastic (GARCH) model of inflation and output growth to examine the causality relationship among nominal uncertainty, real uncertainty and macroeconomic performance measured by the inflation and output growth rates. The application of the constant conditional correlation GARCH(1,1) model leads to a number of interesting conclusions. First, inflation does cause negative welfare effects, both directly and indirectly, i.e. via the inflation uncertainty channel. Secondly, in some countries, more inflation uncertainty provides an incentive to Central Banks to surprise the public by raising inflation unexpectedly. Thirdly, in contrast to the assumptions of some macroeconomic models, business cycle variability and the rate of economic growth are related. More variability in the business cycle leads to more output growth.     

Estimation of fractional integration under temporal aggregation

A result characterizing the effect of temporal aggregation in the frequency domain is known for arbitrary stationary processes and generalized for difference-stationary processes here. Temporal aggregation includes cumulation of flow variables as well as systematic (or skip) sampling of stock variables. Next, the aggregation result is applied to fractionally integrated processes. In particular, it is investigated whether typical frequency domain assumptions made for semiparametric estimation and inference are closed with respect to aggregation. With these findings it is spelled out, which estimators remain valid upon aggregation under which conditions on bandwidth selection.     

BAYESIAN INFERENCE METHODS FOR UNIVARIATE AND MULTIVARIATE GARCH MODELS: A SURVEY

This survey reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The paper makes emphasis on recent Bayesian non‐parametric approaches for GARCH models that avoid imposing arbitrary parametric distributional assumptions. These novel approaches implicitly assume infinite mixture of Gaussian distributions on the standardized returns which have been shown to be more flexible and describe better the uncertainty about future volatilities. Finally, the survey presents an illustration using real data to show the flexibility and usefulness of the non‐parametric approach.     

Thresholds,news impact surfaces and dynamic asymmetric multivariate GARCH

Dynamic Asymmetric Multivariate GARCH (DAMGARCH) is a new model that extends the Vector ARMA‐GARCH (VARMA‐GARCH) model of Ling and Mc Aleer (2003) by introducing multiple thresholds and time‐dependent structure in the asymmetry of the conditional variances. Analytical expressions for the news impact surface implied by the new model are also presented. DAMGARCH models the shocks affecting the conditional variances on the basis of an underlying multivariate distribution. It is possible to model explicitly asset‐specific shocks and common innovations by partitioning the multivariate density support. This article presents the model structure, describes the implementation issues, and provides the conditions for the existence of a unique stationary solution, and for consistency and asymptotic normality of the quasi‐maximum likelihood estimators. The article also presents an empirical example to highlight the usefulness of the new model.     

A general asymptotic theory for time-series models

This paper develops a general asymptotic theory for the estimation of strictly stationary and ergodic time–series models. Under simple conditions that are straightforward to check, we establish the strong consistency, the rate of strong convergence and the asymptotic normality of a general class of estimators that includes LSE, MLE and some M-type estimators. As an application, we verify the assumptions for the long-memory fractional ARIMA model. Other examples include the GARCH(1,1) model, random coefficient AR(1) model and the threshold MA(1) model.