Autoregressive Conditional Heteroscedastic Process Research Articles

Real-Time Traffic Volatility Forecasting in Urban Arterial Networks

A methodology is presented for forecasting traffic volatility in urban arterial networks with real-time traffic flow information. This methodology provides a generalization of the standard modeling approach, in which both the mean, modeled by an autoregressive moving average process, and the variance, modeled by an autoregressive conditional heteroscedastic process, are time-varying. The statistical analysis and forecasting performance of the proposed model are investigated with real-time traffic detector data from a real urban arterial network. The results indicate the potential of the proposed model to improve the accuracy of predicted traffic volatility across different lengths of forecasting horizon in comparison with the standard generalized autoregressive conditional heteroscedastic methodology.

The Mib30 index and futures relationship: econometric analysis and implications for hedging

The interactions between the Mib30 stock market index and its future contract are examined. Using daily data for the 1994–2002 period, it is found that the cost-of-carry model holds as an equilibrium relationship between spot and futures prices. Deviations from equilibrium are corrected by movements in the spot market, but cross-market dynamics are also important in the short run. We model the time-varying volatility of daily returns’ as an autoregressive conditional heteroscedastic process; this model used to estimate minimum-variance hedge ratios. In- and out-of-sample comparisons with static hedging show that, by carefully choosing the ARCH specification, a significant improvement in variance reduction can be achieved.

Modeling speech signals in the time–frequency domain using GARCH

In this paper, we introduce a novel modeling approach for speech signals in the short-time Fourier transform (STFT) domain. We define the conditional variance of the STFT expansion coefficients, and model the one-frame-ahead conditional variance as a generalized autoregressive conditional heteroscedasticity (GARCH) process. The proposed approach offers a reasonable model on which to base the estimation of the variances of the STFT expansion coefficients, while taking into consideration their heavy-tailed distribution.

Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes

We prove the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of pure generalized autoregressive conditional heteroscedastic (GARCH) processes, and of autoregressive moving-average models with noise sequence driven by a GARCH model. Results are obtained under mild conditions.

The efficiency of the estimators of the parameters in GARCH processes

We propose a class of estimators for the parameters of a GARCH(p, q) sequence. We show that our estimators are consistent and asymptotically normal under mild conditions. The quasi-maximum likelihood and the likelihood estimators are discussed in detail. We show that the maximum likelihood estimator is optimal. If the tail of the distribution of the innovations is polynomial, even a quasi-maximum likelihood estimator based on exponential density performs better than the standard normal density-based quasi-likelihood estimator of Lee and Hansen and Lumsdaine. 1. Introduction. The generalized autoregressive conditional heteroscedastic (GARCH) process was introduced by Bollerslev (1986). The GARCH process has received considerable attention from applied as well as from theoretical points of view. We say that {yk, −∞ < k < ∞} is a GARCH(p,q) process if it satisfies the equations

04/00801 Utility charges for extensions: pay us now or pay us later: Switzer, S. The Electricity Journal, 2003, 16, (8), 39–46

This paper evaluates different financial price and time series models, such as mean reversion, autoregressive moving average (ARMA), integrated ARMA (ARIMA) and general autoregressive conditional heteroscedasticity (GARCH) process, usually applied for electricity price simulations. However, as these models are developed to describe the stochastic behaviour of electricity prices, they are extended by a separate data treatment for the deterministic components (trend, daily, weekly and annual cycles) of electricity spot prices. Furthermore price jumps are considered and implemented within a regime-switching model. Since 2008 market design allows for negative prices at the European Energy Exchange, which also occurred for several hours in the last years. Up to now, only a few financial and time series approaches exist, which are able to capture negative prices. This paper presents a new approach incorporating negative prices.The evaluation of the different approaches presented points out that the mean reversion and the ARMA models deliver the lowest mean root square error between simulated and historical electricity spot prices gained from the European Energy Exchange. These models posses also lower mean average errors than GARCH models. Hence, they are more suitable to simulate well-fitting price paths. Furthermore it is shown that the daily structure of historical price curves is better captured applying ARMA or ARIMA processes instead of mean-reversion or GARCH models. Another important outcome of the paper is that the regime-switching approach and the consideration of negative prices via the new proposed approach lead to a significant improvement of the electricity price simulation.

Extremes of Markov Chains with Tail Switching Potential

SummaryA recent advance in the utility of extreme value techniques has been the characterization of the extremal behaviour of Markov chains. This has enabled the application of extreme value models to series whose temporal dependence is Markovian, subject to a limitation that prevents switching between extremely high and extremely low levels. For many applications this is sufficient, but for others, most notably in the field of finance, it is common to find series in which successive values switch between high and low levels. We term such series Markov chains with tail switching potential, and the scope of this paper is to generalize the previous theory to enable the characterization of the extremal properties of series displaying this type of behaviour. In addition to theoretical developments, a modelling procedure is proposed. A simulation study is made to assess the utility of the model in inferring the extremal dependence structure of autoregressive conditional heteroscedastic processes, which fall within the tail switching Markov family, and generalized autoregressive conditional heteroscedastic processes which do not, being non-Markov in general. Finally, the procedure is applied to model extremal aspects of a financial index extracted from the New York Stock Exchange compendium.

Stochastic properties of multivariate time series equations with emphasis on ARCH

Markov chain theory is applied to the nonlinear modelling of conditional variance with focus on the in financial econometrics widely applied class of multivariate autoregressive conditional heteroscedastic (ARCH) processes. The multivariate socalled BEKK-ARCH of Engle and Kroner (1995) as well as other multivariate ARCH processes in the literature are discussed. The results show that an essential regularity condition for the existence of moments is that the largest modulus of the eigenvalues or equivalently, that the spectral radius of a certain matrix Φ parametrizing the conditional heteroscedasticity in the ARCH process is smaller than one. Due to the fact that multivariate systems are considered it is demonstrated that an important step in the derivations is based on changing the measure of size of the matrix Φ from norm to spectral radius.

GARCH model selection criteria

The autoregressive conditional heteroscedasticity (ARCH) family of models has grown to encompass a wide range of specifications, each of which is designed to enhance the ability of the model to capture the characteristics of the data. In this paper, the ability of a number of model selection criteria to correctly identify the data generating process in simulated data is established. The results of this study suggest that the Hannan-Quinn and stochastic complexity criteria provide a superior level of performance for ARCH and generalized ARCH (GARCH) processes compared to the more commonly used criteria. Where leverage and/or power effects are present, however, none of the procedures considered perform well. A new LM based test for the presence of nonlinearity and power effects is introduced and tested.

On adaptive estimation in nonstationary ARMA Models with GARCH errors

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfyinga generalized autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined $M_{\nu}$-estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be asymptotically efficient in the class of $M_{\nu}$-estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterization of the GARCH process. This paper also establishes the locally asymptotic quadratic form of the log-likelihood ratio for nonlinear time series models with ARCH-type errors.

The impacts of Hong Kong’s Currency Board reforms on the interbank market

Among the economies with a Currency Board System (CBS), Hong Kong (HK) is probably the one with the largest and most developed financial sector, as well as the highest capital mobility. Hence, studying HK’s CBS is not only crucial to HK, but also important for the understanding of the modern CBS. This paper outlines the major monetary reforms in HK since the late 1980s. The impacts of these reforms and the 1997–1998 Asian Financial Crisis are then examined empirically. We focus on the differentials between the US and HK interbank interest rates. We assume the conditional-mean equation follows an autoregressive process and the conditional-variance equation follows a generalized autoregressive conditional heteroscedasticity process. This model captures the time-varying level and volatility of the differential. In light of the empirical results we provide an assessment of the reforms in HK.

Maximum likelihood estimation of STAR and STAR‐GARCH models: theory and Monte Carlo evidence

AbstractTheoretical and practical interest in non‐linear time series models, particularly regime switching models, have increased substantially in recent years. Given the abundant research activity in analysing time‐varying volatility through Generalized Autoregressive Conditional Heteroscedasticity (GARCH) processes (see Engle, 1982; Bollerslev, 1986), it is important to analyse regime switching models with GARCH errors. A popular specification in this class is the (stationary) Smooth Transition Autoregressive–GARCH (STAR‐GARCH) model. Little is presently known about the structure of the model, or the consistency, asymptotic normality and finite sample properties of the estimators. The paper develops the structural and statistical properties of the STAR‐GARCH model, and investigates the finite sample properties of maximum likelihood estimation (MLE) of STAR and STAR‐GARCH models through numerical simulation. The effects of fixing the threshold value and/or the transition rate for the STAR model, misspecification of the conditional mean and the transition function of the STAR‐GARCH model, and the finite sample properties of the MLE for the STAR‐GARCH model, are also examined. These numerical results are used as a guide in empirical research, with an application to Standard and Poor's Composite 500 Index returns for alternative STAR‐GARCH models. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Tail dependence of random variables from ARCH and heavy-tailed bilinear models

Discussed in this paper is the dependent structure in the tails of distributions of random variables from some heavy-tailed stationary nonlinear time series. One class of models discussed is the first-order autoregressive conditional heteroscedastic (ARCH) process introduced by Engle (1982). The other class is the simple first-order bilinear models driven by heavy-tailed innovations. We give some explicit formulas for the asymptotic values of conditional probabilities used for measuring the tail dependence between two random variables from these models. Our results have significant meanings in finance.

Estimates of the likelihood of extreme returns in international stock markets

This study applies extreme-value theory to daily international stock-market returns to determine (1) whether or not returns follow a heavy-tailed stable distribution, (2) the likelihood of an extreme return, such as a 20% drop in a single day, and (3) whether or not the likelihood of an extreme event has changed since October 1987. Empirical results reject a heavy-tailed stable distribution for returns. Instead, a Student-t distribution or an autoregressive conditional heteroscedastic process is better able to capture the salient features of returns. We find that the likelihood of a large single-day return diff ers widely across markets and, for the G-7 countries, the 1987 stock-market drop appears to be largely an isolated event. A drop of this magnitude, however, is not rare in the case of Hong Kong. Finally, there is only limited evidence that the chance of a large single-day decline is more likely since the October 1987 market drop; however, exceptions include stock markets in Germany, The Netherlands and the UK.

Bootstrapping forecast intervals in ARCH models

In this paper we develop a bootstrap method for the construction of prediction intervals for an ARMA model when its innovations are an autoregressive conditional heteroscedastic process. We give a proof of the validity of the proposed bootstrap for this process. For this purpose we prove the convergence to zero in probability of the Mallows metric between the empirical distribution function and the theoretical distribution function of the residuals. The potential of the proposed method is assessed through a simulation study.

Asset Price Behaviour in Indian Stock Market: Is Conditional CAPM Relevant?

The present study attempts to test the validity of the Conditional Capital Assel Pricing Model (CAPM) in the context of Indian stock market. The Autoregressive Conditional Heteroscedastic (ARCH) process, as well as, its variants such as ARCH-in-Mean (ARCH-M), Generalised Autoregressive Conditional Heteroscedastic (GARCH) and GARCH-in Mean (GARCH-M), Me employed to ascertain whether the CAPM -with these process can explain asset price behaviour of Indian stock market better. The results based on the monthly return of 53 stocks traded in the Bombay Stock Exchonge (BSE) during the period 1987: 1-1994:5 reveal that the conditional CAPM performs well under empirical scrutiny.

The effect of time-varying covariances on asset risk premia

In this article we examine an intertemporal capital asset pricing model (CAPM) that allows for time-varying conditional covariances that are assumed to follow a multivariate integrated generalized autoregressive conditional heteroscedastic (IGARCH) process. The resulting pricing equation includes idiosyncratic risk premia in addition to the usual market beta. Empirical analysis based on ten size and ten industry portfolios reveals significant idiosyncratic premia for most portfolios. Overall, we reject the static CAPM in favor of the intertemporal CAPM.

Persistence of Stock Return Volatility in Canada

AbstractIn this article, a generalized autoregressive conditional heteroscedastic (GARCH) model is applied to Canadian data covering the period 1976‐1991. The time period is chosen to correspond to distinct monetary regimes, and to the more recent period surrounding the October 1987 crash. The excess price volatility observed over the entire period is reduced by appropriately taking into account three conditional mean and variance shifts corresponding to three distinct monetary regimes and interest rate processes. In spite of the above correction, the hypothesis of a unit root in the variance cannot be rejected for one sub‐period.Subsequently, a substantial reduction in the degree of persistence in volatility is achieved by introducing a dummy variable in the GARCH model to account for extraordinary price movements or “outliers”. The outliers were investigated and found to correspond to the release of business and economic news, in most cases of U.S. origin. Conversely, no reduction in conditional variance is obtained when allowing for seasonal effects, such as the Monday and January effects, in the GARCH equation.The results suggest that, when modelling the conditional variance equation, it is important to account for structural changes in the economic environment and to include in the analysis other sources of non‐stationarity. In doing so, the degree of persistence in market volatility is vastly reduced. The conclusion calls upon further study of the impact of extraordinary price movements and their link to economic information, as well as of other factors likely to affect volatility persistence, such as trading behaviour, in the search to unravel the dynamic behaviour of conditional variances.RésuméUn modèle gééralisé de variance conditionnelle autorégressive (GARCH) est appliqué, dans cet article, à des données canadiennes couvrant la période 1976‐1991. Les années choisies correspondent à trois régimes monétaires distincts et à la période plus récente couvrant la crise boursière d'octobre 1987. L'excès de volatilité de prix constatée durant la période entière est réduit alors qu'on tient compte de trois changements structurels dans les moyennes et les variances correspondant aux trois régimes monétaires et processus de taux d'intérět distincts. En dépit de cela, l'hypothèse de non‐stationnarité dans la variance ne peut pas ětre rejetée pour une souspériode.Par la suite, une réduction substantielle du degré de persistance dans la volatilité est obtenue en introduisant une variable artificielle dans le modèle GARCH pour tenir compte des mouvements de prix extraordinaires. Ces mouvements de prix peu communs ont été étudiés et apparaissent générés par l'arrivée de nouvelles économiques, dans la plupart du temps d'origine américaine. Par contre, aucune réduction dans la variance conditionnelle n'est obtenue quand on introduit les effets sai‐sonniers, tels l'effet “lundi” ou leffet “janvier”, dans l'équation GARCH.Les résultats laissent à penser qu'il est important de prendre en considération lors de la modélisation de l'équation de la variance les changements structurels de l'énvironnement économique et d'inclure dans l'analyse toute autre source de non stationnarité. En faisant cela, le degré de persistance dans la volatilité du marché est fortement réduit. La conclusion recommande l'étude plus approfondie de l'impact des mouvements extraordinaires de prix et de leur relation avec l'arrivée d'information économique, aussi bien que l'étude d'autres facteurs susceptibles d'influencer la persistance en volatilité tel que le comportement des agents sur le marché, dans le but d'approfondir notre connaissance sur le comportement dynamique des variances conditionnelles.

MARKET MODEL CORRECTED FOR GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTICITY AND THE SMALL FIRM EFFECT

AbstractPrior studies show that heteroscedasticity and serial correlation are present in the market model. In this paper I estimate the market model parameters under generalized autoregressive conditional heteroscedastic processes. Employing daily stock rate of return data from the Center for Research in Security Prices for 1985 through 1989, excluding October 1987 and October 1989, I show that previous estimates of beta coefficients are overstated significantly, suggesting model improvements are needed to take these problems into account. The efficiency of the market model parameters is significantly improved across all firms during the period investigated.

The relationship between forecast dispersion and forecast uncertainty: Evidence from a survey data—arch model

AbstractThis paper examines empirically the relationship between measures of forecast dispersion and forecast uncertainty from data on inflation expectations from the Livingston survey series and the Survey Research Center (SRC) survey series. Because the survey series do not provide probabilistic forecasts of inflation, we derive measures of inflation uncertainty by modelling the conditional variance of the inflation forecast errors from the survey series as an autoregressive conditional heteroscedastic (ARCH) process. The analysis is complicated by the fact that the overlap of forecast horizons for the survey series does not preclude the model's disturbance terms from displaying autocorrelation, and also places a restriction on the specification for the ARCH measures of inflation uncertainty. We estimate the model using Hansen's (1982) generalized method of moments (GMM) procedure to account for the presence of serial correlation and conditional heteroscedasticity in the disturbance terms. The results generally support the hypothesis that the measures of forecast dispersion across survey respondents are positively and statistically significantly associated with the measures of inflation uncertainty. However, the appropriateness of using forecast dispersion measures as proxies for inflation uncertainty is sensitive to the choice of the survey series.