Heteroscedastic Time Series Research Articles

A simple additivity test for conditionally heteroscedastic nonlinear autoregression

In this article, we propose a test for the additivity of a nonlinear conditionally heteroscedastic autoregressive model. The test is based on the unequal variance unbalanced design ANOVA scheme. An asymptotic distribution of the test statistic is derived and the test performance in finite samples is studied using simulation. To the best of our knowledge, this is the first additivity test for a conditionally heteroscedastic time series model.

An efficient locally asymptotic parametric test in nonlinear heteroscedastic time series models

Abstrac t. In this paper we deal with a locally asymptotic stringent test for a general class of nonlinear time series heteroscedastic models. Based on the local asymptotic normality (LAN) property of these models, we propose a scoretyp test statistic for testing hypotheses on the parameters appearing in the mean and variance functions of the proposed statistical test with and without nuisance parameters. Its asymptotic null distribution is obtained as well as the local power of the test. Resume . Dans cet article, nous Letudions les propriLetLes asymptotiques dfun test de score traitant simultanLement des hypoth`eses portant sur des fonctions moyennes et variances conditionnelles dans une classe assez gLenLerale de mod`eles hLetLeroscLedastiques non linLeaires de sLeries chronologiques. La suite des alternatives locales considLerLee est paramLetrique portant sur les param`etres intervenant dans les fonctions moyennes et variances du mod`ele. Nous Letablissons dfabord la normalitLe locale asymptotique (LAN) du mod`ele. En se basant sur ce rLesultat la loi limite de la statistique du test proposLee a LetLe obtenue sous lfhypoth`ese nulle et aussi sous des alternatives locales en prLesence ou non des param`etres de nuisance. Key words : ARCH processes; Ergodic processes; LAN; Local power; Nonlinear processes; Score test; Time series.

Non-Invertibility in Some Heteroscedastic Models

In order to calculate the unobserved volatility in conditional heteroscedastic time series models, the natural recursive approximation is very often used. Following Straumann and Mikosch (2006), we will call the model invertible if this approximation (based on true parameter vector) converges to the real volatility. Our main result is the necessary condition for invertibility. We will show that the stationary GARCH(p, q) model is always invertible, but certain types of models, such as EGARCH of Nelson (1991) and VGARCH of Engle and Ng (1993) may indeed be non-invertible. Moreover, we will demonstrate it's possible for the pair (true volatility, approximation) to have a non-degenerate stationary distribution. In such cases, the volatility forecast given by the recursive approximation with the true parameter vector is inconsistent.

연속형-GARCH 시계열의 범주형화(Clipping)를 통한 분석

본 논문에서는 연속형-GARCH 시계열 자료인 금융 시계열 자료에 대해서 클리핑(clipping)을 통해 얻은 이항(binary) 범주형 시계열을 분석하고 응용하는 방안에 대해 연구하고 있다. 모수추정 방법을 소개하고 있으며 이를 이용하여 이분산 시계열과 연관된 확률을 추정하는 방법을 예시하였다. This short article is concerned with a categorical time series obtained after clipping a heteroscedastic GARCH process. Estimation methods are discussed for the model parameters appearing both in the original process and in the resulting binary time series from a clipping (cf. Zhen and Basawa, 2009). Assuming AR-GARCH model for heteroscedastic time series, three data sets from Korean stock market are analyzed and illustrated with applications to calculating certain probabilities associated with the AR-GARCH process.

Prediction intervals in conditionally heteroscedastic time series with stochastic components

Abstract Differencing is a very popular stationary transformation for series with stochastic trends. Moreover, when the differenced series is heteroscedastic, authors commonly model it using an ARMA-GARCH model. The corresponding ARIMA-GARCH model is then used to forecast future values of the original series. However, the heteroscedasticity observed in the stationary transformation should be generated by the transitory and/or the long-run component of the original data. In the former case, the shocks to the variance are transitory and the prediction intervals should converge to homoscedastic intervals with the prediction horizon. We show that, in this case, the prediction intervals constructed from the ARIMA-GARCH models could be inadequate because they never converge to homoscedastic intervals. All of the results are illustrated using simulated and real time series with stochastic levels.

The Application and Modeling for Conditional Heteroscedasticity Time Series

This article mainly presents the fundamental theory, model and application of conditional heteroscedasticity residual sequence. And it also gives detailed, scientific and exact analysis and research on a financial security example. Then summarizing a conclusion: Financial Securities follows specific rules and tracks through above study. The research indicates that ARCH model only applies to a short-term, auto-correlative heteroscedastic function ,whereas the amended GARCH model has the opposite result, that is, GARCH fits a long-term, auto-correlative heteroscedastic function. Meanwhile, SAS program presents more intuitive, exact tables and figures. All analysis and results show that AR (m)-GARCH fits well.

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.

Persistent-threshold-GARCH processes: Model and application

Abstract Threshold models in the context of conditionally heteroscedastic time series have been found to be useful in analyzing asymmetric volatilities. While the effect of current volatility on the future volatility decreases to zero at an exponential rate for standard-threshold-GARCH (TGARCH) processes, here we introduce a class of TGARCH processes exhibiting persistent properties for which current volatility constantly remains in the future volatilities for all-step ahead forecasts. Analogously to IGARCH (cf. [Nelson, D.B., 1990. Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6, 318–334]), our model will be referred to as integrated-TGARCH (I-TGARCH, hereafter). Some theoretical aspects of I-TGARCH processes are discussed. It is also verified that the I-TGARCH class provides a random quantity for limiting cumulative impulse response (cf. [Baillie, R.T., Bollerslev, T., Mikkelsen, H.O., 1996. Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econom. 74, 3–30]), indicating persistency in variance. The Korea stock price index (KOSPI) data are analyzed to illustrate applicability of the first order I-TGARCH model.

Estimation in a class of nonlinear heteroscedastic time series models

Parameter estimation in a class of heteroscedastic time series models is investigated. The existence of conditional least-squares and conditional likelihood estimators is proved. Their consistency and their asymptotic normality are established. Kernel estimators of the noise’s density and its derivatives are defined and shown to be uniformly consistent. A simulation experiment conducted shows that the estimators perform well for large sample size.

A goodness-of-fit test for models

A goodness-of-fit test in the class of conditional heteroscedastic time series models is examined. Due to the nonstandard limiting distribution of the test, we propose to bootstrap the test, showing its asymptotic validity. Moreover, we illustrate the finite sample performance of the test by a small Monte Carlo study.

Volatility Dependence Across Asia-Pacific On-Shore and Off-Shore U.S. Dollar Futures Markets

This paper estimates switching autoregressive conditional heteroscedasticity (SWARCH) time series models for weekly returns of nine Asian forward exchange rates. We find two regimes with different volatility levels, whereby each regime displays considerable persistence. Our analysis provides evidence that the knock-on effects from China's U.S. dollar future rates upon other Asian countries have been modest, in that little evidence exists for co-dependence of volatility regimes.

Estimating the error distribution in multivariate heteroscedastic time-series models

A semiparametric method is studied for estimating the dependence parameter and the joint distribution of the error term in a class of multivariate time-series models when the marginal distributions of the errors are unknown. This method is a natural extension of Genest, C., Ghoudi, K., Rivest, L.-P. [1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82, 543–552.] for independent and identically distributed observations. The proposed method first obtains n -consistent estimates of the parameters of each univariate marginal time series, and computes the corresponding residuals. These are then used to estimate the joint distribution of the multivariate error terms, which is specified using a copula. Our developments and proofs make use of, and build upon, recent results of Koul and Ling [2006. Fitting an error distribution in some heteroscedastic time series model. Ann. Statist. 34, 994–1012.] and Koul [2002.Weighted empirical processes in dynamic nonlinear models, Lecture Notes in Statistics, vol. 166. Springer, New York.] for these models. The rigorous proofs provided here also lay the foundation and collect together the technical arguments that would be useful for other potential extensions of this semiparametric approach. It is shown that the proposed estimator of the dependence parameter of the multivariate error term is asymptotically normal, and a consistent estimator of its large sample variance is also given so that confidence intervals may be constructed. A large-scale simulation study was carried out to compare the estimators particularly when the error distributions are unknown, which is almost always the case in practice. In this simulation study, our proposed semiparametric method performed better than the well-known parametric methods. An example on exchange rates is used to illustrate the method.

국내 금융시계열의 누적(INTEGRATED)이분산성에 대한 사례분석

시계열 자료 분석에서 ARCH류와 같은 조건부 이분산성 모형을 가정하고 분석하는 모형들이 많이 쓰이고 있다. 실제 우리나라 금융 시계열 자료들을 분석해 보면 비정상성을 나타내는 경우가 드물지 않게 나타난다. 즉, 단위근 형태의 비정상 패턴(integrated phenomenon)에 가까운 경우가 자주 나타난다. 본 논문에서는 다양한 국내 금융시계열 15개에(주가지수, 선물지수, 환율, 이자율 등) GARCH(1,1) 모형을 적합시켜 분산의 지속성을 확인하고, 각 데이터에 첨도(Kurtosis)와 적합된 IGARCH(1,1) 모형을 제시하고자 한다. Conditionally heteroscedastic time series models such as GARCH processes have frequently provided useful approximations to the real aspects of financial time series. It is not uncommon that financial time series exhibits near non-stationary, say, integrated phenomenon. For stationary GARCH processes, a shock to the current conditional variance will be exponentially converging to zero and thus asymptotically negligible for the future conditional variance. However, for the case of integrated process, the effect will remain for a long time, i.e., we have a persistent effect of a current shock on the future observations. We are here concerned with providing empirical evidences of persistent GARCH(1,1) for various fifteen domestic financial time series including KOSPI, KOSDAQ and won-dollar exchange rate. To this end, kurtosis and Integrated-GARCH(1,1) fits are reported for each data.

A goodness-of-fit test for ARCH(∞) models

A goodness-of-fit test in the class of conditional heteroscedastic time series models is examined. Due to the nonstandard limiting distribution of the test, we propose to bootstrap the test, showing its asymptotic validity. Moreover, we illustrate the finite sample performance of the test by a small Monte Carlo study.

Underwater Noise Modeling and Direction-Finding Based on Heteroscedastic Time Series

We propose a new method for practical non-Gaussian and nonstationary underwater noise modeling. This model is very useful for passive sonar in shallow waters. In this application, measurement of additive noise in natural environment and exhibits shows that noise can sometimes be significantly non-Gaussian and a time-varying feature especially in the variance. Therefore, signal processing algorithms such as direction-finding that is optimized for Gaussian noise may degrade significantly in this environment. Generalized autoregressive conditional heteroscedasticity (GARCH) models are suitable for heavy tailed PDFs and time-varying variances of stochastic process. We use a more realistic GARCH-based noise model in the maximum-likelihood approach for the estimation of direction-of-arrivals (DOAs) of impinging sources onto a linear array, and demonstrate using measured noise that this approach is feasible for the additive noise and direction finding in an underwater environment.

Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach

This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt , where the unobservable volatility σt is a parametric function of (Xt−1 ,...,X t−p ,σ t−1 ,...,σ t−q ) for some p, q ≥ 0, and (Zt ) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (Xt ) to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models. 1. Introduction. Gaussian quasi-maximum-likelihood estimation, that is, likelihood estimation under the hypothesis of Gaussian innovations, is a popular method which is widely used for inference in time series models. However, it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi-maximum-likelihood estimator (QMLE) applied to specific models and, therefore, an in-depth analysis of the probabilistic structure generated by the model is called for. A classical example of this kind is the seminal paper by Hannan [18] on estimation in linear ARMA time series. In this paper we study the QMLE for a general class of conditionally heteroscedastic time series models, which includes GARCH, asymmetric GARCH and exponential GARCH. Recall that a GARCH(p, q) [generalized autoregressive conditionally heteroscedastic of order (p, q)] process [4 ]i s def ined by

The Doubly-Pareto Uniform Distribution with Applications in Uncertainty Analysis and Econometrics

In this paper we present a novel four parameter continuous univariate distribution that can be motivated from at least two approaches. The first one views the distribution as a generalization of the uniform one that allows for uncertainty specification at the vicinity of its bounds (gradually) represented via two Pareto tails. The second one is that of an asymmetric heavy-tailed, peaked distribution with an unbounded domain with the property that the location of the mode is not uniquely determined but rather is described by a uniform range. Properties of the distribution are described and a maximum likelihood estimation (MLE) procedure for the mode location and the Pareto tails parameters is presented. The procedure is illustrated by means of an i.i.d. sample of standardized log-differences of quarterly monthly US certificate deposit interest rates for the period 1964–2004. The sample is constructed utilizing the Auto-Regressive Conditional Heteroscedastic (ARCH) time series model devised by the Nobel Laureate R.F. Engle (1982).

Fitting an error distribution in some heteroscedastic time series models

This paper addresses the problem of fltting a known distribution to the innovation distribution in a class of stationary and ergodic time series models. The asymptotic null distribution of the usual Kolmogorov-Smirnov test based on the residuals generally depends on the underlying model parameters and the error distribution. To overcome the dependence on the underlying model parameters, we propose that tests be based on a vector of certain weighted residual empirical processes. Under the null hypothesis and under minimal moment conditions, this vector of processes is shown to converge weakly to a vector of independent copies of a Gaussian process whose covariance function depends only on the fltted distribution and not on the model. Under certain local alternatives, the proposed test is shown to have nontrivial asymptotic power. The Monte Carlo critical values of this test are tabulated when fltting standard normal and double exponential distributions. The results obtained are shown to be applicable to GARCH and ARMA-GARCH models, the often used models in econometrics and flnance. A simulation study shows that the test has satisfactory size and power for flnite samples at these models. The paper also contains an asymptotic uniform expansion result for a general weighted residual empirical process useful in heteroscedastic models under minimal moment conditions, a result of independent interest.

Modeling Traffic Volatility Dynamics in an Urban Network

This article discusses the application of generalized autoregressive conditional heteroscedasticity (GARCH) time series models for representing the dynamics of traffic flow volatility. The methods encountered in the literature focus on the levels of traffic flows and assume that variance is constant through time. The approach adopted in this paper concentrates primarily on the autoregressive properties of traffic variability, with the aim to provide better confidence intervals for traffic flow forecasts. The model-building procedure is illustrated with 7.5-min average traffic flow data for a set of 11 loop detectors located at major arterials that direct to the center of the city of Athens, Greece. A sensitivity analysis for coefficient estimates is undertaken with respect to both time and space.

Modeling Traffic Volatility Dynamics in an Urban Network

This article discusses the application of generalized autoregressive conditional heteroscedasticity (GARCH) time series models for representing the dynamics of traffic flow volatility. The methods encountered in the literature focus on the levels of traffic flows and assume that variance is constant through time. The approach adopted in this paper concentrates primarily on the autoregressive properties of traffic variability, with the aim to provide better confidence intervals for traffic flow forecasts. The model-building procedure is illustrated with 7.5-min average traffic flow data for a set of 11 loop detectors located at major arterials that direct to the center of the city of Athens, Greece. A sensitivity analysis for coefficient estimates is undertaken with respect to both time and space.