Pseudo‐likelihood estimation in ARCH models

The author presents asymptotic results for the class of pseudo‐likelihood estimators in the autoregressive conditional heteroscedastic models introduced by Engle (1982). Unlike what is required for the quasi‐likelihood estimator, some estimators in the class he considers do not require the finiteness of the fourth moment of the error density. Thus his method is applicable to heavy‐tailed error distributions for which moments higher than two may not exist.     

Specification testing in nonparametric AR‐ARCH models

In this paper, an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. Tests for the model assumption of independence of innovations from past time series values are suggested. Tests based on weighted L²‐distances of empirical characteristic functions are considered as well as a Cramér–von Mises‐type test. The asymptotic distributions under the null hypothesis of independence are derived, and the consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested, and its performance is shown in a simulation study.     

Asymptotic theory for fractionally integrated asymmetric power ARCH models

Consistency and asymptotic normality of quasi-maximum likelihood estimators (QMLEs) for the fractionally integrated asymmetric power ARCH (FIAPARCH) process are proved. The moment conditions are assumed only for standardized errors. We show the properties for a wide range of QMLEs including Gaussian QMLE.     

On testing for multivariate ARCH effects in vector time series models

Using a spectral approach, the authors propose tests to detect multivariate ARCH effects in the residuals from a multivariate regression model. The tests are based on a comparison, via a quadratic norm, between the uniform density and a kernel‐based spectral density estimator of the squared residuals and cross products of residuals. The proposed tests are consistent under an arbitrary fixed alternative. The authors present a new application of the test due to Hosking (1980) which is seen to be a special case of their approach involving the truncated uniform kernel. However, they typically obtain more powerful procedures when using a different weighting. The authors consider especially the procedure of Robinson (1991) for choosing the smoothing parameter of the spectral density estimator. They also introduce a generalized version of the test for ARCH effects due to Ling & Li (1997). They investigate the finite‐sample performance of their tests and compare them to existing tests including those of Ling & Li (1997) and the residual‐based diagnostics of Tse (2002).Finally, they present a financial application.     

Estimating high-frequency foreign exchange rate volatility with nonparametric ARCH models

High-frequency foreign exchange rate (HFFX) series are analyzed on an operational time scale using models of the ARCH class. Comparison of the estimated conditional variances focuses on the asymmetry and persistence issue. Estimation results for parametric models confirm standard results for HFFX series, namely high persistence and no significance of the asymmetry coefficient in an EGARCH model. To find out whether these results are robust against alternative specifications, nonparametric models are estimated. Local linear estimation techniques are applied to a nonparametric ARCH model of order one (CHARN). The results show significant asymmetry of the volatility function. To allow for both flexibility and persistence, a higher-order multiplicative model is fitted. The results show important asymmetries in volatility. In contrast to the EGARCH specification, the news impact curves have different shapes for different lags and tend to increase slower at the boundaries.     

Optimal lag-length choice in stable and unstable VAR models under situations of homoscedasticity and ARCH

The performance of different information criteria – namely Akaike, corrected Akaike (AICC), Schwarz–Bayesian (SBC), and Hannan–Quinn – is investigated so as to choose the optimal lag length in stable and unstable vector autoregressive (VAR) models both when autoregressive conditional heteroscedasticity (ARCH) is present and when it is not. The investigation covers both large and small sample sizes. The Monte Carlo simulation results show that SBC has relatively better performance in lag-choice accuracy in many situations. It is also generally the least sensitive to ARCH regardless of stability or instability of the VAR model, especially in large sample sizes. These appealing properties of SBC make it the optimal criterion for choosing lag length in many situations, especially in the case of financial data, which are usually characterized by occasional periods of high volatility. SBC also has the best forecasting abilities in the majority of situations in which we vary sample size, stability, variance structure (ARCH or not), and forecast horizon (one period or five). frequently, AICC also has good lag-choosing and forecasting properties. However, when ARCH is present, the five-period forecast performance of all criteria in all situations worsens.     

Rates of convergence of Lp-estimators for a density with an infinity cusp

We study the asymptotics of L_p-estimators, p>0, as estimates of a parameter of location for data coming for a symmetric density with an infinity cusp at the center of symmetry of the distribution. In this situation, the data are more concentrated around the parameter of location than in usual cases. The maximum-likelihood estimator is not defined. The rates of convergence of the L_p-estimators in this situation depend on p and on the shape of the density. For some densities and small values of p, the L_p-estimator converges with a fast rate of convergence.     

Semiparametric ARCH Models

This article introduces a semiparametric autoregressive conditional heteroscedasticity (ARCH) model that has conditional first and second moments given by autoregressive moving average and ARCH parametric formulations but a conditional density that is assumed only to be sufficiently smooth to be approximated by a nonparametric density estimator. For several particular conditional densities, the relative efficiency of the quasi-maximum likelihood estimator is compared with maximum likelihood under correct specification. These potential efficiency gains for a fully adaptive procedure are compared in a Monte Carlo experiment with the observed gains from using the proposed semiparametric procedure, and it is found that the estimator captures a substantial proportion of the potential. The estimator is applied to daily stock returns from small firms that are found to exhibit conditional skewness and kurtosis and to the British pound to dollar exchange rate.     

A mixture integer-valued ARCH model

We propose a mixture integer-valued ARCH model for modeling integer-valued time series with overdispersion. The model consists of a mixture of K stationary or non-stationary integer-valued ARCH components. The advantages of the mixture model over the single-component model include the ability to handle multimodality and non-stationary components. The necessary and sufficient first- and second-order stationarity conditions, the necessary arbitrary-order stationarity conditions, and the autocorrelation function are derived. The estimation of parameters is done through an EM algorithm, and the model is selected by three information criterions, whose performances are studied via simulations. Finally, the model is applied to a real dataset.     

A Markov Model of Switching-Regime ARCH

In this article I present a new approach to model more realistically the variability of financial time series. I develop a Markov-ARCH model that incorporates the features of both Hamilton's switching-regime model and Engle's autoregressive conditional heteroscedasticity (ARCH) model to examine the issue of volatility persistence in the monthly excess returns of the three-month treasury bill. The issue can be resolved by taking into account occasional shifts in the asymptotic variance of the Markov-ARCH process that cause the spurious persistence of the volatility process. I identify two periods during which there is a regime shift, the 1974:2–1974:8 period associated with the oil shock and the 1979:9–1982:8 period associated with the Federal Reserve's policy change. The variance approached asymptotically in these two episodes is more than 10 times as high as the asymptotic variance for the remainder of the sample. I conclude that regime shifts have a greater impact on the properties of the data, and I cannot reject the null hypothesis of no ARCH effects within the regimes. As a consequence of the striking findings in this article, previous empirical results that adopt an ARCH approach in modeling monthly or lower frequency interest-rate dynamics are rendered questionable.     

Estimation Mean Change-Point in ARCH Models with Heavy-Tailed Innovations

The quest of the mean change-point in ARCH models with innovations in the domain of attraction of a κ-stable law appears to still be ongoing. We derive the asymptotic distribution of the residuals CUSUM of squares test (RCUSQ) statistic and find it depends on the stable index κ which is often typically unknown and difficult to estimate. Therefore, the subsampling method is proposed to detect changes without estimating κ. The tests are easy to use and are found to perform well in a Monte Carlo experiment.     

一个新的稳健ARCH检验和YJ-GARCH模型

 众所周知,Engle (1982) 的ARCH检验对于条件均值模型误设并不稳健,特别地,当条件均值是非线性过程而我们仅对之建立线性模型时,它过度地拒绝真实的原假设,导致出现严重的水平扭曲 (size distortion)。因此,本文在文献当中首次利用Yeo-Johnson变换方法来转换均值模型的因变量以排除ARCH 过程中均值部分的非线性,进而提出一个新的稳健ARCH检验以及一个新的GARCH模型——Yeo-Johnson (YJ) GARCH模型。蒙特卡罗模拟结果表明,稳健的ARCH检验在水平 (size) 和势 (power) 方面的表现要显著优于Engle (1982) 的ARCH检验。对上证综指收益率的实证研究结果表明,YJ-GARCH模型的拟合效果要显著优于线性GARCH模型。     

ARCH类模型在风险价值测度中的应用

将ARCH类模型用于计算风险价值,会在很大程度上提高风险价值测度的精度。通过理论分析和实证分析都表明,利用ARCH类模型计算金融资产的风险价值,可以体现其收益率的分布状态和波动性的影响作用,适应风险价值计算的需要,能够提高风险价值的计算精度。     

A Weighted Linear Estimator of Multivariate ARCH Parameters

A weighted linear estimator (WLE) of the parameters of multivariate ARCH models is proposed. The accuracy of WLE in estimating the parameters of multivariate ARCH models is compared with the widely used quasi-maximum likelihood estimator (QMLE) through simulations. Application to real data sets are also presented and forecasts of variance-covariance matrix and value-at-risk (VaR) are obtained. The weighted resampling methods are used to approximate the sampling distribution of the proposed estimator. Our study indicates that the forecasting performance of WLE is not inferior and one-day ahead risk estimates are also found better than the QMLE.     

Statistical estimation errors of VaR under ARCH returns

In this paper we discuss some problems of existing methods for calculating the Value-at-Risk (VaR) in ARCH setting. It should be noted that the commonly used approaches often confuse the true innovations with the empirical residuals, i.e., estimation errors for unknown ARCH parameters are ignored. We adjust this by using the asymptotics of the residual empirical process, and propose a feasible VaR which, according to the spirit of VaR, keeps the assets away from a specified risk with high confidence level. Its meaningfulness in comparison with the usual VaR will be illustrated clearly by numerical studies.     

Semiparametric ARCH Models: An Estimating Function Approach

We introduce the method of estimating functions to study the class of autoregressive conditional heteroscedasticity (ARCH) models. We derive the optimal estimating functions by combining linear and quadratic estimating functions. The resultant estimators are more efficient than the quasi-maximum likelihood estimator. If the assumption of conditional normality is imposed, the estimator obtained by using the theory of estimating functions is identical to that obtained by using the maximum likelihood method in finite samples. The relative efficiencies of the estimating function (EF) approach in comparison with the quasi-maximum likelihood estimator are developed. We illustrate the EF approach using a univariate GARCH(1,1) model with conditional normal, Student-t, and gamma distributions. The efficiency benefits of the EF approach relative to the quasi-maximum likelihood approach are substantial for the gamma distribution with large skewness. Simulation analysis shows that the finite-sample properties of the estimators from the EF approach are attractive. EF estimators tend to display less bias and root mean squared error than the quasi-maximum likelihood estimator. The efficiency gains are substantial for highly nonnormal distributions. An example demonstrates that implementation of the method is straightforward.     

A discrete-time risk model with Poisson ARCH claim-number process

Abstract

In this paper, we propose a discrete-time risk model with the claim number following an integer-valued autoregressive conditional heteroscedasticity (ARCH) process with Poisson deviates. In this model, the current claim number depends on the previous observations. Within this framework, the equation for finding the adjustment coefficient is derived. Numerical studies are also carried out to examine the impact of the Poisson ARCH dependence structure on the ruin probability.