A New Pearson-Type QMLE for Conditionally Heteroscedastic Models

This article proposes a novel Pearson-type quasi-maximum likelihood estimator (QMLE) of GARCH(p, q) models. Unlike the existing Gaussian QMLE, Laplacian QMLE, generalized non-Gaussian QMLE, or LAD estimator, our Pearsonian QMLE (PQMLE) captures not just the heavy-tailed but also the skewed innovations. Under strict stationarity and some weak moment conditions, the strong consistency and asymptotic normality of the PQMLE are obtained. With no further efforts, the PQMLE can be applied to other conditionally heteroscedastic models. A simulation study is carried out to assess the performance of the PQMLE. Two applications to four major stock indexes and two exchange rates further highlight the importance of our new method. Heavy-tailed and skewed innovations are often observed together in practice, and the PQMLE now gives us a systematic way to capture these two coexisting features.     

Normal Mixture Quasi-maximum Likelihood Estimator for GARCH Models

Abstract.  The generalized autoregressive conditional heteroscedastic (GARCH) model has been popular in the analysis of financial time series data with high volatility. Conventionally, the parameter estimation in GARCH models has been performed based on the Gaussian quasi-maximum likelihood. However, when the innovation terms have either heavy-tailed or skewed distributions, the quasi-maximum likelihood estimator (QMLE) does not function well. In order to remedy this defect, we propose the normal mixture QMLE (NM-QMLE), which is obtained from the normal mixture quasi-likelihood, and demonstrate that the NM-QMLE is consistent and asymptotically normal. Finally, we present simulation results and a real data analysis in order to illustrate our findings.     

Inference for Box–Cox Transformed Threshold GARCH Models with Nuisance Parameters

Abstract. Generalized autoregressive conditional heteroscedastic (GARCH) models have been widely used for analyzing financial time series with time‐varying volatilities. To overcome the defect of the Gaussian quasi‐maximum likelihood estimator (QMLE) when the innovations follow either heavy‐tailed or skewed distributions, Berkes & Horváth (Ann. Statist., 32, 633, 2004) and Lee & Lee (Scand. J. Statist. 36, 157, 2009) considered likelihood methods that use two‐sided exponential, Cauchy and normal mixture distributions. In this paper, we extend their methods for Box–Cox transformed threshold GARCH model by allowing distributions used in the construction of likelihood functions to include parameters and employing the estimated quasi‐likelihood estimators (QELE) to handle those parameters. We also demonstrate that the proposed QMLE and QELE are consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Continuous Invertibility and Stable QML Estimation of the EGARCH(1,1) Model

I introduce the notion of continuous invertibility on a compact set for volatility models driven by a stochastic recurrence equation. I prove strong consistency of the quasi‐maximum likelihood estimator (QMLE) when the quasi‐likelihood criterion is maximized on a continuously invertible domain. This approach yields, for the first time, the asymptotic normality of the QMLE for the exponential general autoregressive conditional heteroskedastic (EGARCH(1,1)) model under explicit but non‐verifiable conditions. In practice, I propose to stabilize the QMLE by constraining the optimization procedure to an empirical continuously invertible domain. The new method, called stable QMLE, is asymptotically normal when the observations follow an invertible EGARCH(1,1) model.     

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

This article investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE’s) of the GARCH model augmented by including an additional explanatory variable—the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter d_x; in particular, we allow for both stationary and nonstationary covariates. We show that the QMLE’s of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE’s depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not. Supplementary materials for this article are available online.     

Estimating the Lyapunov exponent from chaotic time series with dynamic noise

In this paper, we propose an estimator of the Lyapunov exponent of the skeleton for chaotic time series with dynamic noise and prove the consistency of the estimator under some assumptions.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

The marked empirical process to test nonlinear time series against a large class of alternatives when the random vectors are nonstationary and absolutely regular

In this paper, we propose a method for testing absolutely regular and possibly nonstationary nonlinear time-series, with application to general AR-ARCH models. Our test statistic is based on a marked empirical process of residuals which is shown to converge to a Gaussian process with respect to the Skohorod topology. This testing procedure was first introduced by Stute [Nonparametric model checks for regression, Ann. Statist. 25 (1997), pp. 613–641] and then widely developed by Ngatchou-Wandji [Weak convergence of some marked empirical processes: Application to testing heteroscedasticity, J. Nonparametr. Stat. 14 (2002), pp. 325–339; Checking nonlinear heteroscedastic time series models, J. Statist. Plann. Inference 133 (2005), pp. 33–68; Local power of a Cramer-von Mises type test for parametric autoregressive models of order one, Compt. Math. Appl. 56(4) (2008), pp. 918–929] under more general conditions. Applications to general AR-ARCH models are given.     

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

We suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.     

Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Linear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.     

Shrinkage and LASSO strategies in high-dimensional heteroscedastic models

ABSTRACT

In this paper, we consider the estimation problem of the parameter vector in the linear regression model with heteroscedastic errors. First, under heteroscedastic errors, we study the performance of shrinkage-type estimators and their performance as compared to theunrestricted and restricted least squares estimators. In order to accommodate the heteroscedastic structure, we generalize an identity which is useful in deriving the risk function. Thanks to the established identity, we prove that shrinkage estimators dominate the unrestricted estimator. Finally, we explore the performance of high-dimensional heteroscedastic regression estimator as compared to classical LASSO and shrinkage estimators.     

Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

M-estimators of some GARCH-type models; computation and application

In this paper, we consider robust M-estimation of time series models with both symmetric and asymmetric forms of heteroscedasticity related to the GARCH and GJR models. The class of estimators includes least absolute deviation (LAD), Huber’s, Cauchy and B-estimator as well as the well-known quasi maximum likelihood estimator (QMLE). Extensive simulations are used to check the relative performance of these estimators in both models and the weighted resampling methods are used to approximate the sampling distribution of M-estimators. Our study indicates that there are estimators that can perform better than QMLE and even outperform robust estimator such as LAD when the error distribution is heavy-tailed. These estimators are also applied to real data sets.     

AMEMIYA'S FORM OF THE WEIGHTED LEAST SQUARES ESTIMATOR

Amemiya's estimator is a weighted least squares estimator of the regression coefficients in a linear model with heteroscedastic errors. It is attractive because the heteroscedasticity is not parametrized and the weights (which depend on the error covariance matrix) are estimated nonparametrically. This paper derives an asymptotic expansion for Amemiya's form of the weighted least squares estimator, and uses it to discuss the effects of estimating the weights, of the number of iterations, and of the choice of the initial estimate. The paper also discusses the special case of normally distributed errors and clarifies the particular consequences of assuming normality.     

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.     

On Mixture Double Autoregressive Time Series Models

This article proposes a mixture double autoregressive model by introducing the flexibility of mixture models to the double autoregressive model, a novel conditional heteroscedastic model recently proposed in the literature. To make it more flexible, the mixing proportions are further assumed to be time varying, and probabilistic properties including strict stationarity and higher order moments are derived. Inference tools including the maximum likelihood estimation, an expectation–maximization (EM) algorithm for searching the estimator and an information criterion for model selection are carefully studied for the logistic mixture double autoregressive model, which has two components and is encountered more frequently in practice. Monte Carlo experiments give further support to the new models, and the analysis of an empirical example is also reported.     

Quasi-maximum exponential likelihood estimation for a non stationary GARCH(1,1) model

ABSTRACT

This article investigates a quasi-maximum exponential likelihood estimator(QMELE) for a non stationary generalized autoregressive conditional heteroscedastic (GARCH(1,1)) model. Asymptotic normality of this estimator is derived under a non stationary condition. A simulation study and a real example are given to evaluate the performance of QMELE for this model.     

Value-at-risk forecasting based on Gaussian mixture ARMA–GARCH model

In this paper, we develop a new forecasting algorithm for value-at-risk (VaR) based on ARMA–GARCH (autoregressive moving average–generalized autoregressive conditional heteroskedastic) models whose innovations follow a Gaussian mixture distribution. For the parameter estimation, we employ the conditional least squares and quasi-maximum-likelihood estimator (QMLE) for ARMA and GARCH parameters, respectively. In particular, Gaussian mixture parameters are estimated based on the residuals obtained from the QMLE of GARCH parameters. Our algorithm provides a handy methodology, spending much less time in calculation than the existing resampling and bias-correction method developed in Hartz et al. [Accurate value-at-risk forecasting based on the normal-GARCH model, Comput. Stat. Data Anal. 50 (2006), pp. 3032–3052]. Through a simulation study and a real-data analysis, it is shown that our method provides an accurate VaR prediction.