Adaptive LASSO for linear regression models with ARMA-GARCH errors

The linear regression models with the autoregressive moving average (ARMA) errors (REGARMA models) are often considered, in order to reflect a serial correlation among observations. In this article, we focus on an adaptive least absolute shrinkage and selection operator (LASSO) (ALASSO) method for the variable selection of the REGARMA models and extend it to the linear regression models with the ARMA-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) errors (REGARMA-GARCH models). This attempt is an extension of the existing ALASSO method for the linear regression models with the AR errors (REGAR models) proposed by Wang et al. in 2007 Wang, H., Li, G., Tsai, C. (2007). Regression coefficient and autoregressive order shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B 69:63–78. [Google Scholar]. New ALASSO algorithms are proposed to determine important predictors for the REGARMA and REGARMA-GARCH models. Finally, we provide the simulation results and real data analysis to illustrate our findings.     

Self-weighted recursive estimation of GARCH models

The generalized autoregressive conditional heteroscedasticity (GARCH) processes are frequently used to investigate and model financial returns. They are routinely estimated by computationally complex off-line estimation methods, for example, by the conditional maximum likelihood procedure. However, in many empirical applications (especially in the context of high-frequency financial data), it seems necessary to apply numerically more effective techniques to calibrate and monitor such models. The aims of this contribution are: (i) to review the previously introduced recursive estimation algorithms and to derive self-weighted alternatives applying general recursive identification instruments, and (ii) to examine these methods by means of simulations and an empirical application.     

Characteristic function-based inference for GARCH models with heavy-tailed innovations

We consider estimation and goodness-of-fit tests in GARCH models with innovations following a heavy-tailed and possibly asymmetric distribution. Although the method is fairly general and applies to GARCH models with arbitrary innovation distribution, we consider as special instances the stable Paretian, the variance gamma, and the normal inverse Gaussian distribution. Exploiting the simple structure of the characteristic function of these distributions, we propose minimum distance estimation based on the empirical characteristic function of properly standardized GARCH-residuals. The finite-sample results presented facilitate comparison with existing methods, while the new procedures are also applied to real data from the financial market.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Robust estimates for GARCH models

In this paper we present two robust estimates for GARCH models. The first is defined by the minimization of a conveniently modified likelihood and the second is similarly defined, but includes an additional mechanism for restricting the propagation of the effect of one outlier on the next estimated conditional variances. We study the asymptotic properties of our estimates proving consistency and asymptotic normality. A Monte Carlo study shows that the proposed estimates compare favorably with respect to other robust estimates. Moreover, we consider some real examples with financial data that illustrate the behavior of these estimates.     

Adaptive LASSO for varying-coefficient partially linear measurement error models

This paper extends the adaptive LASSO (ALASSO) for simultaneous parameter estimation and variable selection to a varying-coefficient partially linear model where some of the covariates are subject to measurement errors of an additive form. We draw comparisons with the SCAD, and prove that both the ALASSO and the SCAD attain the oracle property under this setup. We further develop an algorithm in the spirit of LARS for finding the solution path of the ALASSO in practical applications. Finite sample properties of the proposed methods are examined in a simulation study, and a real data example based on the U.S. Department of Agriculture's Continuing Survey of Food Intakes by Individuals (CSFII) is considered.     

Spectral analysis of Markov switching GARCH models with statistical inference

We derive matrix expressions in closed form for the autocovariance function and the spectral density of Markov switching GARCH models and their powers. For this, we apply the Riesz–Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance function. Under suitable assumptions, we prove that the sample estimator of the spectral density is consistent and asymptotically normally distributed. Further statistical implications in terms of order identification and parameter estimation are discussed. A simulation study confirms the validity of the asymptotic properties. These methods are also well suited for financial market applications, and in particular for the analysis of time series in the frequency domain, as shown in some proposed real-world examples.     

Residual-based CUSUM of squares test for Poisson integer-valued GARCH models

This study considers the problem of testing for a parameter change in integer-valued time series models in which the conditional density of current observations is assumed to follow a Poisson distribution. As a test, we consider the CUSUM of the squares test based on the residuals from INGARCH models and find that the test converges weakly to the supremum of a Brownian bridge. A simulation study demonstrates its superiority to the residual and standardized residual-based CUSUM tests of Kang and Lee [Parameter change test for Poisson autoregressive models. Scand J Statist. 2014;41:1136–1152] and Lee and Lee [CUSUM tests for general nonlinear inter-valued GARCH models: comparison study. Ann Inst Stat Math. 2019;71:1033–1057.] as well as the CUSUM of squares test based on standardized residuals.     

Quasi-maximum likelihood estimation of GARCH models in the presence of missing values

This work presents a new method to deal with missing values in financial time series. Previous works are generally based in state-space models and Kalman filter and few consider ARCH family models. The traditional approach is to bound the data together and perform the estimation without considering the presence of missing values. The existing methods generally consider missing values in the returns. The proposed method considers the presence of missing values in the price of the assets instead of in the returns. The performance of the method in estimating the parameters and the volatilities is evaluated through a Monte Carlo simulation. Value at risk is also considered in the simulation. An empirical application to NASDAQ 100 Index series is presented.     

Data cloning estimation of GARCH and COGARCH models

GARCH models include most of the stylized facts of financial time series and they have been largely used to analyse discrete financial time series. In the last years, continuous-time models based on discrete GARCH models have been also proposed to deal with non-equally spaced observations, as COGARCH model based on Lévy processes. In this paper, we propose to use the data cloning methodology in order to obtain estimators of GARCH and COGARCH model parameters. Data cloning methodology uses a Bayesian approach to obtain approximate maximum likelihood estimators avoiding numerically maximization of the pseudo-likelihood function. After a simulation study for both GARCH and COGARCH models using data cloning, we apply this technique to model the behaviour of some NASDAQ time series.     

Robust and efficient estimation of GARCH models based on Hellinger distance

It is well known that financial data frequently contain outlying observations. Almost all methods and techniques used to estimate GARCH models are likelihood-based and thus generally non-robust against outliers. Minimum distance method, as an important tool for statistical inferences and a competitive alternative for achieving robustness, has surprisingly not been well explored for GARCH models. In this paper, we proposed a minimum Hellinger distance estimator (MHDE) and a minimum profile Hellinger distance estimator (MPHDE), depending on whether the innovation distribution is specified or not, for estimating the parameters in GARCH models. The construction and investigation of the two estimators are quite involved due to the non-i.i.d. nature of data. We proved that the MHDE is a consistent estimator and derived its bias in explicit expression. For both of the proposed estimators, we demonstrated their finite-sample performance through simulation studies and compared with the well-established methods including MLE, Gaussian Quasi-MLE, Non-Gaussian Quasi-MLE and Least Absolute Deviation estimator. Our numerical results showed that MHDE and MPHDE have much better performance than MLE-based methods when data are contaminated while simultaneously they are very competitive when data is clean, which testified to the robustness and efficiency of the two proposed MHD-type estimations.     

Adaptive LASSO for linear mixed model selection via profile log-likelihood

Mixed model selection is quite important in statistical literature. To assist the mixed model selection, we employ the adaptive LASSO penalized term to propose a two-stage selection procedure for the purpose of choosing both the random and fixed effects. In the first stage, we utilize the penalized restricted profile log-likelihood to choose the random effects; in the second stage, after the random effects are determined, we apply the penalized profile log-likelihood to select the fixed effects. In each stage, the Newton–Raphson algorithm is performed to complete the parameter estimation. We prove that the proposed procedure is consistent and possesses the oracle properties. The simulations and a real data application are conducted for demonstrating the effectiveness of the proposed selection procedure.     

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.     

Signed compound poisson integer-valued GARCH processes

Abstract

We propose signed compound Poisson integer-valued GARCH processes for the modeling of the difference of count time series data. We investigate the theoretical properties of these processes and we state their ergodicity and stationarity under mild conditions. We discuss the conditional maximum likelihood estimator when the series appearing in the difference are INGARCH with geometric distribution and explore its finite sample properties in a simulation study. Two real data examples illustrate this methodology.     

Effects of level shifts and temporary changes on the estimation of GARCH models

The aim of this article is to analyse the effect of the level shift and temporary change outliers on the estimation of a model with conditional heteroscedasticity, a concept rarely dealt with up to now, the literature focusing more on additive outliers. To do this, we have conducted various Monte Carlo experiments in which the bias produced by these outliers is analysed.     

The use of GARCH models in VaR estimation

We evaluate the performance of an extensive family of ARCH models in modeling the daily Value-at-Risk (VaR) of perfectly diversified portfolios in five stock indices, using a number of distributional assumptions and sample sizes. We find, first, that leptokurtic distributions are able to produce better one-step-ahead VaR forecasts; second, the choice of sample size is important for the accuracy of the forecast, whereas the specification of the conditional mean is indifferent. Finally, the ARCH structure producing the most accurate forecasts is different for every portfolio and specific to each equity index.     

Zero-truncated compound Poisson integer-valued GARCH models for time series

Starting from the compound Poisson INGARCH models, we introduce in this paper a new family of integer-valued models suitable to describe count data without zeros that we name zero-truncated CP-INGARCH processes. For such class of models, a probabilistic study concerning moments existence, stationarity and ergodicity is developed. The conditional quasi-maximum likelihood method is introduced to consistently estimate the parameters of a wide zero-truncated compound Poisson subclass of models. The conditional maximum likelihood method is also used to estimate the parameters of ZTCP-INGARCH processes associated with well-specified conditional laws. A simulation study that compares some of those estimators and illustrates their finite distance behaviour as well as a real-data application conclude the paper.     

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.     

The LASSO Method for Bilinear Time Series Models

In this article we propose a method called GLLS for the fitting of bilinear time series models. The GLLS procedure is the combination of the LASSO method, the generalized cross-validation method, the least angle regression method, and the stepwise regression method. Compared with the traditional methods such as the repeated residual method and the genetic algorithm, GLLS has the advantage of shrinking the coefficients of the models and saving the computational time. The Monte Carlo simulation studies and a real data example are reported to assess the performance of the proposed GLLS method.     

Smoothed empirical likelihood for GARCH models with heavy-tailed errors

ABSTRACT

This paper proposes an empirical likelihood (EL) method for estimating the GARCH(p, q) models with heavy-tailed errors. Using the kernel smoothing method, we derive a smoothed EL ratio statistic, which yields a smoothed EL estimator. Moreover, we derive a profile EL for the partial parameters in the presence of nuisance parameters. Simulations and empirical results are conducted to illustrate our proposed method.