Taewook Lee

Penalized regression models with autoregressive error terms

Penalized regression methods have recently gained enormous attention in statistics and the field of machine learning due to their ability of reducing the prediction error and identifying important variables at the same time. Numerous studies have been conducted for penalized regression, but most of them are limited to the case when the data are independently observed. In this paper, we study a variable selection problem in penalized regression models with autoregressive (AR) error terms. We consider three estimators, adaptive least absolute shrinkage and selection operator, bridge, and smoothly clipped absolute deviation, and propose a computational algorithm that enables us to select a relevant set of variables and also the order of AR error terms simultaneously. In addition, we provide their asymptotic properties such as consistency, selection consistency, and asymptotic normality. The performances of the three estimators are compared with one another using simulated and real examples.

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.

Tests for serial correlation in mean and variance of a sequence of time series objects

ABSTRACT In this era of Big Data, large-scale data storage provides the motivation for statisticians to analyse new types of data. The proposed work concerns testing serial correlation in a sequence of sets of time series, here referred to as time series objects. An example is serial correlation of monthly stock returns when daily stock returns are observed. One could consider a representative or summarized value of each object to measure the serial correlation, but this approach would ignore information about the variation in the observed data. We develop Kolmogorov–Smirnov-type tests with the standard bootstrap and wild bootstrap Ljung–Box test statistics for serial correlation in mean and variance of time series objects, which take the variation within a time series object into account. We study the asymptotic property of the proposed tests and present their finite sample performance using simulated and real examples.

A new algorithm for maximum likelihood estimation in normal scale-mixture generalized autoregressive conditional heteroskedastic models

In this paper, we propose a new generalized autoregressive conditional heteroskedastic (GARCH) model using infinite normal scale-mixtures which can suitably avoid order selection problems in the application of finite normal scale-mixtures. We discuss its theoretical properties and develop a two-stage algorithm for the maximum likelihood estimator to estimate the mixing distribution non-parametric maximum likelihood estimator (NPMLE) as well as GARCH parameters (two-stage MLE). For the estimation of a mixing distribution, we employ a fast computational algorithm proposed by Wang [On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. J R Stat Soc Ser B. 2007;69:185–198] under the gradient characterization of the non-parametric mixture likelihood. The GARCH parameters are then estimated either using the expectation-mazimization algorithm or general optimization scheme. In addition, we propose a new forecasting algorithm of value-at-risk (VaR) using the two-stage MLE and the NPMLE. Through a simulation study and real data analysis, we compare the performance of the two-stage MLE with the existing ones including quasi-maximum likelihood estimator based on the standard normal density and the finite normal mixture quasi maximum estimated-likelihood estimator (cf. Lee S, Lee T. Inference for Box–Cox transformed threshold GARCH models with nuisance parameters. Scand J Stat. 2012;39:568–589) in terms of the relative efficiency and accuracy of VaR forecasting.

Adaptive lasso for linear regression models with ARMA-GARCH errors

ABSTRACT 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. 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.

Robust estimation for order of hidden Markov models based on density power divergences

In this paper, we study the robust estimation for the order of hidden Markov model (HMM) based on a penalized minimum density power divergence estimator, which is obtained by utilizing the finite mixture marginal distribution of HMM. For this task, we adopt the locally conic parametrization method used in [D. Dacunha-Castelle and E. Gassiate, Testing in locally conic models and application to mixture models. ESAIM Probab. Stat. (1997), pp. 285–317; D. Dacunha-Castelle and E. Gassiate, Testing the order of a model using locally conic parametrization: population mixtures and stationary arma processes, Ann. Statist. 27 (1999), pp. 1178–1209; T. Lee and S. Lee, Robust and consistent estimation of the order of finite mixture models based on the minimizing a density power divergence estimator, Metrika 68 (2008), pp. 365–390] to avoid the difficulties that arise in handling mixture marginal models, such as the non-identifiability of the parameter space and the singularity problem with the asymptotic variance. We verify that the estimated order is consistent and simulation results are provided for illustration.

Bootstrap-Based Test for Volatility Shifts in GARCH against Long-Range Dependence

Volatility is a variation measure in finance for returns of a financial instrument over time. GARCH models have been a popular tool to analyze volatility of financial time series data since Bollerslev (1986) and it is said that volatility is highly persistent when the sum of the estimated coefficients of the squared lagged returns and the lagged conditional variance terms in GARCH models is close to 1. Regarding persistence, numerous methods have been proposed to test if such persistency is due to volatility shifts in the market or natural fluctuation explained by stationary long-range dependence (LRD). Recently, Lee et al. (2015) proposed a residual-based cumulative sum (CUSUM) test statistic to test volatility shifts in GARCH models against LRD. We propose a bootstrap-based approach for the residual-based test and compare the sizes and powers of our bootstrap-based CUSUM test with the one in Lee et al. (2015) through simulation studies.

Skewness of Gaussian Mixture Absolute Value GARCH(1, 1) Model

This paper studies the skewness of the absolute value GARCH(1, 1) models with Gaussian mixture innovations (Gaussian mixture AVGARCH(1, 1) models). The maximum estimated-likelihood estimator (MELE) employed (a two- step estimation method in order to estimate the skewness of Gaussian mixture AVGARCH(1, 1) models. Through the real data analysis, the adequacy of adopting Gaussian mixture innovations is exhibited in reflecting the skewness of two major Korean stock indices.