Changryong Baek

Statistical Tests for a Single Change in Mean Against Long‐Range Dependence

Statistical tests are introduced for distinguishing between short‐range dependent time series with a single change in mean, and long‐range dependent time series, with the former making the null hypothesis. The tests are based on estimation of the self‐similarity parameter after removing the change in mean from the series. The focus is on the GPH (Geweke and Porter‐Hudak, 1983) and local Whittle estimation methods in the spectral domain. Theoretical properties of the resulting estimators are established when testing for a single change in mean, and small sample properties of the tests are examined in simulations. The introduced tests improve on the BHKS (Berkes et al., 2006) test which is the only other available test for the considered problem. It is argued that the BHKS test has a low power against long‐range dependence alternatives and that this happens because the BHKS test statistic involves estimation of the long‐run variance. The BHKS test could be improved readily by considering its R/S‐like regression version which estimates the self‐similarity parameter and which does not involve the long‐run variance. Yet better alternatives are to use more powerful estimation methods (such as GPH or local Whittle) and lead to the tests introduced here.

On distinguishing multiple changes in mean and long-range dependence using local Whittle estimation

Abstract: It is well known that changes in mean superimposed by a shortrange dependent series can be confused easily with long-range dependence. A procedure to distinguish the two phenomena is introduced. The proposed procedure is based on the local Whittle estimation of the long-range dependence parameter applied to the series after removing changes in mean, and comparing the results to those obtained through the available CUSUM-like approaches. According to the proposed procedure, for example, volatility series in finance seem more consistent with the changes-in-mean models whereas hydrology and telecommunication series are more in line with longrange dependence. As part of this work, a new method based on the local Whittle estimation to find the number of breaks is also introduced and its consistency is proved for the changes-in-mean models.

LONG RANGE DEPENDENCE, UNBALANCED HAAR WAVELET TRANSFORMATION AND CHANGES IN LOCAL MEAN LEVEL

Long range dependent (LRD) stationary time series have historically served to model real time series with apparent changes in local mean level. A natural tool to study changes in local mean level is the unbalanced Haar wavelet transformation (UHT). In this work, UHT is used to study changes in local mean level in LRD models and several real and simulated time series exhibiting LRD. In particular, simulations for LRD models suggest that changes in local mean level occur at times essentially governed by a homogeneous Poisson arrival process, and only the local mean levels themselves inherit the LRD property of the original time series. These properties are compared with the analogous ones in several real and simulated time series. The results are mixed though generally in favor of LRD models. The approach based on UHT is also compared to several alternatives such as defining changes in local mean level through kernel smoothing. The interest throughout is mainly in very long time series such as those collected in the studies of data traffic over Internet.

ARMA Cholesky factor models for the covariance matrix of linear models

In longitudinal studies, serial dependence of repeated outcomes must be taken into account to make correct inferences on covariate effects. As such, care must be taken in modeling the covariance matrix. However, estimation of the covariance matrix is challenging because there are many parameters in the matrix and the estimated covariance matrix should be positive definite. To overcomes these limitations, two Cholesky decomposition approaches have been proposed: modified Cholesky decomposition for autoregressive (AR) structure and moving average Cholesky decomposition for moving average (MA) structure, respectively. However, the correlations of repeated outcomes are often not captured parsimoniously using either approach separately. In this paper, we propose a class of flexible, nonstationary, heteroscedastic models that exploits the structure allowed by combining the AR and MA modeling of the covariance matrix that we denote as ARMACD. We analyze a recent lung cancer study to illustrate the power of our proposed methods.

Time Series Modelling of Air Quality in Korea: Long Range Dependence or Changes in Mean?

This paper considers the statistical characteristics on the air quality (PM10) of Korea collected hourly in 2011. PM10 in Korea exhibits very strong correlations even for higher lags, namely, long range dependence. It is power-law tailed in marginal distribution, and generalized Pareto distribution successfully captures the thicker tail than log-normal distribution. However, slowly decaying autocorrelations may confuse practitioners since a non-stationary model (such as changes in mean) can produce spurious long term correlations for finite samples. We conduct a statistical testing procedure to distinguish two models and argue that the high persistency can be explained by non-stationary changes in mean model rather than long range dependent time series models.

Semiparametric, parametric, and possibly sparse models for multivariate long-range dependence

Several available formulations, parametric models and sparsity settings for multivariate long-range dependence (MLRD) are discussed. Furthermore, a new parametric identifiable model for a general formulation of MLRD is introduced in any dimension, and another sparsity setting is identified of potential interest in MLRD modeling. Estimation approaches for MLRD are also reviewed, including some recent progress and open questions about estimation in higher dimensions and sparse settings.

Sparse seasonal and periodic vector autoregressive modeling

Seasonal and periodic vector autoregressions are two common approaches to modeling vector time series exhibiting cyclical variations. The total number of parameters in these models increases rapidly with the dimension and order of the model, making it difficult to interpret the model and questioning the stability of the parameter estimates. To address these and other issues, two methodologies for sparse modeling are presented in this work: first, based on regularization involving adaptive lasso and, second, extending the approach of Davis et al. (2015) for vector autoregressions based on partial spectral coherences. The methods are shown to work well on simulated data, and to perform well on several examples of real vector time series exhibiting cyclical variations.

A modified Lee-Carter model based on the projection of the skewness of the mortality

There have been continuous improvements in human life expectancy. Life expectancy is as a key factor in an aging population and can wreak severe damage on the financial integrity of pension providers. Hence, the projection of the accurate future mortality is a critical point to prevent possible losses to pension providers. However, improvements in future mortality would be overestimated by a typical mortality projection method using the Lee-Carter model since it underestimates the mortality index κt. This paper suggests a mortality projection based on the projection of the skewness of the mortality versus the typical mortality projection of the Lee-Carter model based on the projection of the mortality index, κt. The paper shows how to indirectly estimate future κt trend with the skewness of the mortality and compares the results under each estimation method of the mortality index, κt. The analysis of the results shows that mortality projection based on the skewness presents less improved mortality at an elderly ages than the original projection.

Adaptive lasso in sparse vector autoregressive models

Abstract This paper considers variable selection in the sparse vector autoregressive (sVAR) model where sparsitycomes from setting small coefficients to exact zeros. In the estimation perspective, Davis et al. (2015)showed that the lasso type of regularization method is successful because it provides a simultaneous variableselection and parameter estimation even for time series data. However, their simulations study reports thatthe regular lasso overestimates the number of non-zero coefficients, hence its finite sample performance needsimprovements. In this article, we show that the adaptive lasso significantly improves the performance wherethe adaptive lasso finds the sparsity patterns superior to the regular lasso. Some tuning parameter selectionsin the adaptive lasso are also discussed from the simulations study.Keywords: sparse vector autoregressive model, adaptive lasso, high dimensional time series 1. 서론 현대의급격한 과학 기술의발전은기존에는 상상할 수조차 없는 다양하고도 대용량의데이터를 생산해 내었다. 본 연구에서는 시간에 따라 관측된 고차원의대용량 시계열 자료를 매우 효과적으로 분석할 수있는 벡터자기상관회귀 모형(vector autoregressive model; VAR)의추정을다룬다. VAR 모형은변수들 사이의종속관계(interdependence)를 고려하여 시간에 따른 종속관계(temporal depen-dence)를 선형 종속관계로 나타내는 모형이다. 보다 구체적으로 먼저 차원이

Bootstrap estimation of long-run variance under strong dependence

Abstract This paper considers a long-run variance estimation using a block bootstrap method under strong dependencealso known as long range dependence. We extend currently available methods in two ways. First, it extendsbootstrap methods under short range dependence to long range dependence. Second, to accommodate theobservation that strong dependence may come from deterministic trend plus noise models, we propose toutilize residuals obtained from the nonparametric kernel estimation with the bimodal kernel. The simulationstudy shows that our method works well; in addition, a data illustration is presented for practitioners.Keywords: long-run variance, long range dependence, block bootstrap 1. 서론 정상시계열에 대한 장기적 분산(long-run variance; LRV)은모평균의추정값인표본 평균의점근적 정규성을위한 표준화된극한(scaling limit)으로 정의된다. 따라서LRV는 시계열 분석의추론에서매우중요한 역할을하는 모수이다. 예를 들어 정상시계열의모평균에 대한 추론, 자기 상관성을갖는 회귀분석에서의회귀 모수에 대한 검증, 단위근 검증, 변화점 감지를 위한 CUSUM 통계량 등등그 활용 분야는 무궁무진하다.이러한 중요성에 따라 지난 수십년 동안 LRV에 대한 일치 추정량이활발히 연구되었다. LRV에 대한대표적인시간영역(time domain)에서의추정량은표본 자기 상관 함수들의가중합인heteroskedastic-ity and autocorrelation covariance(HAC)이있다. 가중치를 결정하는 커널(kernel)에 따라 Newey과West (1987)이제안한 바틀렛(Bartlett) 커널을이용한 바틀렛 추정량을비롯해 Andrews (1991)이제안한 quadratic spectral(QS) 커널을이용한 QS 추정량 등이있다. 이는 곧 커널 추정량의특별한 형태로 볼 수 있으므로 띠넓이선택(bandwidth selection)이유한 표본의성능을크게 좌우하며 이와 관련한많은후속연구가 진행되었다. 주파수 영역(frequency domain)에서의추정량은원점 근처에서의스펙트럴 밀도함수의일치 추정과 관련이있으며 자세한 내용은Brillinger (1981)을참조하기 바란다.붓스트랩을이용한 LRV의추정도 활발하게 연구가 되었다. 중요한 점은시계열 자료의경우 그 의존