Influence diagnostics for censored regression models with autoregressive errors

Observations collected over time are often autocorrelated rather than independent, and sometimes include observations below or above detection limits (i.e. censored values reported as less or more than a level of detection) and/or missing data. Practitioners commonly disregard censored data cases or replace these observations with some function of the limit of detection, which often results in biased estimates. Moreover, parameter estimation can be greatly affected by the presence of influential observations in the data. In this paper we derive local influence diagnostic measures for censored regression models with autoregressive errors of order p (hereafter, AR(p)‐CR models) on the basis of the Q‐function under three useful perturbation schemes. In order to account for censoring in a likelihood‐based estimation procedure for AR(p)‐CR models, we used a stochastic approximation version of the expectation‐maximisation algorithm. The accuracy of the local influence diagnostic measure in detecting influential observations is explored through the analysis of empirical studies. The proposed methods are illustrated using data, from a study of total phosphorus concentration, that contain left‐censored observations. These methods are implemented in the R package ARCensReg.     

Regression analysis of interval‐censored failure time data with linear transformation models

The authors consider the estimation of regression parameters in the context of a class of generalized proportional hazards models, termed linear transformation models, in the presence of interval‐censored data. They present an estimating equation approach whose good performance is demonstrated through simulations and which they illustrate in a few concrete cases.     

Asymptotic Inefficiency of Mean-Correction on Parameter Estimation for a Periodic First-Order Autoregressive Model

A common practice in time series analysis is to fit a centered model to the mean-corrected data set. For stationary autoregressive moving-average (ARMA) processes, as far as the parameter estimation is concerned, fitting an ARMA model without intercepts to the mean-corrected series is asymptotically equivalent to fitting an ARMA model with intercepts to the observed series. We show that, related to the parameter least squares estimation of periodic ARMA models, the second approach can be arbitrarily more efficient than the mean-corrected counterpart. This property is illustrated by means of a periodic first-order autoregressive model. The asymptotic variance of the estimators for both approaches is derived. Moreover, empirical experiments based on simulations investigate the finite sample properties of the estimators.     

A Spectral Estimator of Arma Parameters from Thresholded Data

We consider computationally-fast methods for estimating parameters in ARMA processes from binary time series data, obtained by thresholding the latent ARMA process. All methods involve matching estimated and expected autocorrelations of the binary series. In particular, we focus on the spectral representation of the likelihood of an ARMA process and derive a restricted form of this likelihood, which uses correlations at only the first few lags. We contrast these methods with an efficient but computationally-intensive Markov chain Monte Carlo (MCMC) method. In a simulation study we show that, for a range of ARMA processes, the spectral method is more efficient than variants of least squares and much faster than MCMC. We illustrate by fitting an ARMA(2,1) model to a binary time series of cow feeding data.     

Random Level-Shift Time Series Models,ARIMA Approximations,and Level-Shift Detection

The main purpose of this article is to assess the performance of autoregressive integrated moving average (ARIMA) models when occasional level shifts occur in the time series under study. A random level-shift time series model that allows the level of the process to change occasionally is introduced. Between two consecutive changes, the process behaves like the usual autoregressive moving average (ARMA) process. In practice, a series generated from a random level-shift ARMA (RLARMA) model may be misspecified as an ARIMA process. The efficiency of this ARIMA approximation with respect to estimation of current level and forecasting is investigated. The results of examining a special case of an RLARMA model indicate that the ARIMA approximations are inadequate for estimating the current level, but they are robust for forecasting future observations except when there is a very low frequency of level shifts or when the series are highly negatively correlated. A level-shift detection procedure is presented to handle the low-frequency level-shift phenomena, and its usefulness in building models for forecasting is demonstrated.     

Identification of composite (∑+II) arma models by relatively simpler models

In the real world situations, many time series are aggregates of two or more time series. An aggregation may take place due to an addition or the product or both of two or more time series. We are often interested in the study of the properties of aggregates which are, in turn, dependent on the properties of the constituent series. Motivated by this problem, the authors study in this paper the properties of models generated by the operator (Σ+II) on autoregressive-moving-average (ARMA) processes of orders (p_i,q_i), i = l→n . A few practical examples where such models have been used are given in the introduction and an illustrative numerical example is discussed at the end of the paper.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Penalized likelihood regression: General formulation and efficient approximation

The authors consider a formulation of penalized likelihood regression that is sufficiently general to cover canonical and noncanonical links for exponential families as well as accelerated life models with censored survival data. They present an asymptotic analysis of convergence rates to justify a simple approach to the lower‐dimensional approximation of the estimates. Such an approximation allows for much faster numerical calculation, paving the way to the development of algorithms that scale well with large data sets.     

基于图方法的MA和ARMA模型系数检验

应用图模型方法来讨论传统的MA和ARMA模型,证明了MA和ARMA模型的系数为去掉其他时间序列分量线性效应的条件下的偏相关系数,且利用图模型推断算法提出了一种新的参数估计和检验方法。     

DATA-DRIVEN NONPARAMETRIC SPECTRAL DENSITY ESTIMATORS FOR ECONOMIC TIME SERIES: A MONTE CARLO STUDY

Spectral analysis at frequencies other than zero plays an increasingly important role in econometrics. A number of alternative automated data-driven procedures for nonparametric spectral density estimation have been suggested in the literature, but little is known about their finite-sample accuracy. We compare five such procedures in terms of their mean-squared percentage error across frequencies. Our data generating processes (DGP) include autoregressive-moving average (ARMA) models, fractionally integrated ARMA models and nonparametric models based on 16 commonly used macroeconomic time series. We find that for both quarterly and monthly data the autoregressive sieve estimator is the most reliable method overall.     

Testing for Equal Predictability of Stationary ARMA Processes

In this work we use a measure of predictability of a time series following a stationary ARMA process to develop a test of equal predictability of two or more time series. The test is derived by a set of propositions which links the structure of the AR and MA coefficients to the predictability measure. A particular case of this general approach is constituted by time series having a Wold decomposition with weights having the same sign; in this framework the equal predictability is equivalent to parallelism among ARMA models and the null hypothesis of equal predictability is simply a set of linear restrictions. The ARMA representation of the GARCH models presents non-negative weights, so that this test can be extended to verify the equal predictability of squared time series following GARCH structures.     

Modeling nonlinear time series with local mixtures of generalized linear models

The authors consider a novel class of nonlinear time series models based on local mixtures of regressions of exponential family models, where the covariates include functions of lags of the dependent variable. They give conditions to guarantee consistency of the maximum likelihood estimator for correctly specified models, with stationary and nonstationary predictors. They show that consistency of the maximum likelihood estimator still holds under model misspecification. They also provide probabilistic results for the proposed model when the vector of predictors contains only lags of transformations of the modeled time series. They illustrate the consistency of the maximum likelihood estimator and the probabilistic properties via Monte Carlo simulations. Finally, they present an application using real data.     

Bridge Estimation for Linear Regression Models with Mixing Properties

Penalized regression methods have for quite some time been a popular choice for addressing challenges in high dimensional data analysis. Despite their popularity, their application to time series data has been limited. This paper concerns bridge penalized methods in a linear regression time series model. We first prove consistency, sparsity and asymptotic normality of bridge estimators under a general mixing model. Next, as a special case of mixing errors, we consider bridge regression with autoregressive and moving average (ARMA) error models and develop a computational algorithm that can simultaneously select important predictors and the orders of ARMA models. Simulated and real data examples demonstrate the effective performance of the proposed algorithm and the improvement over ordinary bridge regression.     

Theory of Bilinear Time Series Models

ABSTRACT

In this paper, we prove some theoretic properties of bilinear time series models which are extension of ARMA models. The sufficient conditions for asymptotic stationarity and ivertibility of some types of bilinear models are derived. The structural theory of discussed bilinear models is similar to that of ARMA models. For illustration, a bilinear model has been fitted to the Wolfer sunspot numbers and a substantial reduction in sum of squared residuals is obtained as comparing with Box-Jenkins ARMA model.     

On change point test for ARMA–GARCH models: Bootstrap approach

The problem of testing for a parameter change has been a core issue in time series analysis. It is well known that the estimates-based CUSUM test often suffers from severe size distortions in general GARCH type models. The residual-based CUSUM test has been used as an alternative, which, however, has a defect not to detect the ARMA parameter changes in ARMA–GARCH models. As a remedy, one can employ the score vector-based CUSUM test in ARMA–GARCH models as in Oh and Lee (0000). However, it shows some size distortions for relatively small samples. Hence, we consider the bootstrap counterpart for obtaining a more stable test. Focus is made on the verification of the weak consistency of the proposed test. An empirical study is illustrated for its evaluation.     

Cure rate models: A unified approach

The authors propose a novel class of cure rate models for right‐censored failure time data. The class is formulated through a transformation on the unknown population survival function. It includes the mixture cure model and the promotion time cure model as two special cases. The authors propose a general form of the covariate structure which automatically satisfies an inherent parameter constraint and includes the corresponding binomial and exponential covariate structures in the two main formulations of cure models. The proposed class provides a natural link between the mixture and the promotion time cure models, and it offers a wide variety of new modelling structures as well. Within the Bayesian paradigm, a Markov chain Monte Carlo computational scheme is implemented for sampling from the full conditional distributions of the parameters. Model selection is based on the conditional predictive ordinate criterion. The use of the new class of models is illustrated with a set of real data involving a melanoma clinical trial.     

Least squates estimation of missing values in time series

The minimum mean square error linear interpolator for missing values in time series is extended to handle any pattern of nonconsecutive observations. The paper then develops evidence with simple ARMA models that the usefulness of either the"nonparametric"or the parametric form of the least squares interpolator depends on the time series model, the arrangement of the missing data and the objective for completing the series.     

The consistency of model selection for dynamic Semi-varying coefficient models with autocorrelated errors

The consistency of model selection criterion BIC has been well and widely studied for many nonlinear regression models. However, few of them had considered models with lag variables as regressors and auto-correlated errors in time series settings, which is common in both linear and nonlinear time series modeling. This paper studies a dynamic semi-varying coefficient model with ARMA errors, using an approach based on spectrum analysis of time series. The consistency property of the proposed model selection criteria is established and an implementation procedure of model selection is proposed for practitioners. Simulation studies have also been conducted to numerically show the consistency property.     

M‐estimation for general ARMA Processes with Infinite Variance

Abstract. General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non‐Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M‐estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non‐Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996) , we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M‐estimator. We also consider bootstrapping the M‐estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M‐estimation and bootstrap procedures. An empirical example of financial time series is also provided.     

Calculating the autocovariances and the likelihood for periodic V ARMA models

This paper concerns the autocovariance calculation and likelihood evaluation for periodic vector ARMA models (PV ARMA). Based on a state space representation of PV ARMA models, we derive an algorithm for computing the PV ARMA autocovariances. The proposed method computes the autocovariances for distinct seasons separately, thereby facilitating efficient calculation for models with a large period. As a result, the obtained autocovariance calculation procedure is exploited in a periodic Chandrasekhar-type filter to evaluate the exact likelihood for Gaussian PV ARMA series. Empirical evidence shows the superiority of the periodic Chandrasekhar algorithm for likelihood evaluation over the Kalman-based one.