Testing a linear ARMA model against threshold-ARMA models: A Bayesian approach

We introduce a Bayesian approach to test linear autoregressive moving-average (ARMA) models against threshold autoregressive moving-average (TARMA) models. First, the marginal posterior densities of all parameters, including the threshold and delay, of a TARMA model are obtained by using Gibbs sampler with Metropolis–Hastings algorithm. Second, reversible-jump Markov chain Monte Carlo (RJMCMC) method is adopted to calculate the posterior probabilities for ARMA and TARMA models: Posterior evidence in favor of TARMA models indicates threshold nonlinearity. Finally, based on RJMCMC scheme and Akaike information criterion (AIC) or Bayesian information criterion (BIC), the procedure for modeling TARMA models is exploited. Simulation experiments and a real data example show that our method works well for distinguishing an ARMA from a TARMA model and for building TARMA models.     

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.     

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.     

Modelling extremes of time-dependent data by Markov-switching structures

We investigate the extremal clustering behaviour of stationary time series that possess two regimes, where the switch is governed by a hidden two-state Markov chain. We also suppose that the process is conditionally Markovian in each latent regime. We prove under general assumptions that above high thresholds these models behave approximately as a random walk in one (called dominant) regime and as a stationary autoregression in the other (dominated) regime. Based on this observation, we propose an estimation and simulation scheme to analyse the extremal dependence structure of such models, taking into account only observations above high thresholds. The properties of the estimation method are also investigated. Finally, as an application, we fit a model to high-level exceedances of water discharge data, simulate extremal events from the fitted model, and show that the (model-based) flood peak, flood duration and flood volume distributions match their observed counterparts.     

A design for testing clinical strategies: biased adaptive within-subject randomization

We propose a method for assigning treatment in clinical trials, called the 'biased coin adaptive within-subject' (BCAWS) design: during the course of follow-up, the subject's response to a treatment is used to influence the future treatment, through a 'biased coin' algorithm. This design results in treatment patterns that are closer to actual clinical practice and may be more acceptable to patients with chronic disease than the usual fixed trial regimens, which often suffer from drop-out and non-adherence. In this work, we show how to use the BCAWS design to compare treatment strategies, and we provide a simple example to illustrate the method.     

Censored time series analysis with autoregressive moving average models

The authors consider time series observations with data irregularities such as censoring due to a detection limit. Practitioners commonly disregard censored data cases which often result in biased estimates. The authors present an attractive remedy for handling autocorrelated censored data based on a class of autoregressive and moving average (ARMA) models. In particular, they introduce an imputation method well suited for fitting ARMA models in the presence of censored data. They demonstrate the effectiveness of their technique in terms of bias, efficiency, and information loss. They also describe its adaptation to a specific context of meteorological time series data on cloud ceiling height, which are measured subject to the detection limit of the recording device.     

Missing data sensitivity analysis for recurrent event data using controlled imputation

Statistical analyses of recurrent event data have typically been based on the missing at random assumption. One implication of this is that, if data are collected only when patients are on their randomized treatment, the resulting de jure estimator of treatment effect corresponds to the situation in which the patients adhere to this regime throughout the study. For confirmatory analysis of clinical trials, sensitivity analyses are required to investigate alternative de facto estimands that depart from this assumption. Recent publications have described the use of multiple imputation methods based on pattern mixture models for continuous outcomes, where imputation for the missing data for one treatment arm (e.g. the active arm) is based on the statistical behaviour of outcomes in another arm (e.g. the placebo arm). This has been referred to as controlled imputation or reference‐based imputation. In this paper, we use the negative multinomial distribution to apply this approach to analyses of recurrent events and other similar outcomes. The methods are illustrated by a trial in severe asthma where the primary endpoint was rate of exacerbations and the primary analysis was based on the negative binomial model. Copyright © 2014 John Wiley & Sons, Ltd.     

Implementing unit roost tests in ARMA models of unknow order

This paper compares the performance of classical and recent unit root tests based on different estimation procedures, including fitting ARMA models of unknown orders. The article also introduces an estimator of the spectral density function that is based on the estimation of an ARMA model with data previously detrended by GLS. The Monte Carlo experiment shows that tests improve their performance if an ARMA model is estimated, instead of an autoregressive approximation. The best results are obtained by tests based on the estimation of the spectral density function.     

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.     

Real-Time Forecasts of the Real Price of Oil

We construct a monthly real-time dataset consisting of vintages for 1991.1–2010.12 that is suitable for generating forecasts of the real price of oil from a variety of models. We document that revisions of the data typically represent news, and we introduce backcasting and nowcasting techniques to fill gaps in the real-time data. We show that real-time forecasts of the real price of oil can be more accurate than the no-change forecast at horizons up to 1 year. In some cases, real-time mean squared prediction error (MSPE) reductions may be as high as 25% 1 month ahead and 24% 3 months ahead. This result is in striking contrast to related results in the literature for asset prices. In particular, recursive vector autoregressive (VAR) forecasts based on global oil market variables tend to have lower MSPE at short horizons than forecasts based on oil futures prices, forecasts based on autoregressive (AR) and autoregressive moving average (ARMA) models, and the no-change forecast. In addition, these VAR models have consistently higher directional accuracy.     

The effect of temporal aggregation on the estimation accuracy of ARMA models

ABSTRACT

Autoregressive Moving Average (ARMA) time series model fitting is a procedure often based on aggregate data, where parameter estimation plays a key role. Therefore, we analyze the effect of temporal aggregation on the accuracy of parameter estimation of mixed ARMA and MA models. We derive the expressions required to compute the parameter values of the aggregate models as functions of the basic model parameters in order to compare their estimation accuracy. To this end, a simulation experiment shows that aggregation causes a severe accuracy loss that increases with the order of aggregation, leading to poor accuracy.     

On modelling data from degradation sample paths over time

This paper is mainly concerned with modelling data from degradation sample paths over time. It uses a general growth curve model with Box‐Cox transformation, random effects and ARMA(p, q) dependence to analyse a set of such data. A maximum likelihood estimation procedure for the proposed model is derived and future values are predicted, based on the best linear unbiased prediction. The paper compares the proposed model with a nonlinear degradation model from a prediction point of view. Forecasts of failure times with various data lengths in the sample are also compared.     

空气质量指数监控与预测的控制图方法

世界上多数国家都采用空气质量指数这一指标衡量空气质量状况,对空气质量的有效监测和预警是解决空气污染的重要参考依据.本研究使用ARMA模型拟合空气污染指数(API)时序数据,通过模型残差建立控制图,根据控制图的变化监控并预警.研究采用2010年上海世博会API作为可控状态建立控制限,以2011年1～8月上海API数据建立ARMA(1,1)模型,通过2011年9月上海API模型预测与残差控制图证实模型和控制图的有效性.     

Forecasting interest rates: a comparative assessment of some second-generation nonlinear models

Modeling and forecasting of interest rates has traditionally proceeded in the framework of linear stationary methods such as ARMA and VAR, but only with moderate success. We examine here three methods, which account for several specific features of the real world asset prices such as nonstationarity and nonlinearity. Our three candidate methods are based, respectively, on a combined wavelet artificial neural network (WANN) analysis, a mixed spectrum (MS) analysis and nonlinear ARMA models with Fourier coefficients (FNLARMA). These models are applied to weekly data on interest rates in India and their forecasting performance is evaluated vis-à-vis three GARCH models [GARCH (1,1), GARCH-M (1,1) and EGARCH (1,1)] as well as the random walk model. Both the WANN and MS methods show marked improvement over other benchmark models, and may thus hold out several potentials for real world modeling and forecasting of financial 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.     

The effect of temporal aggregation on the estimation accuracy of time series models

In time series analysis, Autoregressive Moving Average (ARMA) models play a central role. Because of the importance of parameter estimation in ARMA modeling and since it is based on aggregate time series so often, we analyze the effect of temporal aggregation on estimation accuracy. We derive the relationships between the aggregate and the basic parameters and compute the actual values of the former from those of the latter in order to measure and compare their estimation accuracy. We run a simulation experiment that shows that aggregation seriously worsens estimation accuracy and that the impact increases with the order of aggregation.     

Estimating linear representations of nonlinear processes

This paper considers the problem of estimating an autoregressive-moving average (ARMA) model when only ergodic and mixing assumptions can be made. The estimation procedure is based on the minimization of a sum of squared deviations about linear conditional expectations. It is shown that the estimator is strongly consistent and asymptotically normal. The results can be used to estimate weak linear representations of some nonlinear processes. Several examples of such linear representations are provided. Other potential areas of applications are inference for noncausal ARMA, aggregation and marginalization of linear processes. A numerical study is also presented. It appears that standard identification routines based on strong hypothesis on the innovation of ARMA models can be seriously misleading when these assumptions do not hold.     

Consistent testing for non‐correlation of two cointegrated ARMA time series

A consistent approach to the problem of testing non‐correlation between two univariate infinite‐order autoregressive models was proposed by Hong (1996). His test is based on a weighted sum of squares of residual cross‐correlations, with weights depending on a kernel function. In this paper, the author follows Hong's approach to test non‐correlation of two cointegrated (or partially non‐stationary) ARMA time series. The test of Pham, Roy & Cédras (2003) may be seen as a special case of his approach, as it corresponds to the choice of a truncated uniform kernel. The proposed procedure remains valid for testing non‐correlation between two stationary invertible multivariate ARMA time series. The author derives the asymptotic distribution of his test statistics under the null hypothesis and proves that his procedures are consistent. He also studies the level and power of his proposed tests in finite samples through simulation. Finally, he presents an illustration based on real data.     

Modelling long-term dependence in measurement errors of plutonium concentration

In this paper the fractional differenced autoregressive-moving average (ARMA) model is applied in order to model the long-term dependence of plutonium concentration measurements of a physical process, and its performance is compared with that of the common ARMA model using a frequency domain based bootstrap approach.     

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.