Efficient inference for parameters of unobservable periodic autoregressive time series

This paper considers estimating the model coefficients when the observed periodic autoregressive time series is contaminated by a trend. The proposed Yule–Walker estimators are obtained by a two-step procedure. In the first step, the trend is estimated by a weighted local polynomial, and the residuals are obtained by subtracting the trend estimates from the observations; in the second step, the model coefficients are estimated by the well-known Yule–Walker method via the residuals. It is shown that under certain conditions such Yule–Walker estimators are oracally efficient, i.e., they are asymptotically equivalent to those obtained from periodic autoregressive time series without a trend. An easy-to-use implementation procedure is provided. The performance of the estimators is illustrated by simulation studies and real data analysis. In particular, the simulation studies show that the proposed estimator outperforms that obtained from the residuals when the trend is estimated by kernel smoothing without taking the heteroscedasticity into consideration.     

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.     

Parameter Estimation Robust to Low-Frequency Contamination

We provide methods to robustly estimate the parameters of stationary ergodic short-memory time series models in the potential presence of additive low-frequency contamination. The types of contamination covered include level shifts (changes in mean) and monotone or smooth time trends, both of which have been shown to bias parameter estimates toward regions of persistence in a variety of contexts. The estimators presented here minimize trimmed frequency domain quasi-maximum likelihood (FDQML) objective functions without requiring specification of the low-frequency contaminating component. When proper sample size-dependent trimmings are used, the FDQML estimators are consistent and asymptotically normal, asymptotically eliminating the presence of any spurious persistence. These asymptotic results also hold in the absence of additive low-frequency contamination, enabling the practitioner to robustly estimate model parameters without prior knowledge of whether contamination is present. Popular time series models that fit into the framework of this article include autoregressive moving average (ARMA), stochastic volatility, generalized autoregressive conditional heteroscedasticity (GARCH), and autoregressive conditional heteroscedasticity (ARCH) models. We explore the finite sample properties of the trimmed FDQML estimators of the parameters of some of these models, providing practical guidance on trimming choice. Empirical estimation results suggest that a large portion of the apparent persistence in certain volatility time series may indeed be spurious. Supplementary materials for this article are available online.     

Integer-valued autoregressive processes with periodic structure

In this paper the periodic integer-valued autoregressive model of order one with period T, driven by a periodic sequence of independent Poisson-distributed random variables, is studied in some detail. Basic probabilistic and statistical properties of this model are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the method of moments, least squares-type and likelihood-based ones. Their performance is compared through a simulation study.     

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.     

Estimation in a bivariate integer-valued autoregressive process

ABSTRACT

A bivariate integer-valued autoregressive time series model is presented. The model structure is based on binomial thinning. The unconditional and conditional first and second moments are considered. Correlation structure of marginal processes is shown to be analogous to the ARMA(2, 1) model. Some estimation methods such as the Yule–Walker and conditional least squares are considered and the asymptotic distributions of the obtained estimators are derived. Comparison between bivariate model with binomial thinning and bivariate model with negative binomial thinning is given.     

PARSIMONIOUS PERIODIC TIME SERIES MODELING

This paper studies techniques for fitting parsimonious periodic time series models to periodic data. Large sample standard errors for the parameter estimates in a periodic autoregressive moving‐average time series model under parametric constraints are derived. Likelihood ratio statistics for hypothesis testing are examined. The techniques are applied in modeling daily temperatures at Griffin, Georgia, USA.     

Estimation in conditional first order autoregression with discrete support

We consider estimation in the class of first order conditional linear autoregressive models with discrete support that are routinely used to model time series of counts. Various groups of estimators proposed in the literature are discussed: moment-based estimators; regression-based estimators; and likelihood-based estimators. Some of these have been used previously and others not. In particular, we address the performance of new types of generalized method of moments estimators and propose an exact maximum likelihood procedure valid for a Poisson marginal model using backcasting. The small sample properties of all estimators are comprehensively analyzed using simulation. Three situations are considered using data generated with: a fixed autoregressive parameter and equidispersed Poisson innovations; negative binomial innovations; and, additionally, a random autoregressive coefficient. The first set of experiments indicates that bias correction methods, not hitherto used in this context to our knowledge, are some-times needed and that likelihood-based estimators, as might be expected, perform well. The second two scenarios are representative of overdispersion. Methods designed specifically for the Poisson context now perform uniformly badly, but simple, bias-corrected, Yule-Walker and least squares estimators perform well in all cases.     

Estimating multivariate autoregressive moving average models by fitting long autoregressions

Some simple methods for the estimation of mixed multivariate autoregressive moving average time series models are introduced. The methods require the fitting of a long autoregression to the data and the computation of consistent initial estimates for the parameters of the model. After these preliminaries the estimators of the paper are obtained by applying weighted least squares to a multivariate auxiliary regression model. Two types of weight matrices are considered. Both of them yield estimators which are strongly consistent and asymptotically normally distributed. The first estimators are also asymptotically efficient while the second ones are not fully efficient but computationally simple. A simulation study is performed to illustrate the behaviour of the estimators in finite samples.     

Moments of Mixture Periodic Autoregressive Models

This article deals with the study of some properties of a mixture periodically correlated autoregressive (MPAR _S ) time series model, which extends the mixture time invariant parameter autoregressive (MAR) model, that has recently received a considerable interest from many economic time series analysts, to mixture periodic parameter autoregressive model. The aim behind this extension is to make the model able to capture, in addition to all features captured by the classical MAR model, the periodicity feature exhibited by the autocovariance structure of many encountered financial and environmental time series with eventual multimodal distributions. Our main contribution here is obtaining of the second moment periodically stationary condition for a MPAR _S (K; 2,…, 2) model, furthermore the closed-form of the second moment is obtained.     

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.     

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.     

Periodic Autoregressive Conditional Heteroscedasticity

Most high-frequency asset returns exhibit seasonal volatility patterns. This article proposes a new class of models featuring periodicity in conditional heteroscedasticity explicitly designed to capture the repetitive seasonal time variation in the second-order moments. This new class of periodic autoregressive conditional heteroscedasticity, or P-ARCH, models is directly related to the class of periodic autoregressive moving average (ARMA) models for the mean. The implicit relation between periodic generalized ARCH (P-GARCH) structures and time-invariant seasonal weak GARCH processes documents how neglected autoregressive conditional heteroscedastic periodicity may give rise to a loss in forecast efficiency. The importance and magnitude of this informational loss are quantified for a variety of loss functions through the use of Monte Carlo simulation methods. Two empirical examples with daily bilateral Deutschemark/British pound and intraday Deutschemark/U.S. dollar spot exchange rates highlight the practical relevance of the new P-GARCH class of models. Extensions to discrete-time periodic representations of stochastic volatility models subject to time deformation are briefly discussed.     

Asymptotic Results for Spatial ARMA Models

Causal quadrantal-type spatial ARMA(p, q) models with independent and identically distributed innovations are considered. In order to select the orders (p, q) of these models and estimate their autoregressive parameters, estimators of the autoregressive coefficients, derived from the extended Yule–Walker equations are defined. Consistency and asymptotic normality are obtained for these estimators. Then, spatial ARMA model identification is considered and simulation study is given.     

THE BIAS IN PERIODOGRAM ORDINATES AND THE ESTIMATION OF ARMA MODELS IN THE FREQUENCY DOMAIN

The paper has its origin in the finding that the frequency-domain estimation of ARh4A models can produce estimates which may be remarkably biased. Both of the frequency-domain estimation methods considered in the paper are based on the frequency-domain likelihood function, which depends on the periodogram ordinates of the time series. It is found that, as estimates of the spectrum ordinates, the corresponding periodogram ordinates may contain a rather remarkable bias, which again causes bias in the estimates of parameters produced by a frequency-domain estimation method of an ARMA model. The bias is reduced by tapering the observed time series. An example is given of estimation experiments for simulated time series from a pure autoregressive process of order two.     

Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.     

Nonparametric vector autoregression

We consider a vector conditional heteroscedastic autoregressive nonlinear (CHARN) model in which both the conditional mean and the conditional variance (volatility) matrix are unknown functions of the past. Nonparametric estimators of these functions are constructed based on local polynomial fitting. We examine the rates of convergence of these estimators and give a result on their asymptotic normality. These results are applied to estimation of volatility matrices in foreign exchange markets. Estimation of the conditional covariance surface for the Deutsche Mark/US Dollar (DEM/USD) and Deutsche Mark/British Pound (DEM/GBP) daily returns show negative correlation when the two series have opposite lagged values and positive correlation elsewhere. The relation of our findings to the capital asset pricing model is discussed.     

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.     

动态面板阈模型的一种序贯两步估计

动态面板阈模型可以刻画经济变量动态调整过程的非对称性,在实证分析中有广泛的运用,但阈值参数的引入同时增加了参数估计的困难,理论上尚有许多问题没有解决。针对此类模型,本文提出了一种简单而实用的序贯两步估计方法,首先利用格点搜索获得阈值参数的一致估计,基于该参数对数据结构进行合理划分并引入不同类型的矩条件,然后利用广义矩方法获得自回归参数的估计。理论研究与模拟结果表明,序贯两步估计具有良好的大样本性质和有限样本表现;与现有文献的方法相比,序贯两步估计能够有效避免不同类型参数估计偏差的相互影响,减小估计量的偏差与均方根误差。     

Estimation of Periodic Bilinear Time Series Models

In this article, a new class of models is proposed for modeling nonlinear and nonstationary time series. This new class of models, referred to as the periodic bilinear models, has a state space representation and can be characterized by a set of recursive equations. Condition for the stationarity is presented. Procedures for parameter estimation using the cumulants of order less than four are described and the accuracy of the proposed method is demonstrated in the Monte Carlo simulations.