Analysis of accumulated rounding errors in autoregressive processes

In considering the rounding impact of an autoregressive (AR) process, there are two different models available to be considered. The first assumes that the dynamic system follows an underlying AR model and only the observations are rounded up to a certain precision. The second assumes that the updated observation is a rounded version of an autoregression on previous rounded observations. This article considers the second model and examines behaviour of rounding impacts to the statistical inferences. The conditional maximum‐likelihood estimates for the model are proposed and their asymptotic properties are established, including strong consistency and asymptotic normality. Furthermore, both the classical AR model and the ordinary rounded AR model are no longer reliable when dealing with accumulated rounding errors. The three models are also applied to fit the Ocean Wave data. It turns out that the estimates under distinct models are significantly different. Based on our findings, we strongly recommend that models for dealing with rounded data should be in accordance with the actions of rounding errors.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

Testing for a Unit Root in Noncausal Autoregressive Models

This work develops maximum likelihood‐based unit root tests in the noncausal autoregressive (NCAR) model with a non‐Gaussian error term formulated by Lanne and Saikkonen (2011, Journal of Time Series Econometrics 3, Issue 3, Article 2). Finite‐sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.     

Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series

Abstract. We provide a direct proof for consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible autoregressive moving‐average (ARMA) time series models, which were initially established by Hannan [Journal of Applied Probability (1973) vol. 10, pp. 130–145] via the asymptotic properties of a Whittle's estimator. This also paves the way to establish similar results for spatial processes presented in the follow‐up article by Yao and Brockwell [Bernoulli (2006) in press].     

Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Thm 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.     

Optimal Detection of Exponential Component in Autoregressive Models

Abstract. In this article, the problem of detecting the eventual existence of an exponential component in an AR(1) model, that is, the problem of testing ordinary AR(1) dependence against the alternative of an exponential autoregression [EXPAR(1)] model, was considered. A local asymptotic normality property was established for EXPAR(1) models in the vicinity of AR(1) ones. Two problems arose in this context, which were quite typical in the study of nonlinear time‐series models. The first was a problem of parameter identification in the EXPAR(1) model. A special parameterization was developed so as to overcome this technical problem. The second problem was related to the fact that the underlying innovation density had to be treated as a nuisance. The problem at hand, indeed, appeared to be nonadaptive. These problems were solved using semi‐parametrically efficient pseudo‐Gaussian methods (which did not require Gaussian observations).     

Estimation in nonstationary random coefficient autoregressive models

Abstract.  We investigate the estimation of parameters in the random coefficient autoregressive (RCA) model X _k  = ( φ  +  b _k ) X _{k −1} +  e _k , where ( φ ,  ω ²,  σ ²) is the parameter of the process,     ,     . We consider a nonstationary RCA process satisfying E  log | φ  +  b ₀| ≥ 0 and show that σ ² cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for ( φ ,  ω ²) is proven so that the unit root problem does not exist in the RCA model.     

On Residual Variance Estimation in Autoregressive Models

In this paper we consider time series models belonging to the autoregressive (AR) family and deal with the estimation of the residual variance. This is important because estimates of the variance are involved in, for example, confidence sets for the parameters of the model, estimation of the spectrum, expressions for the estimated error of prediction and sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. Simulation results are presented for finite samples     

Estimation of Nonparametric Autoregressive Time Series Models Under Dynamical Constraints

Abstract. A method is introduced to estimate nonparametric autoregressive models under the additional constraint that its regression function has a stable cycle. It is based on a penalty approach that chooses a series expansion approximation taking into account both goodness‐of‐fit and fulfillment of the constraint. Consistency of the proposed estimator is obtained under general hypothesis. Feasibility and effective performance of the introduced method are studied through simulated examples and electro‐encephalographic data collected from a subject suffering from epilepsy.     

A Generalized Portmanteau Test For Independence Of Two Infinite‐Order Vector Autoregressive Series

Abstract. In many situations, we want to verify the existence of a relationship between multivariate time series. Here, we propose a semiparametric approach for testing the independence between two infinite‐order vector autoregressive (VAR(∞)) series, which is an extension of Hong's [Biometrika (1996c) vol. 83, 615–625] univariate results. We first filter each series by a finite‐order autoregression and the test statistic is a standardized version of a weighted sum of quadratic forms in the residual cross‐correlation matrices at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel [Journal of Multivariate Analysis (1985) Vol. 16, pp. 393–411], the asymptotic distribution of the test statistic is derived under the null hypothesis and its consistency is also established for a fixed alternative of serial cross‐correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic proposed by Bouhaddioui and Roy [Statistics and Probability Letters (2006) vol. 76, pp. 58–68] that takes into account a fixed number of lags can be viewed as a special case by using the truncated uniform kernel. However, many kernels lead to a greater power, as shown in an asymptotic power analysis and by a small simulation study in finite samples. A numerical example with real data is also presented.     

A p‐Order signed integer‐valued autoregressive (SINAR(p)) model

In this article, we propose an extension of integer‐valued autoregressive INAR models. Using a signed version of the thinning operator, we define a larger class of ‐valued processes, called SINAR, which can have positive as well as negative correlations. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. In particular, it is shown that the autocorrelation function of any real‐valued AR process can be recovered with a SINAR process, which improves INAR modeling.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

Large sample properties of spectral estimators for a class of stationary nonlinear processes

Abstract. We consider the standard spectral estimators based on a sample from a class of strictly stationary nonlinear processes which include, in particular, the bilinear and Volterra processes. It is shown that these estimators, under certain mild regularity conditions are both consistent and asymptotically normal.     

Robust Estimation in Vector Autoregressive Moving-Average Models

Bustos and Yohai proposed a class of robust estimates for autoregressive moving-average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the covariance matrix corresponding to the maximum likelihood estimate. A Monte Carlo study confirms that the RA estimates are efficient under normal errors and robust when the sample contains outliers. A robust multivariate goodness-of-fit test based on the RA estimates is also obtained.     

Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion

Abstract. Fractional Brownian motion is a mean‐zero self‐similar Gaussian process with stationary increments. Its covariance depends on two parameters, the self‐similar parameter H and the variance C. Suppose that one wants to estimate optimally these parameters by using n equally spaced observations. How should these observations be distributed? We show that the spacing of the observations does not affect the estimation of H (this is due to the self‐similarity of the process), but the spacing does affect the estimation of the variance C. For example, if the observations are equally spaced on [0, n] (unit‐spacing), the rate of convergence of the maximum likelihood estimator (MLE) of the variance C is . However, if the observations are equally spaced on [0, 1] (1/n‐spacing), or on [0, n²] (n‐spacing), the rate is slower, . We also determine the optimal choice of the spacing Δ when it is constant, independent of the sample size n. While the rate of convergence of the MLE of C is in this case, irrespective of the value of Δ, the value of the optimal spacing depends on H. It is 1 (unit‐spacing) if H = 1/2 but is very large if H is close to 1.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

On Parameter Estimation for Exponential Dispersion Arma Models

Abstract. A class of autoregressive moving‐average (ARMA) models proposed by Jørgensen and Song [Journal of Applied Probability (1998), vol. 35, pp. 78–92] with exponential dispersion model margins are useful to deal with non‐normal stationary time series with high‐order autocorrelation. One property associated with the class of models is that the projection process takes the exact form of the classical Box and Jenkins ARMA representation, leading to considerable ease to establish theories. This paper focuses on the issue of parameter estimation for such models, which has not been thoroughly investigated in Jørgensen and Song's paper. The key of the proposed approach is to treat the residual process associated with the projection essentially as a measurement error, which enables us to formulate directly an ARMA representation for the observed time series. The parameter estimation therefore becomes straightforward using the existing methods for the Box and Jenkins ARMA models such as the quasi‐likelihood method. The approach is illustrated by simulation studies and by an analysis of myoclonic seizure counts.     

Functional Generalized Autoregressive Conditional Heteroskedasticity

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of autoregressive conditional heteroskedastic and generalized autoregressive conditional heteroskedastic (GARCH) processes. More recently, multivariate variants of these processes have been the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Because of the need for estimation of many parameters, however, these models may not be suitable for modelling now prevalent high‐frequency volatility data. One potentially useful way to bypass these issues is to take a functional approach. In this article, theory is developed for a new functional version of the GARCH process, termed fGARCH. The main results are concerned with the structure of the fGARCH(1,1) process, providing criteria for the existence of strictly stationary solutions both in the space of square‐integrable and continuous functions. An estimation procedure is introduced, and its consistency and asymptotic normality are verified. A small empirical study highlights potential applications to intraday volatility estimation.     

Random effects mixture models for clustering electrical load series

For purposes such as rate setting and long‐term capacity planning, electrical utility companies are interested in dividing their customers into homogeneous groups or clusters in terms of the customers’ electricity demand profiles. Such demand profiles are typically represented by load series, long time series of daily or even hourly rates of energy consumption of individual customers. The high dimension and time series nature inherent in the load series render existing methods of clustering analysis ineffective. To handle the high dimension and to take advantage of the time‐series nature of load series, we introduce a class of mixture models for time series, the random effects mixture models, which are particularly useful for clustering the load series. The random effects mixture models are based on a hierarchical model for individual components. They employ highly flexible antedependence models to effectively capture the time‐series characteristics of the covariance of the load series. We present details on the construction of such mixture models and discuss a special Expectation‐maximization (EM) algorithm for their computation. We also apply these models to cluster the data set which had motivated this research, a set of 923 load series from BC Hydro, a crown utility company in British Columbia, Canada.     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.