A negative binomial integer‐valued GARCH model

This article discusses the modelling of integer‐valued time series with overdispersion and potential extreme observations. For the problem, a negative binomial INGARCH model, a generalization of the Poisson INGARCH model, is proposed and stationarity conditions are given as well as the autocorrelation function. For estimation, we present three approaches with the focus on the maximum likelihood approach. Some results from numerical studies are presented and indicate that the proposed methodology performs better than the Poisson and double Poisson model‐based methods.     

Random environment integer‐valued autoregressive process

An r states random environment integer‐valued autoregressive process of order 1, RrINAR(1), is introduced. Also, a random environment process is separately defined as a selection mechanism of differently parameterized geometric distributions, thus ensuring the non‐stationary nature of the RrNGINAR(1) model based on the negative binomial thinning. The distributional and correlation properties of this model are discussed, and the k‐step‐ahead conditional expectation and variance are derived. Yule–Walker estimators of model parameters are presented and their strong consistency is proved. The RrNGINAR(1) model motivation is justified on simulated samples and by its application to specific real‐life counting data.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

Abstract. In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher‐order integer‐valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004) , we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large.     

On an independent and identically distributed mixture bilinear time‐series model

A class of nonlinear time‐series models in which the underlying process follows a finite mixture of bilinear representations is proposed. The mixture feature appears in the conditional distribution of the process which is given as a finite mixture of distributions evaluated at the normed innovations of diagonal bilinear specifications. This class is aimed at capturing special characteristics exhibited by many observed time series such as tail heaviness, multimodality, asymmetry and change in regime. Some probabilistic properties of the proposed model, namely strict and second‐order stationarity, geometric ergodicity, covariance structure, existence of higher order moments, tail behaviour and invertibility, are first studied. Parameter estimation is then performed through the EM algorithm, performance of which is shown via simulation experiments. Applications to some real‐time‐series data are proposed and through which it is shown how neglecting the mixture framework in a bilinear representation results in a loss in adequacy.     

Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Y_t = ρ_nY_t?1 + ?_t with α‐stable noise {?_t} is studied in this paper. Herein ρ_n = 1 ? γ/n, γ ≥ 0 is a constant, Y₀ is a fixed random variable and ε_t is an α‐stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0 < α < 1 or α > 1 and E?₁ = 0, the limit distribution of the MLE of ρ_n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and E?₁ ≠ 0, the limit distribution of the MLE of ρ_n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.     

Integer‐Valued GARCH Process

Abstract. An integer‐valued analogue of the classical generalized autoregressive conditional heteroskedastic (GARCH) (p,q) model with Poisson deviates is proposed and a condition for the existence of such a process is given. For the case p = 1, q = 1, it is explicitly shown that an integer‐valued GARCH process is a standard autoregressive moving average (1, 1) process. The problem of maximum likelihood estimation of parameters is treated. An application of the model to a real time series with a numerical example is given.     

Non‐stationary autoregressive processes with infinite variance

Consider an AR(p) process , where {?_t} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0<α<2. This time series {Y_t} is said to be a non‐stationary AR(p) process if at least one of its characteristic roots lies on the unit circle. The limit distribution of the least squares estimator (LSE) of for {Y_t} with infinite variance innovation {?_t} is established in this paper. In particular, by virtue of the result of Kurtz and Protter (1991) of stochastic integrals, it is shown that the limit distribution of the LSE is a functional of integrated stable process. Simulations for the estimator of β and its limit distribution are also given.     

Zero‐Modified Geometric INAR(1) Process for Modelling Count Time Series with Deflation or Inflation of Zeros

In this article, we propose a first‐order integer‐valued autoregressive [INAR(1)] process for dealing with count time series with deflation or inflation of zeros. The proposed process has zero‐modified geometric marginals and contains the geometric INAR(1) process as a particular case. The proposed model is also capable of capturing underdispersion and overdispersion, which sometimes are caused by deflation or inflation of zeros. We explore several statistical and mathematical properties of the process, discuss point estimation of the parameters and find the asymptotic distribution of the proposed estimators. We also propose a test based on our model for checking if the count time series considered is deflated or inflated of zeros. Two empirical illustrations are presented in order to show the potential for practice of our zero‐modified geometric INAR(1) process. This article contains a Supporting Information.     

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.     

Wavelet‐Based Tests for Comparing Two Time Series with Unequal Lengths

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.     

Wasserstein autoregressive models for density time series

Data consisting of time-indexed distributions of cross-sectional or intraday returns have been extensively studied in finance, and provide one example in which the data atoms consist of serially dependent probability distributions. Motivated by such data, we propose an autoregressive model for density time series by exploiting the tangent space structure on the space of distributions that is induced by the Wasserstein metric. The densities themselves are not assumed to have any specific parametric form, leading to flexible forecasting of future unobserved densities. The main estimation targets in the order-p Wasserstein autoregressive model are Wasserstein autocorrelations and the vector-valued autoregressive parameter. We propose suitable estimators and establish their asymptotic normality, which is verified in a simulation study. The new order-p Wasserstein autoregressive model leads to a prediction algorithm, which includes a data driven order selection procedure. Its performance is compared to existing prediction procedures via application to four financial return data sets, where a variety of metrics are used to quantify forecasting accuracy. For most metrics, the proposed model outperforms existing methods in two of the data sets, while the best empirical performance in the other two data sets is attained by existing methods based on functional transformations of the densities.     

Periodic autoregressive conditional duration

We propose an autoregressive conditional duration (ACD) model with periodic time-varying parameters and multiplicative error form. We name this model periodic autoregressive conditional duration (PACD). First, we study the stability properties and the moment structures of it. Second, we estimate the model parameters, using (profile and two-stage) Gamma quasi-maximum likelihood estimates (QMLEs), the asymptotic properties of which are examined under general regularity conditions. Our estimation method encompasses the exponential QMLE, as a particular case. The proposed methodology is illustrated with simulated data and two empirical applications on forecasting Bitcoin trading volume and realized volatility. We found that the PACD produces better in-sample and out-of-sample forecasts than the standard ACD.     

Estimation for non‐negative time series with heavy‐tail innovations

For moving average processes where the coefficients are non‐negative and the innovations are positive random variables with a regularly varying tail at infinity, we provide estimates for the coefficients based on the ratio of two sample values chosen with respect to an extreme value criteria. We then apply this result to obtain estimates for the parameters of non‐negative ARMA models. Weak convergence results for the joint distribution of our estimates are established and a simulation study is provided to examine the small sample size behaviour of these estimates.     

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.     

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.     

Estimation in Random Coefficient Autoregressive Models

Abstract. We propose the quasi‐maximum likelihood method to estimate the parameters of an RCA(1) process, i.e. a random coefficient autoregressive time series of order 1. The strong consistency and the asymptotic normality of the estimators are derived under optimal conditions.     

A test for second‐order stationarity of a time series based on the discrete Fourier transform

We consider a zero mean discrete time series, and define its discrete Fourier transform (DFT) at the canonical frequencies. It can be shown that the DFT is asymptotically uncorrelated at the canonical frequencies if and only if the time series is second‐order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic has approximately a chi‐square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a generalized non‐central chi‐square, where the non‐centrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power.     

Infinitely Divisible Distributions in Integer‐Valued Garch Models

We propose an integer‐valued stochastic process with conditional marginal distribution belonging to the class of infinitely divisible discrete probability laws. With this proposal, we introduce a wide class of models for count time series that includes the Poisson integer‐valued generalized autoregressive conditional heteroscedastic (INGARCH) model (Ferland et al., 2006) and the negative binomial and generalized Poisson INGARCH models (Zhu, 2011, 2012a). The main probabilistic analysis of this process is developed stating, in particular, first‐order and second‐order stationarity conditions. The existence of a strictly stationary and ergodic solution is established in a subclass including the Poisson and generalized Poisson INGARCH models.