Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.     

Broadband semi‐parametric estimation of long‐memory time series by fractional exponential models

This article proposes broadband semi‐parametric estimation of a long‐memory parameter by fractional exponential (FEXP) models. We construct the truncated Whittle likelihood based on FEXP models in a semi‐parametric setting to estimate the parameter and show that the proposed estimator is more efficient than the FEXP estimator by Moulines and Soulier (1999) in linear processes. A Monte Carlo simulation suggests that the proposed estimation is more preferable than the existing broadband semi‐parametric estimation.     

Statistical Inference for Unified Garch–Itô Models with High‐Frequency Financial Data

The existing estimation methods for the model parameters of the unified GARCH–Itô model (Kim and Wang, 2014 ) require long period observations to obtain the consistency. However, in practice, it is hard to believe that the structure of a stock price is stable during such a long period. In this article, we introduce an estimation method for the model parameters based on the high‐frequency financial data with a finite observation period. In particular, we establish a quasi‐likelihood function for daily integrated volatilities, and realized volatility estimators are adopted to estimate the integrated volatilities. The model parameters are estimated by maximizing the quasi‐likelihood function. We establish asymptotic theories for the proposed estimator. A simulation study is conducted to check the finite sample performance of the proposed estimator. We apply the proposed estimation approach to the Bank of America stock price data.     

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.     

Robust Wilcoxon‐Type Estimation of Change‐Point Location Under Short‐Range Dependence

We introduce a robust estimator of the location parameter for the change‐point in the mean based on Wilcoxon statistic and establish its consistency for L₁ near‐epoch dependent processes. It is shown that the consistency rate depends on the magnitude of the change. A simulation study is performed to evaluate the finite sample properties of the Wilcoxon‐type estimator under Gaussianity as well as under heavy‐tailed distributions and disturbances by outliers, and to compare it with a CUSUM‐type estimator. It shows that the Wilcoxon‐type estimator is equivalent to the CUSUM‐type estimator under Gaussianity but outperforms it in the presence of heavy tails or outliers in the data.     

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.     

Local Likelihood for non‐parametric ARCH(1) models

Abstract. We propose a non‐parametric local likelihood estimator for the log‐transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non‐parametric estimator is constructed within the likelihood framework for non‐Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real‐data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.     

NORMING RATES AND LIMIT THEORY FOR SOME TIME‐VARYING COEFFICIENT AUTOREGRESSIONS

A time‐varying autoregression is considered with a similarity‐based coefficient and possible drift. It is shown that the random‐walk model has a natural interpretation as the leading term in a small‐sigma expansion of a similarity model with an exponential similarity function as its AR coefficient. Consistency of the quasi‐maximum likelihood estimator of the parameters in this model is established, the behaviours of the score and Hessian functions are analysed and test statistics are suggested. A complete list is provided of the normalization rates required for the consistency proof and for the score and Hessian function standardization. A large family of unit root models with stationary and explosive alternatives is characterized within the similarity class through the asymptotic negligibility of a certain quadratic form that appears in the score function. A variant of the stochastic unit root model within the class is studied, and a large‐sample limit theory provided, which leads to a new nonlinear diffusion process limit showing the form of the drift and conditional volatility induced by sustained stochastic departures from unity. The findings provide a composite case for time‐varying coefficient dynamic modelling. Some simulations and a brief empirical application to data on international Exchange Traded Funds are included. Copyright © 2014 Wiley Publishing Ltd     

A similarity‐based approach to time‐varying coefficient non‐stationary autoregression

We suggest in this article a similarity‐based approach to time‐varying coefficient non‐stationary autoregression. In a given sample, the model can display characteristics consistent with stationary, unit root and explosive behaviour, depending on the similarity between the dependent variable and its past values. We establish consistency of the quasi‐maximum likelihood estimator of the model, with a general norming factor. Asymptotic score‐based hypothesis tests are derived. The model is applied to a data set comprised of dual stocks traded in NASDAQ and the Tokyo Stock Exchange.     

A PARAMETER‐DRIVEN LOGIT REGRESSION MODEL FOR BINARY TIME SERIES

We consider a parameter‐driven regression model for binary time series, where serial dependence is introduced by an autocorrelated latent process incorporated into the logit link function. Unlike in the case of parameter‐driven Poisson log‐linear or negative binomial logit regression model studied in the literature for time series of counts, generalized linear model (GLM) estimation of the regression coefficient vector, which suppresses the latent process and maximizes the corresponding pseudo‐likelihood, cannot produce a consistent estimator. As a remedial measure, in this article, we propose a modified GLM estimation procedure and show that the resulting estimator is consistent and asymptotically normal. Moreover, we develop two procedures for estimating the asymptotic covariance matrix of the estimator and establish their consistency property. Simulation studies are conducted to evaluate the finite‐sample performance of the proposed procedures. An empirical example is also presented.     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

Negative Binomial Quasi‐Likelihood Inference for General Integer‐Valued Time Series Models

Two negative binomial quasi‐maximum likelihood estimates (NB‐QMLEs) for a general class of count time series models are proposed. The first one is the profile NB‐QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two‐stage NB‐QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two‐stage NB‐QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB‐QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well‐known particular classes of count time series models such as the Poisson and negative binomial integer‐valued GARCH model and the INAR(1) model. Application to a real dataset is given.     

Spatio‐temporal models for big multinomial data using the conditional multivariate logit‐beta distribution

We introduce a Bayesian approach for analyzing high‐dimensional multinomial data that are referenced over space and time. In particular, the proportions associated with multinomial data are assumed to have a logit link to a latent spatio‐temporal mixed effects model. This strategy allows for covariances that are nonstationarity in both space and time, asymmetric, and parsimonious. We also introduce the use of the conditional multivariate logit‐beta distribution into the dependent multinomial data setting, which leads to conjugate full‐conditional distributions for use in a collapsed Gibbs sampler. We refer to this model as the multinomial spatio‐temporal mixed effects model (MN‐STM). Additionally, we provide methodological developments including: the derivation of the associated full‐conditional distributions, a relationship with a latent Gaussian process model, and the stability of the non‐stationary vector autoregressive model. We illustrate the MN‐STM through simulations and through a demonstration with public‐use quarterly workforce indicators data from the longitudinal employer household dynamics program of the US Census Bureau.     

Small‐b and Fixed‐b Asymptotics for Weighted Covariance Estimation in Fractional Cointegration

In a standard cointegrating framework, Phillips (1991) introduced the weighted covariance (WC) estimator of cointegrating parameters. Later, Marinucci (2000) applied this estimator to fractional circumstances and, like Phillips (1991), analysed the so‐called small‐b asymptotic approximation to its sampling distribution. Recently, an alternative limiting theory (fixed‐b asymptotics) has been successfully employed to approximate sampling distributions. With the purpose of comparing both approaches, we derive here the fixed‐b limit of WC estimators in a fractional setting, filling also some gaps in the traditional (small‐b) theory. We also provide some Monte Carlo evidence that suggests that the fixed‐b limit is more accurate.     

Change‐Point Detection in Autoregressive Models with no Moment Assumptions

In this paper we consider the problem of detecting a change in the parameters of an autoregressive process where the moments of the innovation process do not necessarily exist. An empirical likelihood ratio test for the existence of a change point is proposed and its asymptotic properties are studied. In contrast to other works on change‐point tests using empirical likelihood, we do not assume knowledge of the location of the change point. In particular, we prove that the maximizer of the empirical likelihood is a consistent estimator for the parameters of the autoregressive model in the case of no change point and derive the limiting distribution of the corresponding test statistic under the null hypothesis. We also establish consistency of the new test. A nice feature of the method is the fact that the resulting test is asymptotically distribution‐free and does not require an estimate of the long‐run variance. The asymptotic properties of the test are investigated by means of a small simulation study, which demonstrates good finite‐sample properties of the proposed method.     

On Asymptotic Theory for ARCH (∞) Models

Autoregressive conditional heteroskedasticity (ARCH)() models nest a wide range of ARCH and generalized ARCH models including models with long memory in volatility. Existing work assumes the existence of second moments. However, the fractionally integrated generalized ARCH model, one version of a long memory in volatility model, does not have finite second moments and rarely satisfies the moment conditions of the existing literature. This article weakens the moment assumptions of a general ARCH( ) class of models and develops the theory for consistency and asymptotic normality of the quasi‐maximum likelihood estimator.     

Inference for the Fourth‐Order Innovation Cumulant in Linear Time Series

The rescaled fourth‐order cumulant of the unobserved innovations of linear time series is an important parameter in statistical inference. This article deals with the problem of estimating this parameter. An existing nonparametric estimator is first discussed, and its asymptotic properties are derived. It is shown how the autocorrelation structure of the underlying process affects the behaviour of the estimator. Based on our findings and on an important invariance property of the parameter of interest with respect to linear filtering, a pre‐whitening‐based nonparametric estimator of the same parameter is proposed. The estimator is obtained using the filtered time series only; that is, an inversion of the pre‐whitening procedure is not required. The asymptotic properties of the new estimator are investigated, and its superiority is established for large classes of stochastic processes. It is shown that for the particular estimation problem considered, pre‐whitening can reduce the variance and the bias of the estimator. The finite sample performance of both estimators is investigated by means of simulations. The new estimator allows for a simple modification of the multiplicative frequency domain bootstrap, which extends its considerable range of validity. Furthermore, the problem of testing hypotheses about the rescaled fourth‐order cumulant of the unobserved innovations is also considered. In this context, a simple test for Gaussianity is proposed. Some real‐life data applications are presented.     

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are non‐stationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non‐fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a Chi‐squared‐test.     

Large‐scale volatility models: theoretical properties of professionals’ practice

Abstract. This article examines the way in which GARCH models are estimated and used for forecasting by practitioners in particular using the highly popular Riskmetrics^TM approach. Although it permits sizable computational gains and provide a simple way to impose positive semi‐definitiveness of multivariate version of the model, we show that this approach delivers non‐consistent parameter’ estimates. The novel theoretical result is corroborated by a set of Monte Carlo exercises. A set of empirical applications suggest that this could cause, in general, unreliable forecasts of conditional volatilities and correlations.     

Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations

Abstract. Methods for parameter estimation in the presence of long‐range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long‐range dependence in the case of heavy‐tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/α for long‐range dependent FARIMA time series with symmetric α‐stable (1 < α < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self‐similar with parameter H. When the parameter α is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H ? 1/α. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet‐based estimation method used here is semi‐parametric, it allows for a more robust treatment of long‐range dependent data than parametric methods.