Empirical likelihood in long‐memory time series models

This article studies the empirical likelihood method for long‐memory time series models. By virtue of the Whittle likelihood, one obtains a score function that can be viewed as an estimating equation of the parameters of a fractional integrated autoregressive moving average (ARFIMA) model. This score function is used to obtain an empirical likelihood ratio which is shown to be asymptotically chi‐square distributed. Confidence regions for the parameters are constructed based on the asymptotic distribution of the empirical likelihood ratio. Bartlett correction and finite sample properties of the empirical likelihood confidence regions are examined.     

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.     

Testing for the Cointegrating Rank of a VAR Process with Level Shift and Trend Break

Abstract. A test for the cointegrating rank of a vector autoregressive (VAR) process with a possible shift and broken linear trend is proposed. The break point is assumed to be known. Our test is not a likelihood ratio test but the deterministic terms including the broken trends are removed first by a generalized least squares procedure. Then, a likelihood ratio‐type test is applied to the adjusted series. The asymptotic null distribution of the test is derived and it is shown by a Monte Carlo experiment that the test has better small‐sample properties in many cases than a corresponding Gaussian likelihood ratio test for the cointegrating rank. Moreover, response surface techniques can be used to easily obtain p‐values of the test for any possible break date.     

CUSUM‐type testing for changing parameters in a spatial autoregressive model for stock returns

The article suggests a CUSUM‐type test for time‐varying parameters in a recently proposed spatial autoregressive model for stock returns and derives its asymptotic null distribution as well as local power properties. As can be seen from Euro Stoxx 50 returns, a combination of spatial modelling and change point tests might allow for superior risk forecasts in portfolio management.     

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.     

Testing for Change in Long‐Memory Stochastic Volatility Time Series

In this article, change‐point problems for long‐memory stochastic volatility (LMSV) models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM‐ and Wilcoxon‐type test statistics is derived. In this context, a limit theorem for the two‐parameter empirical process of LMSV time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long‐range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self‐normalized test statistics is proposed. The theoretical results are accompanied by an analysis of Standard & Poor's 500 daily closing indices with respect to structural changes and by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean and in variance.     

First‐order integer valued AR processes with zero inflated poisson innovations

The first‐order nonnegative integer valued autoregressive process has been applied to model the counts of events in consecutive points of time. It is known that, if the innovations are assumed to follow a Poisson distribution then the marginal model is also Poisson. This model may however not be suitable for overdispersed count data. One frequent manifestation of overdispersion is that the incidence of zero counts is greater than expected from a Poisson model. In this paper, we introduce a new stationary first‐order integer valued autoregressive process with zero inflated Poisson innovations. We derive some structural properties such as the mean, variance, marginal and joint distribution functions of the process. We consider estimation of the unknown parameters by conditional or approximate full maximum likelihood. We use simulation to study the limiting marginal distribution of the process and the performance of our fitting algorithms. Finally, we demonstrate the usefulness of the proposed model by analyzing some real time series on animal health laboratory submissions.     

UNIFIED INTERVAL ESTIMATION FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

The consistency of the quasi‐maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non‐degenerate random variable. In this article, we propose empirical likelihood methods based on weighted‐score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic and whether the process is stationary or non‐stationary, and we present two classes of equations depending on whether a constant trend is included in the model. A simulation study confirms the good finite‐sample behaviour of our resulting empirical likelihood‐based confidence intervals. We also apply our methods to study US macroeconomic data.     

Distributions for residual autocovariances in parsimonious periodic vector autoregressive models with applications

The asymptotic distribution of the residual autocovariance matrices in the class of periodic vector autoregressive time series models with structured parameterization is derived. Diagnostic checking with portmanteau test statistics represents a useful application of the result. Under the assumption that the periodic white noise process of the periodic vector autoregressive time series model is composed of independent random variables, we demonstrate that the finite sample distributions of the Hosking‐Li‐McLeod portmanteau test statistics can be approximated by those of weighted sums of independent chi‐square random variables. The quantiles of the asymptotic distribution can be computed using the Imhof algorithm or other exact methods. Thus, using the (single) chi‐square distribution for these test statistics appears inadequate in general, although it is often recommended in practice for diagnostic methods of that kind. A simulation study provides empirical evidence.     

ON SELF‐NORMALIZATION FOR CENSORED DEPENDENT DATA

This article is concerned with confidence interval construction for functionals of the survival distribution for censored dependent data. We adopt the recently developed self‐normalization approach (Shao, 2010), which does not involve consistent estimation of the asymptotic variance, as implicitly used in the blockwise empirical likelihood approach of El Ghouch et al. (2011). We also provide a rigorous asymptotic theory to derive the limiting distribution of the self‐normalized quantity for a wide range of parameters. Additionally, finite‐sample properties of the self‐normalization‐based intervals are carefully examined, and a comparison with the empirical likelihood‐based counterparts is made.     

Empirical likelihood inference for random coefficient INAR(p) process

In this article, we study the empirical likelihood (EL) method for the pth‐order random coefficient integer‐valued autoregressive process. In particular, the limiting distribution of the log EL ratio statistic is established and the confidence regions for the parameter of interest are derived. Also a simulation study is conducted for the evaluation of the developed approach.     

An approximate likelihood function for panel data with a mixed ARMA(p, q) remainder disturbance model

Abstract. An approximate likelihood function for panel data with an autoregressive moving‐average (ARMA)(p, q) model remainder disturbance is presented and Whittle's approximate maximum likelihood estimator (MLE) is used to yield an asymptotic estimator. Although an asymptotic approach, the power test is quite successful for estimating and testing. In this approach, we do not need to calculate the transformation matrix in exact form. Through the Riemann sum approach, we can construct a simple approximate concentrated likelihood function. In addition, the model is also extended to the restricted maximum likelihood (REML) function, in which the package of Gilmour, Thompson and Cullis [Biometrics (1995) Vol. 51, pp. 1440–1450] is applied without difficulty. In the case study, we implement the model on the characteristic line for the investment analysis of Taiwanese computer motherboard makers.     

On the Consistency of Bootstrap Testing for a Parameter on the Boundary of the Parameter Space

It is well known that with a parameter on the boundary of the parameter space, such as in the classic cases of testing for a zero location parameter or no autoregressive conditional heteroskedasticity (ARCH) effects, the classic nonparametric bootstrap – based on unrestricted parameter estimates – leads to inconsistent testing. In contrast, we show here that for the two aforementioned cases, a nonparametric bootstrap test based on parameter estimates obtained under the null – referred to as ‘restricted bootstrap’ – is indeed consistent. While the restricted bootstrap is simple to implement in practice, novel theoretical arguments are required in order to establish consistency. In particular, since the bootstrap is analysed both under the null hypothesis and under the alternative, non‐standard asymptotic expansions are required to deal with parameters on the boundary. Detailed proofs of the asymptotic validity of the restricted bootstrap are given and, for the leading case of testing for no ARCH, a Monte Carlo study demonstrates that the bootstrap quasi‐likelihood ratio statistic performs extremely well in terms of empirical size and power for even remarkably small samples, outperforming the standard and bootstrap Lagrange multiplier tests as well as the asymptotic quasi‐likelihood ratio test.     

Goodness-of-fit tests for autoregressive processes

Goodness-of-fit tests for autoregressive processes can be based on the difference betwe en the empirical standardized spectral distribution of an observed time series and the standardized spectral distribution of the autoregressive process with parameters estimated from the series. The asymptotic covariance function of this difference, considered as a stochastic process on [0, π], is found. Methods to compute the asymptotic distribution of the Cramer--von Mises statistic are given.     

On composite likelihood estimation of a multivariate INAR(1) model

In several circumstances the collected data are counts observed in different time points, while the counts at each time point are correlated. Current models are able to account for serial correlation but usually fail to account for cross‐correlation. Motivated by the lack of appropriate tools for handling such type of data, we define a multivariate integer‐valued autoregressive process of order 1 (MINAR(1)) and examine its basic statistical properties. Apart from the general specification of the MINAR(1) process, we also study two specific parametric cases that arise under the assumptions of a multivariate Poisson and a multivariate negative binomial distribution for the innovations of the process. To overcome the computational difficulties of the maximum likelihood approach we suggest the method of composite likelihood. The performance of the two methods of estimation, that is, maximum likelihood and composite likelihood, is compared through a small simulation experiment. Extensions of the time‐invariant model to a regression model are also discussed. The proposed model is applied to a trivariate data series related to daily traffic accidents in three areas in the Netherlands.     

Efficient use of higher-lag autocorrelations for estimating autoregressive processes

The Yule–Walker estimator is commonly used in time-series analysis, as a simple way to estimate the coefficients of an autoregressive process. Under strong assumptions on the noise process, this estimator possesses the same asymptotic properties as the Gaussian maximum likelihood estimator. However, when the noise is a weak one, other estimators based on higher-order empirical autocorrelations can provide substantial efficiency gains. This is illustrated by means of a first-order autoregressive process with a Markov-switching white noise. We show how to optimally choose a linear combination of a set of estimators based on empirical autocorrelations. The asymptotic variance of the optimal estimator is derived. Empirical experiments based on simulations show that the new estimator performs well on the illustrative model.     

On modelling and diagnostic checking of vector periodic autoregressive time series models

Abstract.  Vector periodic autoregressive time series models (PVAR) form an important class of time series for modelling data derived from climatology, hydrology, economics and electrical engineering, among others. In this article, we derive the asymptotic distributions of the least squares estimators of the model parameters in PVAR models, allowing the parameters in a given season to satisfy linear constraints. Residual autocorrelations from classical vector autoregressive and moving-average models have been found useful for checking the adequacy of a particular model. In view of this, we obtain the asymptotic distribution of the residual autocovariance matrices in the class of PVAR models, and the asymptotic distribution of the residual autocorrelation matrices is given as a corollary. Portmanteau test statistics designed for diagnosing the adequacy of PVAR models are introduced and we study their asymptotic distributions. The proposed test statistics are illustrated in a small simulation study, and an application with bivariate quarterly West German data is presented.     

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.     

Unit Root Tests Based on Unconditional Maximum Likelihood Estimation for the Autoregressive Moving Average

Unconditional maximum likelihood estimation is considered for an autoregressive moving average that may contain an autoregressive unit root. The limiting distribution of the normalized maximum likelihood estimator of the unit root is shown to be the same as that of the estimator for the first-order autoregressive process. A likelihood ratio test based on unconditional maximum likelihood estimation is proposed. In a Monte Carlo study for the autoregressive moving-average model of order (1, 1), the new test is shown to have better size and power than those of several other tests.     

RESULTS ON ESTIMATION AND TESTING FOR A UNIT ROOT IN THE NONSTATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODEL

Abstract. We review the limiting distribution theory for Gaussian estimation of the univariate autoregressive moving-average (ARMA) model in the presence of a unit root in the autoregressive (AR) operator, and present the asymptotic distribution of the associated likelihood ratio (LR) test statistic for testing for a unit root in the ARMA model. The finite sample properties of the LR statistic as well as other unit root test procedures for the ARMA model are examined through a limited simulation study. We conclude that, for practical empirical work that relies on standard computations, the LR test procedure generally performs better than other standard procedures in the presence of a substantial moving-average component in the ARMA model.