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In this article, we derive the asymptotic distributions of residual and normalized residual empirical autocovariances and autocorrelations of (parsimonious) periodic autoregressive moving‐average (PARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We then deduce the modified portmanteau statistics. We establish the asymptotic behavior of the proposed statistics. It is shown that the asymptotic distribution of the modified portmanteau tests is that of a weighted sum of independent chi‐squared random variables, which can be different from the usual chi‐squared approximation used under independent and identically distributed assumption on the noise. We also propose another test based on a self‐normalization approach to check the adequacy of PARMA models. A set of Monte Carlo experiments and an application to financial data are presented. [ABSTRACT FROM AUTHOR]

Given observations from a stationary time series, permutation tests allow one to construct exactly level α tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null hypothesis of interest is that the underlying process is an uncorrelated sequence, permutation tests are not necessarily level α, nor are they approximately level α in large samples. In addition, permutation tests may have large Type 3, or directional, errors, in which a two‐sided test rejects the null hypothesis and concludes that the first‐order autocorrelation is larger than 0, when in fact it is less than 0. In this article, under weak assumptions on the mixing coefficients and moments of the sequence, we provide a test procedure for which the asymptotic validity of the permutation test holds, while retaining the exact rejection probability α in finite samples when the observations are independent and identically distributed. A Monte Carlo simulation study, comparing the permutation test to other tests of autocorrelation, is also performed, along with an empirical example of application to financial data. [ABSTRACT FROM AUTHOR]

A Bayesian lattice filtering and smoothing approach is proposed for fast and accurate modeling and inference in multivariate non‐stationary time series. This approach offers computational feasibility and interpretable time‐frequency analysis in the multivariate context. The proposed framework allows us to obtain posterior estimates of the time‐varying spectral densities of individual time series components, as well as posterior measurements of the time‐frequency relationships across multiple components, such as time‐varying coherence and partial coherence. The proposed formulation considers multivariate dynamic linear models (MDLMs) on the forward and backward time‐varying partial autocorrelation coefficients (TV‐VPARCOR). Computationally expensive schemes for posterior inference on the multivariate dynamic PARCOR model are avoided using approximations in the MDLM context. Approximate inference on the corresponding time‐varying vector autoregressive (TV‐VAR) coefficients is obtained via Whittle's algorithm. A key aspect of the proposed TV‐VPARCOR representations is that they are of lower dimension, and therefore more efficient, than TV‐VAR representations. The performance of the TV‐VPARCOR models is illustrated in simulation studies and in the analysis of multivariate non‐stationary temporal data arising in neuroscience and environmental applications. Model performance is evaluated using goodness‐of‐fit measurements in the time‐frequency domain and also by assessing the quality of short‐term forecasting. [ABSTRACT FROM AUTHOR]

We construct an asymptotically normal estimator β˜N for the tail index β of a distribution on (0,1) regularly varying at x=1, when its N independent realizations are not directly observable. The estimator β˜N is a version of the tail index estimator of Goldie and Smith (1987) [Goldie CM, Smith RL. 1987. The Quarterly Journal of Mathematics38: 45–71] based on suitably truncated observations contaminated with arbitrarily dependent 'noise' which vanishes as N increases. We apply β˜N to panel data comprising N random‐coefficient AR(1) series, each of length T, for estimation of the tail index of the random coefficient at the unit root, in which case the unobservable random coefficients are replaced by sample lag 1 autocorrelations of individual time series. Using asymptotic normality of β˜N, we construct a statistical procedure to test if the panel random‐coefficient AR(1) data exhibit long memory. A simulation study illustrates finite‐sample performance of the introduced inference procedures. [ABSTRACT FROM AUTHOR]

This article develops statistical methodology for semiparametric models for multiple time series of possibly high dimension N. The objective is to obtain precise estimates of unknown parameters (which characterize autocorrelations and cross-autocorrelations) without fully parameterizing other distributional features, while imposing a degree of parsimony to mitigate a curse of dimensionality. The innovations vector is modelled as a linear transformation of independent but possibly non-identically distributed random variables, whose distributions are nonparametric. In such circumstances, Gaussian pseudo-maximum likelihood estimates of the parameters are typically √ n-consistent, where n denotes series length, but asymptotically inefficient unless the innovations are in fact Gaussian. Our parameter estimates, which we call 'adaptive,' are asymptotically as first-order efficient as maximum likelihood estimates based on correctly specified parametric innovations distributions. The adaptive estimates use nonparametric estimates of score functions (of the elements of the underlying vector of independent random varables) that involve truncated expansions in terms of basis functions; these have advantages over the kernel-based score function estimates used in most of the adaptive estimation literature. Our parameter estimates are also √ n -consistent and asymptotically normal. A Monte Carlo study of finite sample performance of the adaptive estimates, employing a variety of parameterizations, distributions and choices of N, is reported. [ABSTRACT FROM AUTHOR]

For non-stationary time series, the fixed Fourier basis is no longer canonical. Rather than limit our basis choice to wavelet or Fourier functions, we propose the use of a library of non-decimated wavelet packets from which we select a suitable basis (frame). Non-decimated packets are preferred to decimated basis libraries so as to prevent information 'loss' at scales coarser than the finest. This article introduces a new class of locally stationary wavelet packet processes and a method to fit these to time series. We also provide new material on the boundedness of the inverse of the inner product operator of autocorrelation wavelet packet functions. We demonstrate the effectiveness of our modelling and basis selection on simulated series and Standard and Poor's 500 index series. [ABSTRACT FROM AUTHOR]

The traditional and most used measure for serial dependence in a time series is the autocorrelation function. This measure gives a complete characterization of dependence for a Gaussian time series, but it often fails for nonlinear time series models as, for instance, the generalized autoregressive conditional heteroskedasticity model (GARCH), where it is zero for all lags. The autocorrelation function is an example of a global measure of dependence. The purpose of this article is to apply to time series a well-defined local measure of serial dependence called the local Gaussian autocorrelation. It generally works well also for nonlinear models, and it can distinguish between positive and negative dependence. We use this measure to construct a test of independence based on the bootstrap technique. This procedure requires the choice of a bandwidth parameter that is calculated using a cross validation algorithm. To ensure the validity of the test, asymptotic properties are derived for the test functional and for the bootstrap procedure, together with a study of its power for different models. We compare the proposed test with one based on the ordinary autocorrelation and with one based on the Brownian distance correlation. The new test performs well. Finally, there are also two empirical examples. [ABSTRACT FROM AUTHOR]

In stationary time series modelling, the autocovariance function (ACV) through its associated autocorrelation function provides an appealing description of the dependence structure but presupposes finite second moments. Here, we provide an alternative, the Gini ACV, which captures some key features of the usual ACV while requiring only first moments. For fitting autoregressive, moving-average and autoregressive-moving-average models under just first-order assumptions, we derive equations based on the Gini ACV instead of the usual ACV. As another application, we treat a nonlinear autoregressive (Pareto) model allowing heavy tails and obtain via the Gini ACV an explicit correlational analysis in terms of model parameters, whereas the usual ACV even when defined is not available in explicit form. Finally, we formulate a sample Gini ACV that is straightforward to evaluate. [ABSTRACT FROM AUTHOR]

In this article, we show that in times series models with in-mean and level effects, persistence will be transmitted from the conditional variance to the conditional mean and vice versa. Hence, by studying the conditional mean/variance independently, one will obtain a biased estimate of the true degree of persistence. For the specific example of an AR(1)-APARCH(1,1)-in-mean-level process, we derive the autocorrelation function, the impulse response function and the optimal predictor. Under reasonable assumptions, the AR(1)-APARCH(1,1)-in-mean-level process will be observationally equivalent to an autoregressive moving average (ARMA)(2,1) process with the largest autoregressive root being close to one. We illustrate the empirical relevance of our results with applications to S&P 500 return and US inflation data. [ABSTRACT FROM AUTHOR]

This article extends and generalizes the variance-ratio (VR) statistic by employing an estimator of the asymptotic covariance matrix of the sample autocorrelations. The estimator is consistent under the null for general classes of innovations exhibiting statistical dependence including exponential generalized autoregressive conditional heteroskedasticity and non-martingale difference sequence processes. Monte Carlo experiments show that our generalized test statistics have good finite sample size and superior power properties to other recently developed VR versions. In an application to two major US stock indices, our new generalized VR tests provide stronger rejections of the null than do competing VR tests. [ABSTRACT FROM AUTHOR]

In this article, we propose a new joint portmanteau test for checking the specification of parametric conditional mean and variance functions of linear and nonlinear time-series models. The use of a joint test is motivated for complete control of the asymptotic size since marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new test is based on an asymptotically distribution-free transformation on the sample autocorrelations of both normalized residuals and squared normalized residuals. This makes it unnecessary to full detail the asymptotic properties of the estimates used to obtain residuals, which could be inefficient two-step ones, avoiding also choices of maximum lag parameters increasing with sample length to control asymptotic size. The robust versions of the new test also properly account for higher-order moment dependence at a reduced cost. The finite-sample performance of the new test is compared with that of well-known tests through simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The concentration of aerosol particles, largely caused by traffic volume and often enhanced during temperature inversion episodes in the cold season, can be a concern for human health in the urban environment. This particulate matter is typically recorded as PM10, the total mass of particles below 10 μm in diameter. It is suspected that, within the PM10 class, ultrafine particles ( < 100 nm) may be responsible for causing respiratory and cardiovascular diseases. Because of their low mass, ultrafine particles are hard to detect, and researchers try to utilize PM10 in combination with nitrogen oxides NO x and other trace gases to monitor their dynamic evolution. To meet pollution standards set by national government and European Union regulation, the city of Klagenfurt, Austria, began using calcium magnesium acetate as a deicer on 11 January 2012, hoping to literally glue pollutants to the ground and thereby reducing pollution concentrations. With the statistical methodology developed in this article, the dynamic evolution of PM10 and NO x is traced for the time period starting 4 January and ending 25 January 2012, and a change in dynamics is found. The findings are based on on-line monitoring procedures that sequentially detect structural breaks in the mean and the parameter values of an autoregressive moving average process. These are defined in terms of model residuals and one-step ahead predictors. Theoretical properties are studied, and a simulation study shows that the proposed procedures work well in finite samples. [ABSTRACT FROM AUTHOR]

We propose extensions of the Box-Pierce () portmanteau autocorrelation test to allow for two generalizations: (i) time series that exhibit unconditional heteroskedasticity and (ii) to test for the presence of autocorrelation only after a fixed lag q. These extensions involve a generalized quadratic form of the Box-Pierce test that uses the heteroskedasticity autocorrelation consistent-type estimator. While we show that this modified test is robust to unconditional heteroskedasticity, the resulting power loss may be substantial. We therefore develop feasible weighted tests that make use of nonparametric estimates of the unobserved variance process. Simulation experiments show that the weighted tests have good size and superior power properties over the unweighted tests. [ABSTRACT FROM AUTHOR]

Consider an infinite dimensional vector linear process. Under suitable assumptions on the parameter space, we provide consistent estimators of the autocovariance matrices. In particular, under causality, this includes the infinite-dimensional vector autoregressive (IVAR) process. In that case, we obtain consistent estimators for the parameter matrices. An explicit expression for the estimators is obtained for IVAR(1), under a fairly realistic parameter space. We also show that under some mild restrictions, the consistent estimator of the marginal large dimensional variance-covariance matrix has the same convergence rate as that in case of i.i.d. samples. [ABSTRACT FROM AUTHOR]

We present an elaboration of the usual binomial AR(1) process on {0,1, ... , N}that allows the thinning probabilities to depend on the current state N only through the 'density' n ∕ N, a natural assumption in many real contexts. We derive some basic properties of the model and explore approaches to parameter estimation. Some special cases are considered that allow for overdispersion and underdispersion, as well as positive and negative autocorrelations. We derive a law of large numbers and a central limit theorem, which provide useful large- N approximations for various quantities of interest. [ABSTRACT FROM AUTHOR]

We study inference and diagnostics for count time series regression models that include a feedback mechanism. In particular, we are interested in negative binomial processes for count time series. We study probabilistic properties and quasi-likelihood estimation for this class of processes. We show that the resulting estimators are consistent and asymptotically normally distributed. These facts enable us to construct probability integral transformation plots for assessing any assumed distributional assumptions. The key observation in developing the theory is a mean parameterized form of the negative binomial distribution. For transactions data, it is seen that the negative binomial distribution offers a better fit than the Poisson distribution. This is an immediate consequence of the fact that transactions can be represented as a collection of individual activities that correspond to different trading strategies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article explores the problem of estimating stationary autoregressive models from observed data using the Bayesian least absolute shrinkage and selection operator (LASSO). By characterizing the model in terms of partial autocorrelations, rather than coefficients, it becomes straightforward to guarantee that the estimated models are stationary. The form of the negative log-likelihood is exploited to derive simple expressions for the conditional likelihood functions, leading to efficient algorithms for computing the posterior mode by coordinate-wise descent and exploring the posterior distribution by Gibbs sampling. Both empirical Bayes and Bayesian methods are proposed for the estimation of the LASSO hyper-parameter from the data. Simulations demonstrate that the Bayesian LASSO performs well in terms of prediction when compared with a standard autoregressive order selection method. [ABSTRACT FROM AUTHOR]

This paper investigates the joint limiting distribution of the residual autocorrelation functions and the absolute residual autocorrelation functions of ARMA-GARCH models. This leads a mixed portmanteau test for diagnostic checking of the ARMA-GARCH model fitted by using the quasi-maximum exponential likelihood estimation approach in Zhu and Ling (2011). Simulation studies are carried out to examine our asymptotic theory, and assess the performance of this mixed test and other two portmanteau tests in Li and Li (2008). A real example is given. [ABSTRACT FROM AUTHOR]

Continuous-time autoregressive moving average (CARMA) processes with a non-negative kernel and driven by a non-decreasing Lévy process constitute a useful and very general class of stationary, non-negative continuous-time processes which have been used, in particular for the modelling of stochastic volatility. In the celebrated stochastic volatility model of Barndorff-Nielsen and Shephard (2001), the spot (or instantaneous) volatility at time t, V( t), is represented by a stationary Lévy-driven Ornstein-Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, , with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 day.) If instead of the stationary Ornstein-Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A mixed integer-valued autoregressive model of order p is proposed. The existence of this unique, stationary and ergodic process is proved and its autocorrelation structure and some conditional stochastic characteristics are derived. Model parameters are estimated via Yule-Walker, conditional least squares and conditional maximum likelihood methods. Finally, possible application of the model to real data sets is considered. [ABSTRACT FROM AUTHOR]

We provide a self-normalization for the sample autocovariances and autocorrelations of a linear, long-memory time series with innovations that have either finite fourth moment or are heavy-tailed with tail index 2 < α < 4. In the asymptotic distribution of the sample autocovariance there are three rates of convergence that depend on the interplay between the memory parameter d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self-normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non-degenerate distribution. We also treat self-normalization of the autocorrelations. The sampling distributions can then be approximated non-parametrically by subsampling, as the corresponding asymptotic distribution is still parameter-dependent. The subsampling-based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log-squared returns of Merck stock. [ABSTRACT FROM AUTHOR]

Small-angle X-ray scattering (SAXS) is a technique for obtaining low-resolution structural information about biological macromolecules, by exposing a dilute solution to a high-intensity X-ray beam and capturing the resulting scattering pattern on a two-dimensional detector. The two-dimensional pattern is reduced to a one-dimensional curve through radial averaging, that is, by averaging across annuli on the detector plane. Subsequent analysis of structure relies on these one-dimensional data. This article reviews the technique of SAXS and investigates autocorrelation structure in the detector plane and in the radial averages. Across a range of experimental conditions and molecular types, spatial autocorrelation in the detector plane is present and is well-described by a stationary kernel convolution model. The corresponding autocorrelation structure for the radial averages is non-stationary. Implications of the autocorrelation structure for inference about macromolecular structure are discussed. [ABSTRACT FROM AUTHOR]

We extend the concept of distance correlation of Szekely et al. (2007) and propose the auto distance correlation function (ADCF) to measure the temporal dependence structure of time-series. Unlike the classic measures of correlations such as the autocorrelation function, the proposed measure is zero if and only if the measured time-series components are independent. In this article, we propose and theoretically verify a subsampling methodology for the inference of sample ADCF for dependent data. Our methodology provides a useful tool for exploring nonlinear dependence structures in time-series. [ABSTRACT FROM AUTHOR]

In this article we present a method for simulating a multi-variate time series via a vector auto regressive moving average (p, q) process. We also carried out two simulation studies to check the performance of the method and applied the methodology to a real sea condition time series. All results show that the method works very well in practice. [ABSTRACT FROM AUTHOR]

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0, d,0) for d ∈ (−∞, 0), which enjoys non-summable partial autocorrelations and autocorrelations decaying as rapidly as ρ n ≅ n−1+2 d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962). [ABSTRACT FROM AUTHOR]

Many environmental time series have seasonal structures. This article presents an approach to identifying the rare events of such series based on time-series residuals. The methods justify the application of classical peaks over threshold methods to estimated versions of the one-step-ahead prediction errors of the series. Such methods enable the seasonal means, variances and autocorrelations of the series to be taken into account. Even in stationary settings, the proposed strategy is useful as it bypasses the need for blocking runs of extremes. The mathematics are justified via a limit theorem for a periodic autoregressive moving-average time series. A detailed application to a daily temperature series from Griffin, Georgia, is pursued. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce a new class of autoregressive models for integer-valued time series using the rounding operator. Compared with classical INAR models based on the thinning operator, the new models have several advantages: simple innovation structure, autoregressive coefficients with arbitrary signs, possible negative values for time series and possible negative values for the autocorrelation function. Focused on the first-order RINAR(1) model, we give conditions for its ergodicity and stationarity. For parameter estimation, a least squares estimator is introduced and we prove its consistency under suitable identifiability condition. Simulation experiments as well as analysis of real data sets are carried out to attest the model performance. [ABSTRACT FROM AUTHOR]

A Bartlett-type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed. [ABSTRACT FROM AUTHOR]

Testing for first-order autocorrelation in small samples using the standard asymptotic test can be seriously misleading. Recent methods in likelihood asymptotics are used to derive more accurate p-value approximations for testing the autocorrelation parameter in a regression model. The methods are based on conditional evaluations and are thus specific to the particular data obtained. A numerical example and three simulations are provided to show that this new likelihood method provides higher order improvements and is superior in terms of central coverage even for autocorrelation parameter values close to unity. [ABSTRACT FROM AUTHOR]

The topic of serial correlation in regression models has attracted a great deal of research in the last 50 years. Most of these studies have assumed that the structure of the error covariance matrix Ω was known or could be consistently estimated from the data. In this article, we describe a new procedure for generating forecasts for regression models with serial correlation based on ordinary least squares and on an approximate representation of the form of the autocorrelation. We prove that the predictors from this specification are asymtotically efficient under some regularity conditions. In addition, we show that there is not much to be gained in trying to identify the correct form of the serial correlation since efficient forecasts can be generated using autoregressive approximations of the autocorrelation. A large simulation study is also used to compare the finite sample predictive efficiencies of this new estimator vis-à-vis estimators based on ordinary least squares and generalized least squares. [ABSTRACT FROM AUTHOR]

The portmanteau statistic is based on the first m-residual autocorrelations, and is used for diagnostic checks on the adequacy of fit of a model. In this article, we propose a modified portmanteau statistic with a correction term that allows for the use of small values of m for the chi-squared test. For this modification, we take a different approach to that suggested by Ljung [ Biometrika (1986), Vol. 73, pp. 725–30]. Their empirical behaviour is clarified using asymptotic theory. [ABSTRACT FROM AUTHOR]

We compare the performance of the inverse and ordinary (partial) autocorrelations for time series model identification. It is found that, both in terms of Bahadur's slope and Pitman's asymptotic relative efficiency, the inverse partial autocorrelations are more efficient than the ordinary autocorrelations for identification of moving-average models. By duality, the partial autocorrelations turn out to be more powerful than the inverse autocorrelations to identify autoregressive models. Numerical experiments on both simulated and real data sets are presented to highlight the theoretical results. [ABSTRACT FROM AUTHOR]

A new version of the partial autocorrelation plot and a new family of subset autoregressive models are introduced. A comprehensive approach to model identification, estimation and diagnostic checking is developed for these models. These models are better suited to efficient model building of high-order autoregressions with long time series. Several illustrative examples are given. [ABSTRACT FROM AUTHOR]

A symbolic method which can be used to obtain the asymptotic bias and variance coefficients to order O(1/ n) for estimators in stationary time series is discussed. Using this method, the large-sample bias of the Burg estimator in the AR( p) for p = 1, 2, 3 is shown to be equal to that of the least squares estimators in both the known and unknown mean cases. Previous researchers have only been able to obtain simulation results for the Burg estimator's bias because this problem is too intractable without using computer algebra. The asymptotic bias coefficient to O(1/ n) of Yule–Walker as well as least squares estimates is also derived in AR(3) models. Our asymptotic results show that for the AR(3), just as in the AR(2), the Yule–Walker estimates have a large bias when the parameters are near the nonstationary boundary. The least squares and Burg estimates are much better in this situation. Simulation results confirm our findings. [ABSTRACT FROM AUTHOR]

We present some new results on the mutual information between past and future for Gaussian stationary sequences. We provide several formulae to calculate this quantity. As a by-product, we establish the so-called reflectrum identity that links partial autocorrelation coefficients and cepstrum coefficients. So as to obtain these results, we provide an account of several regularity conditions for Gaussian stationary processes in terms of properties of the associated Toeplitz and Hankel operators. We discuss conditions under which the mutual information is finite. These results lead us to an interesting perspective towards the definition of long-memory processes. Our result implies that zeros on the unit circle can cause mutual information to be infinite. Examples include fractional autoregressive integrated moving average (ARIMA) models. In addition, we consider a finite sample from a Gaussian stationary sequence. In the expansion of the determinant of its covariance matrix, the Toeplitz matrix, the first and second term are, entropy and mutual information respectively. A form of approximation to the likelihood using entropy and mutual information is presented. [ABSTRACT FROM AUTHOR]

We analyze, by simulation, the finite-sample properties of goodness-of-fit tests based on residual autocorrelation coefficients (simple and partial) obtained using different estimators frequently used in the analysis of autoregressive moving-average time-series models. The estimators considered are unconditional least squares, maximum likelihood and conditional least squares. The results suggest that although the tests based on these estimators are asymptotically equivalent for particular models and parameter values, their sampling properties for samples of the size commonly found in economic applications can differ substantially, because of differences in both finite-sample estimation efficiencies and residual regeneration methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral properties of a bivariate vector autoregressive VAR( p) model when a root z0 = ρ0ei λ0 of the determinant of the model's characteristic matrixΦ( z) approaches the unit circle, the border of non-stationarity. LetΦ xx( z), Φ xy( z), Φ yx( z), Φ yy( z) be the polynomial elements ofΦ( z). We show that, depending on the relation of the order of z0 as root of det(Φ( z)) with the orders of z0 as root ofΦ ij( z), ( i, j ∈ { x, y}), the two marginal spectra may tend to infinity at λ0, while the coherence may tend to unity at λ0. We investigate the conditions under which any of the above will occur, in detail. In the specific case where z0→1, the marginal series will be near-integrated of certain orders of near-integration, while there will eventually exist a linear combination of them with a lower order of near-integration. We study the possible combinations of their orders of near-integration. Finally, we develop a strategy with the help of which one may define a VAR( p) model with pre-specified extreme spectral features and give some examples. Beyond the benefits of this latter for VAR model simulation, the analysis has, moreover, implications concerning the adequacy of VAR model fitting. [ABSTRACT FROM AUTHOR]

Here we obtain difference equations for the higher order moments and cumulants of a time series{ X t} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method– the Whittle criterion– to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient. [ABSTRACT FROM AUTHOR]

In linear regression models with autocorrelated errors, we apply the residual likelihood approach to obtain a residual information criterion (RIC), which can jointly select regression variables and autoregressive orders. We show that RIC is a consistent criterion. In addition, our simulation studies indicate that it outperforms heuristic selection criteria– the Akaike information criterion and the Bayesian information criterion– when the signal-to-noise ratio is not weak. [ABSTRACT FROM AUTHOR]

This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by squares of time series with a Wold expansion where the input error is a sequence of random variables with mean zero and finite kurtosis. Two important cases are discussed: (i) when the errors are independent and, (ii) when the errors are uncorrelated but their squares are correlated. Both situations are addressed when the process exhibits short or long memory. Consequences of these results on certain models widely used in many disciplines are also discussed. [ABSTRACT FROM AUTHOR]

Recently, as a result of the growing interest in modelling stationary processes with discrete marginal distributions, several models for integer value time series have been proposed in the literature. One of these models is the INteger-AutoRegressive (INAR) model. Here we consider the higher-order moments and cumulants of the INAR(1) process and show that they satisfy a set of Yule–Walker type difference equations. We also obtain the spectral and bispectral density functions, thus characterizing the INAR(1) process in the frequency domain. We use a frequency domain approach, namely the Whittle criterion, to estimate the parameters of the model. The estimation theory and associated asymptotic theory of this estimation method are illustrated numerically. [ABSTRACT FROM AUTHOR]

This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving-average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold. [ABSTRACT FROM AUTHOR]

We consider the problem of making inference for the autocorrelations of a time series in the possible presence of a unit root. Even when the underlying series is assumed to be strictly stationary, the robustness against a unit root is a desirable property to ensure good finite-sample coverage in case the series has a near unit root. In addition to discussing a confidence interval for the autocorrelation at a given lag, we also consider a simultaneous confidence band for the first k autocorrelations. We suggest the use of the subsampling method applied to properly studentized statistics, which results in confidence intervals and bands with asymptotically correct coverage probability. An application to practical model selection is given, while a simulation study examines finite-sample performance. [ABSTRACT FROM AUTHOR]

In analysing time series of counts, the need to test for the presence of a dependence structure routinely arises. Suitable tests for this purpose are considered in this paper. Their size and power properties are evaluated under various alternatives taken from the class of INARMA processes. We find that all the tests considered except one are robust against extra binomial variation in the data and that tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer–valued first–order and second–order autoregressive as well as first–order moving average processes. [ABSTRACT FROM AUTHOR]

Several approaches have been developed for the spectral analysis of nonstationary processes in the literature. Otherwise, it has been shown recently that, as in the stationary case, the partial autocorrelation function characterizes, like the autocovariance function, the second-order properties of the process. Our main result is the introduction of a new time-dependent power spectrum clearly related to this function. At each time, this spectrum describes a stationary situation in which the present is correlated with the past in the same way as our nonstationary process at this time. The properties of this spectrum are analysed. In particular, it is defined for all nonstationary processes and is in a one-to-one correspondence with the autocovariance function. Unfortunately, no spectral representation of the process is actually associated with it. This spectrum is also compared with two similar other spectra. Some examples of theoretical spectra and an estimated spectrum are considered for illustration. [ABSTRACT FROM AUTHOR]