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We suggest specification tests for the conditional mean function in vector multiplicative error models. The test statistics are easy to compute given a suitable estimator of the model parameters. Consistency of the test statistic is proved, the asymptotic distribution of the test under the null hypothesis is studied, while a bootstrap resampling is used in order to approximate critical points and actually carry out the test. Finite-sample results are presented as well as applications of the proposed procedures to real data from the financial markets. [ABSTRACT FROM AUTHOR]

AbstractThis article proposes a minimum distance estimator for long-memory stochastic duration models, which satisfies a central limit theorem. Distinctive features of the proposed method are that it is easy to calculate and implement, allows fast estimation even for huge datasets, and provides asymptotic standard errors for the estimators. Monte Carlo experiments indicate that the proposed estimator performs very well. The proposed method is illustrated with the estimation of a real-life time series of nearly a million observations. [ABSTRACT FROM AUTHOR]

Sufficient dimension reduction (SDR) methods have been extensively studied for regression models with independent data, but options for time series are limited, focusing mainly on scalar responses with TSIR, TSAVE, and TSSH. Although valuable, these SDR methods rely on the slice approach. Extending them to multivariate responses via marginal slicing leads to numerous slices. Furthermore, the slice approach also poses two main questions: how many slices should be chosen and how to divide all samples into different slices. To overcome these, we introduce TMDDM, a slicing-free SDR method for time series, using approximate joint diagonalization of supervised lagged martingale difference divergence matrices (MDDM) to account for the data temporal characteristics. We also discuss lag selection strategies and the dimensionality of dimension reduction space. Simulations and real data analysis demonstrate its favourable finite-sample performance. [ABSTRACT FROM AUTHOR]

The Poisson distribution is a representative distribution for discrete data, much like the normal distribution is for continuous data. Many time series models for count data have been developed based on the Poisson distribution. However, studies examining the validity of the Poisson assumption in these time series models have been relatively scarce. This study addresses the problem of testing for conditional Poissonity in integer-valued conditional autoregressive models. For this purpose, we introduce a test statistic based on the information matrix of the likelihood function. Under regularity conditions, it is shown that the proposed test has an asymptotic null distribution following a chi-square distribution. Simulation results demonstrate the validity of the proposed test. Additionally, a real data analysis is provided for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we consider an integer-valued autoregressive (INAR) model. Based on the likelihood ratio scanning method (LRSM), the numbers and locations of the piecewise stationary INAR process change points are discussed. The minimum description length (MDL) function of the INAR model is given. Moreover, we also study the case when the sum of coefficients in each segment of the model tends to 1 and make adjustments to the MDL criterion in the LRSM. In this way, the accuracy of change point detection is improved. In the end, simulation experiments and real data analysis are conducted to illustrate the efficiency of the LRSM. [ABSTRACT FROM AUTHOR]

AbstractWe make an observation that facilitates exact likelihood-based inference for the parameters of the popular ARFIMA model without requiring stationarity by allowing the upper bound d¯ for the memory parameter d to exceed 0.5: estimating the parameters of a single non-stationary ARFIMA model is equivalent to estimating the parameters of a sequence of stationary ARFIMA models. This allows for the use of existing methods for evaluating the likelihood for an invertible and stationary ARFIMA model. This enables improved inference because many standard methods perform poorly when estimates are close to the boundary of the parameter space. It also allows us to leverage the wealth of likelihood approximations that have been introduced for estimating the parameters of a stationary process. We explore how estimation of the memory parameter d depends on the upper bound d¯ and introduce adaptive procedures for choosing d¯ . We show via simulation how our adaptive procedures estimate the memory parameter well, relative to existing alternatives, when the true value is as large as 2.5. [ABSTRACT FROM AUTHOR]

This paper aims to study the non parametric Negative Binomial Quasi-Maximum Likelihood Estimation (NBQMLE) for locally stationary integer valued processes. So, we have considered two locally stationary integer-valued models of negative binomial type, namely: INARCH (p) and INGARCH (p , q) models. Imposing some contraction arguments, we have extended the stationary negative-binomial QMLE to a localized one in our non-stationary environment. This estimation method is based on a kernel function that achieves the convergence rates of n h n order. Under some regularity assumptions, the consistency, as well as the asymptotic normality of the obtained estimator, are established. The performances of the established estimators are evaluated via a simulation study and an application to real data set. [ABSTRACT FROM AUTHOR]

In this article, we propose a new model to model the spatial count data on a two-dimensional regular grid using binomial thinning operator with dependent Bernoulli counting series. The model is called "first-order spatial dependent count integer-valued autoregressive (Sp-DCINAR(1,1)) model." Some of its properties have been derived and the estimation of the parameters of the model are obtained by the Yule-Walker method, conditional least squares method and conditional maximum likelihood method. Finally, numerical results are presented together with an application to a two-dimensional practical data set. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a first-order Periodic Generalized Poisson Integer-Valued Autoregressive model PGPINAR (1) which has been shown to be useful to describe overdispersion, equidispersion and underdispersion feature encountered in periodically correlated Integer-Valued time series. Some probabilistic and statistical properties are established, such as the periodically correlated stationarity conditions, in the first and the second moments are provided and the closed-forms of these moments are, under these conditions, derived. Moreover, the structure of the periodic autocovariance is obtained. The estimation problem is addressed through the Yule-Walker (YW) , the Two-Stage Conditional Least Squares (CLS) and the Conditional Maximum Likelihood (CML) methods. The performance of these methods is done through an intensive simulation study and an application on real data set is accomplished. Keywords and phrases: Periodic Generalized Poisson, Integer-Valued Autoregressive, Periodically correlated process, periodically stationary condition. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of periodic integer-valued autoregressive BL-PINAR(1) models with Bell innovations distribution based on the binomial thinning operator. The basic probabilistic and statistical properties of this class are studied. Indeed, the first and the second moment periodically stationary conditions are established. The closed forms of these moments are, under the obtained conditions, derived. Furthermore, the periodic autocovariance structure is also considered while providing the closed form of the periodic autocorrelation function. The conditional least squares (CLS), Yule–Walker (YW), weighted conditional least squares (WCLS), and conditional maximum likelihood (CML) methods are applied to estimate the underlying parameters. The asymptotic properties of the CLS and the YW estimators are obtained. The performances of these methods are compared through a simulation study. An application on a real data set is provided. [ABSTRACT FROM AUTHOR]

Interval-valued data have received much attention due to their wide applications in finance, econometrics, meteorology and medicine. However, most regression models developed for interval-valued data assume that the observations are mutually independent, which can not be adapted to scenarios where individuals are spatially correlated. We propose a new linear model to account for areal-type spatial dependencies present in interval-valued data. Specifically, two spatial Durbin models are separately built for the centers and log-ranges of the interval-valued response. To make the new model more robust to heavy-tailed noise, we assume the error term follows a t-distribution. The parameters are obtained using an expectation-maximization (EM) algorithm. Numerical experiments are designed to examine the performance of the proposed method. We also use a weather dataset to demonstrate the usefulness of our model. [ABSTRACT FROM AUTHOR]

This article introduces a new class of volatility models known as Periodic log-Generalized Autoregressive Conditional Heteroscedastic (P-logGARCH) models, which incorporate periodic variations in the parameters. These parameter variations are particularly relevant when incorporating seasonality into economic decision-making theory. The P-logGARCH formulation demonstrates several desirable properties, including unconstrained positivity of parameters and effective management of extreme values, along with a volatility trend. Specifically, in this paper we investigate the probabilistic structure of this model and establish necessary and sufficient conditions for the existence of stationary solutions in a periodic sense. Additionally, we examine the strong consistency and asymptotic normality of the Generalized Quasi-Maximum Likelihood Estimator (GQMLE) under mild assumptions. To assess the performance of our model, we conduct a Monte Carlo study to examine the finite-sample properties of the GQMLE. Finally, we present empirical evidence by applying the P-log GARCH model to analyze the exchange rates of the Algerian Dinar against the U.S. dollar and the Euro, thereby demonstrating its practical utility. [ABSTRACT FROM AUTHOR]

We propose a goodness-of-fit test for a class of count time series models with covariates which includes the Poisson autoregressive model with covariates (PARX) as a special case. The test criteria are derived from a specific characterization for the conditional probability generating function, and the test statistic is formulated as a L 2 weighting norm of the corresponding sample counterpart. The asymptotic properties of the proposed test statistic are provided under the null hypothesis as well as under specific alternatives. A bootstrap version of the test is explored in a Monte–Carlo study and illustrated on a real data set on road safety. [ABSTRACT FROM AUTHOR]

We propose an extension of the double autoregressive (DAR) model: the sign-double autoregressive (SDAR) model, in the spirit of the GJR-GARCH model (also named the sign-ARCH model). Our model shares the important property of DAR models where a unit root does not imply non stationarity and it allows for asymmetry, as other alternatives in the literature such as the GJR-GARCH or asymmetric linear DAR and dual-asymmetry linear DAR models. We establish consistency and asymptotic normality of the quasi-maximum likelihood estimator in the context of the SDAR model. Furthermore, it is shown by simulations that the asymptotic properties also apply in finite samples. Finally, an empirical application shows the usefulness of our model specially in periods of supply/demand crises of oil disruptions, where spikes of volatility are very likely to be predominant. [ABSTRACT FROM AUTHOR]

AbstractAn integer time series is a sequence of discrete integers arranged chronologically to represent a specific event or phenomenon. It may include negative integers, and its behavior can be influenced by variations in covariates. To address these complexities, this paper introduces a p-order signed integer-valued autoregressive model with time-varying coefficients. Considering the presence of zero-inflated phenomena, we specifically employ the zero-inflated Skellam distribution. To estimate the unknown parameters and nonparametric functions in our proposed model, we adopt a two-step iterative process that combines local linear estimation with either conditional least squares or conditional maximum likelihood. Through a comprehensive simulation study, we evaluate the performance of our estimation method and demonstrate the usefulness of the proposed model using a real-life dataset. [ABSTRACT FROM AUTHOR]

This paper provides a least squares regression estimation of the tail index for long memory processes where the innovations are α -stable random sequences. The estimate is based on the property of the characteristic function of the process near the origin. The asymptotics of the estimator are obtained by choosing suitable regression samples with the help of the properties of the α -stable distribution. The numerical simulation and an empirical analysis of financial market data are conducted to assess the finite sample performance of the proposed estimator. [ABSTRACT FROM AUTHOR]

A new non-linear stationary process for time series of counts is introduced. The process is composed of the survival and innovation component. The survival component is based on the generalized zero-modified geometric thinning operator, where the innovation process figures in the survival component as well. A few probability distributions for the innovation process have been discussed, in order to adjust the model for observed series with the excess number of zeros. The conditional maximum likelihood and the conditional least squares methods are investigated for the estimation of the model parameters. The practical aspect of the model is presented on some real-life data sets, where we observe data with inflation as well as deflation of zeroes so we can notice how the model can be adjusted with the proper parameter selection. [ABSTRACT FROM AUTHOR]

One of the most important hyper-parameters in duration-dependent Markov-switching (DDMS) models is the duration of the hidden states. Because there is currently no procedure for estimating this duration or testing whether a given duration is appropriate for a given data set, an ad hoc duration choice must be heuristically justified. In this paper, we propose and examine a methodology that mitigates the choice of duration in DDMS models when forecasting is the goal. The novelty of this paper is the use of the asymmetric Aranda-Ordaz parametric link function to model transition probabilities in DDMS models, instead of the commonly applied logit link. The idea behind this approach is that any incorrect duration choice is compensated for by the parameter in the link, increasing model flexibility. Two Monte Carlo simulations, based on classical applications of DDMS models, are employed to evaluate the methodology. In addition, an empirical investigation is carried out to forecast the volatility of the S&P500, which showcases the capabilities of the proposed model. [ABSTRACT FROM AUTHOR]

Abstract.Conditional least squares estimation is often employed to infer an integer-valued AR(1) model and its convergence rate and asymptotic variance differ for the stable and nearly unstable cases. This article adopts a random weighted bootstrap method to provide a unified interval estimation and hypothesis test regardless of the underlying process being either stable or nearly unstable. A simulation study confirms the good finite sample performance of the proposed inference. We also apply it to test for a unit root test in a COVID-19 dataset. [ABSTRACT FROM AUTHOR]

A wide variety of methods have been proposed for clustering of stochastic processes. However, for clustering of periodically correlated processes (PC) it is demanding to introduce some similarity measures that take into account the inherent periodicity of these processes. The frequency-domain based methods seem more desirable to determine groups of PC processes with similar frequency characterizations. In this article, we present new similarity measures based on Hilbert-Schmidt inner product of finite Fourier transforms of PC processes. Based on simulated stochastic processes and a real gene expression dataset we illustrate the accuracy of the methods. [ABSTRACT FROM AUTHOR]

This paper deals with some probabilistic and statistical properties of periodic generalized Poisson integer-valued autoregressive processes of order p, PG P INAR (p). Necessary and sufficient conditions for the periodic stationarity, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. Moreover, the Wold–Cramér expression of the underlying second-order periodically stationary process is then established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker (YW), the two-stage conditional least squares (CLS) and the conditional maximum likelihood (CML) estimation methods of the underlying parameters are obtained. An intensive simulation study and an application on a real count data consisting of the daily number of daytime road accidents in Schiphol area, in Netherlands are provided. [ABSTRACT FROM AUTHOR]

A test statistic for nonlinearity of a given heavy-tailed time series process is constructed, based on the sub-sample stability of Gini-based sample autocorrelations. The finite-sample performance of the proposed test is evaluated in a Monte Carlo study and compared to a similar test based on the sub-sample stability of a heavy-tailed analogue of the conventional sample autocorrelation function. In terms of size and power properties, the quality of our test outperforms a nonlinearity test for heavy-tailed time series processes proposed by [S.I. Resnick and E. Van den Berg, A test for nonlinearity of time series with infinite variance, Extremes 3 (2000), pp. 145–172.]. A nonlinear Pareto-type autoregressive process and a nonlinear Pareto-type moving average process are used as alternative specifications when comparing the power of the proposed test statistic. The efficacy of the test is illustrated via the analysis of a heavy-tailed actuarial data set and two time series of Ethernet traffic. [ABSTRACT FROM AUTHOR]

This note examines the conditional least squares (CLS) estimators for integer-valued autoregressive (INAR) models, with a focus on the INAR(1) model. Our investigation reveals that the joint limiting distribution of the CLS estimators for the autoregressive coefficient and the mean of the model errors is degenerate in the case of mildly unstable INAR(1) models with moderate deviations from unity. This degeneracy may pose challenges for statistical inference. We provide commenting notes based on the asymptotic results to address this issue. Additionally, we conduct simulation studies and analyze a real-world dataset to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

When modelling unbounded counts, their marginals are often assumed to follow either Poisson (Poi) or negative binomial (NB) distributions. To test such null hypotheses, we propose goodness-of-fit (GoF) tests based on statistics relying on certain moment properties. By contrast to most approaches proposed in the count-data literature so far, we do not restrict ourselves to specific low-order moments, but consider a flexible class of functions of generalized moments to construct model-diagnostic tests. These cover GoF-tests based on higher-order factorial moments, which are particularly suitable for the Poi- or NB-distribution where simple closed-form expressions for factorial moments of any order exist, but also GoF-tests relying on the respective Stein's identity for the Poi- or NB-distribution. In the time-dependent case, under mild mixing conditions, we derive the asymptotic theory for GoF tests based on higher-order factorial moments for a wide family of stationary processes having Poi- or NB-marginals, respectively. This family also includes a type of NB-autoregressive model, where we provide clarification of some confusion caused in the literature. Additionally, for the case of independent and identically distributed counts, we prove asymptotic normality results for GoF-tests relying on a Stein identity, and we briefly discuss how its statistic might be used to define an omnibus GoF-test. The performance of the tests is investigated with simulations for both asymptotic and bootstrap implementations, also considering various alternative scenarios for power analyses. A data example of daily counts of downloads of a TeX editor is used to illustrate the application of the proposed GoF-tests. [ABSTRACT FROM AUTHOR]

We analyse a second-order SPDE model in multiple space dimensions and develop estimators for the parameters of this model based on discrete observations of a solution in time and space on a bounded domain. While parameter estimation for one and two spatial dimensions was established in recent literature, this is the first work which generalizes the theory to a general, multi-dimensional framework. Our approach builds upon realized volatilities, enabling the construction of an oracle estimator for volatility within the underlying model. Furthermore, we show that the realized volatilities have an asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields novel and efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. For proving central limit theorems, we use a high-frequency observation scheme. To showcase our results, we conduct a Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

Forecasting mortality rates is crucial for evaluating life insurance company solvency, especially amid disruptions caused by phenomena like COVID-19. The Lee–Carter model is commonly employed in mortality modelling; however, extensions that can encompass count data with diverse distributions, such as the Generalized Autoregressive Score (GAS) model utilizing the COM–Poisson distribution, exhibit potential for enhancing time-to-event forecasting accuracy. Using mortality data from 29 countries, this research evaluates various distributions and determines that the COM–Poisson model surpasses the Poisson, binomial, and negative binomial distributions in forecasting mortality rates. The one-step forecasting capability of the GAS model offers distinct advantages, while the COM–Poisson distribution demonstrates enhanced flexibility and versatility by accommodating various distributions, including Poisson and negative binomial. Ultimately, the study determines that the COM–Poisson GAS model is an effective instrument for examining time series data on mortality rates, particularly when facing time-varying parameters and non-conventional data distributions. [ABSTRACT FROM AUTHOR]

A useful strategy to examine serial correlation in panel regression models is to adopt serially correlated error terms. The random effect is also present to address the individual-specific heterogeneity. Familiar transformations can restructure the AR(1) process into a serially uncorrelated dynamic model. It may result in unfitting estimation results once the statistical inference is carried on the conditional likelihood. We discuss in the article the consequence of this action. In particular, in the case of large cross-section units and short time-sequences, we provide analytical expressions for the asymptotic bias of maximum likelihood estimates when the analyst naïvely neglects the initial conditions problem. Under several conditions, we compute the size of biases and measure the mean squared errors to illustrate this violation. We also construct a joint regression model of the initial and subsequent responses to handle the initial conditions. Then, we conduct simulation studies to determine the asymptotic behavior. [ABSTRACT FROM AUTHOR]

We provide a new testing procedure to detect serial dependence in time series. Our method is based solely on the ordinal structure of the data. We explicitly allow for ties in the data windows we consider. Consequently, we use generalised ordinal patterns, that is, Cayley permutations. Unlike in the classical case, the pattern distribution is not uniform under the null hypothesis of serial independence. In our new framework, the underlying distribution has to be taken into account and we overcome this problem by a bootstrap procedure. The applicability of our method is supported by a simulation study and two real-world data examples. [ABSTRACT FROM AUTHOR]

The scope of this manuscript is to review some recent developments in statistics for discretely observed semimartingales which are motivated by applications for financial markets. Our journey through this area stops to take closer looks at a few selected topics discussing recent literature. We moreover highlight and explain the important role played by some classical concepts of probability and statistics. We focus on three main aspects: Testing for jumps; rough fractional stochastic volatility; and limit order microstructure noise. We review jump tests based on extreme value theory and complement the literature proposing new statistical methods. They are based on asymptotic theory of order statistics and the Rényi representation. The second stage of our journey visits a recent strand of research showing that volatility is rough. We further investigate this and establish a minimax lower bound exploring frontiers to what extent the regularity of latent volatility can be recovered in a more general framework. Finally, we discuss a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices and its probabilistic and statistical foundation. [ABSTRACT FROM AUTHOR]

In statistical diagnostics, detecting influential observations is pivotal for assessing model fitting. To address parameter restrictions while maintaining necessary properties, the softplus INGARCH model has emerged as an alternative to the INGARCH model and its variants. This paper delves into statistical diagnostics within the softplus INGARCH model using local influence analysis, establishing a framework encompassing first-order diagnostics, second-order diagnostics and stepwise diagnostics. Additionally, we focus on perturbation schemes, refining conventional approaches and offering modifications. To demonstrate the effectiveness and suitability of our proposed methodology, particularly with the inclusion of stepwise diagnostics, we analyze two simulated datasets and two real-world examples. Compared to traditional methods, our approach adeptly handles potential issues such as the "masking effect" and "smearing effect" without necessitating complex calculations. [ABSTRACT FROM AUTHOR]