On Locally Dyadic Stationary Processes

We introduce the concept of local dyadic stationarity, to account for non-stationary time series, within the framework of Walsh-Fourier analysis. We define and study the time varying dyadic ARMA models (tvDARMA). It is proven that the general tvDARMA process can be approximated locally by either a tvDMA and a tvDAR process.Comment: 27 pages, 2 figure

Poisson Autoregression

This paper considers geometric ergodicity and likelihood based inference for linear and nonlinear Poisson autoregressions. In the linear case the conditional mean is linked linearly to its past values as well as the observed values of the Poisson process. This also applies to the conditional variance, implying an interpretation as an integer valued GARCH process. In a nonlinear conditional Poisson model, the conditional mean is a nonlinear function of its past values and a nonlinear function of past observations. As a particular example an exponential autoregressive Poisson model for time series is considered. Under geometric ergodicity the maximum likelihood estimators of the parameters are shown to be asymptotically Gaussian in the linear model. In addition we provide a consistent estimator of the asymptotic covariance, which is used in the simulations and the analysis of some transaction data. Our approach to verifying geometric ergodicity proceeds via Markov theory and irreducibility. Finding transparent conditions for proving ergodicity turns out to be a delicate problem in the original model formulation. This problem is circumvented by allowing a perturbation of the model. We show that as the perturbations can be chosen to be arbitrarily small, the differences between the perturbed and non-perturbed versions vanish as far as the asymptotic distribution of the parameter estimates is concerned.generalized linear models; non-canonical link function; count data; Poisson regression; likelihood; geometric ergodicity; integer GARCH; observation driven models; asymptotic theory

Poisson Network Autoregression

We consider network autoregressive models for count data with a non-random time-varying neighborhood structure. The main methodological contribution is the development of conditions that guarantee stability and valid statistical inference. We consider both cases of fixed and increasing network dimension and we show that quasi-likelihood inference provides consistent and asymptotically normally distributed estimators. The work is complemented by simulation results and a data example

Unveiling Venice’s hotels competition networks from dynamic pricing digital market

We study the dynamic price competition of hotels in Venice using publicly available data scraped from an online travel agency. This study poses two main challenges. First, the time series of prices recorded for each hotel encompasses a twofold time frame. For every single asking price for an overnight stay on a specific day, there is a corresponding time series of asking prices along the booking window on the online platforms. Second, the competition relations between different hoteliers is clearly unknown and needs to be discovered using a suitable methodology. We address these problems by proposing a novel Network Autoregressive model which is able to handle the peculiar threefold data structure of the data set with time-varying coefficients over the booking window. This approach allows us to uncover the competition network of the market players by employing a quick data-driven algorithm. Independent, mixed, and leader–follower relationships are detected, revealing the competitive dynamics of the destination, useful to managers and stakeholders

The R-package PNAR for modelling count network time series

We introduce a new R package for analysis and inference of network count time series. Such data arise frequently in statistics and epidemiology and are modelled as multivariate time series by employing either linear or log-linear models. However, nonlinear models have also been successful in several fields but often excluded from the analysis due to their relative fitting complexity. In this paper, we offer users the flexibility to study and specify non-linear network count time series models by providing them with a toolkit that copes with computational issues. In addition, new estimation tools for (log-)linear network autoregressive models of count time series are also developed. We illustrate the methodology to the weekly number of influenza A & B cases in the 140 districts of the two Southern German states Bavaria and Baden-Wuerttemberg, for the years 2001 to 2008. This dataset is publicly available, so that the analysis is easily reproducible

On categorical time series with covariates

We study the problem of stationarity and ergodicity for autoregressive multinomial logistic time series models which possibly include a latent process and are defined by a GARCH-type recursive equation. We improve considerably upon the existing conditions about stationarity and ergodicity of those models. Proofs are based on theory developed for chains with complete connections. A useful coupling technique is employed for studying ergodicity of infinite order finite-state stochastic processes which generalize finite-state Markov chains. Furthermore, for the case of finite order Markov chains, we discuss ergodicity properties of a model which includes strongly exogenous but not necessarily bounded covariates