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This paper is concerned with a general class of observation driven models for time series of counts whose conditional distributions given past observations and explanatory variables follow a Poisson distribution. These models provide a flexible framework for modeling a wide range of dependence structures. Conditions for stationarity and ergodicity of these processes are established from which the large sample properties of the maximum likelihood estimators can be derived. Simulations are provided to give additional insight into the finite sample behavior of the estimates. Finally an application to a regression model for daily counts of accident and emergency room presentations for asthma at several Sydney hospitals is described.

This paper is concerned with developing a practical approach to diagnosing the existence of a latent stochastic process in the mean of a Poisson regression model. First, a rigorous derivation of the asymptotic distribution of standard GLM estimates is derived for the case that an autocorrelated latent process is present. Simple formulae for the e ect of autocovariance on standard errors of the regression coe cients are also provided. Second, the paper examines tests for the presence of a latent process and considers estimates of the autocovariance of the latent process. Methods for adjusting for the severe bias in previously proposed estimator are derived and their behaviour investigated. Applications of the methods to time series of monthly polio counts in the U.S. and daily asthma presentations at a hospital in Sydney are used to illustrate the results and methods.

This paper studies theory and inference related to a class of time series models that incorporates nonlinear dynamics. It is assumed that the observations follow a one-parameter exponential family of distributions given an accompanying process that evolves as a function of lagged observations. We employ an iterated random function approach and a special coupling technique to show that, under suitable conditions on the parameter space, the conditional mean process is a geometric moment contracting Markov chain and that the observation process is absolutely regular with geometrically decaying coefficients. Moreover the asymptotic theory of the maximum likelihood estimates of the parameters is established under some mild assumptions. These models are applied to two examples; the first is the number of transactions per minute of Ericsson stock and the second is related to return times of extreme events of Goldman Sachs Group stock.

We study generalized linear models for time series of counts, where serial dependence is introduced through a dependent latent process in the link function. Conditional on the covariates and the latent process, the observation is modelled by a negative binomial distribution. To estimate the regression coefficients, we maximize the pseudolikelihood that is based on a generalized linear model with the latent process suppressed. We show the consistency and asymptotic normality of the generalized linear model estimator when the latent process is a stationary strongly mixing process. We extend the asymptotic results to generalized linear models for time series, where the observation variable, conditional on covariates and a latent process, is assumed to have a distribution from a one-parameter exponential family. Thus, we unify in a common framework the results for Poisson log-linear regression models of Davis et al. (2000), negative binomial logit regression models and other similarly specified generalized linear models.

The vector autoregressive (VAR) model has been widely used for modeling temporal dependence in a multivariate time series. For large (and even moderate) dimensions, the number of AR coefficients can be prohibitively large, resulting in noisy estimates, unstable predictions and difficult-to-interpret temporal dependence. To overcome such drawbacks, we propose a 2-stage approach for fitting sparse VAR (sVAR) models in which many of the AR coefficients are zero. The first stage selects non-zero AR coefficients based on an estimate of the partial spectral coherence (PSC) together with the use of BIC. The PSC is useful for quantifying the conditional relationship between marginal series in a multivariate process. A refinement second stage is then applied to further reduce the number of parameters. The performance of this 2-stage approach is illustrated with simulation results. The 2-stage approach is also applied to two real data examples: the first is the Google Flu Trends data and the second is a time series of concentration levels of air pollutants.which satisfies the recursions,see Brockwell and Davis (1991) and Reinsel (1997), which implies that Z t is independent of Y s for s < t. Without loss of generality, we also assume that the vector process {Y t } has mean 0, i.e., µ = 0 in (2.1). Sparse vector autoregressive models (sVAR)The temporal dependence structure of the VAR model (2.1) is characterized by the AR coefficient matrices A 1 , . . . , A p . Based on T observations Y 1 , . . . , Y T from the VAR model, we want to estimate these AR matrices. However, a VAR(p) model, when fully-parametrized, has K 2 p AR parameters that need to be estimated. For large (and even moderate) dimension K, the number of parameters can be prohibitively large, resulting in noisy estimates, unstable predictions and difficult-to-interpret descriptions of the temporal dependence. It is also generally believed that, for most applications, the true model of the series is sparse, i.e., the number of non-zero coefficients is small. Therefore it is preferable to fit a sparse VAR (sVAR) model in which many of its AR parameters are zero. In this paper we develop a 2-stage approach of fitting sVAR models. The first stage selects non-zero AR coefficients by screening pairs of distinct marginal series that are conditionally correlated. To compute direction-free conditional correlation between components in the time series, we use tools from the frequency-domain, specifically the partial spectral coherence (PSC). Below we introduce the basic properties related to PSC. Let {Y t,i } and {Y t,j } (i = j) denote two distinct marginal series of the process {Y t }, and {Y t,−ij } denote the remaining (K − 2)-dimensional process. To compute the conditional correlation between two time series {Y t,i } and {Y t,j }, we need to adjust for the linear effect from the remaining marginal series {Y t,−ij }. The removal of the linear effect of {Y t,−ij } from each of {Y t,i } and {Y t,j } can be achieved by using results of linear filters, e.g., see Brillinge...

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Many real-life time series often exhibit clusters of outlying observations that cannot be adequately modeled by a Gaussian distribution. Heavy-tailed distributions such as the Pareto distribution have proved useful in modeling a wide range of bursty phenomena that occur in areas as diverse as finance, insurance, telecommunications, meteorology, and hydrology. Regular variation provides a convenient and unified background for studying multivariate extremes when heavy tails are present. In this paper, we study the extreme value behavior of the space-time process given bywhere (Zt) t∈Z is an iid sequence of random fields on [0, 1] d with values in the Skorokhod space D([0, 1] d ) of càdlàg functions on [0, 1] d equipped with the J1−topology. The coefficients ψi are deterministic real-valued fields on D([0, 1] d ). The indices s and t refer to the observation of the process at location s at time t. For example, Xt(s), t = 1, 2, . . . , could represent the time series of annual maxima of ozone levels at location s. The problem of interest is determining the probability that the maximum ozone level over the entire region [0, 1] 2 does not exceed a given standard level f ∈ D([0, 1] 2 ) in n years. By establishing a limit theory for point processes based on (Xt(s)), t = 1 . . . , n, we are able to provide approximations for probabilities of extremal events. This theory builds on earlier results of de Haan and Lin [11] and Hult and Lindskog [13] for regular variation on D([0, 1] d ) and Davis and Resnick [7] for extremes of linear processes with heavy-tailed noise.