A goodness-of-fit test for Poisson count processes

We are studying a novel class of goodness-of-fit tests for parametric count time series regression models. These test statistics are formed by considering smoothed versions of the empirical process of the Pearson residuals. Our construction yields test statistics which are consistent against Pitman’s local alternatives and they converge weakly at the usual parametric rate. To approximate the asymptotic null distribution of the test statistics, we propose a parametric bootstrap method and we study its properties. The methodology is applied to simulated and real data.

[1]  Tina Hviid Rydberg,et al.  A Modelling Framework for the Prices and Times of Trades Made on the New York Stock Exchange , 1999 .

[2]  Jiti Gao,et al.  Specification testing in nonlinear and nonstationary time series autoregression , 2009, 0911.3736.

[3]  Mikhail Lifshits.  Absolute continuity of functionals of “supremum” type for Gaussian processes , 1984 .

[4]  Konstantinos Fokianos,et al.  Log-linear Poisson autoregression , 2011, J. Multivar. Anal..

[5]  Michael H. Neumann,et al.  Goodness-of-fit tests for Markovian time series models: Central limit theory and bootstrap approximations , 2008, 0803.0835.

[6]  J. Carlos Escanciano,et al.  Goodness-of-Fit Tests for Linear and Nonlinear Time Series Models , 2006 .

[7]  Timo Teräsvirta,et al.  Modelling nonlinear economic time series , 2010 .

[8]  Bonnie K. Ray,et al.  Regression Models for Time Series Analysis , 2003, Technometrics.

[9]  D. Pollard.  Convergence of stochastic processes , 1984 .

[10]  Hira L. Koul,et al.  Nonparametric model checks for time series , 1999 .

[11]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[12]  Peter C. Young,et al.  Nonlinear and Nonstationary Signal Processing , 1998, Technometrics.

[13]  D. Freedman.  On Tail Probabilities for Martingales , 1975 .

[14]  Juan-Carlos Escanciano,et al.  Data-driven smooth tests for the martingale difference hypothesis , 2010, Comput. Stat. Data Anal..

[15]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[16]  Juan Carlos Escanciano,et al.  Model Checks Using Residual Marked Empirical Processes , 2004 .

[17]  Dag Tjøstheim,et al.  On weak dependence conditions for Poisson autoregressions , 2012 .

[18]  D. Tjøstheim,et al.  Nonlinear Poisson autoregression , 2012 .

[19]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[20]  Stationarity of Count-Valued and Nonlinear Time Series Models , 2010 .

[21]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[22]  Michael H. Neumann.  Absolute regularity and ergodicity of Poisson count processes , 2011, 1201.1071.

[23]  Benjamin Kedem,et al.  Regression models for time series analysis , 2002 .

[24]  Alain Latour,et al.  Integer‐Valued GARCH Process , 2006 .

[25]  Harry Joe,et al.  Modelling Count Data Time Series with Markov Processes Based on Binomial Thinning , 2006 .