Integer-valued Time Series Model via Generalized Linear Models Technique of Estimation

The paper authenticated the need for separate positive integer time series model(s). This was done from the standpoint of a proposal for both mixtures of continuous and discrete time series models. Positive integer time series data are time series data subjected to a number of events per constant interval of time that relatedly fits into the analogy of conditional mean and variance which depends on immediate past observations. This includes dependency among observations that can be best described by Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model with Poisson distributed error term due to its positive integer defined range of values. As a result, an integer GARCH model with Poisson distributed error term was formed in this paper and called Integer Generalized Autoregressive Conditional Heteroscedasticity (INGARCH). Iterative Reweighted Least Square (IRLS) parameter estimation technique type of the Generalized Linear Models (GLM) was adopted to estimate parameters of the two spilt models; Linear and Log-linear INGARCH models deduced from the identity link function and logarithmic link function, respectively. This resulted from the log-likelihood function generated from the GLM via the random component that follows a Poisson distribution. A study of monthly successful bids of auction from 2003 to 2015 was carried out. The Probabilistic Integral Transformation (PIT) and scoring rule pinpointed the uniformity of the linear INGARCH than that of the log-linear INGARCH in describing first order autocorrelation, serial dependence and positive conditional effects among covariates based on the immediate past. The linear INGARCH model outperformed the log-linear INGARCH model with (AIC = 10514.47, BIC = 10545.01, QIC = 34128.56) and (AIC = 37588.83, BIC = 37614.28, QIC = 37587.3), respectively.

Hysteretic Poisson INGARCH model for integer-valued time series

This study proposes a new model for integer-valued time series—the hysteretic Poisson integer-valued generalized autoregressive conditionally heteroskedastic (INGARCH) model—which has an integrated hysteresis zone in the switching mechanism of the conditional expectation. Our modelling framework provides a parsimonious representation of the salient features of integer-valued time series, such as discreteness, over-dispersion, asymmetry and structural change. We adopt Bayesian methods with a Markov chain Monte Carlo sampling scheme to estimate model parameters and utilize the Bayesian information criteria for model comparison. We then apply the proposed model to five real time series of criminal incidents recorded by the New South Wales Police Force in Australia. Simulation results and empirical analysis highlight the better performance of hysteresis in modelling the integer-valued time series.