Efficient Method of Moments Estimators for Integer Time Series Models

Abstract

The parameters of integer autoregressive models with Poisson, or negative binomial innovations can be estimated by maximum likelihood where the prediction error decomposition, together with convolution methods, is used to write down the likelihood function. When a moving average component is introduced this is not the case. To address this problem an efficient method of moment estimator is proposed where the estimated standard errors for the parameters are obtained using subsampling methods. The small sample properties of the estimator are investigated using Monte Carlo methods, while the approach is demonstrated using two well‐known examples from the time series literature.

Modeling digital camera monitoring count data with intermittent zeros for short-term prediction

“…The performance of forecasted values estimated by INMA (1) in this study indicates that an investigation into different estimation procedures may be required to establish a good fit for recreational boating effort data. Some inferential procedures for INMA models have been developed via conditional least squares (CLS), feasible generalized least square (FGLS) and generalized method of moments (GMM) approaches Aleksandrov and Weiß (2020) ; Brä nn ä s and Shahiduzzaman Quoreshi (2010) ; Martin et al. (2014) ; Quoreshi (2008) .…”

“…The performance of forecasted values estimated by INMA (1) in this study indicates that an investigation into different estimation procedures may be required to establish a good fit for recreational boating effort data. Some inferential procedures for INMA models have been developed via conditional least squares (CLS), feasible generalized least square (FGLS) and generalized method of moments (GMM) approaches Aleksandrov and Weiß (2020); Br€ a nn € a s and Shahiduzzaman Quoreshi (2010); Martin et al (2014);Quoreshi (2008). Additionally, the irregular nature of the correlational structure in the data may require the fit of a long-lag INMA model, which has been established to satisfy the modelling restriction of parameter estimates lying within a unit interval (Br€ a nn € a s and Shahiduzzaman Quoreshi, 2010).…”

Improving the Estimation and Predictions of Small Time Series Models

“…The Integer Autoregressive (INAR) class of models was originally proposed in Al-Osh and Alzaid (1987) as a method of modeling dependent series of low counts, and there has been a growing interest in the area. Some recent contributions include Martin et al (2014), Sant'Anna (2017 and Harris and McCabe (2018). As noted in the latter, INAR models have been used for applications in economics, medicine, environmental studies, and commerce.…”

“…A further motivation for addressing estimation of diffusion models is that there are only a few cases where exact ML is possible, moreover we are able to compare the performance of the new approach with results in Tang and Chen (2009) for estimation by Bootstrap, Jackknife and Indirect Inference. A further motivation for addressing estimation of INAR models is that the exact ML estimation of INARMA models more generally is difficult, which motivated the Efficient Method of Moments (EMM) approach in Martin et al (2014), and the new approach is applicable whenever initial consistent estimates of the model parameters are available. Section 5 applies the methodology to prediction of the Effective Federal Funds Rate for overnight lending in the United States, and Section 6 concludes.…”

Approximate Bayesian Forecasting

“…See Jung and Tremayne (2006) for a review, and Drost et al (2009) and McCabe et al (2011) for contributions. Of particular note is the work by Martin et al (2014), in which the IN-ARMA model is estimated 'indirectly' via efficient method of moments (Gallant and Tauchen, 1996), which is similar in spirit to ABC. No investigation of forecasting under this 'approximate' inferential paradigm is however undertaken.…”