Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

Abstract

In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701-722] and develop a general framework for maximum likelihood (ML) analysis of higher-order integer-valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004), we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule-Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd

Likelihood Estimation for the INAR(p) Model by Saddlepoint Approximation

“…When the errors in (1) are assumed to be Poisson with mean λ, then (Bu et al, 2008) the transition probabilities comprising (2) may be written as the (p + 1)-fold convolution…”

Forecast horizon aggregation in integer autoregressive moving average (INARMA) models

“…This is consistent with other studies of INARMA processes which used the same or fewer replications (eg. [22,[25][26][27]) and have been found to give reliable results when compared with findings known from theory.…”

Efficient Method of Moments Estimators for Integer Time Series Models

“…The term after the logarithm on the right‐hand side is, of course, a convolution, and the complete expression for the conditional distribution is given by Bu et al . () as their (13).…”

“…The next model specification entertained is an INAR(2) to determine the properties of the EMM estimator in the case of higher‐order dynamics, and results can be compared with those presented by Bu et al . () for MLE. A comparison of the sampling performance of the EMM and CLS estimators of the INMA(1) model (for which MLE is unavailable) determines whether there are any efficiency gains in finite samples from adopting the EMM estimator over the CLS one.…”

“…The MLE and CLS results are obtained from Bu et al . (), which are based on 1000 replications. The sampling properties of the EMM estimator are based on 5000 replications where the auxiliary model consists of a constant and m = 2 lags.…”