Estimation in integer-valued moving average models (Q2759391)

The authors consider a discrete analogue of the classical stationary q th-order moving average, INMA(q), process defined by NEWLINE\[NEWLINEy_{t}=\theta_{0}\circ \varepsilon_{t}+\theta_{1}\circ \varepsilon_{t-1}+\theta_{2}\circ \varepsilon_{t-2}+\ldots+\theta_{q}\circ \varepsilon_{t-q},\quad t=1,\ldots,T,NEWLINE\]NEWLINE where \(\{\varepsilon_{t}\}\) is an integer-valued i.i.d. sequence with non-negative expectation; \(\theta_{0},\ldots,\theta_{q-1}\in [0,1]\), \(\theta_{q}\in(0,1]\), and usually \( \theta_{0}=1\). The thinning operation is defined by \(\theta\circ y=\sum\limits_{i=1}^{y}u_{i}\) with \(y\) an integer-valued random variable and \(\{u_{i}\}\) a sequence of i.i.d. binary random variables independent of \(y\) and with \(Pr(u_{i}=1)=\theta\). The probability generating function for \(y_{t}\) is of the form \(\phi(z)=E[(1-z)^{y_{t}}]\).NEWLINENEWLINENEWLINEIn this article four models that admit some dependence structure between thinnings are considered. Descriptions of the Yule-Walker and the conditional least squares estimators are presented. A generalized method of moments (GMM) estimator based on the probability generating function is introduced. Small sample performances of estimators are studied using Monte Carlo simulations.