Page 94 - Textos de Matemática Vol. 47

84 I. PEREIRA, M. SCOTTO, AND R. NICOLETTE
case, threshold models are typically characterized by having a linear (ARMA) structure in each regime; see, e.g., Turkman et al. (2014) for details. However, in the field of integer-valued time series modelling little research has been done so far to develop models to cope with time series of counts exhibiting piecewise- type patterns. For this purpose, Monteiro et al. (2012) introduced a class of self-exciting threshold integer-valued autoregressive (SETINAR, in short) models of order one and two regimes, defined by the recursive equation
Xt =⇢↵1 Xt 1+Zt,Xt 1R. ↵2 Xt 1 +Zt, Xt 1 >R
Here, (Zt) constitutes a sequence of integer-valued random variables (r.v’s), and R represents the threshold level. The “↵i ” is the binoPmial thinning operator of Steutel and van Harn (1979). It is defined as ↵ X := Xi=1 Yi, for X with range
N0 = {0, 1, . . .}, where the
(i.i.d.) Bernoulli variables with probability ↵ 2 (0; 1). The authors discussed probabilistic and statistical properties related with this class of models. Note that the SETINAR models fall within the state-dependent thinning class.
In this article, we introduce the periodic self-exciting threshold integer- valued autoregressive model of order one with two regimes (hereafter referred to as PSETINAR(2; 1, 1)T ) which generalizes the SETINAR model by considering periodically varying threshold levels. For this class of models, we investigate some basic probabilistic and statistical properties. Furthermore, parameter es- timation and forecasting are also addressed. Finally some concluding remarks are given.
The PSETINAR(2; 1, 1)T model is defined through the recursive equation
Yi’s are independent and identically distributed
( ↵(1) X + Z(1), X  R
Xt = j t 1 t t d t , t 2 N0, (1.1)
↵(2) X + Z(2), X > R j t 1 t t d t
withRt =rj,fort=j+sT,j=1,...,Tands2N0.Notethatforthe jth-period we have
Xj+sT = (↵(1) Xj+sT 1 + Z(1) )I(1) + (↵(2) Xj+sT 1 + Z(2) )I(2), j j+sT j j j+sT j
with I(k), for k = {1, 2⇢}, defined as j
I(1) := 1, Xj+sT d rj , I(2) =1 I(1). j 0, Xj+sT d >rj j j
(1.2)
(1.3)
The threshold parameter Rt (which is assumed to be known) represents the level of the process and the regime switch is triggered by the lag-d value of the series.