Page 102 - Textos de Matemática Vol. 47

92 I. PEREIRA, M. SCOTTO, AND R. NICOLETTE
4. Point Prediction in the PSETINAR model
In this section we consider the forecasting of future values Xi+NT+h, with h = j + `T, ` 2 N0, given past observations up through time i + NT.
First note that by iterating equation (1.1) it follows that Xt can be expressed
as
0n 1 YXY
where, for t > i
n 1 0i 1 1 @AX@A
j=0 i=1 j=0 n 1
⌘ t,n Xt n + t,i Zt i, i=0
1
d
Xt = t j Xt n + t j
Zt i + Zt
⇢ Qi 1 t j, i>0 ⇢ t,j T`,T, i=j+`T, j=1,...,T t,i:= j=0 = ,
1, i=0 1, i=0 leading to obtain
d hX 1
Xi+NT+h = i+NT+h,h Xi+NT + i+NT+h,m Zi+NT+h m. (4.1)
m=0
Since h = j + `T, it follows that
`T 1 ` 1 T 1
X j+`T,m Zj+`T m =d X X j+`T,m T`,T Zj+(` w)T m,
m=0 w=0 m=0 the expression in (4.1) takes the form
Xi+NT+h =d i+j+(N+`)T,j+`T Xi+NT +
Due to the periodicity of the ’s, i+j+(N+`)T,j+`T = i+j,j+`T = i+j,j T` ,T ,
and considering the relation
with
Xi+NT+h =d i+j,j T`,T Xi+NT +Vi+j+`T i 1 ` 1 T 1
(4.2)
Vi+j+`T := X i+j,m Zi+j m+NT +X X i+j+(N+`)T,m+j+Tw Zi+(N+` w)T m. m=0 w=0 m=0
j + X` T 1 m=0
i+j+(N+`)T,m Zi+j+(N+`)T m.