Munich Personal RePEc Archive
Additional Smoothing Transition
Autoregressive Models
Giovanis, Eleftherios
10 August 2008
Online at
https://mpra.ub.uni-muenchen.de/24657/
!
"
#! "$%
&
'
(
!
" )"
* !
+
+
+
+
'
+
+
+
" )"
,
'
,
,
-&
.
! "$
/
0
#! "$%
1
#2345%
*
%
+
#
%
#
6
.7 .62
#
%
#
%
#.%0#5%
% #2
2
%
+
#
#.%
. % #2
%
+
#
#/%
2
%
2
#
.
%
+
#
. # %#8%
.
%
+
#
#9%
. ..
%
#
%
+
#
#5%
! "$
:
' ( )
)
*)
+#<%
,
,-.
-# $ %
,
/
# $ %
,
(
#2%
(
)
0
(
&
(
'
"
! "$
0
"
"$
"
&
0
'
!=> 977 & !
277 1"1 87
?"@
" )"
2 ,
! "$
,
! "$
2 ,
277
.773
47
.7
0 0
&
0
!
'
! "$
2
,
!=>
?"@
2
1"1 87
9
& ! 277
.
1
!=> 977 & ! 277 ?"@
1"1 87
! "$ )! "$ ! "$ ))A! "$ "B!!A! "$
!=> 977 59 77 99 77 57 77 59 77 57 77
& ! 277 89 77 57 77 57 77 99 77 57 77
1"1 87 <7 77 59 77 99 77 <7 77 59 77
2
! "$
)! "$ &
B!"
,
'
238<0.773
!
0 ?
.773
0
0
0
,
0
1
0
(
"
'
'
2
.
0
0
B!
0
B!"
B!"
1
; 0 0
B!
'
&
C2 /D
7 9
C07 7/ 7 7/D
7 72 "
#
%
C7 9 .D
7 2
,
-0
'
0
"
(
0
.
,
'
1
!=> 977 & ! 277 ?"@
1"1 87
! "$ )! "$ ! "$ ))A! "$ "B!!A! "$!=> 977 59 77 59 77 57 77 59 77 59 77
7 777 7 7.82 7 7..8 7 777 7 727
E 2 777 2 5829 2 9/.5 2 777 2 777
2 5/24
& ! 277 87 77 97 77 57 77 99 77 99 77
7 7/7 7 7..5 7 775 7 777 7 727
E 2 777 2 /.9/ 2 977 2 777 2 977
2 5/24
1"1 87 97 77 /9 77 99 77 57 77 97 77
7 727 07 7/7 7 7/7 7 7/7 07 727
E 2 977 2 977 / 777 2 977 2 727
2 977
?"@ 97 77 57 77 57 77 99 77 99 77
7 727 7 7.9. 7 7./5 07 7/7 07 727
E 2 977 2 8/7< 2 /92< 2 /8// 2 777
. 777
.
"
! "$
)! "$
'
! "$
& !
277
! "$
))A! "$
1"1 87
))A! "$
"B!!A! "$
!=> 977
! "$
&
!
0
;
&
'
!
! "$
'
'
"
!
!
>
#
>
>
/47 1 %
!
6
>
6
&
! "$
"
! "$
'
'
H
I
' H
I
'
!0
J0
&
'
(
)
0
(
! "
1
F !
K #2345% ;
0& 12 % &3 4 5 6 1
2 %
L
<
2<40237
:
#2338% !
!
"
0& 12 % &3 7
5 1
2 6
%
&
&2
L
#
$ %
&
'
(
"
! " "
"
#
$ $ ! "
% ! &
' &
% "
! " $ ! "
"
( ( ! " ( ( ! "
$ $ ( ( ! "
) * + "" ,
- #./ $ "
- #./0 $ 1 $ " 2 ""
3 )
(
! " - #./0 13 ! "
#./ "
#./0 4 ( $ 1 $ """
3 )
(
! " #./0 13 ! "
# #./0 4 ( $ 1 1 $ """ " 3 )
(
! " # #./0 13 ! "
8 9 + :
9:- 0 #./0 4 ( & $ "4 " 2 1& "" 3 )
(
! " 9:- 0 #./0 13 ! "
' 9
9.; 0 #./0 $ $ " 2 4 1 2 ""
3 )
(
! " 9.; 0 #./0 13 ! "
) *) ,
) *) $,
- < = %
& % )>1)"1)>1
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B- #./!& "
! "
! "
& ! "
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B #./!& "
! "
! "
& ! "
5
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B# #./!& "
! "
& ! "
8
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B9:- 0 #./!& "
& ! "
! " ! "
& 5! "
'
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B9.; 0 #./!& "
! "
! "
& ! "
- #./ $ "
- #./0 $ 1 $ " 2 ""
3 )
! "1- #./0 ! " (
! " - #./0 13 ! "
#./ "
#./0 4 ( $ 1 $ """
3 )
! "1 #./0 ! " (
! " #./0 13 ! "
5 #.670 #./ & "
# #./0 4 ( $ 1 1 $ """ "
3 )
! "1# #./0 ! " (
! " # #./0 13 ! "
8 9 + :
9:- 0 #./0 4 ( & $ "4 " 2 1& "" 3 )
! "19:- 0 #./0 ! " (
! " 9:- 0 #./0 13 ! "
9.; 0 #./0 $ $ " 2 4 1 2 "" 3 )
! "19.; 0 #./0 ! " (
! " 9.; 0 #./0 13 ! "
! " *) ! " ! " ! "1 1&
! " *) ! " ! " ) !5" ,1&
! " ( ! "
(
C @<A:"" 9
4
0 ( ! "
)5 *) ,
)51&
0 " ===
0 ===! "D
0 0 0 ===! "
0 ===! "E
0 0 0 ===! "
===
===! "D ===! "
===! "E ===! "
. 0 % 0 0 0 0 0 0 "
. 0 % 0 0 0 0 0 0 "
F 0 % 0 0 0 "
F 0 % 0 0 0 "
# 0 0 0 0 0 0 0 "
F 0 0 # 0 0 0 0 "4 "1
! !> >" !> &>"
- #./ & !$ ! "
& "1$ ! "( & "1$ ! "( & 5"1$ ! "(& 8"1$ ! "" 1
$ & '" 1 $ ! " & G"" 2 ""
#./ & !$ ! "
& "1$ ! "( & "1$ ! "( & 5"1$ ! "(& 8"1$ ! "" 1
4 ( $ & '" 1 $ ! " & G"""""
# #./ & !$ ! "
& "1$ ! "( & "1$ ! "( & 5"1$ ! "(& 8"1$ ! "" 1
4 ( $ 1& '" 1 $ ! " & G"""" "
9:- 0 #./ & !$ ! "
& "1$ ! "( & "1$ ! "( & 5"1$ ! "(& 8"1$ ! "" 1
4 ( & $ ! " & '""4& G"" 2 1& H"""
9.; 0 #./ & !$ ! "
& "1$ ! "( & "1$ ! "( & 5"1$ ! "(& 8"1$ ! "" 1
$ %
&
'
(
"
8 III
0
0 "
"
#
$ $ ! "
% ! &
8 &
% "
! " $ ! "
"
( ( ! " ( ( ! "
$ $ ( ( ! "
) * + "" ,
- #./ $ "
- #./0 $ 1 $ " 2 ""
3 )
(
! " - #./0 13 ! "
#./ "
#./0 4 ( $ 1 $ """
3 )
(
5 #.670 #./ & "
# #./0 4 ( $ 1 1 $ """ "
3 )
(
! " # #./0 13 ! "
8 9 + :
9:- 0 #./0 4 ( & $ "4 " 2 1& "" 3 )
(
! " 9:- 0 #./0 13 ! "
' 9
9.; 0 #./0 $ $ " 2 4 1 2 ""
3 )
(
! " 9.; 0 #./0 13 ! "
) *) ,
) *) $,
- <
& % )>1)"1)>1
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B- #./!& "
! "
! "
& ! "
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B #./!& "
! "
! "
& ! "
5
& *& " & " & 5" & 8" ,
! "
! "
& ! "
8
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B9:- 0 #./!& "
& ! "
! " ! "
& 5! "
'
& *& " & " & 5" & 8" ,
* ! !?!@<A:, ) ! !B9.; 0 #./!& "
! "
! "
& ! "
- #./ $ "
- #./0 $ 1 $ " 2 ""
3 )
! "1- #./0 ! " (
! " - #./0 13 ! "
#./ "
#./0 4 ( $ 1 $ """
3 )
! "1 #./0 ! " (
! " #./0 13 ! "
5 #.670 #./ & "
# #./0 4 ( $ 1 1 $ """ "
3 )
! "1# #./0 ! " (
! " # #./0 13 ! "
8 9 + :
9:- 0 #./0 4 ( & $ "4 " 2 1& "" 3 )
! "19:- 0 #./0 ! " (
! " 9:- 0 #./0 13 ! "
9.; 0 #./0 $ $ " 2 4 1 2 "" 3 )
! "19.; 0 #./0 ! " (
! " 9.; 0 #./0 13 ! "
) *3 ,
*) ! " ! " ) !5" ,1&
* ,
% > >! > >"
)5 *) ,
)51&
C @<A:"" 9
4 8
0
0 0 ( ! "
0 ( ! "
! !> >" !> &>"
$ & >F >" & >A >"
>J 0 >"
>. >!> >! "
! 0 !> >" !> &>"
$ & >F >" & >A >"
>< 0 0 >"
$ %
&
'
(
"
"
"
#
$ $ ! "
% ! &
8 &
% "
! " $ ! "
( ( ! " ( ( ! "
$ $ ( ( ! "
) * + "" ,
- #./ $ "
- #./0 $ 1 $ " 2 ""
3 )
(
! " - #./0 13 ! "
#./ "
#./0 4 ( $ 1 $ """
3 )
(
! " #./0 13 ! "
5 #.670 #./ & "
# #./0 4 ( $ 1 1 $ """ "
3 )
(
8 9 + :
9:- 0 #./0 4 ( & $ "4 " 2 1& "" 3 )
(
! " 9:- 0 #./0 13 ! "
' 9
9.; 0 #./0 $ $ " 2 4 1 2 ""
3 )
(
! " 9.; 0 #./0 13 ! "
) *) ,
) *) $,
- < = %
& % )>1)"1)>1
0 ' 5
0 5 5
0& '
0 + + 0 ""
0 + + 0 ""
0 "
I 0 "
& *& " & " & 5" & 8" 0 " 0 I",
* ! !?!@<A:, ) ! !B- #./!& "
!I" )1 ! "">1 )1 ! ""
0 "
I 0 "
& *& " & " & 5" & 8" 0 " 0 I",
* ! !?!@<A:, ) ! !B #./!& "
!I" )1 ! "">1 )1 ! ""
5
I 0 "
& *& " & " & 5" & 8" 0 " 0 I",
* ! !?!@<A:, ) ! !B# #./!& "
!I" )1 ! "">1 )1 ! ""
8
0 "
I 0 "
0&"
& *& " & " & 5" & 8" 0 I" 0 " 0& ",
* ! !?!@<A:, ) ! !B9:- 0 #./!& "
!I! " )1 5! "">1 )1 5! ""
'
0 "
I 0 "
& *& " & " & 5" & 8" 0 I" 0 ",
* ! !?!@<A:, ) ! !B9.; 0 #./!& "
!I" )1 ! "">1 )1 ! ""
& ! "
! "
$ $ ! "
'
&
" "
( ( ! " ( ( ! "
$ $ ( ( ! "
) * + "" ,
- #./ $ "
- #./0 $ 1 $ " 2 ""
3 )
! "1- #./0 ! "
(
! " - #./0 13 ! "
#./ "
#./0 4 ( $ 1 $ """
3 )
! "1 #./0 ! "
(
! " #./0 13 ! "
5 #.670 #./ & "
# #./0 4 ( $ 1 1 $ """ "
3 )
! "1# #./0 ! "
(
! " # #./0 13 ! "
8 9 + :
9:- 0 #./0 4 ( & $ "4 " 2 1& "" 3 )
! "19:- 0 #./0 ! "
(
! " 9:- 0 #./0 13 ! "
' 9
9.; 0 #./0 $ $ " 2 4 1 2 ""
3 )
! "19.; 0 #./0 ! "
(
) *3 ,
! " *) ! " ! " ) !5" ,1&
! " ( ! "
