Evaluating the forecasting performance
of linear and nonlinear monetary policy
rules for South Africa
Kasai, Ndahiriwe and Naraidoo, Ruthira
Department of Economics, Kigali Independent University, Rwanda,
Department of Economics, University of Pretoria, South Africa
June 2011
Online at
https://mpra.ub.uni-muenchen.de/40699/
! "# ! ! "
#
$ % # &
# "# '
(
# # % # & # # # ) )*
# *++, +) *+)+ )*
-# . #
# ( #
# / 0 !
# # .
# .# &
# % # & . !
# .#
1 # #
#
. # . ! #
# # # 1 / # 9
1 :*+)+; 1 1 :*+)+ ; .
# < # #
# !
. .# #
.#
# # # # # #
# )
= #
# / . # . ! .
# 9 )>>+ = < # :)>>*;
# #
1 2 :*++7; #
# #
( # . 2 :*++7; #
#
& # ?
:*++8; # #
. ( # 1
$ :*+)+;
% # & @ !A #
)& % # & . ! :%& @; # $ :*+)+;
# . #. /
# -# # . # #
# ! # .
( / #
# # '
( .# # # ( #
# % . / ?
:*++8; 1 # 2 :*+)+; $ :*+)+; # . #
% # & # ( #
# . ! ! / .
/ *++, 1+) *+)+ 1)* (
# # .
# . 3 B# :1%$ ;
B# :1 %$ ; . 1
2 ! C
( ' / #
( # . # . % 5
# % 7
# .
&! ! :& 2; % 4
# .
. % , #
!
! # #
:*+)* *+)5; 9 #
. ! ! ! & # :*+));
# ! ( # . . . #
# # # # . # (
:*+)* *+)5; . ! !
"
(
)
{
(
)
1 1}
* 1 0
1 1
)
( − + − + − − + − + −
= i t i t y t f t
t L i y f
i ρ ρ ρ ρπ π π ρ ρ :);
t
i # 1
n 2
1 ...
)
( = i + i + + i n− i L ρ ρ L ρ L
ρ : #
) 1 (
i i ρ
ρ ≡ # ; πt−1
) 1
(t− yt−1 # # (t−1) ft−1 #
/* ( −1)
t # # # εt
$ ρj . j
B# :); ' . (
# . ! ! #
/ ( <# #
# . #
2 :*++8; = 9 :*++*; !
/ . B# . # #
:*+)+; - C :*++>; # #
#
# : / 1
1 *+)+.; & <# #
/ # /
. 5 . ! . . !
: / 1 1 *++>;
2
0 . #
# : *+)* # ; (
/ . . ! A
# # . .#
: @ *++*; ( .
# 3
" !
{
t}
ty t y t t y t y t i t i
t i y M y M
i =ρ −1+(1−ρ) ρ0 +θ ( −1;γ ;τ) 1 +(1−θ )( −1;γ ;τ) 2 +ε :*;
1 1
1 − −
− + +
= j t jy t jf t
jt y f
M ρ ππ ρ ρ <D) * # θ ( 1;γ ;τ)
y t y t y−
. :5; M1t ( # . #
! # .# M2t ( #
. # ! # .# / (
) ; ; ( 1 γ τ
θ y
t y
t y− # # : <!
*++*;
(
)
( ) ( ) 1 / 1 1 1 1; ; 1− − − − + − = − t y t y y e y t y
t y γ τ γ τ σ
θ :5;
:5; γyE +
C = ( F :)>>5; ( F :)>>7; ! γy
. . yt−1
. ( #
:*+)5; #
3
:*+)*; ( #
. %& @
# / % # & 0
B# # :*+)5;
# & # B# :*;
# # ( " #
(
)
(
)
(
)
t t f t y I t t OZD t I t t ZD t i t i t f y i i L i ε ρ ρ π π ρ θ π π ρ θ ρ ρ + + + − − + − + − + = − − − − − − ∗ − 1 1 1 1 1 1 1 1 ) 1 ( ) ( :7; I tπ
(
)
*1
1 π 1 µ π
µ
π = − + −
= t j
n j n I t
(
)
{
δ π π δ}
θ = − ≤ − − I− ≤
t t t
E
pr 1 1 . . '
(
)
{
}
2) 1 1 ( ) 1 1 ( ) 1 1 ( 1 1 1 1 1 I t t I t t I t t e I t t pr − − − − − − − + − − − − + − − − ≤ = − ≤ − = π π σ δ π π δ π π γ δ π π δ θ :4;
% = ( F :)>>5; ( F :)>>7; !
0
>
γ
. .(
I)
t t−1−π−1
π B# :4; # !
(
πt−1−πtI−1)
( # '
C # / # . .
# . .
9 ! # #
B# . # #
( # #
3 B# . # ( #
B# (
! # 4 .
# $
# #
# # % # &
*+++ 1+) *+)+ 1)* C #
/ 7 . *+++ +) *++4 )* #
. *++, +) *+)+ )*
( # : ; # #
# ' # / %& @A
# ( # # # .
# # 9 ! $ :)>>6; (
/ # : ; /
(
REERt)
. 9$/ : ; # /
(
RHt)
. 9$# / # . &@%& . !
. # / : ; !
(
RSt)
# .H .# % ! / & % / . # /
: ;
(
CSt)
. )+. & . 4 : ; # #
4
0 / # / # % !
C :*++4; I # # J
# 0 = C :*++,;
I # # #. # /
J
4 - @# :*++8; # . )+
. 5 # #
( )
Ft# B# #
( ! . # 9$ : 1 *++4 K.;
. / . C
# # *+++ C
# :& : ;; .
# & # #
. . . 9$
( # #
/ : 7 ; ( .
t : ; . (
s # # # # &
#
. #
.
( # / / '
# *+++ #
3 # ) )
& <# ( /
/ . @ ! A - % .
:@ ! *++6;
% &
( . ! !
. # / ( . ) *
( # 1 ) * 5
# *+++ 1) *++4 1)*
- 1 ) 9 A :*++);
. = 'L ' :*++5; #
M 1# * .# , (
# . !
. 1 * 5 & < 1 5 ( . * / .
& 2 # ' # #
/ .#
' ( # #
'C # # = # . 9 A .
GG . # #GN< G O ( #
( & " !
) " * + " ! *, +
0
ρ + 88*PPP
:+ +);
, 86,PPP :+ )>;
1
ρ + 768PPP
:+ +7;
+ 84>PPP :+ +);
π
ρ ) +66PPP
:+ +8;
y
ρ + +*5PP
:+ +);
f
ρ + 88*PPP
:+ +);
1π
ρ + ,>6PPP
:+ +5;
1y
ρ + *8,
:+ *5;
1f
ρ + +4>PPP
:+ +);
2π
ρ + +,*
:+ +>;
2y
ρ + )),
:+ +8;
2f
ρ + +*7PP
:+ +);
τ + ++
:+ ++;
y
γ 4 ++
AIC ) )65 ) *+4
S.E + 7)8 + 7),
2
R + >,> + >,>
J-stat + *78 + *5+
-test + ++)
A-test + +++
g-test + ++)
,
: ; C 1 ) it =ρiit−1+
(
1−ρi)
{
ρ0 +ρππt−1+ρyyt−1+ρf ft−1}
+εt 1 *(
)
{
(
)
(
)(
)
t}
ty t y t t y t y t i t i
t i y M y M
i =ρ −1+ 1−ρ ρ0 +θ −1;γ ;τ 1 + 1−θ −1;γ ;τ 2 +ε
1 1
1 − −
− + +
= j t jy t jf t
jt y f
M ρππ ρ ρ D) * yt .
: ; # . P:PP;QPPPR )+:4;Q)R S
& 2 &! ! 2 H # B# A
:9 )>8*; ( # # ) , > )* #
( ! ( - *" #+
) " #
i
ρ + 85*PPP
:+ +);
π
ρ
ZD
ρ + 5>,
:+ 5+;
OZD
ρ ) )76PPP
:+ +7;
y
ρ + 4*5PPP
:+ +5;
f
ρ + ++8PPP
:+ ++;
µ
+ 45+PPP:+ +5;
δ * +4
% + 5>7
& 2 ./!
2
R + >6*
9+ ρZD =ρOZD: # ; + +++
H : # ; + *7>
, # . P:PP;QPPPR
)+:4;Q)R S % & 2 &! !
H # B# A
:9 )>8*; ( # # ) , > )* #
# # )+ . :15;
/
/ - & &
/ 0
. *+++ +) *++4 )* #
# . . R P
# . . T =R+P & # #
. R T −h #
/ . #
1
−
t h ' ( # .
' h B# P−h+1 (
# . . .
t t
t F X
i = ( −1;θ)+ε :,;
C εt ~iid(0,σ2) Xt (k x 1) / # .
% * ( B#
[
/]
(
;θ)
ˆ
1 /
1 t t t t
t Ei X F X
i+ = + = :6;
B# &
. * )
(
;θ)
ˆ
1 )
( /
2 +
+ = t
n t t F X
i :8;
9 . #.< / .
T A . @ 1 :)>8>; T ! A . ( :)>>+;
( . B# :,; E
(
εt+1 Xt)
=0# . # . M = :)>>7; #.
& T A T ! A # B#
B# ! * #
B# . = ( F :)>>5; - <!
. # . C εˆt
N
{
( )}
1 ) 1 (
1,...,
N t t+ ε+
ε . / *
(
)
= + + + = + N i i t t t B tt F X
N i 1 1 1 2 ; 1 θ ε :>; ( . () ()
1,...,
i h t i t+ ε+
ε i=1,...,N B# #
N it+1t,...,it+ht h≥2 #
. . N :
( F *++,;
- # # 1 %B#
$ :1%$ ; 1 %B# $ :1 %$ ;
-.# # # B# # #
. 1 :DM−t 9 )>>6; ( DM −t
. ) * #
(
)
, ˆ 1 2 1 2 1 dd S d h P tDM− = − +
2 , 2 2 , 1 ˆ ˆ ˆ h t h t h
t e e
d+ = + − + 3 eˆi,t+h. i3
(
)
− = + − − = + −= T h
R
t dt h MSPE MSPE
h P
d 1 2
1 ˆ 1 3
(
)
− = + + − − + − =Γ T h
R
t t h t h j dd(j) P h 1 dˆ dˆ
ˆ 1
0 ≥
j Γˆdd(j)=Γˆdd(−j) 3
(
)
ˆ ( )ˆ K j M j
S dd
j j j
dd = Γ
−
= # dt+h #
! . # K(.) . M / #
# . j C 9 :)>>6; .
) 1 ( ) 1 ( ) 1 ( 2 + − + − − + − = h P h P h h h P ζ ( . 0 ˆ ˆ
: 1, 2,
0 e t+h−e t+h=
H
0 ˆ ˆ
: 1, 2,
1 e t+h−e t+h≠
H
( < # . % # A t .# : V );
.# : 9 )>>6;
( # B# B#
# # # !
h t
d+ B# ' : 1 2 ! *++6; . #
.# . 1 :)>>4;
: 1 2 ! *+++3 2 ! 1 2 ! *++)
( F *++4; 9 = C :*++,;
' .# . 1
: 1 ;
@ < % :*++,; # DM −t . #
# .#
. / . /
& # 2 ! 1 2 !
:*++); :ENC−t; 2 ! C :*++6; @
# B# # ( ENC−t
. 2 1 1 1 2 1 ) ( ) 1 ( − − = − − = + − T R
t ct h c
P
c P
h t h t h t h t h t h t h
t e e e e e e
c+ = + + − + = + − 1,+ 2,+ 2 , 1 , 2 , 1 ,
1 (ˆ ˆ ) ˆ ˆ ˆ
ˆ −
= + −
= T h
R t ct h
P
c 1 ( ENC−t
# DM−t .# H1:eˆ1,t+h−eˆ2,t+h >0
t
DM − H1=eˆ1,t+h−eˆ2,t+h ≠0 - h=1
.# N(0,1) @ 2 ! 1 2 ! :*++);
# :h>1; .# 9
. @ < % :*++,; .# # B#
) 1 , 0 (
N # C :)>86; # h>1 & #
.# . # # # #
: 2 ! 1 2 ! *++) # ;
& B# 2 ! C :2C ;
. ]. ) ˆ ˆ ( ) ˆ [( ) ˆ ( ˆ 2 , 2 , 1 2 , 2 2 ,
1tt h t h tt h tt h tt h h
t h
t i i i i i i
f+ = + − + − + − + − + − +
C t it+h iˆ1t,t+h
h t t
iˆ2,+ t+h it+h−iˆ1t,t+h it+h−iˆ2t,t+h
( B# 1%$ . fˆt+h #
# t ' : 2 ! C *++6; & . #
B# 1%$ * 1%$
) 2 ! C :*++6; # < t
W) *8* : + )+ ; W) ,74 : + +4 ;
/ ! (
0 # .# # #
( . <#
& # #
. ( # . B#
:1%$ ; B# :1 %$ ; .
( &
. . '
. . ( ! 5 A
( . 5 7 (
' & ( . 5 7
1 * 1%$ 1 %$
' ! 2
. 1 ) ! .
' 9 # :h>3; #
# 5 1%$ !
. # #
. ( . # # . #
: # *++7; 1 %$ 1 ) !
& ! . 1%$ 1 %$
( # " 1 2 * +
" 3 3! 3# 3% 3/ 3' 34 35 36 3 . 3 3 !
2
* + * * * 5 5 5 5 5 5 5 5 5 * 64
! * + ) ) ) ) ) ) ) ) ) ) ) ) )
# * + 5 5 5 * * * * * * * * * * *4
, ( ( . # ! 1 ) * 5 #
' D) X )* # 1 %B# $ :1%$ ; ( #
! 1 ) 1 * # #
. 1 5 #
( % " 1 2 * +
" 3 3! 3# 3% 3/ 3' 34 35 36 3 . 3 3 !
2
* + * * * * * 5 5 * * * * * * ),
! * + ) ) ) ) ) ) ) ) ) ) ) ) )
# * + 5 5 5 5 5 * * 5 5 5 5 5 * 85
, ( ( . # ! 1 ) * 5 #
' D) X )* # 1 %B# $ :1 %$ ; ( #
( . 1 : ; # ( . 4 (
/ 1%$ # #
( ( . # .
# ( . 4 . 1 :)>>4;
# 1 * #
' :2≤h≤8; .# # '
:h≥9; 0 1 5 .
(# # B# #
<# . ENC_t CW _t
( . , 6 # . 1%$ .
# ( . , 6 #
1 ) 1 * .
# ' . : ENC_t;
2 ! C :CW _t; 2 # )
1 5 # :h>3; 1
5 . <# . ! ' 9 1 )
# # :h≤3; . 1 5
& 1 * . '
0 ! 1 5 . #
' & # # . 2
:*++7; ! .
( / 0
(
DM−t)
! # % / ' 4 5 6 . !
"
" ! + +* ) 7,P ) >,PP * +,PP * +6PP ) 68PP ) 4+P ) 5)P ) *+ ) )7 ) +> ) )5
" # ) *8 ) +8 + 6) + 57 + *5 + )8 + )) + +8 + +* + +) + +5 + +,
" !
" # + 64 * +8 * )> ) 8* ) 64 ) ,+ ) 7* ) *, ) )5 ) +6 ) +) ) +4
, ( . )* . . 1
(
DM−t)
' / ) )* (. # 1 A # . 1%$ # 1 <
4S )+S . ! : < # ;
( ' 0
(
ENC−t)
! # % / ' 4 5 6 . !
"
" ! ) 58P * +6PP * 75PP * ,)PP * >)PP 5 **PP 5 47PP 5 87PP 7 *+PP 7 66PP 4 )7PP 4 6+PP
" # + 65 + 76 + ,, ) 5)P ) ,6PP ) >*PP * *4PP * 7*PP * 46PP * 88PP 5 +8PP 5 74PP
" !
" # ) 68PP ) *5 + >8 ) *+ ) ), ) +> ) +* + >5 + 66 + 6+ + ,4 + 44
, ( . )5 . 2 ! 1 2 ! :*++);
(
ENC−t)
' / ) )* (. # 1 A # . 1%$ # 1 <
4S )+S . ! : < # ; - 1
( 4 0
(
CW−t)
! # % / ' 4 5 6 . !
"
" ! ) 65*P 5 >**PP 7 ,>5PP 7 4,8PP 7 6>,PP 7 >>>PP 4 +)6PP 7 >),PP 7 6+4PP 7 45,PP 7 5+>PP 7 *+7PP
" # + 6*) + 76+ + ,46 ) 5+5P ) ,54P ) 8,8PP * )65PP * 5)*PP * 75*P * ,>6PP * 844PP 5 ),5PP
" !
" # ) 6,)PP ) **) + >,> ) )>* ) )7+ ) +,7 + >8+ + 88> + 65) + ,4* + 4>8 + 4+5
, ( . )7 . 2 ! C
(
CW−t)
' / ) )* ( .# 1 A # . 1%$ # 1 < 4S )+S
. ! : < # ; - 1
' )
# ( # . #
/ . . . 0#
# . . .
# ( # . # 3
B# :1%$ ; B# :1 %$ ;
. 1 2 ! C
2 # . !
#
. # ! #
# . #
.# & #
7
) @ ! :*++6; - % . *) :& ; ) 68
* @ - 1 @ % - 2 :*++*; I&
# % - = . ! J
Y , * & 5
5 @ < = % :*++,; I& # #
& - 1& J
Y )5) ) * 45> 468
7 @ @ U % 1 :)>8>; I$
. J Y 4 5 75+ 74*
4 @# $ :*++8; I1 % # & J
! I( # # M
= % # & J
2 (
, 2 ! ( 1 C 1 2 ! :*++); I( B# #
J Y )+4 ) 84 ))+
6 2 ! ( K C :*++6; & / B# #
Y )58 : ; *>) 5))
8 2 Y :*++8; I& 2 @ ! : # ; ( # ZJ
C ! $ % 8,+
> 2 1 $ 9 $ - % :*++7;
I-J " # Y *+ * ),>V)85
)+ 2 1 = = 'L ' :*++5; I(
. J Y ))7 ) )7)
),7
)) = < H = # :)>>*; I%
J " # Y 8 )54V )4,
)* . - [ % 1 :)>>4; I2 # J $
Y )5 5 *45 *,5
)5 - H U ? :*++5; % %
)7 - $ 9 <! :*+++; 2 .
$ 2 .
)4 = 9 C :*++,; I( . J
Y 67 , )474\)468
), = 2 @ 9 :*++*; & #
M %
)6 = 2 C H ( ( F :)>>5; 0/
$ 0/
)8 9 H :*++); I& / . J
Y ,> 5 456 465
)> 9 M :)>8*; IM = ' 1 1 J
Y 4+ : ; )+*> )+47
*+ 9 % M . # $ . :)>>6; I( B#
B# J " # Y )5 * *8) *>)
*) 9 & % 1 :*+)+;
I-( # J Y *6 4 )*>* )5+)
** 9 ! H 2 $ :)>>6; I$ % .# &
J & ! & $ ' Y *> ) )V),
*5 # & M :*++7; I # . C
# # ZJ ( ) * Y *5 : ; 56) 7+*
*7 :*+)5; I( 0 #
! J Y 74 )8 *456 *474 !
GG / G)+ )+8+G+++5,87, *+)* ,,>7,7
*4 :*+)*; I- # &
# % # & @ !J
&Y 5> * ),) ))6
*, :*+)); I& # % # & J %
$
*6 M H M = 2 C H :)>>7; I- J
# &Y )5 ) )V>
*8 1 2 2 1 :*+)+ ; I( # J
*> 1 2 2 1 :*+)+.; I- . J 1 & *+)+
5+ 1 2 2 1 :*++>; I( %#. $ 2 1 $ J "
! $ ' : ;
5) 1 2 ! 1 C :*++6; I& # = # J
Y )7+ * 6)> 64*
5* 1 2 ! 1 C :*+++; I .# # J Y >>
* )>4 **5
55 1 2 ! 1 C :)>>>; I& # # J + ' %
M # %
57 1 ( 9 $ . :*++4 ; I0 %#. 0 9 ! $
- J Y *6 ) 45 ,6
54 1 ( 9 $ . :*++4.; I2
9 ! $ - & J 1 & .
GG #!G G G >*
5, 1 # 1 M 2 & :*+)+; IC ( # . # M
& ZJ Y 5* ) 5>* 7+7
56 $ :*+)+; I- 1 $ # % # & J + '
,-.- -/ $
58 M # :*+)+; I]
! / . # %& @
J + ' ,-.--0 $
5> C C :)>86; ^9 ( 1 1
^" ( ) * *8 5 666 686
7+ ? ( C :*++8; I #
( # J &Y 5+ ) 7*8V775
7) 1 % # $ :*+)+; IM # Z 2 %& @ ! ZJ
Y 68 5 *,> *8,
7* % ! H 9 1 C C :*++4; I&
# J + ' 9
75 ( F ( :*++,; I- . J =
= 2 C H ( & : ; 1 2 ' # &
77 ( F ( :*++4; I- . J %% G
-+ ' % - 4>8 %
% ! % %
74 ( F ( :)>>7; I% # % (
&# 1 J Y 8> 7*4 *+8
*)8
7, ( 9 :)>>+; 3 0/ $
0/
76 <! ( ( F $ 9 - :*++*; I % #
V # J ( ) * Y *) ) ) 76
78 C 2 :*++>; I 1 $ % . ' & - % .