Patrick Marsh

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
Discussion Papers in Economics 
 
 
 
 
 
 
 
 
 
 
No. 2000/62 
 
Dynamics of Output Growth, Consumption and Physical Capital 
in Two-Sector Models of Endogenous Growth 
 
by 
 
Farhad Nili 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Department of Economics and Related Studies 
University of York 
Heslington 
York, YO10 5DD 
 
No. 2000/58 
 
Edgeworth Expansions in Gaussian Autoregression 
 
by 
 
Patrick Marsh 
 Hgjhzruwk H{sdqvlrqv lq Jdxvvldq Dxwruhjuhvvlrq
Sdwulfn Pduvk
Ghsduwphqw ri Hfrqrplfv dqg
Ghsduwphqw ri Pdwkhpdwlfv
Xqlyhuvlw| ri \run
X1N1
Qryhpehu 5:/ 5333
Devwudfw
Zh frqvlghu wkh frqvwuxfwlrq ri ydolg Hgjhzruwk h{sdqvlrqv iru vwdwlvwlfv
dulvlqj lq wkh frqwh{w ri Jdxvvldq dxwruhjuhvvlrq1 E| h{sorlwlqj wkh surshuwlhv
ri h{srqhqwldo idplolhv +wr zklfk wkhvh prghov ehorqj,/ ydolglw|/ ri dq| rughu/
lv urxwlqho| hvwdeolvkhg iru d zlgh fodvv ri vwdwlvwlfv1
￿Dgguhvv iru fruuhvsrqghqfh= Ghsduwphqw ri Hfrqrplfv/ Xqlyhuvlw| ri \run/ Khvolqjwrq/ \run/
\R43 8GG/ X1N1 h0pdlo= szqp4C|run1df1xn1
44L q w u r g x f w l r q
Wkh dssolfdwlrq ri kljkhu0rughu dv|pswrwlf whfkqltxhv iru dxwrfruuhodwlrq hvwlpd0
wruv lq Jdxvvldq dxwruhjuhvvlrq kdv uhfhlyhg frqvlghudeoh dwwhqwlrq1 H{dpsohv lq0
foxgh ghulydwlrq ri JE?3￿ Hgjhzruwk vhulhv frqwdlqhg lq Skloolsv +4<::,/ Rfkl +4<;6,/
Vdwfkho +4<;7,/ Ervh +4<;;,/ Wdqljxfkl +4<<4, dqg pruh uhfhqwo| Ndnl}dzd +4<<<,/
zkloh Skloolsv +4<:;, dqg Gxuelq +4<;3, +iru wkh flufxodu fdvh, ghulyh yduldqwv ri wkh
vdggohsrlqw dssur{lpdwlrq1 Iru wkh sxusrvhv ri wklv sdshu/ zh vxssrvh wkdw d vdpsoh
+ 'E +￿cc+￿￿ zdv jhqhudwhg e| wkh iroorzlqj Jdxvvldq -ER surfhvv/
+￿ ' k￿+￿3￿ n k2+￿32 n  n kR+￿3R n 0￿ +4,
0￿  UEfcj
2(+3Rn￿ ' +3Rn2 '  ' +f 'f c
dqg zh dvvxph wkdw wkh urrwv ri wkh sro|qrpldo E5R nk￿5R3￿ nnkR o l hl q v l g hw k h
xqlw flufoh/ vr wkdw +4, lv vwdwlrqdu|1
Wkh pdmru vwxpeolqj eorfn iru dssolfdwlrqv ri kljkhu0rughu dv|pswrwlf wrrov lq
prghov vxfk dv +4, lv/ dv rqh pljkw h{shfw/ ghprqvwudwlqj ydolglw|1 Lq jhqhudo/ ohw
f￿ eh d &   udqgrp yhfwru/ kdylqj fxpxodqwv VU￿c iru  ' c2cc +iru ghwdlov ri
wkh clqgh{ qrwdwlrq* vhh PfFxoodjk +4<;:,/ Fkdswhu 5, dqg vxssrvh wkdw dv  $4 c
f￿ $_ &EfcV U2 H{sdqvlrq dqg whup0e|0whup lqyhuvlrq ri wkh fxpxodqw jhqhudwlqj
ixqfwlrq ri f￿ |lhogv dq Hgjhzruwk vhulhv iru wkh glvwulexwlrq/ 8￿E% ' hEf￿ 	% c
dv
	 8￿E%'x E %(V
U2  E%(V
U2
+
K [
￿’￿
S￿c￿EV^￿E%
,
c +5,
5zkhuh wkh S￿c￿EV duh frh!flhqwv lqyroylqj wkh vdpsoh vl}h dqg wkh fxpxodqwv ri f￿/
wkh ^￿E% duh wkh whqvrudo Khuplwh sro|qrpldov dqg x dqg  uhsuhvhqw wkh &0glphqvlrq
qrupdo FGI dqg SGI uhvshfwlyho|1 Wkh vhulhv lq +5, lv ydolg li
tT
%

 8￿E%  	 8￿E%

  ' JE
3EK32￿*2
Dowkrxjk qhfhvvdu| frqglwlrqv iru ydolglw| duh zhoo hvwdeolvkhg/ +vhh Ekdwwdfkdu|d dqg
Jkrvk +4<:;, ru Gxuelq +4<;3,,/ wr ghprqvwudwh ydolglw| lq qrq0l1l1g1 gdwd vhwwlqjv
wkh iroorzlqj lv vx!flhqw +iru h{dpsoh/ vhh Kdoo +4<<5,/ Fkdswhu 5 dqg Wdqljxfkl
+4<<4,/ s147,1
Dvvxpswlrq 4 Ohw V
U￿
f eh wkh |￿ fxpxodqw ri f￿c wkhq
V
U￿
f ' 
3E￿32￿*2
" [
,’￿
V
U￿
fc,
3E,3￿￿*2c +6,
zkhuh wkh fxpxodqw frh!flhqwv V
U￿
fc, d u hi u h hr i dqg zh dovr dvvxph V
U￿
fc￿ 'f 
Iru wkh vshfldo fdvh ri wkh dxwruhjuhvvlyh prgho +4,/ zlwk R ' c ydolglw| xs wr rughu
JE3￿ +dv/ iru h{dpsoh/ lq Skloolsv +4<::, dqg Ndnl}dzd +4<<<,, lv ghprqvwudwhg e|
qglqj orz rughu fxpxodqw frh!flhqwv/ dqg vkrzlqj wkdw/ iru  ' ccec


V
U￿
f  
3E￿32￿*2V
U￿
fc￿  
3￿*2V
U￿
fc2


 ' JE
3￿*2
Lq wklv sdshu zh ghprqvwudwh wkdw lq rughu wr suryh ydolglw|/ vxfk fdofxodwlrqv
duh lq idfw qrw qhfhvvdu|1 Wkdw lv e| ixoo| h{sorlwlqj wkh surshuwlhv ri wkh prgho/
ydolglw|/ xs wr rughu JE3EK32￿*2c qrw mxvw JE3￿ lv dvvxuhg/ iru d zlgh fodvv ri
vwdwlvwlfv dulvlqj iurp wklv prgho1 Pruhryhu wkh dssurdfk ri wklv sdshu vhhpv pruh
6vwudljkwiruzdug dqg jhqhudo wkdq wkdw ri Wdqljxfkl dqg Zdwdqdeh +4<<7,/ zkr ghulyh
JE3￿ dssur{lpdwlrqv iru wkh POH lq fxuyhg h{srqhqwldo prghov1 Wkh nh| lv wkh
idfw wkdw mrlqw glvwulexwlrqv ri vdpsohv jhqhudwhg e| Jdxvvldq dxwruhjuhvvlrq duh
vlpso| phpehuv ri wkh h{srqhqwldo idplo| dqg frqvhtxhqwo| wkh pdmrulw| ri lqihuhqfh
zloo eh frqgxfwhg wkurxjk vlpsoh ixqfwlrqv ri wkh vx!flhqw vwdwlvwlf1 Wkhuhiruh/ lq
rughu wr hvwdeolvk ydolglw| zh qhhg rqo| suryh ydolglw| iru +d, wkh vx!flhqw vwdwlvwlf
lwvhoi dqg +e, vlpsoh ixqfwlrqv ri lw1 Rqfh ydolglw| kdv ehhq hvwdeolvkhg iru Hgjhzruwk
vhulhv d ixoohu udqjh ri kljkhu rughu whfkqltxhv vxfk dv Errwvwuds +Kdoo +4<<5,, dqg
wudqvirupdwlrq phwkrgv +Qlnl dqg Nrqlvkl +4<;9,, pd| eh xvhg1
5P d l q U h v x o w v
Ohw + 'E +￿cc+￿ kdyh wkh xvxdo fxuyhg h{srqhqwldo ghqvlw| +vhh iru h{dpsoh/
Eduqgru0Qlhovhq dqg Fr{ +4<;<,,/ yl}1
sE+(w ' i Ti|
￿
￿#  g￿E#nE|￿jc +7,
zlwk &0glphqvlrq vx!flhqw vwdwlvwlf |￿ ' |E+ dqg fdqrqlfdo sdudphwhu # ' #Ewc
d vprrwk ixqfwlrq ri wkh _0glphqvlrq sdudphwhu w dqg fxpxodqw ixqfwlrq g￿E#'
g￿E#Ew Ehiruh vshfldolvlqj wr dxwruhjuhvvlrq zh h{dplqh wkh ydolglw| ri dv|ps0
wrwlf h{sdqvlrqv ri wkh w|sh +5,/ iru wkh glvwulexwlrq ri |￿ Ohw r￿ ' ￿*2|￿c dqg
ghqrwh wkh fxpxodqwv ri r￿ e| V
U￿
r /w k h qr￿ lv dovr plqlpdo vx!flhqw dqg +7, pd| eh
7uhsdudphwhulvhg/ zlwk  ' Ewc dv
sE+(w ' i Tir
￿
￿  g￿En
WEr￿j
Pruhryhu/ wkh fxpxodqw jhqhudwlqj ixqfwlrq ri |￿ lv g|Eb'g￿E# n b  g￿E#c
dqg vr |￿ kdv fxpxodqwv
V
U￿
| 'E 
3￿ Y￿g￿E# n b
Yb
￿￿
￿ Yb
￿&
&



 
b’f
'E 
3￿
3￿*2 Y￿g￿E n b
Yb
￿￿
￿ Yb
￿&
&



 
b’f
' 
3￿*2V
U￿
r c
zlwk
S&
,’￿ , ' c vr wkdw li V
U￿
r ' E iru doo c wkhq V
U￿
| ' E3E￿32￿*2c zklfk
vdwlvhv Dvvxpswlrq 4/ zlwk V
U￿
|c, 'firu ,  2 Wkdw lv ydolglw| iru wkh glvwulexwlrq ri
|￿ iroorzv li wkh fxpxodqwv ri ￿*2|￿ duh doo E
Qrz/ ohw }￿ ' }E%￿ eh d 6 ixqfwlrq ri wkh vx!flhqw vwdwlvwlfv %￿ ' r￿* '
E%￿cc%&c vdwlvi|lqj wkh iroorzlqj dvvxpswlrq1
Dvvxpswlrq 5 }E%￿ lv  wlphv glhuhqwldeoh/ zkhuh :K / zlwk ghulydwlyhv
}U￿ '
Y￿}E%￿
Y%
￿￿
￿ Y%
￿&
&
(
[&
,’￿ , ' 
zlwk wkh }U￿ frqwlqxrxv lq d qhljkerxukrrg ri  ' VU￿
r ' .d%￿o dqg doo plqruv
ri }U￿ erxqghg dzd| iurp }hur1
Xqghu Dvvxpswlrq 5/ }￿ shuplwv wkh iroorzlqj vwrfkdvwlf h{sdqvlrq
}
o '7 }
o
Uf n
[￿
￿’￿ 7 }
o
U￿~
U￿ n RE
3E￿n￿￿*2(o ' cc6c +8,
zkhuh 7 }o
U￿ '
Y￿}E%￿￿
Y%
￿￿
￿ ￿￿Y%
￿&
&




%￿’￿
dqg ~ 'E %￿  'RE3￿*2 Zh frqvlghu dssur{lpdw0
lqj wkh glvwulexwlrq ri wkh vwdqgduglvhg vwdwlvwlf
￿ ' T
3￿*2
￿ E}E%￿  }Ec
8zkhuh T￿ ' @od}E%￿o1
Wkhruhp 4 Dvvxph wkdw wkh fxpxodqwv ri r￿ ' ￿*2|￿ duh E dqg wkdw Dvvxps0
wlrq 5 krogv iru }￿ ' }E%￿ dqg ohw wkh glvwulexwlrq ri ￿ eh 8￿Ec zlwk Hgjhzruwk
vhulhv dssur{lpdwlrq/ dqdorjrxv wr +5,/ 	 8￿Ec wkhq
tT
￿


8￿E  	 8￿E


 ' JE
3EK32￿*2
Surri1 Wkh surri ri Wkhruhp 4 lv jlyhq lq wkh dsshqgl{1
D frqvhtxhqfh ri Wkhruhp 4 lv wkdw wkh rqo| frqglwlrq uhtxluhg iru ydolglw| lv wkh
h{lvwhqfh ri d vx!flhqw vwdwlvwlf zlwk fxpxodqwv ri rughu E Khqfh/ lq Jdxvvldq
dxwruhjuhvvlrqv/ wklv lv wkh rqo| frqglwlrq zklfk qhhgv wr eh fkhfnhg1 Dv d frp0
sdulvrq/ li wkhuh h{lvwv vxfk d vx!flhqw vwdwlvwlf/ wkhq Dvvxpswlrqv 4 dqg 6 lq wkh
dssurdfk ri Wdqljxfkl dqg Zdwdqdeh +4<<7, duh lpphgldwho| vdwlvhg1
Vshflfdoo|/ wkh vhw ri & 'E R n ER n2  *2 vwdwlvwlfv/ r￿ 'E r￿ccr &￿c lv wkh
plqlpdo vx!flhqw vwdwlvwlf/ vhh Dqghuvrq +4<<7/ s168;,/ zkhuh
r￿ '
qSA3R
￿’Rn￿ +2
￿c+ 2
￿ n +2
￿c c +2
R n +2
￿3Rn￿c
SA3Rn￿
￿’Rn￿ +￿+￿3￿c+ ￿+2 n +￿3￿+￿c c +R3￿+R n +￿3Rn￿+￿3Rn2c
G
SA
￿’Rn￿ +￿+￿3R
r
+9,
Vlqfh xqghu qrupdolw| +  EfcPEwc zkhuh w 'E k￿cckRcj2 dqg
PEw
3￿ '
[&
￿’￿ #￿Ew￿c
9iru d vhtxhqfh ri frqvwdqw pdwulfhv ￿ dqg vprrwk ixqfwlrqv ri wc #￿Ew +vhh ydq
Jdughuhq +4<<:, dqg Dqghuvrq +4<<7,/ Vhfwlrq 9,/ wkhq wkh mrlqw ghqvlw| ri + pd| eh
zulwwhq
sE+(w'i  T



2

￿*2|
￿
￿#Ewn

2
*?
 

[&
￿’￿ #￿Ew￿
 
 

2
*?2Z

c +:,
zkhuh #Ew'E #￿Ewcc#&Ew￿ Frqvhtxhqwo|/ iru wkh Jdxvvldq dxwruhjuhvvlrq prgho
ghqhg e| +4,/ zh kdyh wkh iroorzlqj wkhruhp1
Wkhruhp 5 Ohw wkh plqlpdo vx!flhqw vwdwlvwlf r￿ iru prgho +4,/ eh ghqhg dv lq
+9,/ wkhq wkh fxpxodqwv ri r￿ vdwlvi| V
U￿
r ' E iru doo 
Surri1 Wkhruhp 5 lv suryhg lq wkh dsshqgl{1
Dsso|lqj/ uvw Wkhruhp 5/ wkhq Wkhruhp 4/ |lhogv wkh iroorzlqj uhvxowv1 Iluvw/ zh
pd| dssur{lpdwh/ wr dq| rughu/ wkh ghqvlw| ri wkh vx!flhqw vwdwlvwlfv yld dq dssur0
suldwh Hgjhzruwk vhulhv1 Vhfrqg/ zh pd| wkhq wudqvirup wr dq| ixqfwlrq/ vd| }￿c ri
wkrvh vx!flhqw vwdwlvwlfv/ surylghg rqo| wkdw Dvvxpswlrq 5 krogv/ dqg dssur{lpdwh
wr dq| rughu/ wkh ghqvlw| ri }￿ Dv d frqvhtxhqfh/ wkh uhvxowv ri wkh sdshuv phqwlrqhg
lq wkh uvw sdudjudsk ri wkh lqwurgxfwlrq pd| eh rewdlqhg/ lq sulqflsoh/ dv vshfldo
fdvhv ri wkh uhvxowv khuh1
:6 Dssolfdwlrq
Dowkrxjk zh kdyh suryhg ydolglw| ri kljkhu0rughu Hgjhzruwk dssur{lpdwlrqv iru
Jdxvvldq dxwruhjuhvvlrq/ ydolglw| lq lwvhoi/ lv e| qr phdqv d jxdudqwhh ri uhdvrqdeoh
dffxudf|1 Iru frpsdulvrq zlwk suhylrxv vwxglhv/ wdnh wkh vlpsohvw DU+4, surfhvv>
+￿ ' k+￿3￿ n 0￿c0 ￿  Efc dqg  ' ccc zlwk vx!flhqw vwdwlvwlfv
r '
3
E
E
C
r￿
r2
4
F
F
D '
3
E
E
C
S
￿ +2
￿3￿
S
￿ +￿+￿3￿
4
F
F
Dc
zlwk phdq yhfwru .Er' 'E ￿ck ￿c zkhuh ￿ 'E  Ek2￿*Ek2*Ek2
Zh zloo frqvlghu wkh glvwulexwlrq ri wkh 50wudqvirupdwlrq/ vhh Wdqljxfkl +4<<4,/ ri
wkh +eldv fruuhfwhg, POH iru kc  k ' ￿n￿
￿
r2
r￿
￿E k'
s


*L}

n k
   k

 *L}

nk
  k


Qrwlfh wkdw dowkrxjk *L}

￿n￿ k
￿3￿ k

lv ghqhg rqo| iru  k 5 Ecc wkh suredelolw|  
hd k 5 Eco lv ri h{srqhqwldoo| vpdoo rughu lq c dqg khqfh grhv qrw dhfw rxu
fdofxodwlrqv khuh1 Ghqrwh wkh fxpxodqwv ri ￿ e| V
￿
￿c' c2c/ wkhq diwhu vrph
dojheud/ wkh fxpxodqw frh!flhqwv lq +6, duh
V
￿
￿c2 '
k
s
E  k2
( V
2
￿c￿ ' ( V
￿
￿c￿ 'f
V
2
￿c2 '
k2
E  k2
( V
e
￿c￿ '
E2  Sk2
E  k2
Wkhq/ dsso|lqj Wkhruhpv 4 dqg 5 dq Hgjhzruwk dssur{lpdwlrq wr hd￿  o'8E
lv
	 8E'x E   E
[K
￿’￿ 
3E￿32￿*2@￿E

n JE
3EK32￿c +;,
;zlwk x dqg  wkh vwdqgdug qrupdo FGI dqg SGI/ uhvshfwlyho| dqg/ iru h{dpsoh/
@￿E'V
￿
￿c2
@eE'

2
EEV
￿
￿c2
2 n V
2
￿c2M￿En

2e
V
e
￿c￿M￿E +<,
zkhuh M￿E lv wkh |￿ Khuplwh sro|qrpldo1 Vlqfh kljkhu0rughu Hgjhzruwk dssur{l0
pdwlrqv duh surqh wr qrq0prqrwrqh ehkdylrxu/ fdxvhg e| wkh kljkhvw0rughu Khuplwh
sro|qrpldo lq wkh h{sdqvlrq/ vhh Qlnl dqg Nrqlvkl +4<;9,/ wkhq uhprylqj wkh whup
lqyroylqj V￿
￿c￿c plqlplvhv wkh ulvn ri qrq0prqrwrqlflw|1
Wr looxvwudwh wkh sureohp ri qrq0prqrwrqlflw| dqg wr kljkoljkw wkh fduh qhhghg
zkhq frqvwuxfwlqj kljkhu0rughu dv|pswrwlf dssur{lpdwlrqv zh zloo h{dplqh wkh glv0
wulexwlrq ri erwk  k dqg ￿E k Lq sduwlfxodu/ wkh hpslulfdo glvwulexwlrqv ri  k dqg
￿E k zhuh vlpxodwhg iru  'D fdqg k 'f b zlwk ffcfff uhsolfdwlrqv1 Wkhq
JE3￿ dssur{lpdwlrqv zhuh frqvwuxfwhg iru  k +xvlqj wkh h{sdqvlrq frqwdlqhg lq
Rfkl +4<;6, dqg dovr Ndnl}dzd +4<<<,/ s1679, dqg iru ￿E k xsrq vxevwlwxwlrq ri
+<, lqwr +;,1 Frpsdulvrqv ehwzhhq wkh vlpxodwhg dqg dssur{lpdwh glvwulexwlrqv duh
frqwdlqhg lq Iljxuhv 4 dqg 5 lq wkh dsshqgl{1
Lpsruwdqwo|/ iurp Wkhruhpv 4 dqg 5/ erwk dssur{lpdwlrqv duh ydolg/ exw wkdw iru
wkh POH  kc hyhq lq d prghudwh vdpsoh vl}h/ lv fohduo| dq xqvxlwdeoh edvh iru lqihuhqfh
derxw k1 Wkhuhiruh/ li zh zlvk wr xvh kljkhu0rughu dv|pswrwlf dssur{lpdwlrqv iru/
iru h{dpsoh/ rqh0vlghg frqghqfh lqwhuydov iru xqnqrzq sdudphwhuv/ wkhq lw lv fuxfldo
wkdw zh dsso| vxfk whfkqltxhv wr vxlwdeoh vwdwlvwlfv1
<Dfnqrzohgjphqwv= Wklv sdshu lv ghulyhg iurp p| SkG wkhvlv/ cKljkhu0Rughu
Dv|pswrwlfv iru Hfrqrphwulf Hvwlpdwruv dqg Whvwv*/ iru zklfk wkdqnv iru sdwlhqw dqg
khosixo vxshuylvlrq jr wr Judqw Kloolhu1 Frpphqwv e| Ndulp Dedglu/ Iudqfhvfr Eudyr/
Jlrydqql Irufklql/ Vruhq Mrkdqvhq/ Sdxo Pduulrww/ Shwhu Skloolsv/ Pdun Vdoprq dqg
Vwhyh Vdwfkhoo/ sduwlflsdqwv dw cGlhuhqwldo Jhrphwulf Phwkrgv lq Hfrqrphwulfv*/ khog
dw HXL/ Ioruhqfh dqg e| dq dqrq|prxv uhihuhh suryhg prvw khosixo1 Ilqdqfldo vxssruw
lq wkh irup ri d Ohyhukxoph Vshfldo Uhvhdufk Ihoorzvkls lv judwhixoo| dfnqrzohgjhg1
43Uhihuhqfhv
Dqghuvrq/ W1Z1 +4<<7,= Wkh Vwdwlvwlfdo Dqdo|vlv ri Wlph Vhulhv1 Zloh| Fodvvlfv
Oleudu|/ Zloh|/ Qhz \run1
Eduqgru0Qlhovhq/ R1H1 dqg Fr{ / G1 +4<;<,= Dv|pswrwlf Whfkqltxhv iru xvh lq Vwd0
wlvwlfv1 Fkdspdq dqg Kdoo/ Orqgrq1
Ekdwwdfkdu|d/ U1Q1 dqg Jkrvk/ M1N1 +4<:;,= Rq wkh ydolglw| ri wkh irupdo Hgjhzruwk
h{sdqvlrq1 Dqqdov ri Vwdwlvwlfv/ 9/ 76707841
Ervh/ D1 +4<;;,= Hgjhzruwk fruuhfwlrq e| errwvwuds lq dxwruhjuhvvlrqv1 Dqqdov ri
Vwdwlvwlfv/ 49/ 4:3<0551
Gxuelq/ M1 +4<;3,= Dssur{lpdwlrqv iru wkh ghqvlwlhv ri vx!flhqw hvwlpdwhv1 Elr0
phwulnd/ ::/ 64406661
Kdoo/ S1 +4<<5,= Wkh Errwvwuds dqg Hgjhzruwk H{sdqvlrq/ Vsulqjhu vhulhv lq vwdwlvwlfv/
Vsulqjhu0Yhuodj1
Ndnl}dzd/ \1 +4<<<,= Ydolg Hgjhzruwk h{sdqvlrqv ri vrph hvwlpdwruv dqg errwvwuds
frqghqfh lqwhuydov lq uvw0rughu dxwruhjuhvvlrq1 Mrxuqdo ri Wlph Vhulhv Dqdo|vlv/
53/ 67668<1
PfFxoodjk/ S1 +4<;:,= Whqvru Phwkrgv lq Vwdwlvwlfv1 Fkdspdq dqg Kdoo/ Orqgrq1
Qlnl/ Q1 dqg Nrqlvkl/ V1 +4<;9,= Hhfwv ri wudqvirupdwlrqv lq kljkhu rughu dv|pswrwlf
h{sdqvlrqv1 Dqqdov ri wkh Lqvwlwxwh ri Vwdwlvwlfdo Pdwkhpdwlfv/ 6;/ 6:406;61
Rfkl/ \1 +4<;6,= Dv|pswrwlf h{sdqvlrqv iru wkh glvwulexwlrq ri dq hvwlpdwru lq wkh
uvw0rughu dxwruhjuhvvlyh surfhvv1 Mrxuqdo ri Wlph Vhulhv Dqdo|vlv/ 7/ 8:9:1
44Skloolsv/ S1F1E1 +4<::,= Dssur{lpdwlrqv wr vrph qlwh Vdpsoh glvwulexwlrqv dvvrfl0
dwhg zlwk d uvw0rughu vwrfkdvwlf glhuhqfh htxdwlrq1 Hfrqrphwulfd/ 78/ 79607;81
 +4<:;,= Hgjhzruwk dqg vdggohsrlqw dssur{lpdwlrqv lq d uvw rughu qrq0
flufxodu dxwruhjuhvvlrq1 Elrphwulnd/ 98/ <40<;1
Vdwfkhoo/ V1 +4<;7,= Dssur{lpdwlrq wr wkh qlwh vdpsoh glvwulexwlrq iru qrq0vwdeoh
uvw0rughu vwrfkdvwlf glhuhqfh htxdwlrq1 Hfrqrphwulfd/ 85/ 45:4045;<1
Wdqljxfkl/ P1 +4<<4,= Kljkhu Rughu Dv|pswrwlf Wkhru| iru Wlph Vhulhv Dqdo|vlv1
Ohfwxuh qrwhv lq vwdwlvwlfv/ Vsulqjhu0Ehuolq1
Wdqljxfkl/ P1 dqg Zdwdqdeh/ \1 +4<<7,= Vwdwlvwlfdo Dqdo|vlv ri Fxuyhg Suredelolw|
Ghqvlwlhv1 Mrxuqdo ri Pxowlyduldwh Dqdo|vlv/ 7;/ 55;057;1
ydq Jdughuhq/ N0M1 +4<<:,= Fxuyhg h{srqhqwldo prghov lq vwdwlvwlfv1 Hfrqrphwulf
Wkhru|/ 48/ ::40:<31
45Dsshqgl{
Surri ri Wkhruhp 4
Wkh surri uhtxluhv rqo| wkdw zh vkrz wkdw Dvvxpswlrq 4 krogv iru ￿1 Zlwkrxw
orvv ri jhqhudolw|/ dvvxph  ' }E'fdqg ohw } ' }￿ dqg % ' %￿c wkh mrlqw ghqvlw|
ri + p d |e hz u l w w h q /i r uq ' qEwc dv
sE+(w'i  T iq
￿%  g￿EqnoE%j
dqg wkh fkdudfwhulvwlf ixqfwlrq ri } lv
}El'
]
U&
i Til
￿} n q
￿%  g￿EqnoE%j_%
' e
3g￿Eq￿
]
U&
i Til
￿} n q
￿% n oE%j_% +43,
Wkh lqwhjudo rq wkh UKV ri +43, lv htxdo wr wkh frpsoh{ Odsodfh wudqvirup=
OEi Til
￿}ji TioE%jc +44,
zkhuh OEE% '
U
U& E%e3#￿%_%c ghqrwhv wkh Odsodfh wudqvirup rshudwru1 Qrz zh
nqrz
OEi TioE%j'i  T ig￿Eqjc
dqg h{sdqglqj i Til￿}j rq dffrxqw ri +8,/
i Til
￿}j 'n
[￿
￿’￿ KU￿%
U￿ n RE
3E￿n￿￿*2c
zkhuh %U￿ ' %￿￿%￿&/d q gw k hKU￿ duh E whqvru frh!flhqwv lq wkh hohphqwv ri l
dqg wkh ghulydwlyhv ri }Ec wkhq
O
q
n
[￿
￿’￿ KU￿%
U￿
r
i TioE%j

'
[￿
￿’fE
￿KU￿
Y￿ i Tig￿Eqj
Yq
￿￿
￿ Yq
￿&
&
c +45,
46z k h u hl q+ 4 5 , /KUf ' c d q gz hk d y hx v h g
O

%
U￿ i TioE%j

'E 
￿Y￿ i Tig￿Eqj
Yq
￿￿
￿ Yq
￿&
&
dqg wkh olqhdulw| dqg frqwlqxlw| ri wkh Odsodfh wudqvirup1 Frqvhtxhqwo|/ wkh Odsodfh
wudqvirup +44, lv +qrwlqj wkdw Fudpìu*v frqglwlrq lv vdwlvhg dxwrpdwlfdoo| lq wkh
h{srqhqwldo fdvh/ dqg khqfh wuxqfdwlrq ri wkh lqwhjudo vhulhv lv shuplwwhg,
OEi Til
￿}ji TioE%j'
[￿
￿’fE
￿KU￿
Y￿ i Tig￿Eqj
Yq
￿￿
￿ Yq
￿&
&
n E
3E￿n￿￿*2c
dqg vr/
}El'ne
3g￿Eq￿ [￿
￿’￿E
￿KU￿V
U￿
% n E
3E￿n￿￿*2c +46,
wkdw lv }Eb shuplwv d vhulhv h{sdqvlrq lq whupv ri wkh fxpxodqwv ri %1D v v x p l q j
:K / wdnlqj orjv/ dqg h{sdqglqj whup e| whup/ wkh fxpxodqw jhqhudwlqj ixqfwlrq ri
} lv
g}El 
["
&’￿E
&3￿

e
3g￿Eq￿ [￿
￿’￿E
￿KU￿V
U￿
%
&
c
zkhuh  lpsolhv dv|pswrwlf htxlydohqfh ri rughu E3E￿n￿￿*2 Vlqfh VU2
% ' E3￿c
wkhq T￿  3￿*2Tc zkhuh T ' Ec dqg vr wkh fxpxodqw jhqhudwlqj ixqfwlrq ri ￿
lv
g￿￿El 
["
&’￿E
&3￿

e
3g￿E￿￿ [￿
￿’￿E
￿
￿*2_U￿V
U￿
%
&
c
zkhuh _U￿ ' T M￿cU￿KM￿ vr wkdw/ vlqfh e| ghqlwlrq/ e3g￿E￿￿ ' E dqg VU￿
% '
E3￿n￿c wkhq
YKg￿￿El
Yl
￿￿
￿ Yl
￿&
&


 
l’f
 
3EK32￿*2
￿ [
,’￿
_
W
U￿
3E,3￿￿*2c
47zkhuh wkh _W
U￿ duh Ec dv uhtxluhg1
Surri ri Wkhruhp 5
Wkh fxpxodqw jhqhudwlqj ixqfwlrq ri wkh plqlpdo vx!flhqw vwdwlvwlf lv
g|Eb'g￿E# n b  g￿E#c
dqg iurp +:, wkh fxpxodqw ixqfwlrq lv
g￿E#'

2
*?
 

[&
￿’￿ #￿￿
 
c +47,
dqg khqfh wkh fxpxodqwv duh jlyhq e|
V
U￿
| Ew'
E3￿
2
Y￿ *?



S&
￿’￿E#￿ n b￿￿



Yb
￿￿
￿ Yb
￿&
&



 

b’f
c
[&
￿’￿ ￿ ' c
zkhuh w ' ik￿cckRcj 2j irupv wkh qdwxudo sdudphwhu lq wkh -ER prgho1 Ghqlqj
S&
￿’￿E#￿ n b￿￿ ' Eb vd|/ wkhq qrwh wkh iroorzlqj lghqwlwlhv
E@G
Y *?￿￿Eb￿￿
_b￿ ' Ao
k
3￿Eb
Y￿Eb￿
_b￿
l
EKG
Y￿3￿Eb￿
_b￿ ' 3￿Eb
Y￿Eb￿
_b￿ 3￿Eb
ESG
YAod￿Eb￿o
_b￿ ' Ao
k
Y￿Eb￿
_b￿
l
E_ G *4b<f 3￿Eb'P E w
Dsso|lqj +d,/ +e, dqg +f, jlyhv
Y￿ *?

 
S&
￿’￿E#￿ n b￿￿

 
Yb
￿￿
￿ Yb
￿&
&
2 Ao
%
& \
￿’￿


3￿Eb￿
￿￿
&
dqg khqfh iurp +g,
Y￿ *?



S&
￿’￿E#￿ n b￿￿



Yb
￿￿
￿ Yb
￿&
&


 


b’f
2 Ao
%
& \
￿’￿
EPEw￿
￿￿
&

48Vlqfh wkh ￿ duh frqvwdqw pdwulfhv wkhq
Ao
%
& \
￿’￿
EPEw￿
￿￿
&
'
[&
￿’￿ o￿E,￿Ewc
zkhuh o￿E'o@?&EPEw￿
￿￿ dqg wkh ,￿Ew duh sro|qrpldov lq wc dqg duh wkxv E
iru doo  dqg a Frqvhtxhqwo|/ ghqlqj
o4@ Ew ' 4@ 
￿ca
o@?&EPEw￿
￿￿
wkhq
Ao
k\&
￿’￿ EPEw￿
￿￿
l

[&
￿’￿ o4@ E,￿Ewc
dqg ixuwkhu/ vlqfh o4@ E'E   2c wkhq
Ao
k\&
￿’￿ EPEw￿
￿￿
l
 E  2
[&
￿’￿ ,￿Ew'Ec
dqg wkh wkhruhp lv suryhg1
49Iljxuhv
Ilj1 4= Vlpxodwhg +vrolg, dqg JE3￿ Hgjhzruwk
dssur{lpdwlrq +grwwhg, iru wkh glvwulexwlrq ri  k
-4 -2 2
0.2
0.4
0.6
0.8
1
Ilj1 5= Vlpxodwhg +vrolg, dqg JE3￿ Hgjhzruwk
dssur{lpdwlrq +grwwhg, iru wkh glvwulexwlrq ri ￿E k
-2 2 4
0.2
0.4
0.6
0.8
1
4:

Edgeworth Expansions in Gaussian Autoregression

http://www.york.ac.uk/media/economics/documents/discussionpapers/2000/0058.pdf

Edgeworth Expansions in Gaussian Autoregression

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics