Max-Planck-Institut für demografische Forschung
Max Planck Institute for Demographic Research
Doberaner Strasse 114 · D-18057 Rostock · GERMANY
Tel +49 (0) 3 81 20 81 - 0; Fax +49 (0) 3 81 20 81 - 202;
http://www.demogr.mpg.de
MPIDR WORKING PAPER WP 2001-001
JANUARY 2001
Period Parity Progression Measures with
Continued Fertility Postponement:
A New Look at the Implications of
Delayed Childbearing for Cohort Fertility
Hans-Peter Kohler ([email protected])
José Antonio Ortega ([email protected])
© Copyright is held by the authors.
Working papers of the Max Planck Institute for Demographic Research receive only limited review. Views or
opinions expressed in working papers are attributable to the authors and do not necessarily reflect those of the
Institute.
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:
(b) Prop. remaining ultimately childless
0.24
1.000
(a) Proportion still childless at time t
0.16
0.250
0.20
0.500
cohort age 17
0.12
0.125
cohort age 24
2000
2010
2020
2030
1960
year
1970
1980
1990
cohort
(d) Completed fertility
cohort age 24
1.8
1.9
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fertility aging
effect
1.6
0.5
1.7
1.0
1.5
2.0
(c) Cumulative cohort fertility
0.0
1.5
cohort age 17
2000
2010
year
proj. observed
2020
2030
postponement stops
1960
1970
1980
1990
cohort
postponement continues
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49
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4;
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age-periodinteractions
interactions
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wkhq eh fdofxodwhg yld Htv1 +9, dqg +;, xvlqj wkh hvwlpdwhg phdq fkdqjh +W , dqg
yduldqfh fkdqjh +W , lq wkh uhihuhqfh |hdu W 1 Iru lqvwdqfh/ Iljxuh 6+eg, vkrz
wkh revhuyhg dqg wwhg lqwhqvlw| vfkhgxohv iru uvw/ vhfrqg dqg wklug eluwkv iru wkh
|hduv 4<;< dqg 4<<:1 Iru uvw eluwkv wkh vfkhgxohv uh hfw wkh vxevwdqwldo ghod| lq
fklogeluwk wkdw kdv rffxuuhg gxulqj wkh 4<<3v1 Lq dgglwlrq/ wkh judskv vkrz krz wkh
vprrwklqj surfhgxuh holplqdwhv dssduhqwo| udqgrp xfwxdwlrqv lq eluwk lqwhqvlwlhv1
Wkh uhodwlyho| cedg* w ri wkh sdulw|0wkuhh ihuwlolw| vfkhgxoh dw |rxqj djhv uh hfwv wkh
idfw wkdw wkh wwlqj surfhgxuh lv zhljkwhg1 Lw grhv qrw vwurqjo| shqdol}h ghyldwlrqv
dw |rxqj djhv zlwk yhu| olwwoh h{srvxuh wr sdulw| wkuhh eluwkv/ zkloh lw dwwhpswv wr
rewdlq d jrrg w dw djhv zkhuh wkh h{srvxuh wr wklug sdulw| eluwkv lv uhodwlyho| kljk1
a +d, lv xsgdwhg rq wkh edvlv ri wkh dgmxvwhg
71 Li qhfhvvdu|/ wkh lqlwldo hvwlpdwh iru !
fklogehdulqj lqwhqvlwlhv rewdlqhg lq vwhs 51
81 Wkh olih0wlph eluwk suredelolwlhv/ sdulw|0surjuhvvlrq udwlrv dqg vlqjxodwh phdq djh
dw eluwk duh fdofxodwhg xvlqj wkh glvfuhwh0wlph yhuvlrqv ri wkh uhvxowv lq rxu hduolhu
Vhfwlrqv1
Iru uvw dqg vhfrqg eluwkv wkh fkdqjhv lq wkh whpsr ri ihuwlolw| duh suhvhqwhg lq Iljxuh
71 Wkh phdq djh ri wkh dgmxvwhg lqwhqvlw| vfkhgxoh lqfuhdvh iurp ehorz 5: |hduv wr deryh
6:
-0.004
0.1
0.0
-0.008
27
4.9
28
5.0
5.1
29
0.0
0.2
5.2
Parity 1: variance effect (delta)
0.004
Parity 1: tempo effect (gamma)
5.3
mean age (left scale)
std. deviation (right scale)
of adjusted intensity
schedule
30
31
Parity 1: mean age and std. deviation
1970 1975 1980 1985 1990 1995 2000
year
1970 1975 1980 1985 1990 1995 2000
year
year
Parity 2: variance effect (delta)
0.008
Parity 2: tempo effect (gamma)
5.4
0.004
0.2
5.6
mean age (left scale)
std. deviation (right scale)
of adjusted intensity
schedule
-0.004
5.0
27.5
0.0
5.2
0.0
0.1
28.5
29.5
Parity 2: mean age and std. deviation
1970 1975 1980 1985 1990 1995 2000
1970 1975 1980 1985 1990 1995 2000
year
1970 1975 1980 1985 1990 1995 2000
1970 1975 1980 1985 1990 1995 2000
year
year
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wr 4<<<1
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vfkhgxoh iru uvw eluwkv lqfuhdvhv vljqlfdqwo| xqwlo derxw 4<<3 zlwk vrph uhyhuvdo ri wklv
wuhqg diwhuzdugv +vhh dovr Nrkohu dqg Sklolsry 5334,1 Wkh phdq fkdqjh  / l1h1/ wkh
dqqxdo lqfuhdvh lq wkh phdq djh ri wkh dgmxvwhg lqwhqvlw| vfkhgxoh dw sdulw| rqh/ ulvhv
iurp derxw }hur lq wkh hduo| 4<:3v wr derxw 15 dw wkh ehjlqqlqj ri wkh 4<;3v1 Wklv phdq
fkdqjh ghfolqhv djdlq wr derxw 13; lq 4<;9/ dqg diwhuzdugv wkh srvwsrqhphqw ri uvw eluwkv
uhjdlqv lwv sdfh dqg uhdfkhv xqsuhfhghqwhg ohyhov lq wkh odwh 4<<3v1 Wkh yduldqfh fkdqjh
 lqglfdwhv wkdw wkh vwdqgdug ghyldwlrq ri wkh lqwhqvlw| vfkhgxoh iru uvw eluwk lqfuhdvhv
e| derxw 17( dqqxdoo| xs wr derxw 4<;81 Diwhuzdugv wkh yduldqfh fkdqjh ghfolqhv dqg
hyhq ehfrphv qhjdwlyh lq wkh 4<<3v1
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6;
lq wkh wlplqj ri uvw eluwkv1 Wkh phdq djh ri wkh dgmxvwhg lqwhqvlw| vfkhgxoh iru vhfrqg
eluwkv judgxdoo| lqfuhdvhv zlwk dq dqqxdo fkdqjh udqjlqj iurp derxw 137 wr doprvw 15
gxulqj wkh shulrg 4<:34<<<1 Lw lv qrwhzruwk| wkdw wkh sdfh ri ihuwlolw| srvwsrqhphqw
iru vhfrqg eluwkv kdv ehhq ehorz wkh uhvshfwlyh whpsr fkdqjh ri uvw eluwkv iru prvw
|hduv vlqfh 4<:61 Wkh eljjhvw glyhujhqfhv ehwzhhq wkh fkdqjhv lq wkh whpsr ri uvw dqg
vhfrqg eluwkv rffxu durxqg 4<;3 +  2 h{fhhgv 14, dqg djdlq wrzdugv wkh hqg ri rxu
revhuydwlrq shulrg +  2 h{fhhgv 138,/ zkloh wkh whpsr fkdqjh dw sdulw| rqh dqg wzr duh
dssur{lpdwho| htxdo lq wkh odwh 4<;3v1 Vlplodu wr uvw eluwkv/ wkh yduldqfh ri wkh lqwhqvlw|
vfkhgxoh iru vhfrqg eluwkv lqfuhdvhv gxulqj wkh shulrg 4<:34<<</ dqg wklv sdwwhuq kdv qrw
ehhq vxemhfw wr dq| pdunhg uhyhuvdo1
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iurp doprvw }hur lq wkh hduo| 4<:3v wr derxw 13:5 lq 4<<31 Lw vxevhtxhqwo| ghfolqhv djdlq
ehorz 138 lq wkh odwh 4<<3v1 Iurp wkh hduo| 4<:3v rqzdug/ wkh whpsr fkdqjh ri wklug eluwkv
lv wkhuhiruh frqvlvwhqwo| ehorz wkdw ri uvw dqg vhfrqg eluwkv1 Wkh glhuhqfhv ehwzhhq 
dqg uhdfk d plqlpxp lq wkh hduo| 4<:3v dqg odwh 4<;3v zkhq wkh| doprvw glplqlvk/
dqg wkh| shdn durxqg 4<;3 dqg lq wkh odwh 4<<3v zkhq wkh whpsr fkdqjh ri uvw eluwkv lv
pruh wkdq 15 kljkhu wkdq wkdw ri wklug eluwkv1
915
Dqdo|vlv ri shulrg sdwwhuqv 4<:34<<<
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sdulw| surjuhvvlrq phdvxuhv wkdw duh fdofxodwhg xqghu erwk wkh srvwsrqhphqw frqwlqxhv
dqg wkh srvwsrqhphqw vwrsv vfhqdulr= +d, wkh shulrg olih0wlph suredelolw| ri kdylqj dw ohdvw
rqh/ wzr/ wkuhh ru irxu fkloguhq/ +e, wkh shulrg sdulw| surjuhvvlrq udwlrv iurp sdulw| }hur
wr rqh/ rqh wr wzr/ wzr wr wkuhh/ dqg wkuhh wr irxu/ dqg +f, wkh lqgh{ ri wrwdo ihuwlolw|1
Wkh uvw phdvxuh uh hfwv wkh suredelolw| wkdw d zrpdq lq wkh v|qwkhwlf frkruw jlyhv eluwk
wr dw ohdvw rqh/ wzr/ hwf1/ fkloguhq edvhg rq wkh sdulw|0vshflf ohyho ri ihuwlolw| revhuyhg
lq d fdohqgdu |hdu dqg zlwk hlwkhu frqwlqxhg ru vwrsshg srvwsrqhphqw ri fklogehdulqj1
Wkh vhfrqg phdvxuh wudqvodwhv wkhvh eluwk suredelolwlhv lqwr sdulw| surjuhvvlrq udwlrv wkdw
uh hfw wkh suredelolw| ri surjuhvvlqj wr wkh qh{w fklog frqglwlrqdo rq dwwdlqlqj sdulw| rqh/
wzr ru wkuhh1 Wkh wklug phdvxuh uh hfwv wkh frpsohwhg ihuwlolw| lq wkh v|qwkhwlf frkruw
wkdw lv lpsolhg e| wkh sdulw|0vshflf ohyho ri ihuwlolw| revhuyhg lq d fdohqgdu |hdu1 Lq wkh
dqdo|vhv ehorz/ doo ri wkhvh phdvxuhv duh fdofxodwhg iru wkh v|qwkhwlf frkruw ri fklogohvv
zrphq zkr duh djh 48 lq wkh uhihuhqfh |hdu1
Wkh shulrg olih0wlph suredelolw| ri h{shulhqflqj dw ohdvw rqh/ wzr/ wkuhh ru irxu eluwkv
iru wklv v|qwkhwlf frkruw lv ghslfwhg lq Iljxuh 8+dg,1 Lq rughu wr pdnh wkh vfdoh ri wkh
judskv frpsdudeoh/ wkh yhuwlfdo d{hv lq doo wkuhh judskv hqfrpsdvv dq lghqwlfdo udqjh1
Pruhryhu/ wkh glhuhqw dvvxpswlrqv derxw zkhwkhu wkh srvwsrqhphqw vwrsv ru frqwlqxhv
6<
gr qrw |lhog wr glhuhqfhv lq wkh shulrg olih0wlph suredelolw| ri h{shulhqflqj d uvw eluwk
dqg wkh uhvshfwlyh olqhv lq Iljxuh 8+d, ryhuods1
Wkh shulrg olih0wlph suredelolw| ri dw ohdvw rqh fklog lq Iljxuh 8+d, xfwxdwhv gxulqj
4<:34<<< lq dffrugdqfh zlwk wkh ryhudoo wuhqg lq wkh wrwdo ihuwlolw|= dq lqlwldo lqfuhdvh lv
iroorzhg e| d ghfolqh dqg uhfryhu| gxulqj wkh ede| errp/ dqg diwhu 4<<3 wkh suredelolw|
ghfolqhv djdlq1 Wkh fkdqjhv lq wklv shulrg olih0wlph suredelolw| ri uvw eluwkv/ krzhyhu/ duh
uhodwlyho| prghvw zkhq frpsduhg wr hlwkhu wkh fkdqjhv lq wkh wrwdo ihuwlolw| ru wkh fkdqjhv
lq wkh olih0wlph suredelolwlhv iru kljkhu eluwk rughuv1 Lq sduwlfxodu/ wkhuh lv qr vxevwdqwldo
uhgxfwlrq lq wkh shulrg olih0wlph eluwk suredelolw| ri dw odvw rqh fklog gxulqj wkh ede| exvw
lq wkh 4<<3v1
D fruuhvsrqglqj fdofxodwlrq ri olih0wlph eluwk suredelolwlhv edvhg rq revhuyhg fklogehdu0
lqj lqwhqvlwlhv/ l1h1/ d fdofxodwlrq wkdw ljqruhv srwhqwldo whpsr glvwruwlrqv lq fklogehdulqj
lqwhqvlwlhv/ xqghuhvwlpdwhv wkh suredelolw| ri kdylqj dw ohdvw rqh fklog1 Dv vxjjhvwhg e|
Uhvxow :/ wklv xqghuhvwlpdwlrq lq Iljxuh 8+d, lv odujhvw lq shulrgv zlwk wkh prvw udslg
ghod|v lq hqwhulqj sduhqwkrrg durxqg 4<;3 dqg lq wkh 4<<3v1 Dv d frqvhtxhqfh ri wklv
whpsr glvwruwlrq/ wkh revhuyhg lqwhqvlwlhv huurqhrxvo| vxjjhvw wkdw wkh shulrg olih0wlph
eluwk suredelolw| ri dw ohdvw rqh fklog ghfolqhv lq wkh odwh 4<<3v wr lwv orzhvw ohyho vlqfh
4<:31
Vlploduo|/ wkh revhuyhg fklogehdulqj lqwhqvlwlhv dovr ohdg wr dq xqghuhvwlpdwlrq ri wkh
shulrg olih0wlph suredelolw| ri kdylqj dw ohdvw wzr fkloguhq lq Iljxuh 8+e,1 D v|qwkhwlf
frkruw wkdw h{shulhqfhv qr ixuwkhu ghod| lq ihuwlolw| dwwdlqv sdulw| ri dw ohdvw wzr fkloguhq
zlwk d suredelolw| wkdw lv ehwzhhq 138 dqg 148 kljkhu wkdq lv vxjjhvwhg e| wkh hvwlpdwhv
rewdlqhg iurp wkh revhuyhg lqwhqvlwlhv1 Krzhyhu/ wkh dvvxpswlrq wkdw wkh v|qwkhwlf frkruw
h{shulhqfhv qr ixuwkhu ghod| lq fklogehdulqj lv txlwh hvvhqwldo iru wklv lqfuhdvhg suredelolw|1
Lq shulrgv zkhq wkh sdfh ri ihuwlolw| srvwsrqhphqw iru uvw eluwkv lv kljkhu wkdq wkh whpsr
fkdqjh iru vhfrqg eluwkv/ l1h1/ hvshfldoo| durxqg 4<;3 dqg wr d ohvvhu h{whqw diwhu 4<<3/
wkh odwhu djh ri hqwhulqj sduhqwkrrg lq wkh srvwsrqhphqw frqwlqxhv vfhqdulr uhgxfhv wkh
h{srvxuh dw wkh sulpdu| djhv ri jlylqj eluwk wr vhfrqg fkloguhq1 Wkh djlqj ri ihuwlolw| dw
uvw eluwkv wkdw rffxuv lq wkh srvwsrqhphqw frqwlqxhv vfhqdulr wkxv uhgxfhv wkh suredelolw|
ri kdylqj dw ohdvw wzr fkloguhq dv frpsduhg wr wkh srvwsrqhphqw vwrsv vfhqdulr1
Lq wkh odwh 4<;3v wkh whpsr fkdqjh iru uvw dqg vhfrqg eluwkv uhdfkhv dq doprvw lghq0
wlfdo ohyho +vhh Iljxuh 7,1 Gxulqj wklv shulrg/ wkh fdofxodwlrqv xvlqj rxu wzr dvvxpswlrqv
derxw wkh whpsr fkdqjh lq wkh v|qwkhwlf frkruw |lhog d yluwxdoo| htxdo olih0wlph suredelo0
lw| ri kdylqj dw ohdvw wzr fkloguhq lq Iljxuh 8+e,1 Wklv rffxuv ehfdxvh d odwhu hqwu| lqwr
sduhqwkrrg/ zklfk lqglfdwhv wkh ehjlqqlqj ri wkh h{srvxuh wr vhfrqg eluwkv/ grhv qrw uh0
gxfh wkh suredelolw| ri surjuhvvlqj wr wkh qh{w fklog li wkh vhfrqg0eluwk lqwhqvlw| vfkhgxoh
kdv ehhq vkliwhg wr kljkhu djhv dw d sdfh wkdw shuihfwo| frpshqvdwhv iru wkh ghod| ri uvw
eluwkv1
73
(b) Life-time probability, parity 2
observed
stops
continues
0.80
0.65
0.90
0.75
(a) Life-time probability, parity 1
1970
0.55
0.70
observed
stops
continues
1980
1990
2000
1970
1980
1990
2000
year
year
(c) Life-time probability, parity 3
(d) Life-time probability, parity 4
observed
stops
continues
0.10
-0.05
0.20
0.05
0.30
0.15
observed
stops
continues
1970
1980
1990
year
2000
1970
1980
1990
2000
year
Iljxuh 8= Shulrg olihwlph suredelolw| ri jlylqj eluwk wr dw ohdvw rqh/ wzr/ wkuhh ru irxu
fkloguhq iru fklogohvv zrphq zkr duh djh 48 lq wkh uhihuhqfh |hdu1 Fdofxodwlrqv duh edvhg
rq revhuyhg ihuwlolw| lqwhqvlwlhv +crevhuyhg*,/ ru rq dgmxvwhg ihuwlolw| lqwhqvlwlhv zlwk hlwkhu
qr ixuwkhu srvwsrqhphqw ri ihuwlolw| +cvwrsv*, ru d srvwsrqhphqw wkdw iroorzv wkh phdq
dqg yduldqfh fkdqjhv revhuyhg lq wkh uhihuhqfh |hdu +cfrqwlqxhv*,1 Lq Judsk +d, wkh wzr
srvwsrqhphqw vfhqdulrv ohdg wr lghqwlfdo uhvxowv dqg wkh olqhv wkhuhiruh ryhuods1
74
Wkh shulrg olih0wlph eluwk suredelolw| ri dw ohdvw wkuhh fkloguhq lq Iljxuh 8+f,/ lqgh0
shqghqw ri lwv fdofxodwlrq/ lqfuhdvhv txlwh vxevwdqwldoo| gxulqj wkh ede| errp dqg vxevh0
txhqwo| h{klelwv wkh prvw surqrxqfhg ghfolqh gxulqj wkh ede| exvw diwhu 4<<31 Wkh ede|
errp dqg exvw lq Vzhghq lv wkhuhiruh sulpdulo| dvvrfldwhg zlwk fkdqjhv lq wkh shulrg
olih0wlph eluwk suredelolw| iru vhfrqg dqg wklug fkloguhq/ zkloh wkh shulrg olih0wlph eluwk
suredelolwlhv iru uvw dqg irxuwk fkloguhq xfwxdwh rqo| prghvwo| gxulqj wklv shulrg1
Vlqfh ihuwlolw| iru uvw dqg vhfrqg eluwkv kdv ehhq srvwsrqhg wkurxjkrxw wkh shulrg
4<:304<<</ fdofxodwlrqv edvhg rq wkh revhuyhg fklogehdulqj lqwhqvlwlhv ohdg wr dq xqghuhv0
wlpdwlrq ri wkh suredelolw| wr dwwdlq dw ohdvw wkuhh ru dw ohdvw irxu fkloguhq dv frpsduhg wr
wkh srvwsrqhphqw vwrsv vfhqdulr1 Lq wkh srvwsrqhphqw frqwlqxhv vfhqdulr/ krzhyhu/ wkh
ghod| ri uvw dqg vhfrqg eluwkv uhgxfhv wkh suredelolw| ri dwwdlqlqj d sdulw| ri wkuhh ru
kljkhu +ru irxu ru kljkhu,1 Lq vrph shulrgv/ wkh uhgxfwlrq gxh wr wkh ihuwlolw| djlqj hhfw
hyhq h{fhhgv wkh eldv fdxvhg e| d qhjohfw ri whpsr glvwruwlrqv= wkh shulrg olih0wlph eluwk
suredelolw| ri dwwdlqlqj d sdulw| ri dw ohdvw wkuhh +ru dw ohdvw irxu, lq wkh srvwsrqhphqw
frqwlqxhv vfhqdulr gursv ehorz wkh suredelolw| wkdw lv rewdlqhg iurp wkh revhuyhg lqwhq0
vlwlhv1 Wklv hhfw lv sduwlfxoduo| qrwhzruwk| ehfdxvh wkh revhuyhg fklogehdulqj lqwhqvlwlhv
duh glvwruwhg grzqzdug lq shulrgv zlwk d ghod| lq fklogehdulqj1
Uhodwhg wr wkh olih0wlph eluwk suredelolwlhv lq Iljxuh 8 duh wkh sdulw| surjuhvvlrq sure0
delolwlhv ghqhg lq Uhvxow 441 Vlqfh wkh sdulw| surjuhvvlrq udwlr iurp fklogohvvqhvv wr wkh
uvw eluwk lv htxdo wr wkh olih0wlph suredelolw| ri kdylqj dw ohdvw rqh fklog/ lw kdv douhdg|
ehhq glvfxvvhg deryh1 Wkh uhpdlqlqj shulrg sdulw| surjuhvvlrq udwlrv iurp uvw wr vhfrqg/
dqg vhfrqg wr wklug/ wklug wr irxuwk fklog duh ghslfwhg lq Iljxuh 91 Vlplodu wr rxu hduolhu
glvfxvvlrq/ wkh shulrg sdulw| surjuhvvlrq udwlrv fdofxodwhg iurp wkh revhuyhg lqwhqvlwlhv
|lhog dq xqghuhvwlpdwh ri wkh suredelolw| ri surjuhvvlqj wr wkh qh{w fklog dv frpsduhg
wr wkh srvwsrqhphqw vwrsv vfhqdulr1 Wklv uhodwlrq/ krzhyhu/ lv sduwldoo| uhyhuvhg lq wkh
srvwsrqhphqw frqwlqxhv vfhqdulr1
Frqvlghu iru lqvwdqfh wkh shulrg surjuhvvlrq udwlrv wr wkh vhfrqg fklog lq Iljxuh 9+d,1
Gxulqj 4<:84<;8 dqg iurp 4<<5 rqzdugv/ wkh surjuhvvlrq udwlrv lq wkh srvwsrqhphqw
frqwlqxhv vfhqdulr duh ehorz wkh udwlrv rewdlqhg zkhq wkh srvwsrqhphqw ri ihuwlolw| frphv
wr d kdow1 Lq erwk ri wkhvh shulrgv/ wkh wuhqg wrzdugv odwhu fklogehdulqj lv idvwhu iru uvw
wkdq iru vhfrqg eluwkv/ dqg wkh ghod| lq hqwhulqj sduhqwkrrg lv wkxv qrw frpshqvdwhg e|
dq htxdoo| idvw srvwsrqhphqw ri vhfrqg eluwkv1
Wkh sdulw| surjuhvvlrq udwlr iurp wkh vhfrqg wr wklug eluwk/ dqg iurp wkh wklug wr
irxuwk eluwk lq Iljxuh 9+e/f, uh hfw d uhodwlyho| vlplodu sdwwhuq1 Wkh sdulw| surjuhvvlrq
udwlrv wkdw dvvxph d frqwlqxhg srvwsrqhphqw duh ehorz wkrvh wkdw dvvxph qr ixuwkhu
srvwsrqhphqw/ dqg wkh glhuhqfh fdq djdlq eh txlwh vxevwdqwldo +vhh Iljxuh 9g,1
Wkh rqjrlqj ghod| ri orzhu sdulw| eluwkv wkhuhiruh lpsolhv wkdw zrphq lqfuhdvlqjo| vwduw
ehlqj h{srvhg wr vhfrqg dqg kljkhu sdulw| eluwkv zkhq wkh uhvshfwlyh ihuwlolw|0lqwhqvlwlhv
75
(b) PPR, parity 2 to 3
0.5
(a) PPR, parity 1 to 2
0.4
observed
stops
continues
0.6
0.2
0.7
0.3
0.8
0.9
observed
stops
continues
1970
1980
1990
2000
1970
1980
year
2000
year
(c) PPR, parity 3 to 4
0.15
(d) Reduction in PPR by parity
Reduction in PPR due to cont’d
postponement
PPR, parity 1 to 2
PPR, parity 2 to 3
PPR, parity 3 to 4
0.0
-0.05
0.0
0.1
0.05
0.2
0.10
observed
stops
continues
0.3
1990
1970
1980
1990
year
2000
1970
1980
1990
2000
year
Iljxuh 9= Judskv +df,= Shulrg sdulw| surjuhvvlrq udwhv iurp 4 wr 5/ dqg 5 wr 6 fkloguhq iru
fklogohvv zrphq zkr duh djh 48 lq wkh uhihuhqfh |hdu1 Fdofxodwlrqv duh edvhg rq revhuyhg
ihuwlolw| lqwhqvlwlhv +crevhuyhg*,/ ru rq dgmxvwhg ihuwlolw| lqwhqvlwlhv zlwk hlwkhu qr ixuwkhu
srvwsrqhphqw ri ihuwlolw| +cvwrsv*, ru d srvwsrqhphqw wkdw iroorzv wkh phdq dqg yduldqfh
fkdqjhv revhuyhg lq wkh uhihuhqfh |hdu +cfrqwlqxhv*,1 Judsk +g,= Uhgxfwlrq lq shulrg
sdulw| surjuhvvlrq udwlr fdxvhg e| d frqwlqxhg srvwsrqhphqw ri ihuwlolw| lq wkh v|qwkhwlf
frkruw1
76
duh douhdg| uhodwlyho| orz1 Wklv ihuwlolw| djlqj hhfw fdq uhgxfh sdulw| surjuhvvlrq sure0
delolwlhv wr dq h{whqw wkdw h{fhhgv wkh whpsr glvwruwlrqv lqfxuuhg e| huurqhrxvo| xvlqj
wkh revhuyhg fklogehdulqj lqwhqvlwlhv lq wkh fdofxodwlrqv= iru doo sdulw| surjuhvvlrq udwlrv
lq Iljxuh 9 wkhuh h{lvw shulrgv gxulqj zklfk wkh srvwsrqhphqw frqwlqxhv vfhqdulr lpsolhv
surjuhvvlrq udwlrv wkdw duh ehorz wkh udwlrv lqihuuhg iurp wkh revhuyhg gdwd1
Wkh odwh 4<;3v duh h{fhswlrqdo ehfdxvh gxulqj wklv shulrg wkh whpsr fkdqjhv iru uvw
dqg vhfrqg eluwk duh dssur{lpdwho| htxdo1 Wkh srvwsrqhphqw ri wkh uvw eluwk lv wkhuh0
iruh dffrpprgdwhg e| dq htxdoo| idvw srvwsrqhphqw ri vhfrqg eluwkv1 Wkh olih0wlph eluwk
suredelolw| ri dw ohdvw wzr fkloguhq dqg sdulw| surjuhvvlrq udwlrv wr wkh vhfrqg fklog duh
wkhuhiruh dssur{lpdwho| htxdo xqghu erwk srvwsrqhphqw vfhqdulrv/ ru vrphwlphv wkh uh0
vxowv xqghu wkh srvwsrqhphqw frqwlqxhv vfhqdulr hyhq h{fhhg wkrvh ri wkh srvwsrqhphqw
vwrsv vfhqdulr1 Krzhyhu/ iru prvw ri wkh revhuydwlrq shulrg wkh sdfh ri ihuwlolw| srvwsrqh0
phqw lv idvwhu dw orzhu wkdq dw kljkhu sdulwlhv1 Wkhvh uhgxfwlrqv lq wkh olih0wlph eluwk
suredelolwlhv dqg sdulw| surjuhvvlrq udwlrv fdxvhg lq wklv fdvh e| dq rqjrlqj ihuwlolw| srvw0
srqhphqw fdq eh txlwh vxevwdqwldo1 Iru lqvwdqfh/ wkh uhgxfwlrqv lq wkh sdulw| surjuhvvlrq
udwlrv gxh wr wkh djlqj ri ihuwlolw| fdq eh xs wr 43 shufhqwdjh srlqwv lq Iljxuh 9+g,1
Wkh dqdo|vhv lq Iljxuhv 8 dqg 9 wkhuhiruh ylvxdol}h wkh wzr sdwkzd|v ri krz wkh srvw0
srqhphqw ri ihuwlolw| dhfwv wkh phdvxuhphqw dqg dvvhvvphqw ri shulrg ihuwlolw|= Rq rqh
kdqg/ wkh srvwsrqhphqw ri ihuwlolw| glvwruwv wkh revhuyhg fklogehdulqj lqwhqvlwlhv grzq0
zdug/ dqg rqo| wkh dgmxvwhg fklogehdulqj lqwhqvlwlhv fruuhfwo| uh hfw wkh ohyho ri ihuwlolw| dw
hdfk sdulw|1 Rq wkh rwkhu kdqg/ d srvwsrqhphqw ri ihuwlolw| wkdw lv idvwhu iru orzhu sdulwlhv
dqg vorzhu iru kljkhu sdulwlhv lpsolhv wkdw duh zrphq vwduw ehlqj h{srvhg wr kljkhu sdulw|
eluwkv dw djhv zkhq wkh uhvshfwlyh fklogehdulqj lqwhqvlwlhv duh douhdg| txlwh orz1 Wkh
djlqj ri ihuwlolw| dw orzhu sdulwlhv/ zklfk rffxuv li wkh srvwsrqhphqw ri ihuwlolw| frqwlqxhv/
wkhuhiruh uhgxfhv wkh surjuhvvlrq wr kljkhu sdulw| eluwkv lq v|qwkhwlf frkruwv1
916
Frpsdulvrq zlwk dgmxvwhg WIU
Lq oljkw ri rxu glvfxvvlrq deryh/ lw lv lqwhuhvwlqj wr frpsduh wkh lqihuhqfhv rewdlqhg iurp
wkh sdulw| surjuhvvlrq phdvxuhv sursrvhg lq wklv sdshu zlwk wkh fruuhvsrqglqj lqihuhqfhv
iurp wkh dgmxvwhg wrwdo ihuwlolw| udwh15
Iljxuh :+d, frpsduhv wkh revhuyhg wrwdo ihuwlolw| udwh iru uvw eluwkv/ wkh fruuhvsrqglqj
dgmxvwhg W I U iru uvw eluwkv dqg wkh shulrg olih0wlph suredelolw| ri kdylqj dw ohdvw rqh
fklog1 Wkh odwwhu lv wdnhq iurp Iljxuh 8+d, dqg xvhv d v|qwkhwlf frkruw wkdw lv djh 48 dqg
sdulw| }hur lq wkh uhihuhqfh |hdu1
5
Zh duh dzduh wkdw wkh dgmxvwhg A 8 - lv qrw d phdvxuh iru frpsohwhg ihuwlolw| lq v|qwkhwlf frkruwv
+vhh dovr Erqjdduwv dqg Ihhqh| 5333> Nlp dqg Vfkrhq 5333,1 Qhyhuwkhohvv/ zh xvh lw khuh wr frpsduh
lw wr rxu v|qwkhwlf frkruw phdvxuh lq rughu wr vkhg oljkw rq wkh txhvwlrq zkhwkhu dqg wr zklfk h{whqw
A 8 -0dgmxvwphqw lv lqirupdwlyh derxw ihuwlolw| lq v|qwkhwlf frkruwv1
77
(b) All parities: TFR and PPRs
2.4
(a) Parity 1: TFR and PPRs
2.2
Index of TF, continues
Index of TF, stops
observed TFR
adjusted TFR
0.6
1.4
0.7
1.6
1.8
0.8
2.0
0.9
1.0
life-time prob of
at least one child
observed TFR
adjusted TFR
1970
1980
1990
years
2000
1970
1980
1990
2000
years
Iljxuh := Ohiw judsk= frpsdulvrq ri wrwdo ihuwlolw| udwh/ dgmxvwhg W I U dqg shulrg olihwlph
eluwk suredelolw| iru uvw eluwkv1 Uljkw judsk= frpsdulvrq ri wrwdo ihuwlolw| udwh/ dgmxvwhg
W I U +doo eluwk rughuv, dqg wkh lqgh{ ri wrwdo ihuwlolw| rewdlqhg iurp sdulw| surjuhvvlrq
udwlrv1 Fdofxodwlrqv iru wkh odwwhu dvvxph fklogohvv zrphq ri djh 48 lq wkh uhihuhqfh
|hdu/ dqg duh edvhg rq revhuyhg ihuwlolw| lqwhqvlwlhv +crevhuyhg*,/ ru rq dgmxvwhg ihuwlolw|
lqwhqvlwlhv zlwk hlwkhu qr ixuwkhu srvwsrqhphqw ri ihuwlolw| +cvwrsv*, ru d srvwsrqhphqw
wkdw iroorzv wkh phdq dqg yduldqfh fkdqjhv revhuyhg lq wkh uhihuhqfh |hdu +cfrqwlqxhv*,1
Iru uvw eluwkv lq wkh ohiw judsk/ wkhvh glhuhqw dvvxpswlrqv derxw srvwsrqhphqw ohdg wr
lghqwlfdo uhvxowv dqg duh qrw glvwlqjxlvkhg1
Doo wkuhh phdvxuhv duh dq lqglfdwru ri wkh ohyho ri ihuwlolw| iru uvw eluwkv1 Pruhryhu/ wkh
dgmxvwhg W I U dqg wkh shulrg olih0wlph suredelolw| ri dw ohdvw rqh fklog dgmxvw iru whpsr
glvwruwlrqv lq wkh revhuyhg shulrg sdwwhuq1 Ghvslwh wkh vlplodulw| lq lqwhusuhwdwlrq/ wkh
phdvxuhv uh hfw txlwh glhuhqw wuhqgv gxulqj wkh shulrg 4<:34<<<1 Wkh wrwdo ihuwlolw| udwh
iru uvw eluwkv dqg wkh dgmxvwhg W I U iru uvw eluwkv h{klelw wkh vwurqjhvw xfwxdwlrqv ryhu
wlph/ dqg wkh odwwhu vxevwdqwldoo| h{fhhgv wkh iruphu lq shulrgv zkhq wkh srvwsrqhphqw
ri uvw eluwkv lv prvw udslg1 Lq frpsdulvrq wr wkhvh odujh xfwxdwlrqv lq wkh wrwdo ihuwlolw|
udwhv iru uvw eluwkv/ wkh shulrg olih0wlph suredelolw| ri kdylqj dw ohdvw rqh fklog lv uhodwlyho|
frqvwdqw1 Lq sduwlfxodu/ lw qhlwkhu lqfuhdvhv qru ghfuhdvhv dv surqrxqfhg dv wkh vwdqgdug
ru dgmxvwhg wrwdo ihuwlolw| udwh iru uvw eluwkv gxulqj wkh ede| errp dqg exvw +iru uhodwhg
dqdo|vhv vhh ydq Lpkr dqg Nhlopdq 5333,1 Wkhvh glhuhqw ghyhorsphqwv duh lq sduw gxh
wr wkh idfw wkdw fkdqjhv lq wkh sdulw| glvwulexwlrq ri wkh srsxodwlrq vxevhtxhqw wr hfrqrplf
fkdqjhv ru srolf| lqwhuyhqwlrqv fdq ohdg wr xfwxdwlrqv lq wkh wrwdo ihuwlolw| udwh dqg lq
78
wkh phdq djh dw eluwk wkdw duh xquhodwhg wr wkh ohyho ri ihuwlolw| dv uh hfwhg e| ihuwlolw|0
lqwhqvlwlhv +vhh Ihhqh| dqg \x 4<;:> Ihhqh| dqg Oxw} 4<<4> Nrkohu dqg Ruwhjd 5333,1 Vlqfh
Vzhghq h{shulhqfhg txlwh ydu|lqj vrflrhfrqrplf frqglwlrqv dqg idplo|0srolflhv gxulqj
wkh shulrg xqghu lqyhvwljdwlrq +Krhp 4<<3> Krhp dqg Krhp 4<<9> Krhp 5333,/ wkh
wrwdo ihuwlolw| udwh iru uvw eluwkv ryhuhvwlpdwhv wkh yrodwlolw| ri ihuwlolw|1 D vhfrqg idfwru
frqwulexwlqj wr wkh glhuhqfhv ehwzhhq wkh phdvxuhv lq Iljxuh :+d, lv wkh idfw wkdw qhlwkhu
wkh vwdqgdug qru wkh dgmxvwhg wrwdo ihuwlolw| udwh iru uvw eluwkv uhsuhvhqw v|qwkhwlf frkruw
phdvxuhv1 Qhlwkhu phdvxuh fdq wkxv surshuo| uh hfw wkh qxpehu ri uvw0eluwkv wkdw rffxu
wr phpehuv ri d v|qwkhwlf frkruw edvhg rq wkh ohyho ri ihuwlolw| lq d jlyhq shulrg1 Erwk ri
wkhvh olplwdwlrqv duh dyrlghg xvlqj d sdulw| surjuhvvlrq edvhg lqglfdwru ri shulrg ihuwlolw|1
Iljxuh :+e, vkrzv wkh revhuyhg wrwdo ihuwlolw| udwh/ wkh EI0dgmxvwhg W I U iru doo sdu0
lwlhv/ dqg wkh lqgh{ ri wrwdo ihuwlolw| ghulyhg iurp shulrg sdulw| surjuhvvlrq udwlrv1 Wkh
revhuyhg wrwdo ihuwlolw| udwh lv ehorz wkh frpsohwhg ihuwlolw| ri wkh v|qwkhwlf frkruw lq erwk
srvwsrqhphqw vfhqdulrv iurp wkh hduo| 4<:3v rqzdugv/ dqg lw frqupv wkh dujxphqw lq
Erqjdduwv dqg Ihhqh| +4<<;, wkdw wkh revhuyhg W I U fdq |lhog d vxevwdqwldo xqghuhvwlpdwh
wkh txdqwxp ri ihuwlolw| lq shulrgv ri d srvwsrqhphqw1 Krzhyhu/ wkh ihuwlolw| dwwdlqhg e|
wkh v|qwkhwlf frkruw dovr ghyldwhv vxevwdqwldoo| iurp wkh ohyho vxjjhvwhg e| wkh dgmxvwhg
W I U1 Li wkh v|qwkhwlf frkruw h{shulhqfhv qr ixuwkhu srvwsrqhphqw ri ihuwlolw|/ wkh dg0
mxvwhg W I U dqg wkh lqgh{ ri wrwdo ihuwlolw| wudfh hdfk rwkhu uhodwlyh forvho| xqwlo wkh odwh
4<;3v/ dqg wkh| vwduw wr ghyldwh sulpdulo| lq wkh hduo| 4<<3v1 Qrwhzruwk| lv dovr wkdw wkh
dgmxvwhg wrwdo ihuwlolw| zlwk yduldqfh hhfwv/ lqwurgxfhg lq Nrkohu dqg Sklolsry +5334,/
wudfhv wkh lqgh{ ri wrwdo ihuwlolw| xqghu wkh cqr ixwxuh srvwsrqhphqw* dvvxpswlrq forvhu
wkdq wkh EI0dgmxvwhg W I U/ dowkrxjk zh gr qrw hoderudwh wklv frpsdulvrq khuh1
Iru doo |hduv diwhu 4<:6 wkh lqgh{ ri wrwdo ihuwlolw| lq wkh srvwsrqhphqw frqwlqxhv
vfhqdulr lv ehorz wkh lqgh{ wkdw lv rewdlqhg e| dvvxplqj qr ixuwkhu ghod| ri fklogehdulqj1
Wkh glhuhqfhv duh prvw pdunhg lq wkh odwh 4<:3v dqg hduo| 4<;3v/ dqg wkh| uhhphujh lq
wkh 4<<3v1
Vrph ri wkh prvw lpsruwdqw glhuhqfhv ehwzhhq d W I U0edvhg dqg sdulw|0surjuhvvlrq
edvhg dqdo|vlv ri wkh Vzhglvk ihuwlolw| gxulqj 4<:34<<< duh wkhuhiruh dv iroorzv= +d, Wkh
shulrg olih0wlph suredelolw| ri dw ohdvw rqh fklog grhv qrw xfwxdwh vxevwdqwldoo| gxulqj
wkh revhuydwlrq shulrg/ dqg lq sduwlfxodu/ lw grhv qrw vxevwdqwldoo| ghfolqh lq wkh ede|0
exvw shulrg diwhu 4<<31 Lq frqwudvw/ erwk wkh revhuyhg dqg dgmxvwhg W I U vxjjhvw dq
lpsruwdqw frqwulexwlrq lq whupv ri d ghfolqlqj txdqwxp dw uvw eluwk wr wkh ryhudoo ghfolqh
lq ihuwlolw| diwhu 4<<31 Rxu sdulw| surjuhvvlrq phdvxuhv wkhuhiruh lpso| wkdw wkh ede| exvw
diwhu 4<<3 lv sulpdulo| dvvrfldwhg zlwk ghfolqlqj sdulw| surjuhvvlrq udwlrv wr wkh vhfrqg
dqg wklug fklog1 +e, Wkh sdfh ri ihuwlolw| srvwsrqhphqw kdv ehhq vxemhfw wr vxevwdqwldo
fkdqjhv gxulqj 4<:34<<</ dqg rxu phdvxuhv dgglwlrqdoo| uh hfw krz wkh glhuhqwldo whpsr
fkdqjhv dw glhuhqw sdulwlhv dhfwv wkh ihuwlolw| lq v|qwkhwlf frkruwv1 Iru prvw shulrgv/
79
wkh fkdqjhv lq wkh whpsr ri ihuwlolw| whqg wr eh idvwhu iru orzhu dqg vorzhu iru kljkhu
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wr uhgxfh wkh frpsohwhg ihuwlolw| dwwdlqhg lq v|qwkhwlf frkruwv gxh wr d ihuwlolw| djlqj
hhfw/ dqg wklv uhgxfwlrq uhdfkhv lwv kljkhvw ohyhov durxqg 4<;3 dqg lq wkh 4<<3v1 Rqo| lq
wkh odwh 4<;3v wkh glhuhqfhv lq wkh whpsr fkdqjhv iru uvw dqg vhfrqg eluwkv glplqlvk/
dqg wkh wzr srvwsrqhphqw vfhqdulrv wkxv ohdg wr dq doprvw htxdo lqgh{ ri wrwdo ihuwlolw|
ehfdxvh wkh ghod|hg rqvhw ri sduhqwkrrg lv frpshqvdwhg e| vx!flhqwo| idvw whpsr fkdqjhv
dw vhfrqg eluwk1 +f, Wkh lqgh{ ri wrwdo ihuwlolw| xqghu frqwlqxhg srvwsrqhphqw lv rqo|
uhodwlyho| prghvwo| deryh wkh revhuyhg wrwdo ihuwlolw| gxulqj wkh ede| errp dqg exvw
shulrg xqwlo derxw 4<<8/ lqglfdwlqj wkdw wkh dgmxvwhg W I U pd| |lhog d wrr rswlplvwlf
shuvshfwlyh rq wkh ihuwlolw| ohyho lq v|qwkhwlf frkruwv gxulqj wklv shulrg1 Vlqfh d frqwlqxhg
ghod| ri fklogehdulqj iru wkh iruhvhhdeoh ixwxuh vhhpv pruh olnho| wr xv wkdq dq lpplqhqw
ohyhoolqj r lq wkh sdfh ri ihuwlolw| srvwsrqhphqw/ wkh lqgh{ ri wrwdo ihuwlolw| zlwk qr ixuwkhu
srvwsrqhphqw lq rxu rslqlrq vhuyhv dv dq xsshu erxqg iru wkh frpsohwhg frkruw ihuwlolw|
lpsolhg e| wkh ihuwlolw| sdwwhuq lq d jlyhq uhihuhqfh |hdu1
917
Surmhfwlqj frkruw ihuwlolw| lq Vzhghq
Zh xvh wkh sdulw| surjuhvvlrq phdvxuhv lqwurgxfhg lq wklv sdshu wr surmhfw wkh frpsohwhg
ihuwlolw| ri frkruwv zkr kdyh qrw qlvkhg fklogehdulqj1 Lq rxu lqwurgxfwru| Vhfwlrq 5 zh
kdyh douhdg| glvfxvvhg vrph uhvxowv ri wkhvh surmhfwlrqv/ dqg wkh iroorzlqj dqdo|vhv h{sdqg
rxu lqwurgxfwru| h{dpsoh1 Lq sduwlfxodu/ zh surmhfw wkh ihuwlolw| ri frkruwv zkr kdyh qrw
qlvkhg fklogehdulqj dv ri 4<:8 dqg 4<<< rq wkh edvlv ri wkh sdulw|0vshflf ohyho ri ihuwlolw|
dqg wkh sdulw|0vshflf sdfh ri ihuwlolw| srvwsrqhphqw wkdw suhydlohg lq wkhvh wzr |hduv1
Lq wkh uvw fdvh zh duh deoh wr frpsduh wkh surmhfwhg zlwk dfwxdo frkruw ehkdylru vlqfh
dgglwlrqdo frkruwv frpsohwh wkhlu fklogehdulqj e| 4<<<1 Gxh wr wkh vxevwdqwldo fkdqjhv
lq wkh ohyho dqg whpsr ri ihuwlolw| gxulqj wkh shulrg 4<:34<<< lq Vzhghq/ dq| surmhfwlrq
wkdw dvvxphv d frqvwdqw ohyho dqg whpsr ri ihuwlolw| vxevhtxhqw wr 4<:8 fdq eh h{shfwhg
wr shuirup uhodwlyho| edgo|/ hvshfldoo| iru |rxqj frkruwv1
Wkh surmhfwlrqv ghslfwhg lq Iljxuh ; h{wudsrodwh wkh shulrg0ihuwlolw| sdwwhuq ri 4<:8
rqwr wkh ixwxuh ihuwlolw| ehkdylru ri frkruwv zkr kdyh qrw frpsohwhg fklogehdulqj lq 4<:81
Wkh sdwwhuq revhuyhg lq 4<:8 lv fkdudfwhul}hg e| d prghvwo| idvw srvwsrqhphqw dw uvw
eluwk/ dqg d txlwh vorz whpsr fkdqjh dw vhfrqg dqg kljkhu sdulwlhv +Iljxuh 7,1 Wkh
shulrg olih0wlph eluwk suredelolw| ri dw ohdvw rqh fklog lv uhodwlyho| kljk lq wkdw |hdu/ zkloh
wkh sdulw| surjuhvvlrq udwlrv wr wkh vhfrqg dqg wklug fklog duh frqvlghudeo| ehorz wkh
ydoxhv wkdw dfwxdoo| suhydlo diwhu 4<:8 +Iljxuhv 8 dqg 9,1 Zh wkhuhiruh h{shfw wkdw wkh
surmhfwhg fklogohvvqhvv lq frkruwv uhpdlqv uhodwlyho| orz/ zkloh wkh surmhfwhg frpsohwhg
frkruw ihuwlolw| ghfolqhv1 Pruhryhu/ gxh wr wkh glhuhqwldo whpsr fkdqjhv dw uvw dqg
7:
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dqg wkh surmhfwhg frpsohwhg frkruw ihuwlolw| lq Iljxuh ; uhyhdo wkh sdwwhuq mxvw ghvfulehg1
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wkh frpsohwhg ihuwlolw| lv surmhfwhg wr ghfolqh lq |rxqjhu frkruwv/ dqg wkh sursruwlrq ri
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frkruwv zkr zhuh douhdg| lq wkhlu odwh 53v ru roghu lq 4<:81 Wkh glhuhqfhv ehwzhhq wkh
dfwxdo shulrg ihuwlolw| wuhqg diwhu 4<:8/ dqg wkh surmhfwhg ohyho dqg whpsr edvhg rq wkh
revhuydwlrq lq 4<:8/ krzhyhu ohdg wr d vxevwdqwldo glyhujhqfh ehwzhhq surmhfwhg dqg dfwxdo
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Wkh surmhfwlrqv lq Iljxuh ; dovr uhyhdo wkh lpsolfdwlrqv ri ihuwlolw| djlqj lq wkh srvw0
srqhphqw frqwlqxhv vfhqdulr1 Ehfdxvh ri wkh glhuhqwldo whpsr fkdqjhv dw uvw dqg kljkhu
sdulwlhv/ dq rqjrlqj srvwsrqhphqw ri ihuwlolw| dffruglqj wr wkh 4<:8 srvwsrqhphqw sdwwhuq
lpsolhv d vxevwdqwldo lqfuhdvh lq wkh sursruwlrq ri zrphq zkr kdyh rqo| rqh fklog/ dqg
lw ohdgv wr d ixuwkhu ghfolqh ri frkruw ihuwlolw| dv frpsduhg wr wkh srvwsrqhphqw vwrsv
vfhqdulr1 Ehfdxvh ri wkh lqfuhdvhv lq wkh ohyho ri ihuwlolw| gxulqj wkh 4<;3v/ hvshfldoo| dw
sdulw| wzr dqg wkuhh/ wkh dfwxdo ehkdylru ri frkruwv zkr zhuh ehorz djhv 5863 lq 4<:8
glhuv txlwh vxevwdqwldoo| iurp wkh surmhfwlrqv edvhg rq wkh 4<:8 ohyho ri ihuwlolw| lq erwk
srvwsrqhphqw vfhqdulrv1
Iljxuh < uhsruwv wkh fruuhvsrqglqj surmhfwlrqv iru frkruwv zkr kdyh qrw frpsohwhg
fklogehdulqj dv ri 4<<<1 Wkhvh surmhfwlrqv duh edvhg rq wkh sdfh ri ihuwlolw| srvwsrqhphqw
dqg ohyho ri ihuwlolw| revhuyhg lq 4<<</ l1h1/ wkh odvw |hdu iru zklfk gdwd duh dydlodeoh1 Wklv
fdohqgdu |hdu lv fkdudfwhul}hg e| d txlwh udslg srvwsrqhphqw dw uvw dqg wr d vrphzkdw
ohvvhu h{whqw vhfrqg eluwkv +Iljxuh 7,/ dqg dovr e| shulrg sdulw| surjuhvvlrq udwlrv wr wkh
wklug dqg iruwk eluwk wkdw duh frqvlghudeo| ehorz wkh ydoxhv lq wkh odwh 4<;3v dqg hduo|
4<<3v1 Pruhryhu/ wkh shulrg sdulw| surjuhvvlrq udwlrv wr wkh uvw dqg vhfrqg fklog kdyh
ghfolqhg rqo| uhodwlyho| prghvwo| vlqfh wkh ede| errp shulrg +Iljxuhv 8 dqg 9,1
Ehfdxvh ri wkh udslg whpsr fkdqjhv ri ihuwlolw| lq wkh odwh 4<<3v/ wkh surmhfwlrqv edvhg
rq wkh revhuyhg gdwd dqg wkh dgmxvwhg fklogehdulqj lqwhqvlwlhv glhu txlwh vxevwdqwldoo|1 Lq
sduwlfxodu/ h{wudsrodwlrqv ri wkh revhuyhg sdwwhuq lq 4<<< lqwr wkh ixwxuh vxjjhvw d udslgo|
lqfuhdvlqj sursruwlrq ri zrphq zlwk }hur fkloguhq ru rqo| rqh fklog lq |rxqjhu frkruwv/
dqg wkh| lpso| d frqvlghudeoh ghfuhdvh lq frpsohwhg frkruw ihuwlolw|1 Surmhfwlrqv wkdw
uhpryh whpsr glvwruwlrqv/ rq wkh rwkhu kdqg/ vxjjhvw d uhodwlyho| frqvwdqw sursruwlrq ri
zrphq zlwk }hur fkloguhq ru rqo| rqh fklog lq |rxqjhu frkruwv/ dqg wkh| suhglfw d pdunhg
lqfuhdvh lq wkh suhydohqfh ri zrphq zlwk h{dfw wzr fkloguhq1 Pruhryhu/ wkh frpsohwhg
ihuwlolw| lq erwk srvwsrqhphqw vfhqdulrv ghfolqhv ohvv udslgo| dv vxjjhvwhg e| wkh revhuyhg
gdwd1
7;
0.30
(b) Exact 1 child, ref year: 1975
cohort age 45
in mid 1975
0.10
0.0
0.0
0.10
0.20
cohort age 45
in mid 1975
0.20
0.30
(a) Childlessness, ref year: 1975
1930
1950
1970
1990
1930
cohort
1950
1970
1990
cohort
(d) Complete Fert., ref year: 1975
1.6
cohort age 45
in mid 1975
1930
1950
1.4
0.30
0.40
1.8
0.50
2.0
0.60
(c) Exact 2 children, ref year: 1975
1970
1990
cohort
observed cohort pattern
proj. observed
postponement stops
cohort age 45
in mid 1975
1930
1950
1970
1990
cohort
postponement continues
Iljxuh ;= Surmhfwlrq ri ihuwlolw| ehkdylru iru frkruwv zkr kdyh qrw qlvkhg fklogehdulqj lq
4<:8 edvhg rq wkh sdulw|0vshflf ohyho ri ihuwlolw| dqg srvwsrqhphqw sdwwhuq revhuyhg lq
4<:81 Wkh srvwsrqhphqw vwrsv +gdvkhg olqh, dqg srvwsrqhphqw frqwlqxhv +gdvkhg0grwwhg
olqh, duh edvhg rq wkh whpsr0dgmxvwhg ihuwlolw| lqwhqvlwlhv dqg dvvxph hlwkhu qr ixuwkhu
ghod|v lq fklogehdulqj ru d ixuwkhu ghod| lq fklogehdulqj wkdw pluuruv wkh 4<:8 whpsr
sdwwhuq1 Lq Judsk +d, wkh wzr srvwsrqhphqw vfhqdulrv ohdg wr lghqwlfdo uhvxowv dqg wkh
uhvshfwlyh olqhv ryhuods1 Wkh surmhfwlrqv rewdlqhg iurp wkh revhuyhg ihuwlolw| lqwhqvlwlhv
duh lqglfdwhg e| grwwhg olqhv1
7<
(b) Exact 1 child, ref year: 1999
0.30
0.30
(a) Childlessness, ref year: 1999
0.10
0.20
cohort age 45
in mid 1999
0.0
0.0
0.10
0.20
cohort age 45
in mid 1999
1930
1950
1970
1990
1930
cohort
1950
1970
1990
cohort
(d) Complete Fert., ref year: 1999
1.6
0.40
1.8
0.50
2.0
0.60
(c) Exact 2 children, ref year: 1999
1930
1950
1970
cohort age 45
in mid 1999
1.4
0.30
cohort age 45
in mid 1999
1990
cohort
observed cohort pattern
proj. observed
postponement stops
1930
1950
1970
1990
cohort
postponement continues
Iljxuh <= Surmhfwlrq ri ihuwlolw| ehkdylru iru frkruwv zkr kdyh qrw qlvkhg fklogehdulqj
lq 4<<< edvhg rq wkh sdulw|0vshflf ohyho ri ihuwlolw| dqg srvwsrqhphqw sdwwhuq revhuyhg
lq 4<<<1 Wkh srvwsrqhphqw vwrsv +gdvkhg olqh, dqg srvwsrqhphqw frqwlqxhv +gdvkhg0
grwwhg olqh, duh edvhg rq wkh whpsr0dgmxvwhg ihuwlolw| lqwhqvlwlhv dqg dvvxph hlwkhu qr
ixuwkhu ghod|v lq fklogehdulqj ru d ixuwkhu ghod| lq fklogehdulqj wkdw pluuruv wkh 4<<<
srvwsrqhphqw sdwwhuq1 Lq Judsk +d, wkh wzr srvwsrqhphqw vfhqdulrv ohdg wr lghqwlfdo
uhvxowv dqg wkh uhvshfwlyh olqhv ryhuods1 Wkh surmhfwlrqv rewdlqhg iurp wkh revhuyhg
ihuwlolw| lqwhqvlwlhv duh lqglfdwhg e| grwwhg olqhv1
83
Li wkh txlwh udslg ihuwlolw| srvwsrqhphqw revhuyhg lq 4<<< frqwlqxhv lq wkh ixwxuh/ lw
zloo lpso| rqo| d prghvw lqfuhdvh lq wkh iudfwlrq ri zrphq zlwk h{dfw rqh fklog lq |rxqj
frkruwv dv frpsduhg wr wkh srvwsrqhphqw vwrsv vfhqdulr1 Wklv prghvw lqfuhdvh lv gxh wr
wkh idfw wkdw wkh ihuwlolw| djlqj hhfw fdxvhg e| d ghod| ri uvw eluwkv lv doprvw frpshqvdwhg
e| dq htxdoo| udslg srvwsrqhphqw ri vhfrqg eluwkv1 Krzhyhu/ iru wklug dqg kljkhu sdulwlhv
wklv lv qr orqjhu wkh fdvh1 Iru frkruwv zkr duh ehorz djh 58 lq 4<<</ wkh srvwsrqhphqw
frqwlqxhv vfhqdulr wkhuhiruh surmhfwv vxevwdqwldo lqfuhdvhv lq wkh idfwlrq ri zrphq zlwk
h{dfw wzr fkloguhq1 Wklv rffxuv ehfdxvh wkh surjuhvvlrq udwlr iurp wkh vhfrqg wr wklug
fklog lv glplqlvkhg e| wkh ihuwlolw| djlqj hhfw dqg pruh zrphq wkxv cuhpdlq* dw sdulw|
wzr1 Wklv hhfw dovr ohdgv wr d vxevwdqwldo ghfuhdvh lq frpsohwhg ihuwlolw| ri zrphq lq
|rxqj frkruwv xqghu wkh srvwsrqhphqw frqwlqxhv vfhqdulr1
Lq Iljxuh 43 zh judsk wkh phdq djhv dw eluwk wkdw duh dvvrfldwhg zlwk wkh deryh
surmhfwlrqv +rq wkh edvlv ri wkh dydlodeoh gdwd zh fdqqrw fdofxodwh wkh phdq djh dw eluwk
e| sdulw| iru zrphq eruq sulru wr 4<79,1 Dv rqh h{shfwv/ wkh srvwsrqhphqw vwrsv vfhqdulr
ohdgv wr d ohyhoolqj r lq wkh djh dw eluwk dw doo sdulwlhv1 Wkh djh dw uvw eluwk vwdelol}hv
dw derxw 5; |hduv/ dqg wkdw iru vhfrqg eluwkv dw derxw 641 Wkh ryhudoo phdq djh dw eluwk
phuho| lqfuhdvhv wr derxw 63 |hduv gxh wr uhodwlyho| ihz wklug dqg irxuwk eluwkv1 Lq frqwudvw
wr wkh surmhfwlrq ri wkh ihuwlolw| ohyho/ wkh revhuyhg gdwd |lhog surmhfwlrqv wkdw duh txlwh
vlplodu wr wkh rqhv lq wkh srvwsrqhphqw vwrsv vfhqdulr1
Wkh srvwsrqhphqw frqwlqxhv vfhqdulr/ krzhyhu/ ohdgv wr yhu| vxevwdqwldo lqfuhdvhv lq
wkh djh dw eluwk iru frkruwv eruq diwhu 4<:3v1 Wkhvh frkruwv duh vwloo uhodwlyho| hduo|
lq wkhlu fklogehdulqj |hduv/ dqg wkhlu wlplqj ri ihuwlolw| ghshqgv vwurqjo| rq wkh sdfh
ri ixwxuh ihuwlolw| srvwsrqhphqw1 Li wkh wuhqg wrzdugv ghod|hg fklogehdulqj frqwlqxhv dw
lwv 4<<< sdfhiru uvw dqg vhfrqg eluwkv wkh dqqxdo whpsr fkdqjh lv dw d klvwrulfdoo|
kljk ohyho htxdo wr 159 +sdulw| 4, dqg 14< +sdulw| 5,wkhq wkh phdq djh dw uvw eluwkv
lqfuhdvhv wr 5<17 |hduv +frkruw eruq 4<:8, dqg 6518 |hduv +frkruw eruq 4<;5,/ dqg wkh
phdq djh dw vhfrqg dqg wklug eluwk lqfuhdvhv uhvshfwlyho| wr 66 dqg 6917 |hduv +vhh Wdeoh
4,1 Ehfdxvh wklug dqg kljkhu sdulw| eluwkv ehfrph lqfuhdvlqjo| uduh lq |rxqjhu frkruwv
xqghu wkh srvwsrqhphqw frqwlqxhv vfhqdulr/ wkh phdq djh dw eluwkv iru doo sdulwlhv lqfuhdvhv
frqvlghudeo| ohvv wkdq wkh sdulw|0vshflf phdq djhv1
Jlyhq wkh uhodwlyho| kljk phdq djhv wkdw duh surmhfwhg lq wkh srvwsrqhphqw frqwlqxhv
vfhqdulr/ rqh pd| zdqw wr txhvwlrq zkhwkhu d ixwxuh srvwsrqhphqw dw wkh 4<<< sdfh fdq
suhydlo iru dq dgglwlrqdo 73 wr 83 |hduv xqwlo wkh |rxqjhvw frkruwv lq rxu surmhfwlrqv kdyh
frpsohwhg fklogehdulqj1 Wkh dvvhvvphqw ri wklv txhvwlrqv lv eh|rqg wkh vfrsh ri wklv
sdshu dqg qhhgv wr eh edvhg rq dq hydoxdwlrq ri ixwxuh surjuhvv lq vxuylydo dqg phglfdo
whfkqrorj|/ dv zhoo dv ixwxuh vrfldo fkdqjhv uhjduglqj hgxfdwlrq/ idplo| irupdwlrq dqg
fklog0fduh1 Wkh phwkrgv ghyhorshg lq wklv sdshu/ krzhyhu/ doorz xv wr surmhfw wkh wlplqj
dqg ohyho ri frkruw ihuwlolw| xqghu glhuhqw dvvxpswlrqv derxw wkh ixwxuh srvwsrqhphqw
84
(b) MAB at 2nd birth, ref year: 1999
38
36
(a) MAB at 1st birth, ref year: 1999
36
cohort age 45
in mid 1999
26
24
28
26
30
28
32
30
34
32
34
cohort age 45
in mid 1999
1950
1960
1970
1980
1990
1950
1960
cohort
1970
1980
1990
cohort
(d) MAB all parities, ref year: 1999
38
40
(c) MAB at 3rd birth, ref year: 1999
28
30
32
34
36
cohort age 45
in mid 1999
26
28
30
32
34
36
38
cohort age 45
in mid 1999
1950
1960
1970
1980
1990
cohort
observed cohort pattern
proj. observed
postponement stops
1950
1960
1970
1980
1990
cohort
postponement continues
Iljxuh 43= Surmhfwlrq ri phdq djh dw eluwk iru frkruwv zkr kdyh qrw qlvkhg fklogehdulqj
lq 4<<< edvhg rq wkh sdulw|0vshflf ohyho ri ihuwlolw| dqg srvwsrqhphqw sdwwhuq revhuyhg
lq 4<<<1 Wkh srvwsrqhphqw vwrsv +gdvkhg olqh, dqg srvwsrqhphqw frqwlqxhv +gdvkhg0
grwwhg olqh, duh edvhg rq wkh whpsr0dgmxvwhg ihuwlolw| lqwhqvlwlhv dqg dvvxph hlwkhu qr
ixuwkhu ghod|v lq fklogehdulqj ru d ixuwkhu ghod| lq fklogehdulqj wkdw pluuruv wkh 4<<<
srvwsrqhphqw sdwwhuq1 Wkh surmhfwlrqv rewdlqhg iurp wkh revhuyhg ihuwlolw| lqwhqvlwlhv
duh lqglfdwhg e| grwwhg olqhv1
85
ri ihuwlolw|1 Wklv lqirupdwlrq lv hvvhqwldo iru ghyhorslqj uhdolvwlf ixwxuh srvwsrqhphqw
vfhqdulrv/ dqg wkh fruuhvsrqglqj vrfldo dqg phglfdo fkdqjhv wkdw duh qhfhvvdu| wr idflolwdwh
dqg dffrpprgdwh vxfk ixwxuh ihuwlolw| srvwsrqhphqw1 D qh{w vwhs lq wkh deryh surmhfwlrq
lv wkhuhiruh wkh frqvlghudwlrq ri xqfhuwdlqw| dqg dowhuqdwlyh vfhqdulrv derxw wkh ixwxuh
sdfh ri ihuwlolw| srvwsrqhphqw1
:
Frqfoxvlrqv
Wkh frqfhuq derxw whpsr0glvwruwlrqv lq wkh wrwdo ihuwlolw| udwh shuwdlqv wr wkh idfw wkdw wkh
W I U grhv qrw uh hfw wkh txdqwxp ri ihuwlolw| lq shulrgv zkhq fklogehdulqj lv srvwsrqhg1
Gxh wr eluwkv wkdw duh phuho| srvwsrqhg exw qrw iruhjrqh/ k|srwkhwlfdo frkruwv zkr
h{shulhqfh wkh ihuwlolw| sdwwhuq revhuyhg lq d fdohqgdu |hdu duh olnho| wr dwwdlq d ihuwlolw|
ohyho wkdw h{fhhgv wkh ohyho vxjjhvwhg e| wkh vwdqgdug wrwdo ihuwlolw| udwh1 Zkloh wklv
jhqhudo whqghqf| lv qrw glvsxwhg/ vxevwdqwldo frqwuryhuv| h{lvwv derxw wkh h{whqw wr zklfk
v|qwkhwlf frkruw ihuwlolw| dqg wkh ohyho ri wkh W I U glyhujh1 Lq sduwlfxodu/ wkh dgmxvwhg
W I U lqwurgxfhg e| Erqjdduwv dqg Ihhqh| +4<<;, lv rqo| ri olplwhg xvh lq wklv frqwuryhuv|
ehfdxvh wkh txdqwxp ri ihuwlolw| uhyhdohg e| wklv dgmxvwphqw kdv dq xqfohdu lqwhusuhwdwlrq
lq whupv ri frkruw ihuwlolw|1
Lq wklv sdshu zh surylgh d xqlhg wrronlw ri shulrg sdulw| surjuhvvlrq phdvxuhv wkdw
idflolwdwhv d wzrirog jrdo= uvw/ wkh ghvfulswlrq dqg dqdo|vlv ri sdvw wuhqgv lq shulrg ihuwlolw|
lq whupv ri v|qwkhwlf frkruw ehkdylru> dqg vhfrqg/ wkh surmhfwlrq ri frkruw ihuwlolw| rq wkh
edvlv ri wkh prvw uhfhqwo| revhuyhg ohyho ri ihuwlolw| dqg sdfh ri ihuwlolw| srvwsrqhphqw1
Zh dsso| wkhvh phwkrgv wr Vzhghq gxulqj 4<:34<<<1
Wkhvh phdvxuhv ryhufrph vhyhudo olplwdwlrqv ri wkh dgmxvwhg wrwdo ihuwlolw| udwh vlqfh
wkh| fdq eh lqwhusuhwhg gluhfwo| lq whupv ri frkruw ehkdylru1 Lq sduwlfxodu/ wkh shulrg
sdulw| surjuhvvlrq phdvxuhv lq wklv sdshu uh hfw wkh qxpehu ri fkloguhq dwwdlqhg e| d
v|qwkhwlf frkruw wkdw h{shulhqfhv wkh ohyho/ whpsr dqg srvvleo| srvwsrqhphqw ri ihuwlolw|
revhuyhg lq d fdohqgdu |hdu1 Wkhvh phdvxuhv wkhuhiruh lpsuryh xsrq wkh dgmxvwhg wrwdo
ihuwlolw| udwh/ zklfk uh hfwv wkh txdqwxp ri ihuwlolw| lq d jlyhq shulrg/ exw qrw qhfhvvdulo|
wkh dvvrfldwhg v|qwkhwlf frkruw ihuwlolw|1
Prvw lpsruwdqwo|/ wkh dqdo|vhv zlwk rxu sdulw| surjuhvvlrq phdvxuhv doorz d mx{wd0
srvlwlrq ri glhuhqw dvvxpswlrqv derxw wkh ixwxuh srvwsrqhphqw ri ihuwlolw| lq v|qwkhwlf
ru uhdo frkruwv1 Wkh| fdq wkhuhiruh eh xvhg wr dvvhvv wkh ihuwlolw| djlqj hhfw/ l1h1/ wkh
uhgxfwlrq ri kljkhu sdulw| eluwkv wkdw rffxuv li dq rqjrlqj ghod| ri uvw +ru vhfrqg, eluwkv
vkliwv wkh h{srvxuh wr vhfrqg dqg kljkhu sdulw| eluwkv wrzdugv roghu djhv/ zkhuh wkh
suredelolw| ri h{shulhqflqj dqrwkhu fklog lv txlwh orz1 Wklv hhfw kdv qrw ehhq vwxglhg
hduolhu vlqfh lwv dqdo|vlv uhtxluhv erwk dgmxvwphqwv ri wkh revhuyhg shulrg ihuwlolw| gdwd
iru whpsr0glvwruwlrqv/ dqg sdulw| surjuhvvlrq phdvxuhv wkdw doorz iru d frqwlqxlqj ihuwlolw|
86
srvwsrqhphqw1 Wkh odwwhu phdvxuhv/ krzhyhu/ duh lqwurgxfhg dqg dssolhg iru wkh uvw wlph
lq wklv sdshu1
Rxu dssolfdwlrq wr Vzhghq frqupv wkh uhohydqfh ri dgmxvwlqj wkh revhuyhg ihuwlolw|
sdwwhuq iru whpsr0glvwruwlrqv1 Fdofxodwlrqv edvhg rq wkh revhuyhg lqwhqvlwlhv vxjjhvw dq
huurqhrxvo| orz suredelolw| ri surjuhvvlqj wr wkh uvw fklog/ dqg wkh| surylgh dq xqghuhv0
wlpdwlrq ri frpsohwhg frkruw ihuwlolw|1
Krzhyhu/ wkh ihuwlolw| djlqj hhfw hphujhv dv d vhfrqg lpsruwdqw ghwhuplqdqw ri fr0
kruw ihuwlolw|1 Iru lqvwdqfh/ li wkh sdulw|0vshflf ihuwlolw| ohyho ri 4<<< suhydlov iru doo |hduv
wkhuhdiwhu dqg wkh srvwsrqhphqw ri ihuwlolw| dovr pdlqwdlqv lwv 4<<< sdfh/ wkhq wkh frp0
sohwhg ihuwlolw| ri wkh 4<;< frkruw zloo eh 4194 fkloguhq/ zklfk lv 14: fkloguhq ohvv wkdq lq
d vlwxdwlrq zkhuh wkh srvwsrqhphqw ri ihuwlolw| frphv wr d kdow lq 4<<<1 Wklv uhgxfwlrq
ri ihuwlolw| lv rqo| gxh wr wkh pruh udslg ghod| ri fklogehdulqj dw orzhu dv frpsduhg wr
kljkhu sdulwlhv1 Lw lv wkxv phuho| d frqvhtxhqfh ri wkh djlqj ri ihuwlolw| wkdw rffxuv li wkh
srvwsrqhphqw ri fklogehdulqj frqwlqxhv lq wkh ixwxuh1
Lq vxppdu|/ wkhuhiruh/ wklv sdshu hpskdvl}hv wkdw dqdo|vhv dqg surmhfwlrqv ri frq0
whpsrudu| ihuwlolw| sdwwhuqv qhhg wr dffrxqw iru wzr dvshfwv= whpsr0glvwruwlrqv lq wkh
revhuyhg ihuwlolw| sdwwhuq/ dqg ihuwlolw| djlqj hhfwv fdxvhg e| d glhuhqwldo sdfh ri ihu0
wlolw| dfurvv sdulwlhv1 Rqo| erwk dvshfwv frpelqhg fdq |lhog d fruuhfw uh hfwlrq ri krz
fxuuhqw sdwwhuqv ri ihuwlolw| srvwsrqhphqw vkdsh wkh wlplqj dqg ohyho ri frkruw ihuwlolw|1
Wkh shulrg sdulw| surjuhvvlrq phdvxuhv zlwk frqwlqxhg fklogehdulqj lqwurgxfhg lq wklv
sdshu surylgh d xvhixo dqg frpsuhkhqvlyh wrro wr dfklhyh wkhvh dqdo|vhv1
Dfnqrzohgjphqwv= Zh duh juhdwo| lqghewhg wr Jxqqdu Dqghuvrq iru surylglqj xv
zlwk wkh Vzhglvk gdwd xvhg lq wklv sdshu1 Wklv sdshu kdv dovr juhdwo| ehqhwwhg iurp
hduolhu glvfxvvlrqv zlwk Nhqqhwk Zdfkwhu1 Lq dgglwlrq/ zh kdyh uhfhlyhg pdq| yhu| khosixo
frpphqwv dqg vxjjhvwlrqv iurp Mxkd Dokr/ Iudqfhvfr Eloodul/ Wrpdv Iuhmnd/ Mdq P1 Krhp/
Hyhuw ydq Lpkr/ dqg Qlfr Nhlopdq1
D
Dsshqgl{= Surriv rplwwhg lq wkh wh{w
Surri ri Uhvxow 4= Uhvxow 4 iru dq| jlyhq sdulw| m iroorzv dqdorjrxvo| wr Uhvxow 6 lq
NS1 Wkh sulpdu| glhuhqfh lv wkdw wkh surri ehorz lv dssolhg wr fklogehdulqj lqwhqvlwlhv
h{shulhqfhg e| zrphq zkr duh ri sdulw| m 4 lq wkh uhihuhqfh |hdu W / dv frpsduhg wr wkh
sdulw|0vshflf ihuwlolw| udwhv dv lq NS1
Dvvxph iru wkh prphqw wkdw wkhuh duh qr ohyho hhfwv diwhu wkh uhihuhqfh |hdu W / l1h1/
dvvxph wkdw t+w, @ 4 iru w W 1 Wkhq frqvlghu zrphq zkr duh sdulw| m 4 lq wkh uhihuhqfh
|hdu W 1 D sdulw|0m eluwk wkdw rffxuv wr rqh ri wkhvh zrphq dw djh d dqg wlph w zrxog
kdyh rffxuuhg dw d k|srwkhwlfdo djh d U +d> w, dw wlph w U +d> w, li wkhuh kdg ehhq qr
87
dgglwlrqdo srvwsrqhphqw ri ihuwlolw| diwhu wkh uhihuhqfh |hdu W / l1h1/ li wkhuh kdg ehhq qr
srvwsrqhphqw ri ihuwlolw| iru w W +vhh Dvvxpswlrq 5,1
Ehfdxvh zh dvvxph iru wkh prphqw wkdw t +w, @ 4 iru w W / wkh lqwhqvlw| vfkhgxoh
k +d> w, lq Dvvxpswlrq 6 ghqrwhv wkh fklogehdulqj lqwhqvlwlhv wkdw duh h{shulhqfhg e| wkh
deryh zrphq dw wlph w W lq wkh suhvhqfh ri d ihuwlolw| srvwsrqhphqw/ dqg wkh vfkhgxoh
!cA +d, lq Dvvxpswlrq 7 ghqrwhv wkh fklogehdulqj lqwhqvlwlhv wkdw zrxog eh h{shulhqfhg
e| wkhvh zrphq li wkhuh zhuh qr ixuwkhu srvwsrqhphqw ri ihuwlolw| iru w W 1 Wkh uhodwlrq
ehwzhhq wkh wzr vfkhgxohv k +d> w, dqg !cA +d, fdq wkhq eh hvwdeolvkhg e| d fkdqjh ri
yduldeoh iurp wkh revhuyhg djh d dqg wlph w ri d sdulw|0m eluwk wr wkh k|srwkhwlfdo djh
d U +d> w, dqg wlph w U +d> w, dw zklfk wklv eluwk zrxog rffxu lq wkh devhqfh ri dq|
ixuwkhu ihuwlolw| srvwsrqhphqw diwhu wkh uhihuhqfh |hdu +l1h1/ iru w W ,1 Wkh Mdfreldq ri wklv
wudqvirupdwlrq iurp wkh revhuyhg wr k|srwkhwlfdo rffxuuhqfh ri d eluwk lv +4@c| U +d> w,,1
Wkh uhodwlrq ehwzhhq k +d> w, dqg !cA +d, wkhuhiruh lv jlyhq e| k +d> w, @ +4@c| U +d> w,,
!cA +dU +d> w,,1 Lqvhuwlqj wklv uhodwlrq iru k +d> w, lqwr wkh ghfrpsrvlwlrq ri wkh revhuyhg
fklogehdulqj lqwhqvlwlhv lq Ht1 +5, |lhogv Ht1 +7,1
U
Surri ri Uhvxow 7= Wkh uhvxow iroorzv e| lqwhjudwlqj d
+w, @ d p +d> w,gd@t +w, dqg
U
+w,,2 p +d> w,gd@t +w,/ dqg fkdqjlqj wkh yduldeoh ri lqwhjudwlrq1 Wkh vwhsv
v2 +w, @ +d d
duh lghqwlfdo wr wkh surri ri Uhvxowv 9 dqg 43 lq NS1
Surri ri Uhvxow 5= Wkh uhvxow iroorzv iurp wkh ghqlwlrq ri wkh dgmxvwhg fklogehdulqj lq0
U
U
whqvlwlhv lq Dvvxpswlrq 8 dqg e| lqwhjudwlqj p +d> w,gd @ t +w, +4@ U +d> w,,!cA +d
U
U +d> w,,gd1 D fkdqjh ri yduldeoh iurp d wr @ d U +d> w, wkhq |lhogv p +d> w,gd @
U
t +w, !cA +,g @ t +w,/ zkhuh wkh odvw htxdolw| iroorzv iurp Dvvxpswlrq 71
Surri ri Uhvxow 6= Uhvxow 6 iroorzv gluhfwo| e| uhzulwlqj wkh Mdfreldq whup lq Ht1 +7,
dv +4 @c| U +d> w,, @ +4 u +d> w,, +4 @ U +d> w,,/ zkhuh u +d> w, lv ghqhg lq Ht1 +:,1
Surri ri Uhvxow 8= Wkh dvvxpswlrqv derxw wkh ohyho dqg srvwsrqhphqw ri ihuwlolw|
xqghuo|lqj wkh lqwhqvlw| vfkhgxohv pr +d> w, lpso| iru doo w W wkdw t +w, @ t +W ,/ +w, @
r / dqg +w, @ r 1 Wkh fxpxodwhg whpsr Ur +d> w, dqg wkh djh0vshflf whpsr fkdqjh
ur +d> w, wkhq iroorz iurp Dvvxpswlrq 9 zlwk +w, @ r dqg +w, @ r 1 Htxdwlrq +44, wkhq
iroorzv gluhfwo| iurp Ht1 +7, diwhu lqvhuwlqj wkh deryh dvvxpswlrqv derxw wkh ghwhuplqdqwv
ri wkh fklogehdulqj lqwhqvlwlhv iru wkh v|qwkhwlf frkruw1 Htxdwlrq +46, wkhq iroorzv gluhfwo|
iurp Htv1 +44, dqg +9,1
Wkh vwdwhphqwv derxw wkh phdq dqg yduldqfh ri wkh dgmxvwhg lqwhqvlw| vfkhgxoh pr +d> w,
U
iroorz gluhfwo| iurp Uhvxow 71 Wkh vwdwhphqw pr +d> w,gd @ +4 r , t +W , iroorzv e|
dq dqdorjrxv lqwhjudwlrq dv lq wkh surri ri Uhvxowv : dqg 43 lq NS/ dqg wkh uhodwlrq
U r
p +d> w,gd @ t +W , iroorzv gluhfwo| iurp Uhvxow 51
Surri ri Uhvxow 9= Frqvlghu wkh zrpdq vshflhg lq Uhvxow 91 Ohw \ @ id> w m d |
dqg w @ W . +d {,j ghqrwh wkh olqh vhjphqw lq wkh Oh{lv gldjudp dorqj zklfk wkh
88
olih0frxuvh ri wklv zrpdq rffxuv diwhu vkh dwwdlqv djh |1 Wklv zrpdq lv vxemhfw wr wkh
fklogehdulqj lqwhqvlwlhv pr +d> w,/ zlwk d> w 5 \ / ghqhg lq Ht1 +44, dv orqj dv vkh kdv
qrw surjuhvvhg wr wkh m0wk fklog1 Vlqfh $ lv wkh pd{lpxp ihdvleoh djh dw fklogehdulqj
dqg fklogehdulqj lqwhqvlwlhv duh }hur dw djhv deryh $/ wklv xsshu olplw fdq eh ljqruhg lq
wkh vhw \ wkdw ghqhv wkh frkruw olqh lq wkh Oh{lv gldjudp1 Vwdqgdug vxuylydo dqdo|vlv
lpsolhv wkdw wkh suredelolw| ri qrw h{shulhqflqj d sdulw|0m eluwk iru wklv zrphq lv jlyhq
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e| h{s^ t pr +d> w,,gd`/ zkhuh t ghqrwhv wkh olqh lqwhjudo dorqj wkh olqh lq wkh Oh{lv
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sAc r cBr +{> |,
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judwlrq iurp wkh revhuyhg djh d lq wkh v|qwkhwlf frkruw wr wkh k|srwkhwlfdo djh @
d U +d> w,/ dw zklfk wkh eluwkv zrxog rffxu lq wkh devhqfh ri d ihuwlolw| srvwsrqhphqw/
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|lhogv sAc r cBr +{> |, @ 4h{s^t +W , t !cA +,g` zlwk \ @ i> m |Ur +|> W .|{,
dqg @ W .+{,j1 Wkh Mdfreldq ri wkh wudqvirupdwlrq iurp d wr lv +4@c| Ur +d> w,,3
ehfdxvh wkh lqwhjudwlrq lv dorqj wkh frkruw olqh lq wkh Oh{lv gldjudp1 Wkh whup lqyroylqj
wkh sduwldo ghulydwlyhv lq Ht1 +56, wkhuhiruh ydqlvkhv e| fkdqjlqj wkh yduldeoh ri lqwhjud0
wlrq1 Vlqfh p +d> W , @ t +W ,!cA +d, +vhh Dvvxpswlrq 7,/ vlpsoli|lqj wkh olqh lqwhjudo dqg
fkdqjlqj wkh qrwdwlrq iru wkh yduldeoh ri lqwhjudwlrq |lhogv Ht1 +47, lq Uhvxow 91
Surri ri Uhvxow := Ghqrwh dv sAc r cBr +{> |, wkh frqglwlrqdo sdulw| surjuhvvlrq sured0
elolw| wkdw lv rewdlqhg iurp wkh fdofxodwlrq lq Ht1 +47, zkhq wkh revhuyhg fklogehdulqj
lqwhqvlwlhv p +d> W ,/ lqvwhdg ri wkh dgmxvwhg fklogehdulqj lqwhqvlwlhv p +d> W ,/ duh xvhg
lq wkh lqwhjudwlrq1 Wkh huuru wkdw lv lqfxuuhg e| xvlqj wkh revhuyhg lqvwhdg ri wkh dg0
mxvwhg fklogehdulqj lqwhqvlwlhv wkhq iroorzv iurp wkh idfw wkdw wkh revhuyhg dqg dgmxvwhg
fklogehdulqj lqwhqvlwlhv dw wlph W duh uhodwhg dv p +d> W , @ +4 u +d> W ,,p +d> W ,/ zkhuh
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u +d> W , lv jlyhq lq Ht1 +43,1 Wkhq sAc r cBr +{> |, @ 4h{s^ +3-r E%cA nE+3% p +d> W ,gd` @
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cA ,p +d> W ,gd 31 Iru +W , A 3 dqg +W , 3 zh wkhuhiruh
odvw whup vdwlvhv + +d d
rewdlq +4 sAc r cBr +{> |,, A +4sAc r cBr +{> |,,/ ru htxlydohqwo| sAc r cBr +{> |, ? sAc r cBr +{> |,1
Wkh fruuhfw ydoxh ri sAc r cBr +{> |, lv wkxv xqghuhvwlpdwhg zkhq wkh fdofxodwlrq ri wklv frq0
glwlrqdo sdulw| surjuhvvlrq suredelolw| lq Ht1 +47, lv edvhg rq wkh revhuyhg lqvwhdg ri wkh
dgmxvwhg fklogehdulqj lqwhqvlwlhv1 Lq wkh vshfldo fdvh zkhq +W , @ 3/ wkh whupv lqyroylqj
+W , ydqlvk dqg wkh fdofxodwlrq ri wkh frqglwlrqdo sdulw| surjuhvvlrq suredelolw| edvhg rq
wkh revhuyhg fklogehdulqj lqwhqvlwlhv |lhogv sAc r cBr +{> |, @ 4 +4 sAc r cBr +{> |,,3 EA 1
Surri ri Uhvxow ;= Frqvlghu d zrpdq lq wkh shulrg0W v|qwkhwlf frkruw ri djh { dqg
sdulw| l vshflhg lq Uhvxow ;1 Zh uvw ghulyh wkh uhodwlrq iru vAc r cBr +d> |,/ zklfk ghqrwhv
wkh suredelolw| ri cvxuylylqj* lq sdulw|0m xqwlo djh | frqglwlrqdo rq ehlqj dw sdulw| m dw
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ri h{shulhqflqj d sdulw| m eluwk dw djh d1
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rqg htxdolw| iroorzv e| glhuhqwldwlrq/ wkh wklug htxdolw| iroorzv e| lqvhuwlqj p +d> W , @
t +W ,!cA +d, +vhh Dvvxpswlrq 8,/ dqg wkh qdo htxdolw| iroorzv iurp wkh ghqlwlrq ri
pr +d> w, lq Ht1 +44,1
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lv wkh pd{lpxp ihdvleoh djh dw fklogeluwk1
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sdulw| l vshflhg lq Uhvxow 441 Wkh sdulw| surjuhvvlrq udwlr wr wkh l . 4vw fklog htxdov
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lv wkh suredelolw| ri surjuhvvlqj wr sdulw| m frqglwlrqdo rq dwwdlqlqj sdulw| m 4/ dqg wkh
sdulw| surjuhvvlrq suredelolw| wkhuhiruh iroorzv iurp wkh olih0wlph eluwk suredelolw| dv lq
Ht1 +4:,1 Wkh lqgh{ ri wrwdo ihuwlolw| wkhq iroorzv e| fdofxodwlqj wkh dgglwlrqdo qxpehu ri
fkloguhq eruq wr d zrphq zkr lv djh { dqg sdulw| l lq wkh uhihuhqfh |hdu W iurp hlwkhu
wkh shulrg eluwk olih0wlph suredelolwlhv ru shulrg sdulw| surjuhvvlrq udwlrv dqg dgglqj wkh
sdulw| ri wkh zrpdq lq wkh uhihuhqfh |hdu1
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lv wkh suredelolw| ri cvxuylylqj* lq sdulw| l xqwlo djh d dqg lv fdq eh zulwwhq lq whupv
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r cB r +{> d, @
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Surri ri Uhvxow 46= Wkh ryhudoo phdq djh dw eluwk lv mxvw wkh zhljkwhg dyhudjh ri wkh
sdulw|0vshflf phdq djhv/ zkhuh zhljkwv duh jlyhq e| wkh olihwlph eluwk suredelolwlhv iru
S
wkh uhvshfwlyh sdulwlhv1 Wkh ghqrplqdwru W IAc
r cB r +{, l @
: IAc r cB r +{> $, lq wkh
h{suhvvlrq iru wkh ryhudoo phdq djh lv wkh h{shfwhg qxpehu ri eluwkv ri sdulw| l . 4 dqg
kljkhu wkdw duh frqvlghuhg lq wkh phdq djh fdofxodwlrqv1
Surri ri Uhvxow 47= Wkh surri ri Uhvxow 48 surfhhgv lq irxu vwhsv1 Iru doo vwhsv zh
frqvlghu wkh shulrg0W v|qwkhwlf frkruw ri djh d dqg sdulw| l wkdw lv vshflhg lq Uhvxow 481
8;
Iluvw/ zh frqvlghu wkh uhodwlrq ehwzhhq wkh sursruwlrq ri zrphq vxuylylqj lq wkh
fklogohvv vwdwhv lq wkh wzr srvwsrqhphqw vfhqdulrv/ l1h1/ zh frqvlghu wkh uhodwlrq ehwzhhq
fcf
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r cBr 'f +{> d, dqg GAc r 'fcB r 'f +{> d,1 Wklv uhodwlrq iroorzv e| frqvlghulqj wkh fklogehdu0
lqj lqwhqvlwlhv h{shulhqfhg lq wkh wzr srvwsrqhphqw vfhqdulrv1 Li wkh dqqxdo phdq fkdqjhv
r suhydlov gxulqj wkh olih0frxuvh ri wkh v|qwkhwlf frkruw/ wklv frkruw dw djh d lv vxemhfw wr
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iurp Ht1 +44, e| lqvhuwlqj wkh vshflf dvvxpswlrqv derxw wkh phdq fkdqjh r dqg wkh
yduldqfh fkdqjh r @ 31 Lq wkh srvwsrqhphqw vwrsv vfhqdulr/ wkh v|qwkhwlf frkruw dw djh
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