Migration Age Patterns: Measurement and Analysis (1979)

8/3/2019 Migration Age Patterns: Measurement and Analysis (1979)

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NOT F OR QUOT AT I ONW I T H O U T P E R M I S S I O NO F T H E A U T H O R

M I G R A T I O N A G E P A T T E R N S :MEASUREMENT AND ANALYSIS

L u i s J . C a s t r oA n d r e i R o g e r s

F e b r u a r y 1 9 7 9W P - 7 9 - 1 6

W o r k i n g P a p e r s a r e i n t e r i m r e p o r t s o n w o r k of t h eI n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i sa n d have r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s o ro p i n i o n s e x p r e s s e d h e r e i n do n o t n e c e s s a r i l y r ep r e -s e n t t h o s e of t h e I n s t i t u t e o r of i t s N a t i o n a l M e m b e rO r g a n i z a t i o n s .I N TE RN A TI O NA L I N S T I T U T E F O R A P P L I E D S Y ST EM S A N A L Y SI SA - 2 3 6 1 L a x e n b u r g , A u s t r i a


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T h i s p a pe r was o r i g i n a l l y pr e pa r ed u nd er t h e t i t l e " M od el li ngf o r Management" f o r p r e s e n t a t i o n a t a N a t e r R e se a rc h C e n t r e(U.K. ) C o nf er en c e on " R i v e r P o l l u t i o n C o n t r o l " , O x fo rd ,9 - 1 1 A s r i l , 1979.


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PREFACE

Interest in human settlement systems and policies has beena central part of urban-related work at IIASA since its inception.From 1975 through 1978 this interest was manifested in the workof the Migration and Settlement Task, which was formally conclud-ed in November 1978. ' Since then, attention has turned to dissemina-tion of the Task's results and to the conclusion of its compara-tive study, which is carrying out a comparative quantitativeassessment of recent migration patterns and spatial populationdynamics in all of IIASArs 17 NMO countries.

This paper is a part of the ask's dissemination effort andis the first of several to focus on the age patterns of migrationexhibited in the data bank assembled for the comparative study.It focuses on the mathematical description of observed migrationschedules, the analysis of their age profiles and the study ofhow these profiles are influenced by the age composition of thepopulation in the region of origin.Reports, summarizing previous work on migration and settle-ment at IIASA, are listed at the back of this paper.

Andrei RogersChairmanHuman Settlementsand Services Area


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ACKNOWLEDGEMENTS

The d a t a u se d i n t h i s p a p e r were c o l l e c t e d a s p a r t o f ac o m p ar a t iv e m i g r a t i o n a nd s e t t 1 , em e nt s t u d y c o n du c t ed a t t h eI n t e r n a t i o n a l I n s t i t u t e f o r A pp l i ed S ys te ms A n a l ys i s . Thea u t h o r s g r a t e f u l l y a ck no wl ed ge t h e g e ne r ou s p r o v i s i o n o f d e-t a i l e d S w ed is h p o p u l a t i o n d a t a by Ar ne A r vi d s s on o f t h e Sw ed is hC e n t r a l B ur ea u o f S t a t i s t i c s . Our t h a nk s a l s o g o t o L a r r y L o n go f t h e U .S .C en su s B u re au , K a zi m ie r z D z i e w o n s k i a n d P i o t r Ko r c e l l io f t h e G e og r ap h ic a l I n s t i t u t e o f t h e P o l i s h Academy o f S c i e n c e s ,a n d P a u l D r e w e o f t h e U n i v e r s i t y o f D e l f t , f o r m ig r a t i o n d a t a o nt h e Un i t ed S t a t e s , P o la n d, a nd t h e N et h e r l a n ds , r e s p e c t i v e l y .F i n a l l y , e a r l i e r work on t h i s t o p i c b e n e f i t e d g r e a t l y f r om t h ee f f o r t s o f o u r c o l l e a g u e a t IIASA, R i c h a r d R a q u i l l e t .


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ABSTRACT

This paper develops support for three principal points.First, the profiles of age-specific gross migration rates allover the world have a fundamental regularity that can be cap-tured and expressed in mathematical form. Second, this mathe-matical "model" schedule summarizes the empirical regularity ina way that permits analytical examinations to be carried outregarding the fundamental properties of the migration age pro-files. Finally, migration rate schedules may be convenientlydecomposed to illuminate the influences on migration patternsof migration level, the age composition of migrants, and theage composition of the population in the region of origin.


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CONTENTS

MIGRATION AGE PATTERNS: MEASUREMENT AND ANALYSIS, 1M i g r a t i o n R a t e s a n d S c h e d u l e s , 2M odel M i g r a t i o n S c h e d u l e s , 1 3P r o p e r t i e s o f t h e Model M i g r a t i o n S ch e d u l e , 26M i g r a t i o n P r o p o r t i o n s a nd S c h e d u l e s , 36C o n c l u s i o n , 46R e f e r e n c e s , 47A p p e n d i x A : A g g r e g a t i o n o f S w ed is h C o u n t i e s i n t o R e gi o n s , 49A ppend i x B : T a b l e s I , 11, a n d 111, 50A ppend i x C : N o n l i n e a r P a r a m e t e r E s t i m a t i o n i n M od el M i g r a t i o nS c h e d u l e s , 53


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MIGRATION AGE PATTERNS: MEASUREMENT AND ANALYSIS

Miqration studies have in the past exhibited acuriouslyambi-valent position with reqard to the measurement of migration.This ambivalence is particularly striking because of the contrastit poses with respect to the corresponding studies of mortalityand fertility (natality)--literature that is richly endowedwith detailed discussions of measurement problems. Haenszel(1967) attributes this paradox to the stronq influence ofRavenstein's early contributions to miqration analysis:

Work on miqration and population redistribution appearsto have been stronqly influenced by the early successesof Ravenstein in formulating "laws of miqration". Sub-sequent papers have placed a premium of the developmentand testing of new hypotheses rather than on descriptionsof facts and their collation. ..This is in contrast tothe history of vital statistics. While Graunt more thantwo centuries before Ravenstein, had made several import-ant generalizations from the study of "bills of mortality"in London, his successors continued to concentrate ondescriptions of the forces of mortality and natality bymeans of rates based on populations at risk (Haenszel,1967, p. 260).It is natural to look to the state of mortality and fertili-

ty measurement for guidance in developing measures of migration.Like mortality, migration may be described as a process of inter-state transfer; however, death can occur but once, whereasmigration is a potentially repetitive event. This suggests theadoption of a fertility analog; but migration's definitionaldependence on spatial boundaries introduces measurement diffi-culties that do not arise in fertility analysis.

Miqration measurement can usefully apply concepts borrowedfrom both mortality and fertility analysis, modifying themwhere necessary to take into account aspects that are peculiarto spatial rrobility. From mortality analysis, miqration canborrow the notion of the life table, extending it to includeincrements as well as decrements, in order to reflect the mutual


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interaction of several regional cohorts (Rogers, 1973a,b, and1975; Rogers and Ledent, 1976). From fertility analysis,migration can borrow well-developed techniques for graduatingage-specific schedules (Rogers, Raquillet, and Castro, 1978).Fundamental to both "borrowings" is a workable definition ofmigration rate.

Migration Rates and SchedulesDuring the course of a year, or some such fixed interval

of time, a number of individuals living in a particular com-munity change their regular place of residence. Let us callsuch persons mobiles to distinguish them from those individualswho did not change their place of residence, i.e., the-on-mobiles. Some of the mobiles will have moved to a new community--f residence; others will simply have transferred their house-hold to another residence within the same community. Theformer may be called movers, the latter are relocators. A fewin each category will have died before the end of the unittime interval.

Assessing the situation with respect to the start and theend of the unit time interval, we may divide movers who -ur-vived to the end of the interval into two groups: those livingin the same community of residence as at the start of the inter-val and those living elsewhere. The first group of movers willbe referred to as surviving returnees, the second will becalled surviving migrants. An analogous division may be madeof movers who died before the end of the interval to definenonsurviving returnees and nonsurviving migrants.

A move,then is an event: a separation from a community.A mover is an individual who has made a move at least onceduring a given interval of time. A migrant (i.e., a survivingor nonsurviving migrant), on the other hand, is an individualwho at the end of a given time interval no longer inhabits thesame community of residence as at the start of the interval.


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(The act of separation from one state is linked to an additionto another). Thus paradoxically, a multiple mover may be anonmigrant by our definition. This is illustrated by life lineC in the multiregional Lexis diagram in Figure 1 . Because thisparticular mover returned to the initial place of residencebefore the end of the unit time interval, no "migration" tookplace.

The focus on migrants instead of on movers reflects theneed at some point to calculate probabilities. As Haenszel(1967) has observed:

the label "migration" had been applied to two related,but different, universes of discourse--a population of"moves" andapopulation of "people who move". A uni-verse of "moves" can be generated by simultaneous clas-sification of individuals by initial and subsequentplace of residence, and the data provide useful des-criptions of population redistribution. Such results,however, do not lend themselves to probability state-ments. Probabilities can be computed only for denum-erable populations at risk, whether they be people,telephone poles, or transistors (Haenszel, 1967, p.254).The simplest and most common measure of migration is the

crude migration rate, defined as the ratio of the number of--migrants, leaving a particular population located in spaceand time, to the average number of persons (more exactly, thenumber of person-years) exposed to the risk of becoming mi-grants. *

Because migration is highly age selective, with a largefraction of migrants being the young, our understanding of mi-gration patterns and dynamics is aided by computing migrationrates for each single year of age. Summing these rates overall ages of life gives the gross migraproduction rate GMR ) , themigration analog of fertility's gross reproduction rate.

* We define migration to be the transition between states ex-perienced by a migrant.* * Because data on nonsurviving migrants are generally unavail-able, the numerator in this ratio generally excludes them.


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REGION i : t t* t + l TIME

TIME

Figure 1. Two-region Lexis diagram.


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Figure 2 indicates that age-specific annual rates of resi-dential mobility among whites and blacks in the U.S. during1966-1971 exhibited a common profile. Mobility rates amonginfants and young children mirrored the relatively high ratesof their parents--young adults in their late twenties. Themobility of adolescents was lower, but exceeded that of youngteens, with the latter showing a local low point around agefifteen. Thereafter mobility rates increased, attaining a highpeak at about age twenty-two and then declining monotonicallywith age to the ages of retirement. The mobility levels ofboth whites and blacks were roughly similar, with whites show-ing a gross migraproduction rate of about 14 moves and blacksone of approximately15, over a lifetime undisturbed by mortalitybefore the end of the mobile ages.

Although it has been frequently asserted that migrationisstron2ly sex selective, with males being more mobile thanfemales, recent research indicates that sex selectivity ismuch less pronounced than age selectivity,and that it is lessuniform across time and space. Nevertheless, because mostmodels and studies of population dynamics distinguish betweenthe sexes, most migration measures do also.

Figure 3 illustrates the age profiles of male and femalemigration schedules in four different countries at about thesame point in time between roughly comparable areal units:communes in the Netherlands and Sweden, voivodships in Poland,and counties in the U.S. The migration levels for all butPoland are similar, varying between 3.5 and 5.3 moves per life-time; and the levelsfor males and females are roughly the same.The age profiles, however, show a distinct, and consistent,difference. The high peak of the female schedule always pre-cedes that of the male schedule by an amount that appears toapproximate the difference between the average ages at marriageof the two sexes.


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AGEIhite- l a ckFigure 2. Observed annual migration rates by color and singleyears of age: The United States, 1966-1971.


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Under n orm al s t a t i s t i c a l c o n d i t i o n s , p o i n t - t o - p o i n t move-m ents a r e a g g r e ga t e d i n t o s t r e a m s be tw een o n e c i v i l d i v i s i o nand a n o t h e r ; c o n se q u en t ly , t h e l e v e l o f i n t e r r e g i o n a l m ig ra -t i o n d ep en ds o n t h e s i z e o f t h e a r e a l u n i t s e l e c t e d . Thus , i ft h e a r e a l u n i t c ho se n i s a m inor c i v i l d i v i s i o n s u ch a s ac o u n t y o r a commune, a g r e a t e r p r o p o r t i o n o f r e s i d e n t i a l l o c a -t i o n w i l l b e i n cl ud e d a s m i g r a t i o n t ha n i f t h e a r e a l u n i tc h o s e n i s a m aj o r c i v i l d i v i s i o n s uc h a s a s t a t e o r a p r o v in c e.

F i g u r e 4 p r e s e n t s t h e a g e p r o f i l e s o f f em al e m o b i l i t ya nd m i g r a t i o n s c h e d u l e s a s me as ur ed by d i f f e r e n t s i z e s o f a r e a lu n i t s : 1 ) a l l moves from o ne r e s i d e n c e t o a n o t h e r , 2 ) c h a ng e so f r e s i d e n c e w i t h i n c o u nt y b o u n d a ri e s , 3 ) m i g r a t i o n b et we enc o u n t i e s , a nd 4 ) m i g r a t i o n be tw een s t a t e s . The r e s p e c t i v e f o u rg r o s s m i g ra p ro d u ct i on r a t e s ( GMR s ) a r e 1 4 . 3 , 9 . 3 , 5 . 0, a nd 2 . 5 ,r e s p e c t i v e l y . The f o u r a g e p r o f i l e s ap p e ar t o b e r em ar ka bl ys i m i l a r , i n d i c a t i n g t h a t t h e r e g u l a r i t y i n a g e p a t t e r n p e r s i s t sa c r o s s a r e a l d e l i n e a t i o n s o f d i f f e r e n t s i z e s .

F i n a l l y , m i g r a t i o n o c c u r s o v e r t i m e a s w e l l a s a c r o s s s p a c e ;t h e r e f o r e , s t u d i e s o f i t s p a t t e r n s must t r a c e i t s o c c u r r e n c ew i t h r e s p e c t t o a t i m e i n t e r v a l , a s w e l l a s o v e r a s y st em o fg e og r a ph i ca l a r e a s . I n g e n e r a l , t h e l o ng e r t h e t i m e i n t e r v a l ,t h e l a r g e r w i l l b e t h e number o f r e t u r n m ov er s a nd n o n s u r v i v i n gm i g r a n ts a n d, h e nc e , t h e m ore t h e c o u n t on m i g r a n t s w i l l u nd er -s t a t e t h e number o f i n t e r - a r e a m ov ers ( a nd , o f c o u r s e , a l s o o fm ov es ). P h i l i p R e e s , f o r e xample , a f t e r ex am in ing t h e r a t i o so f o n e - ye a r t o f i v e - y e a r m i g r a n t s b et we en t h e S t a n d a r d R e gi on so f G r e a t B r i t a i n , fo und t h a t

t h e number o f m i g r a n t s r e c o r d e d o v e r f i v e y e a r s i n a ni n t e r r e g i o n a l f lo w v a r i e s f rom f o u r t i m e s t o two t i m e st h e number o f m i g r a n t s r e c o r d e d o v e r o n e y e a r .( R e e s , 1 9 7 7 , p . 2 4 7 ) .A f un da m en ta l a s p e c t o f m i g r a t i o n i s i t s c h a n g e o v e r t i m e .

A t i m e s e r i e s o f a g e - s p e c i f i c m i g r a t i o n r a t e s may b e u s e f u l l ys e t o u t i n t h e f orm o f a t a b l e w i t h a g e s f o r rows and c a l e n d a r


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AGE- OTAL--+-+- W/COUNTY- /COUNTY- /STATEFigure 4. Observed female annual migration rates by levels of

areal aggregation and single years of age: the UnitedStates, 1966-1971.


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Ages 0 - 5 00123456783

1 0111 21 31 41 51 6171 8192 021222 32425262 72 8293 031323 3343 53 63 73 83Y4 0414 74:44 5rl C,440

495 0

Table 1. Observed femal e annual migrat ion rates by single yearsof age: the Netherlands, migra tion between commu nes,1972-1976.

R A T E S P E R T H O U S A N D1'372 1 9 7 3 1 9 7 4 1 9 7 5 1 9 t ,

R A T E S P E R T H O U S A N D1 9 7 2 1 9 7 3 1 9 7 4 1 9 7 5 1 9 7 6

I

Ages 51-945 15 25 35 45 55 6575 85 96 0

616 26 36 46 56 66 76 86 97 0717 27 374757 67 77 8798 08 18 2838 4e.5

8 6870 88 99 09 19 29 39 4


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a g e of m i g r a t i o n d e c l i n e s m o no t o ni c a l l y fro m c o h o r t t o c o h o r t .And, i f d e c l i n i n g economic c o n d i t i o n s l e a d p o t e n t i a l m i g r an t st o d e l a y t h e i r m ig r a t i o n , c u r r e n t p e r i o d i n d i c e s o f m i g ra t i o nl e v e l may d e c l i n e o n l y t o b e f o l l o w ed by a c o m p e n s a t o r y i n c r e a s ei n t h e f u t u r e .

The u s e f u l n e s s o f a c o h o r t ap pr oa ch i n m i g r a t i o n a s i n f e r t -i l i t y a n a l y s i s l i e s i n t h e i mp or ta nc e o f h i s t o r i c a l e x p e r i en c et o t h e e x p l a n a t i o n o f c u r r e n t b eh a v i o r . A s M o r r is o n ( 19 70 ) p o i n t so u t , m i g r a t i o n i s i nd u ce d by t r a n s i t i o n s f ro m o n e s t a g e o f t h el i f e c y c l e t o a n o t h e r , a n d " c h r o n ic " m i g r a n t s may a r t i f i c i a l l yi n f l a t e t h e m ig ra t i on r a t e s of o r i g i n a r e a s h e a v i l y p o p u la t e dw i t h m i g ra t io n - pr o n e i n d i v i d u a l s . B oth i n f l u e n c e s o n p e r i o dm i g r a t i o n r a t e s a r e r e a d i l y a s s e s s e d by a c o h o r t a n a l y s i s .

I t i s t h e m i g r a t i o n of a p e r i o d , how ever , a nd n o t t h a t o fa c o h o r t , t h a t de t e r m i ne s t h e s ud den r e d i s t r i b u t i o n o f a n a t i o n a lp o p u la t i o n i n r e s po n s e t o e conomic f l u c t u a t i o n s , a nd i t i s p e r i o dm i g r a t i o n r a t e s t h a t a r e ne ed ed t o c a l c u l a t e s p a t i a l p o p u l a t i o np r o j e c t i o n s .

C u r r e n t p e r i o d m i g ra t i o n i n d i c e s d o n o t d i s t i n g u i s h t r e n dfro m f l u c t u a t i o n a nd t h e r e f o r e may be d i s t o r t e d ; c u r r e n t c o h o r tm i g r a t i o n i n d i c e s a r e i n c om p l et e . Thus it may b e u s e f u l t o dr awon R y d e r ' s ( 19 64 ) t r a n s l a t i o n t e c h n i qu e t o c ha ng e f ro m o n e t o t h eo t h e r . A s K e y f i t z ( 19 77 , p . 25 0 ) o b s e r v e s , t h e c o h o r t a n d p e r i o dmoments i n R y d e r ' s f o rm u l ae c a n " b e i n t e r p r e t e d , n o t a s c h i l d -b e a r i n g , b u t a s m o r t a l i t y , m a r r i a g e , s c h o o l a t t e n d a n c e , in co me , o rsome o t h e r a t t r i b u t e o f i n d i v i d u a l s " . M i g ra t i o n i s c l e a r l y s u c ha n a t t r i b u t e .

The i mp or ta nc e of h i s t o r i c a l e xp e r i e n ce i n i n t e r p r e t i n g andu n d e r s ta n d i n g c u r r e n t m i g r a t i o n b e h a v i o r l e d P e t e r M or ri so n ( 19 70 ,p . 9 ) t o d e f i n e t h e n o t i o n o f s t a g i n g a s b e i n g " an y l i n k a g e b e t -w een a p r i o r s e q u e n c e a nd s u b s e q u e n t m i g r a t i o n b e h a v i o r " . Morri-s o n r e c o g n i z e s f o u r k i n d s o f s t a g i n g : g e o gr a p hi c , l i f e - c y c l e ,s oc io ec on om ic , an d e x p e r i e n t i a l . G e og r ap h ic a l s t a g i n g r e f e r s t or e t u r n m i g r a t i o n a nd t o w h a t i s c o n v e n t i o n a l l y u n d e r s t oo d t o mean


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" s t a g e m i g r a t i o n " , i . e . , t h e i d e a t h a t m ig r a n t s t e nd t o movet o p la c e s n o t v e r y d i s s i m i l a r f rom t h o s e th e y l e f t be h i nd .L i fe - c yc l e s t a g i n g v i ew s m i g r a t i o n a s a r i s i n g o u t o f b r e a k s i na n i n d i v i d u a l ' s o r ho us eh ol d ' s l i f e c y c l e , s uc h a s e n t r y i n t ot h e l a b o r f o r c e , m a rr i a g e, c h i l d r e a r i n g , r e t i r e m e n t . S oc io -e c o n o m i c s t a g i n g sees m i g r a t i o n s e q ue n c e s a s b e i n g c o n d i t i o n e dby s o c i o s t r u c t u r a l f a c t o r s s uc h a s o c c up a t i o n , e d u c a t i o n a l a t -t a i n m e nt , and income l e v e l . F i n a l l y , e x p e r i e n t i a l s t a g i n g re-f e r s t o movement e x p e r i e n c e i n t e r m s o f number o f p r e v i o u smoves a nd d u r a t i o n s i n c e t h e l a s t move. I t i s t h e " p a r i t y "d im e ns io n o f m i g r a t i o n a n a l y s i s a nd w i l l b e r e f e r r e d t o a s" m i g r a t i v i t y " .

C a l c u l a t i o n s o f m i g ra t io n r a t e s o f i n c r e a s i n g s p e c i f i c i t ys e e k t o unconfound t h e " t r u e " m i g r a t i o n r a t e s f ro m w e i g h t s t h a tr e f l e c t t h e a r i t h m e t i c a l i n f l u e n c e o f t h e p a s t . T h i s p r o c e s so f m ea su r i n g m ig r a t i o n " a t d i f f e r e n t l e v e l s o f s p e c i f i c i t y o fo c c u r r e nc e a nd e x p o s u r e y i e l d s p r o d u c t s w h ic h dra w e v e r f i n e rd i s t i n c t i o n s be tw ee n c u r r e n t b e h av i or a nd t h e r e s i d u e o f p a s tb eh av io r r e f l e c t e d i n t h e e xp os ur e d i s t r i b u t i o n a t a ny t i me "( R y d er , 1 9 7 5, p . 1 0 ) .

S uc h p r o d u c t s may b e w e i g h t e d a nd a g g r e g a t e d t o p r od u c et h e "c ru d e" r a t e s o f h i g h e r l e v e l s o f a g g r e g a t i o n . F or exa mp le ,t h e ag e- se x s p e c i f i c m i g ra t io n r a t e i s a w e i g h t e d a g g r e g a t i o nw i t h r e s p e c t t o t h e m i g r a t i on " p a r i t y - d u r a t i o n" d i s t r i b u t i o nj u s t a s t h e c ru de m ig r a t i on r a t e i s a w e i gh te d a g g r e g a t i o n w i t hr e s p e c t t o t h e ag e -s e x d i s t r i b u t i o n .

?.lode1 M ig r a t io n Sch edu lesI t a p p e a r s t h a t t h e mos t p ro mi ne nt r e g u l a r i t y foun d i n

e m p i r i c a l s c h e d u le s o f a g e - s p e c i f i c m i g r a t io n r a t e s i s t h e s e-l e c t i v i t y o f m i g r a t i o n w i t h r e s p e c t t o ag e. Young a d u l t s i nt h e i r e a r l y t w e n t i e s g e n e r a l l y show t h e h i g h e s t m ig r a t i o n r a t e sa n d yo un g t e e n a g e r s t h e l o w e s t . The m i g r a t io n r a t e s o f c h i l d r e nm i r r o r t h o s e o f t h e i r p a r e n t s ; h en ce t h e m ig r a t i on r a t e s of i n -f a n t s e xc ee d t h o s e o f a d o l e s c e n t s . F i n a l l y , m i g r a t i o n s t re a m s


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directed toward regions with warmer climates and into or out oflarge cities with relatively hiqh levels of social services andcultural amenities often exhibit a "retirement peak" at aqes inthe mid-sixties or beyond.

Fiqure 5 illustrates a typical observed age-specific mig-ration schedule (the jagged outline) and its graduation by amodel schedule (the superimposed smooth outline) defined as the--sum of four components:

1) a single negative exponential curve of the pre-laborforce ages, with its rate of descent, al:

2) a left-skewed unimodal curve of the labor force ageswith its rates of ascent and descent, h 2 and a2, res-pectively;

3) an almost bell-shaped curve of the post-labor forceages with its rates of ascent and descent, X 3 and a3,respectively; and

4) a constant curve c, the inclusion of which improvesthe quality of fit provided by the mathematical ex-pression of the schedule.

The decomposition described above suggests the followingsimple sum of four curves (Rogers, Raquillet, and Castro, 1978)*:

* Both the labor force and the post-labor force components inequation (1) are described by the "double exponential" curveformulated by Coale and McNeil (1972) for their studies ofnuptiality and fertility.


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a

aeod

opea

cccv

xp=hlowpn

A=aeoa

oa

occv

x=hhgp

a2aeod

oa

occv

xi=heremep

A=aeoa

op

a

occv

X=ha

ocs

a3aeod

op

a

occv

A=hpeas

h1GRATIONRATE,Mo

I

c=ca

B=hump

Figure5

Theodelmigrationschedule.


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The "full" model schedule in equation (1) has eleven parameters:a h3, and c. The profile ofa l I l I 2, u2, a2, h2, a3, u3, 3 I

the full model schedule is defined by seven of the eleven param-eters: ul, u2, a2, h2, 9,3, and h3. Its level is determined

ay the remaining four parameters: al, 2, a3, and c. A changein the value of the gross migraproduction rate of a particularmodel schedule alters proportionally the values of the latterbut does not affect the former. However, as we shall see in thenext section, certain aspects of the profile also depend on theallocation of the schedule's level among the labor, pre-labor,and post-labor force age components, and on the share of thetotal level accounted for by the constant term, c. Finally,migration schedules without a retirement peak may be representedby a "reduced" model with seven parameters, because in such in-stances the third component of equation (1) is omitted.

Table 2 sets out illustrative values of the basic and derivedmeasures presented in Figure 5. The data refer to 1974 migrationschedules for an eight-region disaggregation of Sweden (seeAppendix A). The method chosen for fitting the model scheduleto the data was a functional-minimization procedure known asthe modified Levenberg-Marquardt algorithm.* Minimum chi-square estimators were used instead of least squares estimators.The differences between the two parametric estimates tend to besmall, and because the former give more weight to age groupswith smaller rates of migration, we use minimum chi-square esti-mators in the remainder of the paper.

To assess the quality of fit that the model schedule pro-vides when it is applied to observed data, we calculated the"mean absolute error as a percentage of the observed mean":

This measure indicates that the fit of the model to the Swedishdata is reasonably good, the eight indices of goodness-of-fitbeing 6.87, 6.41, 12.15, 11.01, 9.31, 10.77, 11.74, and 14.82,for males and 7.30, 7.23, 10.71, 8.78, 9.31, 11.61, 11.38,and 13.28 for females. Figures 6 and 7 illustrate graphically*See Appendix C and Brown and ~ennis 1 972) Levenberg (1 44)and Marquardt (1963).


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Table 2. Parameters and variables defining observed modelmigration schedules: Swedish regions, 1974.

Parameters 2. East Mid- 3. South Mid-1. Stockholmand dl -Sweden dle-Sweden 4. SouthVariables M F M F M F M F

*The GMR, its percentage distribution across the three major age categories(i.e., 0-14, 15-64, 65+), and iT all are calculated with a model schedulespanning an age range of 95 years.


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Table 2. Parameters and variables defining observed modelmigration schedules: Swedish regions, 1974,(continued) .

P aram e t e r s 6. No rth Mid- 7. Lower5 . West 8. Upperand dle-Sweden North-Sweden North-Sweden

V a r i a b l e s M F M F M F M F

GMRa1

a 2u2a2

2c-n% (0-14)%(15-64)% (6 5 + )6 l c&12612O2X~

==hXAB

0 .82 0 .84-0 2 1 -0 2 2. 0 8 9 . l o 6-046 -055

20.36 19.36- 0 9 1 - 1 1 4-416 -442.001 .002

28.46 28.3623.55 23.1970.38 69.06

6.07 7.7514 .50 10 .10

-458 .395-979 .9264.56 3.88

1 6 . 1 1 1 5 . 2 423.80 22.29

7.69 7.0529.57 27.42

-024 -028

1 .25 1 .37-031 .028-1 0 4 . l o 0.084 .114

19 .75 18 .13. l o 3 . 1 3 6-437 .572.002 .004

28.11 27.9821.52 19-4872.49 72.66

5.99 7.8513 .40 7 .64

-3 6 9 -2 4 11 .00 .7304.23 4 .20

15.56 14.7422.93 20.58

7.37 5.8429.92 27.09

.044 .061

1 . 3 8 1 .52.034 -031.123 -119. l o 9 . 1 4 1

19 .62 17 .93.I18 .148.427 .701-003 -004

28.27 27.9719 .83 18 .2573.57 73.60

6 .59 8 .141 1 . 3 9 7 . 4 0

.309 .2191 . 0 4 -8 0 13.63 4 .74

15.19 15.0722.56 20.12

7.37 5 .0530.15 26.94

.054 .080

1 . 0 8 1 . 2 9.024 -023-136 .126.079 -115

19 .48 17 .60-114 .142.448 .720.003 .004

29.94 28.8818 .28 16 .4573.43 74.67

8 .29 8 .888 .24 '5 .99.304 .2001 .19 -8893.93 5 .08

15.20 14.7922.47 19.83

7.27 5.0431.61 28.36

- 0 4 1 - 0 6 7


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'5L 2. .d$ 5 .& L o -


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this goodness-of-fit of the model schedule to the observed re-gional migration data for Swedish females.

Model migration schedules of the form specified in equation(1) may be classified into families according to the ranges ofvalues taken on by their principal parameters. For example, wemay order schedules according to their migration levels asdefined by the values of the four level parameters in equation( 1 ) , i. e., a a a and c (or by their associated gross migra-1' 2' 3 'production rates). Alternatively, we may distinguish scheduleswith a retirement peak from those without one, or we may refer toschedules with relatively low or high values for the rate ofascent A 2 or the mean age n. In many applications, it is alsomeaningful to characterize migration schedules in terms of severalof the fundamental measures illustrated in Figure 5, such as thelow point, xz, the high peak, xh, and the retirement peak, x .rAssociated with the first pair of points is the labor force shift,X, which is defined to be the difference in years between theages of th eh igh peak and the low point, i.e., X = xh - X ~ ' The

,. increase in the migration rate of individuals aged xh over thoseaged xZ will be called the jump, B.

The close correspondence between the migration rates ofchildren and those of their parents suggests another importantshift in observed migration schedules. If, for each point xon the post-high-peak part of the migration curve, we obtain(where it exists) by interpolation the age, x - Ax say, withthe identical rate of migration on the pre-low-point part ofthe migration curve, then the average of the values of Ax,calculated incrementally for the number of years between zeroand the low-point xZ, will be defined to be the observedparental shift, A.

An observed (graduated) age-specific migration schedulemay be described in a number of useful ways. For example,references may be made to the heights at particular ages, tolocations of important peaks or troughs, to slopes along theschedule's age profile, to ratios between particular heights orslopes, to areas under parts of the curve, and to both horizontal


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and vertical distances between important heights and locations.The various descriptive measures characterizing an age-specificmodel migration schedule may be conveniently grouped into thefollowing categories and sub-categories:

Basic measures (the11 fundamental parameters and their ratios)heights : al, a2, a3, clocations: p2, p3slopes : al, a2, h2, a3, h 3ratios : d I c = al/cf d 1 2 = al/a2, dj2 = a a 2'

012 = al/a2, o2 = h /a o = h /a2 2' 3 3 3Derived measures (properties of the model schedule)

areas : GMR, % (0-1 )' $ (15-64) X (65+)-locations: n, xL, xh, xrdistances: X, A, B

A convenient approach for characterizing an observed modelmigration schedule (i.e., an empirical schedule graduated byequation (1)) is to begin with the central labor force curve andthen to "add-on" the pre-labor and post-labor force components,and the constant component. This approach is represented graph-ically in Figure 8.

One can imagine describing a decomposition of the model mi-gration schedule along the vertical and horizontal dimensions,e.g., allocating a fraction of its level to the constant compon-ent and then dividing the remainder among the other three (ortwo) components. The ratio d l c = al/c measures the former al-location, and d l = al/a2 and d32 = a /a reflect the latter3 2division.


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'a 56l a 0u o aW M E !0 0 0a r u o

M U2 5k O Oa r u o


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The heights of the labor force and pre-labor force componentsare reflected in the parameters a2 and al, respectively, thereforethe ratio a2/al indicates the degreeof "labor dominance", and itsreciprocal, 612 = al/a2, the index of child dependency, measuresthe level at which children migrate with their parents. Thusthe lower the value of 612, the lower is the degree of childdependency exhibited by a migration schedule and, correspondingly,the greater is its labor dominance. This suggests a dichotomousclassification of migration schedules into child dependent andlabor dominant categories.

An analogous argument applies to the post-labor force curve,and 632 = a3/a2 suggests itself as the appropriate index. Howeverit will be sufficient for our purposes to rely simply on the valuetaken on by the parameter A3, with positive values pointing outthe presence of a retirement peak and a zero value indicatingits absence. High-values f A3 will be interpreted as identifyingretirement dominance.

Labor dominance reflects the relative migration levels ofthose in the working ages relative to those of children andpensioners. Labor asymmetry refers to the shape of the skewedbell-shaped curve describing the profile of labor-force-age mi-gration. Imagine that a perpendicular line, connecting the highpeak with the base of the bell-shaped curve (i.e., the jump, B),divides the base into two segments X and Y as, for example, inthe schematic diagram: -


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Clearly, the ratio Y/X is an indicator of the degree of asymmetryof the curve. A more convenient index, using only two parametersof the model schedule is the ratio a2 = A2/a2, the index of laborasymmetry. Its movement is highly correlated with that of Y/X,because of the approximate relation:

A2 B B Y0 = - q , - + - = -2 "2 X Y X

where a denotes proportionality. Thus a2 may be used to classifymigration schedules according to their degree of labor asymmetry.

Again, an analogous argument applies to the post-labor forcecurve, and a3 = A3/a3 may be defined to be the index of retirementasymmetry.

When "adding-on" a pre-labor force curve of a given level,to the labor force component, it is also important to indicatesomething of its shape. For example, if the migration rates ofchildren mirror those of their parents, then al should be approx-imately equal to a2, and B1 2 = a1/a2, the index of parental-shift regularity, should be close to unity.

The Swedish regional migration patterns described inFigures 6 and 7, and in Table 2, may be characterized in termsof the various basic and derived measures defined above. Webegin with the observation that the outmigration levels in allof the regions are similar, ranging from a low of 0.82 for malesin Region 5 to a high of 1.85 for females in Region 2. This sim-ilarity permits a reasonably accurate visual assessment and char-acterization of the profiles in Figures 6 and 7.

Large differences in gross migraproduction rates give riseto slopes and vertical relationships among schedules that arenon-comparable when examined visually. Recourse then must bemade to a standardization of the areas under the migration curves,for example, a general re-scaling to a GMR of unity. Note that


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this difficulty does not arise in the numerical data in Table 2,because, as we pointed out earlier, the principal slope and lo-cation parameters and ratios used to characterize the schedulesare not affected by changes in levels. Only heights, areas, andvertical distances, such as the jump, are level-dependent measures.

Among the eight regions examined, only the first two exhibita definite retirement peak, the male peak being the more dominantone in each case. The index of child dependency is highest inRegion 1 and lowest in Region 8, distinguishing the latter re-gion's labor dominant profile from Stockholm's child dependentoutmigration pattern. The index of labor asymmetry varies froma low of 2.37, in the case of males in Region 4 to a high of5.08 for the female outmigration profile of Region 8. Finally,with the possible exception of males in Region 1 and females inRegion 6, the migration rates of children in Sweden do indeedseem to mirror those of their parents. The index of parental-shift regularity is 1.26 in the former case and .730 in thelatter; for most of the other schedules it is close to unity.

Table 2 describes interregional migration flows betweenSwedish regions, Tables I, 11, and I11 in Appendix B providecomparable descriptions for the migration schedules previouslyillustrated in Figures 2,3, and 4. They present the necessarybasic and derived measures with which to carry out a comparativeanalysis of the differences in levels and age profiles exhibitedby those schedules, an analysis that is beyond the scope ofthis paper.

Properties of the Kodel Migration ScheduleThe age profiles of model migration schedules without a re-

tirement peak are determined by the four parameters al, p2, a*,h 2 and the ratio 612 = a1/a2. TO simplify our analysis of theproperties of such schedules we shall assume that their index ofparental-shift regularity is sufficiently close to unity for usto set al = a2. Consequently, "pure" profile measures such asxZ, xh, and A, will vary only as a function of the four parameters:


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P2, a2, X 2 t and 6 . The ,6 0 Ts 100


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0hlhlIIurlw

5d3aJrl3aaJCU[I]

[I]c0drn Urn aE -5M aaU 0

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L4[I] (da~ arl3 ua I2aJ aJ5 :[I] wwI2 .d0 a.u acd I2L4 aM.d rne:rl cdaJ L4a0 I2.rl U(d uU 3.d aU 0aJ L45 30 L4a M&'-smaJL43M.rl!&


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40/67

The s i x s c h e d u l e s i n F i g u r e 9B d e p i c t t h e c o r r e s p o n d i n g twof a m i l i e s o f c h i l d d e pe nd en t = . 8 ) p r o f i l e s . The r e s u l t sa r e g e n e ra l l y s i m i l a r t o t ho s e i n F i gu r e 9A, w i t h t h e e x ce p t io nt h a t t h e r e l a t i v e i m po rt an ce o f m ig ra t i on i n t h e p r e- l ab o r f o r c ea g e g r o u p s i s i n c r e a s e d c o n s i de r a b ly . The p r i n c i p a l e f f e c t s o ft h e c ha ng e i n 6 1 2 a r e : ( 1 ) a r a i s i n g of t h e i n t e r c e p t a l + ca l on g t h e v e r t i c a l a x i s , and ( 2 ) a s i m ul ta ne ou s r e d u c t i o n i n t h eh e i g h t o f t h e l a b o r f o r c e component i n o r d e r t o m a i n t a i n a con-s t a n t a r e a of u n i t y u n de r e a ch c ur ve .

F i n a l l y , t h e do ze n s c h e d u l e s i n F i g u r e s 9C a nd 9D d e s c r i b es i m i l a r f a m i l i e s of m i gr a t io n c u r v es , b u t i n t h e s e p r o f i l e s t h er e l a t i v e c o n t r i b u t i o n of t h e c o n s t a n t component t o t h e u n i t GMR

-- --h as be en i nc r e a se d s i g n i f i c a n t l y ( i . e . , 6 1 c = 2 . 6 ) . I t i s i m p o r -t a n t t o n o t e t h a t s u ch " p ur e" m ea su re s o f p r o f i l e a s x t , x h, X ,and A r em a in u n a f f e c t e d by t h i s ch a n ge , w h e re a s " i mp u re " p r o f i l em e a su r e s s u c h a s n a n d % (O -1 4) now t a k e o n a d i f f e r e n t s e t o fv a l u e s .

--.-- -I t i s d i f f i c u l t t o ex am in e, i n F ig u r e 9 , how ch an ge s i n t h e

v a l u e s o f t h e f un da me nt al f o u r p ar a m et e rs a f f e c t p r o f i l e m ea su re ss u ch a s A , x t , a nd x h. F i gu r e 10 i l l u s t r a t e s , t h e r e f o r e , how t h ep a r e n t a l s h i f t , f o r example , v a r i e s a s a f u n c t i o n o f p 2 , a 2 , X 2 ,and 6 1 2 . I t s v a r i a t i o n a p p ea r s t o b e d i r e c t l y c o r r e l a t e d w i t ht h e v a r i a t i o n o f p 2, i n v e r s e l y a s s o c i a t e d w i t h ch an ge s i n a and26 1 2 , a nd v e r y w ea kl y i n f l u e n c e d b y v a l u e s o f h 2 . T h i s p a t t e r no f r e l a t i o n s h i p s may b e i l l u m i n a t e d by a n a n a l y t i c a l a rg um en t.

F o r a g e s i m m ed i at e ly f o l l o w i n g t h e h i g h pe ak x h , t h e l a b o rf o r c e c om pon ent o f t h e m od el m i g r a t i o n s c h e d u l e i s c l o s e l y a pp ro x-i m a te d b y t h e f u n c t i o n

R e c a l l i n g t h a t t h e p r e - l a bo r f o r c e c u rv e i s g i v e n b y


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Figure 10. Variation of the parental sh i f t asa function of a2, or a l ternat ivef ixed values of p 2 , X 2 , and 612 .


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when a = a 2 , w e may e q u a t e t h e t wo f u n c t i o n s t o s o l v e f o r t h e1d i f f e r e n c e i n a g e s t h a t w e have c a l l e d t h e p a r e n t a l s h i f t , i . e . ,

T a b l e 3 com pares t h e v a l u e s o f t h i s a n a l y t i c a l l y d e f i n e d" t h e o r e t i c a l " p a r e n t a l s h i f t w i t h t h e c o r r es p on d in g o bs er ve dp a r e n t a l s h i f t s p r e s en t ed e a r l i e r i n T ab le 2 f o r Swedish m al esa nd f e m a le s . The two d e f i n i t i o n s a p p e a r t o p r od u ce s i m i l a r nu-m e r i c a l v a lu e s , b u t t h e a n a l y t i c a l d e f i n i t i o n h a s t h e a d va nt ag eo f b e i n g s i m p l er t o c a l c u l a t e an d an a l yz e .

E q ua t i on ( 2 ) shows t h a t t h e p a r e n t a l s h i f t w i l l i n c r e a s ew i t h i n c r e a s i n g v a l u e s o f p2 a nd w i l l d e c r e as e w i t h i n c r e a s i n gv a l u e s o f a 2 an d d 1 2 . I f t h r e e p a ra m e t er s a ssum e v a l u e s w i t h i nt h e r a n ge s s e t o u t a b ov e , t h e n A sh o u l d v a r y b e t w e e n a l ow o f

a n d a h i g h o f

However, b e c au s e o f t h e p a t t e r n s o f j o i n t v a r i a t i o n among p a2 ' 2'and t h e p a r e n t a l s h i f t v a r i e s w i t h i n a much n a rr o we r r a n geo f v a l u e s i n t h e o bs er ve d d a t a s e t o u t i n t h i s p ap er : *

*S t o t o (1 97 7) h a s s u g g es t ed t h a t t h e p a r e n t a l s h i f t may b e c l o s e l ya p p ro x i m at e d by t h e mean a g e o f c h i l d b e a r i n g .


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Te3.

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26.67

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p4


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I n a d d i t i o n t o t h e p a r e n t a l s h i f t , t h r e e o t h e r m ea s u r e su s e f u l l y c h a r a c t e r i z e t h e p r o f i l e o f a " r e g u l a r " ( i . e . , n l = n 2 )m ode l m i g r a t i o n s c h e d u l e w i t h o u t a r e t i r e m e n t p e a k. They a r e :

.t h e low p o i n t , x z , t h e h i g h p ea k, x h , and t h e h o r i z o n t a l d i s -t a n c e b etw een them , t h e l a b o r f o r c e s h i f t , X . F i g u r e 1 1 s h o w show t h e f i r s t two me as ure s v ar y a s a f u n c t i o n o f a 2 , f o r a l t e r -n a t i v e f i x e d v a l u e s o f p 2 , X 2 , an d 6 1 2 . The d i a g ra m s i n d i c a t et h a t t h e i n f lu e n c e of a 2 and 6 1 2 on t h e l o c a t i o n o f t h e l owp o i n t a nd t h e h i g h p ea k i s n e g l i g i b l e , f o r h ig h v a l u e s o f h 2 ,a nd t h a t i n s u c h i n s t a n c e s x Z a nd x may be e x p r e s s e d a s s im p l ehf u n c t i o n s o f p 2:

T hu s f o r X 2 = .8, X G 5 y e a r s , d e c l i n i n g s l i g h t l y a s a 2 i n c r e a s e s .F o r r e l a t i v e l y low v a l u e s o f h 2 , b o t h a 2 a n d 6 1 2 i n f l u e n c e

t h e l o c a t i o n o f t h e low p o i n t a n d t h e h i g h p e a k. H ow ever, t h ei m p ac t o f a c h a n g i n g 6 1 2 on x n e v e r e x c e e d s a r a n g e o f tw oZy e a r s , i n o u r 32 s c h e d u l e s , a nd i t s i n f l u e n c e on t h e h i g h pe aki s n e g l i g i b l e . I nd ee d , f o r X 2 = . 2 , w e may ad op t t h e app rox ima-t o n

An an a l o g o u s , b u t s om ew ha t c r u d e r , ap p r o x i m a t i o n a l s o may b ea d op te d f o r t h e low p o i n t :

Thus f o r X 2 = . 2 , X 13 - 5a2 .


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A. Variation of the low point

a2B. Variation of the high peak

Figure 11. Variation of the low point and high peakas a function of a for alternative fixed2values of p2, h2, and 612.


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Migration Proportions and SchedulesThe age profile of a schedule of migration rates reflects

the influences of two age distributions: the age composition ofmigrants and that of the population of which they were a part(Rogers, 1976). This can be easily demonstrated by decomposingthe numerator and denominator of the fraction that defines anage-specific migration rate, M(x).

If O(x) denotes the number of outmigrants of age x, leavinga region with a population of K(x) at that age, then

where

0 = total number of outmigrants;N(x) = proportion of migrants aged x years at

the time of migration;K = total population;

C(x) = proportion of total population agedx years at mid-year;

o = crude outmigration rate.

We define the collection of N(x) values to be the migrationproportion schedule (MPS) that is associated with a migrationrate schedule (MRS) of M(x) values. Figure 12A illustrates bothschedules for Swedish female outmigration from Stockholm in 1974(highest d 1 2 value). Figure 12B presents comparable data forUpper North-Sweden (lowest d 1 2 value). Both figures also showthe age composition of the origin region's population.

A number of observations follow from the decomposition setout in equation (3). First, it is clear that different popula-tion age compositions, in a region of origin, will give rise todifferent age-specific migration rate schedules for the same


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migration proportion schedule. The level of the former deter-mines whether it is situated above or below the latter. Sincethe GMRs of the migration rate schedules in Figure 12A and 12Bexceed unity, both M(x) profiles lie above the N(x) profiles.Finally, for the same N(x) schedule, an "old" population agecomposition (e.g., that of a stationary population) will pro-duce relatively high pre-labor force migration, whereas a "young"age composition will instead generate relatively high post-laborforce migration.

The observed migration proportion schedules in Figures 12Aand 12B appear to have the same fundamental profile as the modelmigration schedule defined by equation (1). Figures 12C and 12Dillustrate that this indeed is the case, and Table 4 presentsthe estimated parameters and descriptive measures that resultfrom fits of that model schedule to the eight-region data forSweden. A comparison of the values in Table 4 with correspond-ing values in Table 2 is instructive, and reveals that the rangesof variation are similar for most profile measures.

Having shown that the age composition of migrants may bedescribed by the model migration schedule in equation (I), wenow shall consider how to describe the age composition of thepopulation of which they were a part. This will then permitus to examine how these two age distributions combine to in-fluence the age profile of a migration rate schedule.

The age composition of a stable population with an intrin-sic rate of growth, r, and survivorship function Z(x) is given by

Substituting this expression into equation ( 3 ) , with N(x) takingon the profile of the model migration schedule defined by equa-tion (1) gives an analytical decomposition of the migration rateschedule that permits an exploration of how changes in N(x) and


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Table 4. Parameters and variables defining observed modelmigration proportion schedules: Swedish regions,1974.

2. East Mid- 3. South Mid-Parameters 1. Stockholm dle-Sweden dle-Sweden 4. SouthandVariables M F M F M F M F

GMR

al

a21-12C12

a31-13C13A3C-n% (0-14)% (15-64)% (65+)61612632B12u2u32"hXrXAB


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Table 4. Parameters and variables defining observed modelmigration proportion schedules: Swedish regions,1974 (continued) .

6. North Mid- 7. LowerParameters 5 . West 8. Upperdle-Sweden North-Sweden North-SwedenandVariables M F M F M F M F


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C ( x ) c om bin e t o i n d u c e c o r r e s p o n d i n g ch a n g es i n M ( x ) . N u me ri ca lc om pu ta t i on s u s i n g t h i s a n a l y t i c a l d e c om p os it io n r e q u i r e v a l u e sf o r t h e s u r v i v o rs h i p f un c t i o n Z ( x ) .

D u ri n g t h e p a s t d e c a d e i n c r e a s i n g u s e h a s b e e n made o f m ode ll i f e t a b l e s and s t a b l e p o pu la t i on s t o e s t im a t e t h e c h a r a c t e r i s -t i c s of p o p u la t i on s h av in g i na d eq u a t e b i r t h a nd d e a t h s t a t i s t i c sa nd i n a c c u r a t e o r i n c om p le te d a t a o n a ge a nd s e x d i s t r i b u t i o n s .The ex tr em e f l e x i b i l i t y o f t h e B ra s s l o g i t s ys te m ( B r a s s , 1 97 1)i n c om pu te r a p p l i c a t i o n s m akes it p a r t i c u l a r l y a t t r a c t i v e f o r o u rp u r po s e s . T h i s scheme s u b j e c t s t h e s u r v i v o r s h i p f u n c t i o n o f a" s ta n da r d" l i f e t a b l e t o t h e s o - ca l le d l o g i t t r a ns f or m a t i o n , a ndt h e n c o n s id e r s l i f e t a b l e s w i t h l o g i t s t h at . a r e l i n e a r l y r e l a t e dt o t h e l o g i t s o f t h e s t a n d a r d t a b l e . The fu nda me nta l l i n e a r re -l a t i o n i s

where

an d Z s ( x ) i s t h e s u r v i v o r s h ip p r o b a b i l i t y o f t h e s t a n d a r d l i f et a b l e .

T h e p a r am e t e r s a an d 8 o f t h e B r a s s s y s te m sh o u ld n o t b eco n f u s e d w i t h o u r s am e tw o t e r m s t h a t d e s c r ib e c h a r a c t e r i s t i c so f t h e m ode l m i g r a t i o n s c h e d u l e . V a r i a t i o n o f a h e r e r e f e r s t och an g es i n m o r t a l i t y l e v e l s , a nd v a r i a t i o n i n B r e f l e c t s c h an ge si n t h e r e l a t i o n betw een c hi ld h oo d a nd a d u l t m o r t a l i t y . A do pt in gt h e Y s(x ) v a l u e s of t h e s t a n da r d l i f e t a b l e s e t o u t i n H i l l andT r u s s e l l ( 1 9 7 7) , a nd s e t t i n g a = -1 .5 and 8 = 1 . 0 , g i v e s a m ode ll i f e t a b l e w i t h a n e x p e c ta t io n o f l i f e a t b i r t h of 78 .9 y e a r s( B r a s s , 1 97 1, p . 1 0 9 ) . The Z (x ) v a l u e s may b e c a l c u l a t e d u s i n g


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w i t h Z ( 0 ) s e t e q u a l t o 1 a nd Z (100) t o ze r o . F i g u r e 1 3 i l l u s -t r a t e s t h e a ge co mp os it io ns o f s t a b l e p o pu l a t io n s w i t h t h i sm o r t a l i t y r eg im e and f o u r d i f f e r e n t r a t e s o f g ro wt h, r = - .01 ,0 , . 0 1 , an d . 0 2, r e s p e c t i v e l y .

I n 1974 t h e c r u d e f em a le o u tm i g r a t i o n r a t e s f o r t h e S to ck -holm and t h e Upper-Nor th r e g i on s o f Sweden w e r e .0191 and .0168,r e s p e c t i v e l y . T h e i r m odel m i g r a t i o n p r o p o r t i o n s c h e d u l e s w e r ed e f i ne d i n T a bl e 4 a n d i l l u s t r a t e d i n F i g u r e s 12C a n d 12D. Com-b i n i n g t h e s e two com po ne nts o f e q u a t i o n ( 3 ) w i t h t h e s t a b l e a gec o mp os i t io n s i n F i g u r e 1 3 g i v e s t h e h y p o t h e t i c a l m i g r at i o n r a t es c h e d u l e s s e t o u t i n F i g u r e s 14A and 14B b e lo w .

The s c h e d u le s i n F i g u r e 14 c o n fi r m o u r e a r l i e r o b s e r v a t i o nt h a t a n o l d p o p ul a t io n a g e co m po si t i on pr od uc e s a r e l a t i v e l yh i g h p r e - l a b o r f o r c e m i g r a t i o n , w h e r ea s a y oun g a g e co m p o s i ti o ng e n e r a t e s a r e l a t i v e l y h i g h p o s t - l a b o r f o r c e m i g r a t i o n . I nd ee dt h e r a t e s o f t h e l a t t e r t e n d t o " c u r l " upwards a t t h e o l d e r a g e st o what a p p ea r t o b e u n r e a l i s t i c a l l y h ig h l e v e l s .

To ex am in e n u m e r i ca l l y t h e p a t t e r n s o f v a r i a t i o n i n d uc e dby c h a n ge s i n a g e c o m p o s i t i o n , t h e model m i g r a t i o n s c h e d u l e o fe q ua t io n ( 1 ) was f i t t e d t o t h e h y p o t h e t i c a l s c h e du l es i n F i g u re1 4.* The r e s u l t s a r e s e t o u t i n T ab le 5 , a nd s u g g e s t s e v e r a lo b s e r v a t i o n s .

F i r s t , it i s a p p a r e n t t h a t i n c r e a s i n g t h e r a t e of g ro wths h i f t s t h e l ow p o i n t o f t h e s c h e d u le to w ar d y ou ng er a g e s , w h i l es i m u l t a n e o u s l y m oving t h e h i g h p o i n t t o w ar d o l d e r a g e s . Con-s eq u e n tl y , t h e l a b o r f o r c e s h i f t , X , i n c r e a s e s a s t h e p o pu l a t io nb ecom es y o u n g e r . The sam e p a t t e r n o f v a r i a t i o n i s e x h i b i t e d byt h e p a r e n t a l s h i f t and t h e mean a g e , i . e . , y o u n g e r p o p u l a t i o n sha ve a l a r g e r v a l u e o f A and o f n. S in c e a l l o f t h e s e m easu re sa r e d i r e c t l y a s s o c i a t e d w i t h p2, it i s n o t s u r p r i s i n g t o f i n dt h a t t h e l a t t e r p ar am e te r a l s o i n c r e a s e s a s r i n c r e a s e s . F i n a l l y ,a move t o w ar d y o u n g e r ag e co m p o s i t i o n s p r o d u ces l o w er v a l u e s o f- - -*To r e d u c e t h e i n f l u e n c e o f t h e v e r y o l d a ge g ro u p s , o n l y t h er ~ i g r a t i o n a t e s o f t h o s e un de r 75 y e a r s o f a g e w e r e u se d i n

t h e f i t t i n g p ro ce ss .


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AGE

Figure 13 . Age composi t ions of s t ab le popula t ions wi than e xp e ct a t io n of l i f e a t b i r t h of 78 .9 y e a r sand d i f f e r e n t r a t e s of i n c r e a s e.


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Table 5. Param eters and variables defining hypotheticalmodel migration schedules for Swedish femaleswith an expectation of life at birth of 78.9 years.

Param e t e r sand

V a r i a b l e sGMRa1a1a 2u2a2A 2a 3p3Ci3A 3c

-n% (0-14)% (15-64)% (65+)& l C&12&32El,O2O3*"hxrXAB

M i g r a t i o n P r o p o r t i o n s and Crude Rate of

r = - . O 12.16-082-099.2 06

24.70.199.196. 0 0 1

85.32.224.050. 0 0 1

24.3830.5365.89

3.5765.62

.398

.004

.496

.982

.22514.9224.3753.899.45

24.71.063

8 .r = - 0 1

1 . 9 5-046-0 9 8.194

17.50. I 4 7-7 4 5------------

. 00324.3420.6679 .91

4.4415 .90

- 236---

-6 6 95 .06---

14 .841 9 . 6 3---

4.7925.44

.112

Stockholmr = . 0 1

1 . 5 2. 0 3 1-0 8 8-136

24.99.187.192. 0 0 1

85.76. 2 1 3.051.003

31.6919 .7271.18

9 .1012 .14

- 232- 0 1 1- 4 6 81 . 0 3.240

14 .6624.9756.941 0 . 3 128 .73

.044

1.r = O1 . 7 2-0 5 0.092.161

24.80.192-195- 0 0 1

8 4 . 8 3.2 16.051-002

27 .7324.976 9 . 2 3

5 .8028.64

-307-0 0 8.4771 . 0 1.237

1 4 . 8 024.6655.449.86

26.63.051

r =.021 . 5 1- 0 2 1-0 8 8-1 2 3

25.30-185.189.001

89 .81.2 10.048.004

36.1415.0471.5013.46

5.75.168.011.4741.02.229

14 .4725.3058.551 0 . 8 333.54

.041

Upperr = O

1 . 5 3-027-0 9 8.145

17 .55-142-726------------

. 0 0 327.0517 .0676.25

6 . 6 97 . 8 1- 1 8 5---

.6905.12---

1 4 . 7 819 .76

---4 .98

27.65.084

North-Swedenr = . 0 1

1 . 3 4. 16. l o 8.117

17 .63.139.700------------

.00430.341 3 . 7 376.44

9 .843.62.136---

.7805.04---

1 4 . 6 819 .92

---5.24

30.66.068

r = . 0 21 . 2 8-009-1 6 5. l o 3

17.76-140.670------------

-00634.1910 .7775.3613.87

1 . 5 0.087---

1 . 1 84.77---

14 .5220.09---

5.5736.51. 59


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of h 2 and h I 2 . The decline in the latter reflects the dampeningeffect on pre-labor force migration rates that a younger popula-tion induces if the migration proportion schedule is held fixed.

ConclusionThis paper develops support for three principal points.

First, the profiles of age-specific gross migration rates allover the world have a fundamental regularity that can be cap-tured and expressed in mathematical form. Second, this math-ematical model schedule summarizes the empirical regularity ina way that permits analytical examinations to be carried outregarding the fundamental properties of the migration age pro-files. Finally, migration rate schedules may be convenientlydecomposed to illuminate the influences on migration patternsof migration level, the age composition of migrants, and theage composition of the population in the region of origin.

In a subsequent paper we plan to examine further the regu-larities exhibited by a large number of migration schedules in-- - -- . . . -.. .-order to develop a family of model schedules for use in situa-tions where observed migration data are scarce, inadequate, orinaccurate. The use of the model migration schedule to studycause- and status-specific age profiles will also be explored.


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REFERENCES

Bard, Y. ( 1 9 7 4 ) Nonlinear Parameter Estimation. New York:Academic Press.

Benson, M. ( 1 9 7 9 ) Parameter Fitting in Dynamic Models.Ecological ~odelling : 9 7 - 1 1 5 .

Brass, W. ( 3 9 7 3 ) Biological Aspects of Demography. London:Taylor & ranc cis 6 9 - 1 1 0 .

Brown, K.11. and J.E. Dennies ( 1 9 7 2 ) ~erivative ree Analoguesof the Levenberg-Marquardt and Gauss Algorithms for Non-linear Least Squares Approximations. ~umerische athe-matik 1 8 : 2 8 9 - 2 9 7 .

Coale, A.J. and D.R. McNeil ( 1 9 7 2 ) The Distribution by Ageof the Frequency of First Marriage in a Female Cohort.Journal of the American Statistical Association 6 7 i7437 .749 .

Fiacco, A . and G. llccormick ( 3 9 6 8 ) Nonlinear Programming;Sequential Unconstrained Minimization Techniques.New York: Wiley.

Haenszel, W. ( 1 9 6 7 2 Concept, Measurement and Data inMigration Analysis. ~eho~raphy; $53- .26 1.

Hill, K. and J. Trussell ( 1 9 7 7 ) Further Developments inIndirect Mortality Estimation. Population Studies3 3 ( 2 ) : 3 3 3 - 3 3 4 .

Keyfitz, N. ( 1 9 7 7 ) Applied Mathematical Demography.New York: Wiley.


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Levenberg, K. (1944) A Method for the Solution of CertainNonlinear Problems in Least Squares. Q. Appl. Math 2:164-168.

Marquardt, D.W. (1963) An Algorithm for Least-Squares Estimationof onl linear Parameters. SIAM, J. Nurner. Anal. 1 1 : 431-441.

Morrison, P.M. (1970) Implications of Migration Histories forModel Design. P-4342. The Rand Corporation, Santa Monica,~alifornia.

Rees, P.H. (197.7) he Measurement of Migration, from CensusData and Other Sources. Environment and Planning A (1):247-260.

Rogers, A. (1973a) The Multiregional Life Table. The Journalof Mathematical Sociology (3): 127-137.

Rogers, A. (1973b) The Mathematics of Multireaional DemographicGrowth. Environment and Flanning (5): 3-29.Rogers, A. (1975) ~ntroduction o Multiregional Mathematical

Demography. New York; Wiley.Rogers, A. (1976) Two Methodological Notes on Spatial

Population Dynamics in the Soviet Union. InternationalInstitute for Applied Systems Analysis, Laxenburg,Austria. RM-76-48.

Rogers, A. and J. Ledent (1 97 ) Increment-Decrement LifeTables A Comment. Demography (1 3) 2877.290.Rogers A., R. Raquillet and L.J. Castro (1 978) Model MigrationSchedules and Their Applicationsl Environment and Planning

A (1 01 : 475-502.Ryder, N.B. (1964) The Process of Demographic Translation.

Demography (31; 74-82.Ryder. N.B. (1975) Fertility Measurement Through Cross-.

Sectional Surveys. Social Forces (54) 7-35Stoto, M. (1977) On the Relationship of Childhood to LaborForce ~igration ates. International ~nstitute or

Applied Systems Analysis, Laxenburg, Austria. RM-77-55.


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APPENDIX A

AGGREGATION OF SWEDISH COUNTIES INTO REGIONS

Reaion (viksomrsden)1. Stockholm2. 0stra mellansverige(East Middle-Sweden)

3. ~msland ch oarna(South Middle-Sweden)

4. Sydsverige(South)

5. Vastsverige(West)

6. Norra mellansverige(North Middle-Sweden)

7. Mellersta norrland(Lower North-Sweden)

8. Ovre norrland(Upper North-Sweden)

Counties (lan)StockholmUppsalaSodermanlandste ergot landOrebroVastmanlandJonkopingKronobergKalmarGotlandBlekingeKristianstadMalmohusHallandGoteborg och Bohus~ l v s b o r ~SkaraborgVarmlandKopparberg~avleborg~asternorrlandJamtlandVasterbottenNorrbotten


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APPENDIX B

T a b l e I . P a r a m e t er s a nd v a r i a b l e s d e f i n i n g o b s e r v e d modelm i g r a t i o n s c h e d u l e s : t h e N e t h e r l a n d s , P o l an d ,Sw eden, a n d t h e - U n i t e d S t a t e s , a r o u n d 1 9 7 4 . *

Nether- Sweden,1968-Poland,1973 United States,lands,1972 7 1966-71

' In te rc om m un al m i g r a t i o n i n t h e N e t h e r l a n d s a nd Sweden; i n t e r -v o i vo d sh i p m i g r a t io n i n P o la nd ; i n t e r c o u n t y m i g r a t io n i n t h eU n i t e d S t a t e s .

GMRa1

a2p2a22c-n% (0-14)% (15-64)% (65+)61c612B1202X~

"hXAB

1.10 1.13-028 .026-253 -327-090 .092

24.31 21.94.212 -272-306 .359-006 .008

36.47 38.1218.49 17.3564.10 62.0217.42 20.634.32 3.36-311 .2861.19 1.201.44 1.3217.00 15.4225.51 22.738.51 7.3131.22 28.13.034 .035

4.10 4.19.065 -068.205 -247.286 -339

23.08 19.77-179 .202-212 .399.024 .026

36.74 36.5116.18 15.5766.26 66.1517.56 18.282.69 2.67.226 -2021.14 1.221.18 1.9813.29 14.3223.85 21.4910.56 7.1732.12 29.41.lo2 -143

3.52 3.62.lo1 .lo1.I16 -124.210 .236

21.27 19.05.lo0 -115.371 -535-008 -010

29.41 28.7123.85 22.9968.62 68.537.53 8.4812.55 10.08-482 .4261.15 1.073.70 4.6316.44 15.5724.65 21.838.21 6.2630.06 27.38.lo2 -128

5.28 4.99-095 .094.I10 .lo5-228 .233

19.37 18.44-103 -119-657 .654.025 .024

34.20 33.3820.22 21.4165.36 63.9714.41 14.623.83 3.89-416 .4021.07 .8876.40 5.5016.50 15.5722.12 20.965.62 5.3928.95 25.82.I38 -134


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Table 11. Parameters and variables defining observed nodelmigrat ion schedules: the United States, 1966-1971.

P a r a m e t e r s FEMALE MOBILITY OR MIGRATIONand MOBILITY r e s i d en - w i t h i n b e t ween b e tween

V a r ia b le s w h i te b l a c k t i a l c o un ty c ou n ty s t a t e s

GMRa1"1a21-12"2A 2c-n% (0-14)% (15-64)% (65+)61&1281202* Z"hXAB

14.37 15.40.251 .249-124 -074-594 .390

18.86 18.34.115 -067-5 3 3 , 5 5 3-080 .070

35.56 35.1420.26 21.4062.80 63.4316.94 15.173.13 3 .55.423 .6371 .08 1 .104.64 8.20

15 .37 15 .2821.65 21.32

6.28 6.5427.33 27.87

.322 -234

14 .27 9 .30.248 -156-1 1 8 . I3 0.583 .357

18.30 18.23.122 .126.587 -528.082 -058

35.44 36.6120.83 20.4361.85 60.7017 .32 18 .873.02 2.67-425 -437-967 1 .034 .83 4 .19

15 .13 14 .7220.89 20.86

5.76 6.1425.52 25.39

-320 .186

4.99 2 .53.094 .052. l o 5 . 0 9 9-233 -127

1 8 4 4 1 8. 62-119 .113.654 -602-024 -010

33.38 31.6921.41 22.2663.97 65.3914 .62 12 .353.89 5 .10.402 .414.887 .8715 .50 5 .31

15 .57 15 .5520.96 21.29

5 .39 5 .7425.82 26.12

.134 .071


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Table111.Parametersandvariablesdefiningobservedodel

migrationschedules:theetherlands,1972-1976.

Pand

MALES

FEMALES

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APPENDIX C

NONLINEAR PARAMETER ESTIMATION IN MODEL MIGRATION SCHEDULES

This appendix will briefly illustrate the mathematicalprogramming procedure used to estimate the parameters ofthe model migration schedule. The nonlinear estimation problemmay be defined as the search for the "best" parameter values forthe function:

best in the sense that a pre-defined objective function is mini-mized when the parameters take on these values.

This problem is the classical one of nonlinear parameterestimation in unconstrained optimization. All of the availablemethods start with a set of given initial conditions, or initialguesses of the parameter values, from which they begin a searchfor better estimates following specific convergence criteria.The iterative sequence ends after a finite number of iterations,and the solution is accepted as giving the "best" estimates forthe parameters.

The problem of selecting a "good" method has been usefullysummarized by Bard (1974, p.84) as follows:... o single method has emerged which is best for

the solution of all nonlinear programming problems.One cannot even hope that a "best" method will everbe found, since problems vary so much in size and


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nature. For parameter estimation problems we mustseek methods which are particularly suitable to thespecial nature of these problems which may be char-acterized as follows:

1. A relatively small number of unknowns,rarely exceeding a dozen or so2. A highly nonlinear (though continuousand differentiable) objective function,whose computation is often very timeconsuming

3. A relatively small number (sometimeszero) of inequality constraints. (Thoseare usually of a very simple nature,e . , upper and lower bounds )

4. No equality constraints, except in thecase of exact structural models (where,incidentally, the number of unknowns islarge)

For computational convenience, we have chosen the Marquardtmethod (Levenberg, 1944; Marquardt, 1963). This method seeks outa parameter vector P* that minimizes the following objectivefunction:

2where fp is the residual vector and ) ) - I 1 represents the known2Euclidean vector norm. For the case of a model schedule with aretirement peak, vector P has the following elements:

pT = [ al, ,, a2, a2, u2, A2, a3, a3, 3, 3, cl (A2The elements of the vector f p can be computed by either of thefollowing two expressions:

A

where M(x) is the observed value at age x and Mp(x) is the esti-mated value using equation(1) and a given vector P of parameterestimates.


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By i n t r o d u c i n g e q u a t i o n (A3) i n t h e o b j e c t i v e f u n c t i o n s e to u t i n e q u a t i o n ( A l l , t h e sum of s q u a r e s i s m in im iz ed ; i f , o n t h eo t h e r h an d, e q u a t i o n ( A 4 1 i s i n t r o d u ce d i n s t e a d , t h e ch i -s q ua r es t a t i s t i c i s m i n i m i z e d .

I n m a t r i x n o t a t i o n , t h e L ev en be rg -M a rq ua dt m eth od f o l l o w st h e n e x t i t e r a t i v e s eq ue n c e :

where X i s a n o n- ne ga tiv e p a ra m et e r a d j u s t e d t o e n s u r e t h a t a te ac h i t e r a t i o n t h e v a l ue o f f u n c t i o n ( Al ) i s r e d u c e d , J d e n o t e s t h eqJ a c o bi a n m a t r i x o f g(P) e v a l u a t e d a t t h e q t h - i t e r a t i o n , a nd D i s a

Td i a go n a l m a t r i x e q u a l t o t h e d i a g o n a l o f J J .The p r i n c i p a l d i f f i c u l t y i n n o n l i n e a r pa ra m et er e s t i m a t i o n

i s t h a t o f c o nv e rg e nc e , a n d t h i s meth od i s n o e x c e p t i o n . T hea l g o r i th m s t a r t s o u t by as su mi ng a s e t o f i n i t i a l p a r am e te r s , a n dt h e n a new v e c t o r P i s e s t i m a t e d a c c o r d in g t o t h e v a l u e o f A ,which i n t u r n i s a l s o m o d if ie d f o l lo w i n g g r a d i e n t c r i t e r i a .Once g i v e n s t o p p i n g v a l u e s a r e a c h ie v e d , v e c t o r P* i s assumedt o b e t h e optim um . However, i n m o st c a s e s , t h i s P* r e f l e c t s l o -c a l minima t h a t may b e im pro ved w i t h b e t t e r i n i t i a l c o n d i t i o n sand a d i f f e r e n t s e t o f g r a d i e n t c r i t e r i a .

U si ng t h e d a t a d e s c r i be d i n t h i s p a p e r , s e v e r a l e x p er im e nt sw e r e c a r r i e d o u t t o e xam ine t h e v a r i a t i o n i n p a ra m et er e s t i m a t e st h a t c a n r e s u l t f rom d i f f e r e n t i n i t i a l co n d i t i o n s ( a s s u m in g New-- --t o n ' s g r a d i e n t c r i t e r i a ) . * Among t h e c a s e s s t u d i e d , t h e m o s ts i g n i f i c a n t d i f f e r e n c e s were f ou nd f o r t h e v e c to r P w i t h e l e v enp a r am e t e rs , p r i n c i p a l l y among t h e p ar a m e te r s o f t h e r e t i r e m e n tco mp on en t. F o r s c h e d u l e s w i t h o u t t h e r e t i r e m e n t pe a k, t h e v e c t o rP* s ho ws n o v a r i a t i o n i n m o s t c a s e s .

The i mp ac t o f t h e g r a d i e n t c r i t e r i a on t h e o p t i m a l v e c t o rP* was a l s o a n a l y z e d , u s i n g t h e Newton a n d t h e S t e e p e s t D e sc e ntm eth od s. The e f f e c t s o f t h e s e two a l t e r n a t i v e s w e r e r e f l e c t e d i n

* Fo r a c o m p l et e d e s c r i p t i o n o f g r a d i e n t me th o ds , see F i a c c o( 1 9 6 8 ) a n d B a rd ( 1 9 7 4 ) .


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the computing times but not in the values of the vector P*.Nevertheless, Bard ( 1974 ) has suggested that both methods cancreate problems in the estimation, and therefore they should beused with caution, in order to avoid unrealistic parameter esti-mates. It appears that the initial parameter values may be im-proved by means of an interactive approach suggested by Benson( 1 9 7 9 ) .


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RELATED PAPERS ON MIGRATION AND SETTLEMENT

Rogers, A., ed. (1978) Migration and Settlement: SelectedEssays. RR-78-6. Laxenburg, Austria; InternationalInstitute for Applied Systems Analysis.

Rogers, A. and F. Willekens (1978) Migration and Settlement:Measurement and Analysis. RR-78-13. Laxenburg, Austria:International Institute for Applied Systems Analysis.

Rogers, A. (1978) The Formal Demography of ~igration ndRedistribution: Measurement and Dynamics. RM-78-15.Laxenburg, ~ustria: International Institute forApplied Systems Analysis.

Willekens, F. and A. Rogers (3978) Spatial Population Analysis:Methods and Computer Programs. RR-.78r.18. Laxenburg,Austria: ~nternational nstitute for ~pplied ystemsAnalysis.

Migration Age Patterns: Measurement and Analysis (1979)

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