Model Migration Schedules (1981)

8/3/2019 Model Migration Schedules (1981)

1/160

MODEL MIGRATION SCHEDULES

Andrei Rogers and Luis J . CastroInternational Institute for Applied S y s t e m Analysis, Austria

RR-8 1-30November 1981

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSISLaxenburg, Austria


2/160

International Standard Book Number 3-7045-00224

Research Reports , which record research conducted at IIASA, are independently reviewed beforepublication. However, the views and opinions they express are not necessarily those of the Institu te orthe National Member Organizations that support it.

Copyright O 1981Internat ional Institu te for Applied Systems Analy sisAll rights reserved. No part of this publication may be reproduced or transmitted in any form or byany means, electronic or mechanical, including photocopy, recording, or any information storage orretrieval system, without permission in writing from the publisher.


3/160

PREFACE

Interest in hum an settlement systems a nd policies has been a central part of urb an-related work a t IIASA since its inception. From 197 5 thro ugh 1 97 8 his interest was mani-fested in th e work of th e Migration an d S ettlem ent Task, which was formally concludedin November 197 8. Since the n, atten tion has turn ed to th e dissemination of the Task'sresults and to t he conclusion of its comparative stu dy: a quantitative assessment of recen tmigration patterns and spatial population dynamics in all of IIASA's 17 NMO countries.This report is par t of the Task's dissemination effo rt, focusing on t he age patterns

of migration exhibited in the data bank assembled for the comparative study. It beginswith a comparative analysis of over 5 00 observed migration schedules and th en develops,on the basis of this analysis, a family of hypothetical schedules for use in instances wheremigration da ta are unavailable o r inaccurate.Rep orts summarizing previous work on m igration and settlemen t a t IIASA are listedat the back of this repo rt. They should be consulted for further details regarding the database that underlies this study.

ANDRE1 ROGERSChairmanHum an Settlem ents and Services Area


4/160


5/160

CONTENTS

SUMMARY1 INTRODUCTION2 AGE PATTERNS OF MIGRATION

2.1 Migration Rates an d Migration Schedules2.2 Model Migration Schedules

3 A COMPARATIVE ANALYSIS OF OBSERVED MODEL MIGRATIONSCHEDULES3.1 Data Preparation, Parameter Estimation, and Sum mary Statistics3.2 National Contrasts3.3 Families of Schedules3.4 Sensitivity Analysis

4 ESTIMATED MODEL MIGRATION SCHEDULES4.1 Introdu ction: Alternative Perspectives4.2 The Correlational Perspective: The Regression Migration System4.3 The Relational Perspective: The Logit Migration System

5 CONCLUSIONREFERENCESAPPENDIX A Nonlinear Parameter Estimation with Model MigrationSchedulesAPPENDIX B Summary Statistics of National Parameters and Variables ofthe Reduced Sets of O bserved Model Migration SchedulesAPPENDIX C National Parameters and Variables of th e Full Sets of ObservedModel Migration Schedules


6/160


7/160

Research Report RR-8 1-30 , November 198 1

MODEL MIGRATION SCHEDULES

Andrei Rogers and Luis J . CastroInternational Institute for Applied Systems A nalysis, Austria

SUMMARY7% report draws on the fundamental regulan'ty exhibited by age profiles o f migra-tion all over the world t o develop a system o f hypothetical model schedules that can be

used in multiregional population analyses cam-ed ou t in countn'es that lack adequa te migra-tion data.

1 INTRODUCTIONMost hum an pop ulation s experience rates of age-specific fertility and mo rtality th at

exhibit remarkably persistent regularities. Consequently, demographers have found it pos-sible to summarize and codify such regularities by means of mathematical expressionscalled model schedules. Although the development of model fertility and mortality sched-ules has received considerable a tten tion in demographic studies, the con struction of modelmigration schedules has no t, even though the techn iques tha t have been successfully appliedto treat th e former can be readily extended t o deal with the latte r.We begin this rep ort with an exam ination of regularities in age profile exhib ited byempirical schedules of migration rates and go on to ado pt the notio n of model migrationschedules t o express these regularities in math ema tical form. We the n use model schedulesto examine patterns of variation present in a large data bank of such schedules. Drawingon this comparative analysis of "o bserved" mod el schedules, we develop several "families"of schedules and conclude by indicating how they m ight be used t o generate hypo thetical"estimated" schedules for use in Third World migration studies - ettings where the avail-able migration data are often inadequ ate or inaccurate.2 AGE PAITERNS OF MIGRATION

Migration measurement can usefully apply concepts borrowed from both mortalityand fertility analysis, modifying them where necessary to take into account aspects that


8/160

2 A . Rogers, L.J. Cnstroare peculiar t o spatial mobility. From mortality analysis, migration studies can borrow th enotio n of the life table, extending it to include increments as well as decremen ts, in orderto reflect the mu tual interaction of several regional cohorts (Rogers 197 3a, b, 19 75 , Rogersand Ledent 1976). From fertility analysis, migration stud ies can borrow well-developedtechniques for graduating age-specific schedules (Rogers et al. 1978). Fundam ental t o bot h"borrowings" is a workable def inition of the migration rate.

2.1 Migration Rates and Migration SchedulesThe simplest and most common measure of migration is the crude migration rate,defmed as the ratio of the number of migrants, leaving a particular population located inspace and time, to the average number of persons (more exactly, the number of person-years) exposed to the risk of becoming migrants. Data o n nonsurviving migrants are often

unavailable, therefore the n umerator in this ratio generally excludes them.Because migration is highly age selective, with a large fraction of migrants beingyou ng, our understanding of m igration pattern s and dynamics is aided by computing migra-tion rates for each single year of age. Summing these rates over all ages of life gives thegross migraproduction rate (GMR), the migration analog of fertility's gross reproductionrate. This rate reflects th e level at which migration occurs out of a given region.Th e age-specific migration schedules of multiregional populatio ns exhi bit remarkablypersistent regularities. For example, when comparing the age-specific annual rates of resi-dential migration am ong whites and blacks in the United States during 1966-1971, onefinds a comm on profile (Figure 1). Migration rates amo ng infants and yo ung childrenmirrored the relatively high rates of their parents, young adults in their late twenties. Themobility of adolescents was lower but exceeded that of young teens, with the latt er show-ing a local low p oint arou nd age 1 5. Thereafter migration rates increased, attaining a highpeak at abo ut age 22 and the n declining mono tonically w ith age to th e ages of retirem ent.The m igration levels of bo th whites and blacks were roughly similar, with w hites showinga GMR of a bo ut 1 4 migrations and blacks one of approximately 15 over a lifetime undis-turbe d by mortality before the end of the mobile ages.

Although it has freq uently been asserted th at migration is strongly sexselective, withmales being more mobile than females, recent research indicates that sex selectivity ism uch less prono unce d tha n age selectivity and is less unifo rm across time and space. Never-theless, because most models and studies of population dynamics distinguish between thesexes, most migration measures d o also.

Figure 2 illustrates the age profdes of male and female migration schedules in fourdifferent coun tries at about the same point in time between roughly comparable arealunits: communes in the Netherlands and Sweden, voivodships in Poland, and counties inthe United States. Th e migration levels for all but P oland are similar, varying between 3.5and 5 .3 m igrations per lifetime; and th e levels for males and females are roughly the same.The age profiles, however, show a distinct, and consistent, difference. The high peak ofthe female schedule precedes that of the male schedule by an amount that appears toapproxim ate the difference between the average ages at marriage of the tw o sexes.Under normal statistical con ditions, point-to- poin t movem ents are aggregated in tostreams between one civil division and another; consequently, the level of interregionalmigration depe nds on the size of th e areal unit se lected . Thu s if the areal unit chosen is a


9/160

Model migration schedules

FIGURE 1 Observed annual migration rates by color (- -- white,- lack) and single years o fage: the United States, 1966-1971.minor civil division such as a county or a commune, a greater proportion of residentiallocati on will b e included as migration th an if th e areal unit cliosen is a major civil divisionsuch as a state o r a province.

Figure 3 presents the age profiles of female migration schedules as measured by dif-ferent sizes of areal units: (1 ) all migrations from one residence t o ano ther, (2) changes ofresidence within county boundaries, (3) migration between counties, and (4) migrationbetween states. The respective four GMRs are 14.3, 9.3, 5.0, and 2.5. The four age pro-files appear to be remarkably similar, indicating that the regularity in age pattern persistsacross areal delineations of different size.

Finally, migration occurs over tim e as well as across space; there fore , studies of itspatte rns mu st trace its occurrence with respect t o a time interval, as well as over a systemof geographical areas. In general, the longer th e time interval, the larger t he num ber ofreturn movers a nd nonsurviving migrants and , henc e, the more th e count of migrants willunderstate the number of interarea movers (and, of course, also of moves). Philip Rees,for example, after examining the ratios of one-year to five-year migrants between theStandard Regions of Great Britain, found that

. . . he number of migrants recorded over five years in an interregional flowvaries from four times to two tim es the number of migrants recorded over oneyear. (Rees 197 7, p. 247 )


10/160

0.25 1 Netherlands, 1972

0.251 Sweden, 1968-1 9730.20

0.25 { Poland, 1973

0.251 United States, 1966-1 971

FIGURE 2 Observed annual migration rates by sex (--- females,- ales) and single years of age: the Netherlands (intercommunal), Poland (inter-voivodship), Sweden (intercommunal), and the United States (intercou nty); around 197 0.


11/160


0.45

0.40

0.35

V) 0.30a3ICg 0.25.-CIC. 0.205

0.15

0.10Within counties

0.05 ntieses

0 10 20 30 40 50 60 70 80 90

FIGURE 3 Observed average annual migration rates of females by levels of areal aggregation a nd singleyears of age: the United States, 1966-1971.

2.2 Model Migration SchedulesFrom the preceding section it appears that the most prominent regularity found in

empirical schedules of age-specific migration rates is the selectivity of migration withrespect t o age. Young adul ts in their early twen ties generally show the highest migrationrates and young teenagers the lowest. The migration rates of children mirror those of theirparents; hence the migration rates of infants exceed those of adolescents. Finally, migra-tion streams directed toward regions with warmer climates and into or ou t of large citieswith relatively high levels of social services and cu ltura l amenities o fte n exhib it a "retire-me nt peak" at ages in the m id-sixties or beyond .

Figure 4 illustrates a typical observed age-specific migration schedule (the jaggedoutline) an d its graduation by a model schedule (the superimposed smo oth o utline) definedas the sum of four components :

1. A single negative exponential curve of the pre-labor force ages, with its rate ofdescent a,

2. A left-skewed unirnodal curve of the labor force ages positioned at mean age p ,on the age axis and exhibiting rates of ascent h, and descent a,


12/160

A. Rogers, L.J. Castro

a, = rate of descent of pre-labor force componentA, = rate of ascent of labor force compone nta, = rate of descent of labor force com ponentA , = rate of ascent of post-labor force componenta, = rate of descent of post-labor force com ponent

c = constant

x , = low pointxh = high peakx r = retirement peakX = labor force shiftA = parental shiftB = jump

x X I x, , x + AAge, x

FIGURE 4 The model migration schedule.

3 . An almost bell-shaped curve of the post-labor force ages positioned a t /.I, onthe age axis and exhibiting rates of ascent A, and descent a,4. A constan t curve c , he inclusion o f which improves the fit of the ma thema tical

expression to the observed scheduleThe decom position described above suggests the following simple sum of four curves(Rogers et al. 197 8):


13/160

Model migration schedules 7The labor force and the post-labor force components in eq. (1) adopt the "doubleexponential" curve formulated by Coale and McN eil(1972) for their studies of nuptialityand fertility.The "full" model schedule in eq. (I ) has 1 1 parameters: a , , a , ,a , , p , , a , , A,, a , , p , ,

a , , A,, and c . The profile of th e full model schedule is defined by 7 of th e 1 1 parameters:cu,,p2,cu2,A, , p , , a , , an d A,. Its level is determined by the remaining 4 parameters: a , , a , ,a , , and c . A c hange in th e value of th e GMR of a particular m odel sched ule alters prop or-tionally the values of th e latter but does not affect th e form er. As we shall see in the nextsection, howev er, certain aspects of the profile also depend on t he allocation of the sched-ule's level among the pre-labor, labor , and post-labor force age compo nent s and o n theshare of the tot al level accounted for by the con stant term c. Finally, migration scheduleswith out a retiremen t peak may be represented by a "reduced" model with seven param-eters, because in such instances the third compo nent of eq . (1) is omitted .

Table 1 sets o ut illustrative values of th e basic and derived measures presented inFigure 4 . The 19 74 data refer t o migration schedules for an eight-region disaggregation ofSweden (Andersson and Holmberg 198 0). The metho d chosen for fitting the model sched-ule t o th e da ta is a functional-minimization procedure k nown as the modified Levenberg---Marquardt algorithm (see A ppendix A, Brown and Dennis 197 2, Levenberg 194 4, Mar-quar dt 1963). M inimum chi-square estimators are used to give mo re weight t o age groupswith smaller rates of migration.To assess the goodness-of-fit tha t the m odel schedule provides when it is applied t oobserved da ta , we calculate E, the mean of the absolute differences between estimatedand observed values expressed as a percentage of the observed m ean:

This measure indicates that the fit of the model to the Swedish data is reasonably good,th e eight regional indices of goodness-of-fit E being 6.87,6.41,12.15,11.01,9.3 1, 10.77,11.74, and 14.82 for males and 7.30, 7.23, 10 .71, 8.7 8,9 .31 , 11.61, 11.38, and 13.28for females. Figure 5 illustrates graphically this goodness-of-fit of the model schedule tothe observed regional migration data for Swedish females.Model migration schedules of the form specified in eq. (1) may be classified intofamilies according to the ranges of values taken on by their principal parameters. Forexample, we may order schedules according to their migration levels as defined by thevalues of t he four level parameters in eq. (I ), i.e., a , , a , , a , , an d c (or b y their associatedGMRs). Alternatively, we may distinguish schedules with a retirement peak from thosewith out on e, or we m ay refer to schedules with relatively low or high values for the rateof ascent of the labor force curve A, or the mean age 5. In many applications, it is alsomeaningful to characterize migration schedules in terms of several of the fundamentalmeasures illustrated in Figure 4, such as the low point x,, he high peak xh, nd the retire-ment peak x,. Associated with the first pair of points is the labor force shift X , which isdefined to be th e difference in years between th e ages of the high peak and t he low po int ,i.e., X = xh - xl. he increase in the migration rate of individuals aged xh over thoseaged xl ill be called th e jum p B.


14/160

8 A . Rogers, L.J. C~ s r r c ,TABLE 1 Parameters and variables defining observed mod el migration schedules: ou tmigration from the 8

Parametersand variablesaGMR0 1a 101ka1A 10 3M3a 3A3C-n%(O-14)%(IS-64)%(65+)61c611'3 119110 20 3

Region1 . Stockholm 2. East Middle 3. South Middle 4. SouthMale Female Male Female

1.44 1.480.035 0.0390.088 0.1080.079 0.096

20.27 18.520.090 0.1090.406 0.491

Male Female1.33 1.410.032 0.0330.096 0.1060.091 0.112

19.92 18.490.104 0.1270.404 0.560

Male Female0.840.02 10.1040.067

19.880.1290.442

' ~ 1 1 parameters and variables are briefly defined in Ap pendix B and discussed more comprehensively in the%he GMR , its percentage distribution across the three major age categories (i.e., 0-14, 15-64, 65+), and

The close correspondence between t he m igration rates of children and those o f theirparents suggests anoth er im port ant shift in observed migration schedules. If, for each poin tx on the post-high-peak part of the migration curve, we obtain by interpolation the age(where it exists), x - A, say, with the identical rate of migration on the pre-low-pointpart of the migration curve, then t he average of the values of A,, calculated incrementallyfor the num ber of years between zero and the low point x l , will be defined as the observedparental shift A .

An observed (or a graduated) age-specific migration schedule may be described in anumb er of useful ways. For exam ple, references may be made t o the heights at particularages, to lo cations of im port ant peaks or troughs, to slopes along the schedule's age prof de,to ratios between particular heights or slopes, to areas under parts of the curve, and tobo th horizontal and vertical distances between im porta nt heights and locations. The vari-ous descriptive measures characterizing an age-specific model migration schedule may beconveniently g rouped into the following categories and subcategories:


15/160

Model migration schedulesSwedish regions, 1974 observed data by single years o f age.-

. West 6. North Middle 7. Lower North 8. Upper NorthMale Female Male Female Male Fema le Male Female

7.69 7.07 7.37 5.89 7.37 5.05 7.26 5.0829.57 27.42 29.92 27.01 30.15 26.94 31.61 28.30

0.023 0.027 0.042 0.059 0.053 0.077 0.040 0.063following text.the mean age ii are all calculated with a mod el schedule spanning an age range of 95 years.

1. Basic measures (the 11 fundam ental parameters a nd their ratios)heights: a , , a ,, a , , clocations: p , , p ,slopes: a , , a , , A,, a , , A,-atios: S I C - a , / c , S , , = a , / a , , a = a , / a , , b,, = a , / a , , a , = A , /a , ,

= A,/%2 . Derived measures (properties of the m odel schedule)areas: GMR, %(O-14), %(15 --64), %( 65+ )

locations: ?i,I, xh, ,distances: X , A ,B

A convenien t ap proach for characteriz ing an observed mode l migration schedule (i.e.,an empirical schedule gra duated by eq . (1)) is to begin with the central labor force curve


16/160

A. Rogers, I. .J. Castro

2. E m Middle

FIGURE 5 continued on facing page.


17/160


0.09- 5.w-Iom.0.07

1

6. Nonh Middla

FIGURE 5 Observed (jagged line) and model (smooth line) migration schedules: females, Swedishregions, 1974.


18/160

12 A. Rogers, L.J. Castroand then t o "add on" the pre-labor force, post-labor force , and con stant compo nents. Thisapproach is represented graphically in Figure 6.

I Ilabor force pre-labor post-labor constant model schedulecomponent force force componentcomponent component

FIGURE 6 A schematic diagram of th e fundamental components o f the full model migration schedule.

One can imagine describing a decompo sition of th e model migration schedule alongthe vertical an d horizon tal dimensions; e.g., allocating a fraction of its level to the con stantcomponent and then dividing the remainder among the other three (or two) components.The ratio S , , = a , / c measures the former allocation, and S , , = a , / a , and S , , = a , / a ,reflect t he latter division.

The heights of the labor force and pre-labor force components are reflected in theparameters a , an d a , , espectively, therefore the ratio a , / a , indicates the degree of "labordominance", and its reciprocal, S , , = a , / a , , the index of child dependency, measures thepace at which children migrate with their parents. Thus the lower the value of 6 , , , thelower the degree of child d ependency exh ibited by a migration schedule an d, correspond-ingly, th e greater its labor do min ance. This suggests a dich oto mo us classification of migra-tion schedules into child dependent and labor dominant categories.An analogous argument applies to the post-labor force curve, and S , , = a , / a , sug-gests itself as the appropriate index. It will be sufficient for our purposes, however, torely simply on the value taken on by the parameter a,,with positive values pointing outthe presence of a retirement peak and a zero value indicating its absence.Labor dominance reflects the relative migration levels of those in the working agesrelative to those of children and pensioners. Labor asymmetry refers to the shape of theleft-skewed unim odal curve describing th e age profile of labor force migration. Imaginethat a perpendicular line, connecting the high peak with the base of the bell-shaped curve(i.e., the jump B), divides the base into two segments g an d h as in Figure 7 . Clearly, theratio h/g is an indicator of the degree of asymm etry of the curve. A mo re convenient index,using only two parameters of the model schedule is the ratio a , = A , / a , , the index oflabor asymmetry. Its movement is highly correlated with that of h / g , because of theapproxim ate relation


19/160


FIGURE 7 A schematic diagram o f the curve describing the age profde o f labor force m igration.

where o: denotes proportionality. Thus o, may be used to classify migration schedulesaccording t o their degree of labor asymmetry.Again, an analogous argument applies to the post-labor force curve, and o, = h,/a,may be defined as the index of retirement asym metry.When "adding on" a pre-labor force curve of a given level to the labor force com-pon ent, it is also impor tant to indicate something of its shape. For example, if the mgr a-tion rates of children mirror those of their parents, then a, should be approxim ately equalt o a,, and p,, = CY,/CY,,he index of parental-shift regularity, should be close to unity.The Swedish regional migration patte rns described in Figure 5 and in Table 1 may

be characterized in terms of the various basic and derived measures defined above. Webegin with the observation that the outmigration levels in all of the regions are similar,with GMRs ranging from a low of 0.80 for males in Region 5 t o a high of 1.48 for femalesin Region 2 . This similarity permits a reasonably accu rate visual assessment an d chara cter-ization of the profiles in Figure 5.Large differences in GMRs, however, give rise to slopes and vertical relationshipsamong schedules that are nonc omparab le when examined visually. Recourse the n mus t bemade t o a standardization of the areas under th e migration curves, for example, a generalrescaling to a GMR of unity . Note th at this difficulty does not arise in the numerical datain Table 1 , because, as we pointed ou t earlier, the principal slope and location parametersand ratios used to characterize the schedules are not affected by changes in levels. Onlyheights, areas, and vertical distances, such as the jump, are level-dependent measures.

Among the eight regions examined, only the first two exhibit a definite retirementpeak, t he male peak being the more d ominant one in each case. The index of child depen-dency S , , is highest in Region 1 and lowest in Region 8 , distinguishing the latter region'slabor dom inan t profile fro m Stockholm's child dependent outm igration patter n. Th e indexof labor asymmetry o, varies from a low of 2.34 , in the case of males in Region 4 to ahig hof 4 .95 for t he female outmigration profile of Region 8 . Finally, with the possible excep-tion of males in Region 1 and females in Region 6, the migration rates of children inSweden do indeed seem t o mirror those of their paren ts. The inde x of parental-shift regu-larity P I , is 1.26 in the former case and 0.73 0 in the latter; for most of the other sched-ules it is close to unit y.


20/160

14 A . Rogers, L.J. Castro3 A COMPARATTVE ANALYSIS O F OB SERVED M ODEL MIGRATIONSCHEDULES

Section 2 demonstrated that age-specific rates of migration exhibit a fundamentalage profile, which can be expressed in mathe matical f orm as a model migration scheduledefined by a to tal of 11 parameters. In this section we seek t o establish the ranges of val-ues typically assumed by each of these parameters a nd their associated derived variables.This exercise is made possible by the availability of a relatively large data base collectedby the C omparative Migration and S ettlement St ud y, recently concluded at IlASA (Rogers197 6a, 19 76 b, 19 78 , Rogers and Willekens 19 78 , Willekens an d Rogers 1978). The m igra-tion data for each of the 1 7 countries included in this stud y are set ou t in individual casestudies, which are listed at th e end of this report.

3.1 Data Re para tion, Parameter Estimation, and Summary StatisticsThe age-specific migration rates tha t were used to d em onst rate the fits of the m odel

migration schedule in the last section were single-year rates. Such data are scarce a t th eregional level and, in our comparative analysis, are available only for Sweden. AU otherregion-specific migration data are reported for five-year age groups only an d, therefor e,must be interpolated to provide th e necessary inpu t data by single years of age. In all suchinstances t he region-specific migration schedules were first scaled to a GMR of unity (GMR= 1) before being subjected to a cubic-spline interpolation (McNeil e t al. 19 77).

Starting w ith a migration schedule with a GMR of unity and rates by single years ofage, the nonlinear parameter estimation algorithm ultimately yields a set of estimates forthe model schedule's parameters (see Appendix A for details). Table 1 in section 2 pre-sented the results that were obtained using the data for Sweden. Since these data wereavailable for single years of age, the influence of the interpolation procedure could be

TABLE 2 h a m e te r s defining observed model m igration schedules and parameters obtained after a cubic-Region and width of age group1. Stockholm 2. East Middle 3. South Middle 4. South

Parameters 1 ~r 5 v 1 ~r 5 yr 1 ~r 5 Yr 1 ~r 5 ~r0 1 0.029 0.028 0.026 0.026 0.023 0.023 0.025 0.025a 1 0.091 0.089 0.108 0.106 0.106 0.105 0.104 0.106a 1 0.047 0.049 0.065 0.070 0.080 0.087 0.080 0.085Cla 19.32 19.69 18.52 18.99 18.49 18.93 19.88 20.23a 1 0.094 0.098 0.109 0.117 0.127 0.136 0.129 0.135A1 0.369 0.313 0.491 0.351 0.560 0.375 0.442 0.367c 0.002 0.002 0.003 0.003 0.003 0.003 0.003 0.003a3 0.000 0.000Cl n 85.01 81.20a 3 0.369 0.364A, 0.072 0.080a~ bs er ve d ata are for single yearsof age (1 yr) ; the cubic-spline-interpolated nputsare obtained from observed


21/160

Model migration schedules 15assessed. Table 2 contrasts the estima tes for female schedules in Table 1 with those obtainedwhen the same d ata are first aggregated to five-year age groups and then disaggregated tosingle years of age by a cubic-spline interpolation. A comparison of the param eter estimatesindicates t ha t t he interpolation procedu re gives generally satisfactory results.

Table 2 refers t o results for rates of migration from each of eight regions to th e restof Sw eden. If these rates are disaggregated by region of d estin ation , then 8' = 64 inter-regional schedules need t o be examined for each sex , which will comp licate comparisonswith othe r na tions. T o resolve this difficulty we shall associate a "typical" schedule witheach collection of national rates by calculating the mean of each parameter and derivedvariable. Ta ble 3 illustrates the results for th e Sw edish data .

To avoid the influence of unrepresentative "outlier" observations in the com puta tionof averages defining a typical national schedule, it was decided to delete approximately10 percent of the "extreme" schedules. Specifically, the param eters and derived variableswere ordered from low value to high value; the lowest 5 percent and th e highest 5 percentwere defined to be extreme values. Schedules with the largest number of low and higliextreme values were discarded, in sequence, until only ab ou t 9 0 percent of th e originalnumbe r of schedules remained. This reduced se t then served as the population of schedulesfor the calculation o f various summary statistics. Table 4 illustrates the average paramete rvalues obtained with the Swedish data. Since the median, mod e, standard deviation-to-mean ratio, and lower and upper bounds are also of interest, they are included as part ofthe more detailed c ompu ter outp uts reproduced in Appendix B.The comparison, in Table 2, of estimates obtained using one-year and five-year ageintervals for the same Swedish data indicated that the interpolation procedure gave satis-factory results. It also suggested, however, that the parameter A, was consistently unde r-estimated with five-year da ta. To confirm this , the results of Table 4 were replicated wit11the Swedish data b ase, using an aggregation w ith five-year age intervals. The results, setout in Table 5 , show once again that A, is always underestimated by the interpolationprocedure. This tendency should be noted and kept in mind.

spline interpolation: Sweden, 8 regions, females, 197 4.'

5 . West 6. North Middle 7. Lower North 8. Upper North

data by five-year age groups (5 yr).


22/160

16 A. Rogers, L.J. CastroTABLE 3 Mean values of parameters defining the full set of observed model migration schedules:Sweden, 8 regions, 1974 observed data by single years of age until 84 years and over.'

Males FemalesWithout retirement With retirement Without retirement With retirementParameters peak (52 schedules) peak (1 1 schedules) peak (58 schedules) peak (5 schedules)

01 0.029 0.025 0.027 0.02301 0.1 26 0.080 0.114 0.0870' 0.066 0.050 0.078 0.051Pa 21.09 21.52 19.13 19.20(2.1 0.113 0.096 0.133 0.101Aa 0.459 0.439 0.525 0.377c 0.003 0.002 0.003 0.003a s 0.0012 0.0017P3 75.45 72.070 3 0.797 0.688A3 0.294 0.192'Region 1 (Stockholm) is a singlecommune region; hence there exists no intraregional schedule for it,leaving 8' - 1 = 63 schedules.TABLE 4 Mean values of parameters defining the reduced set of observed model migration schedules:Sweden, 8 regions, 1974 observed data by single years of age until 84 years and over.'

Males FemalesWithout retirement With retirement Without retirement With retirementParameters peak (48 schedules) peak (9 schedules) peak (54 schedules) peak (3 schedules)

a1 0.029 0.026 0.026 0.02401 0.1 24 0.085 0.108 0.093a 1 0.067 0.05 1 0.076 0.055P I 20.50 21.25 19.09 18.870 1 0.104 0.093 0.127 0.106A1 0.448 0.416 0.537 0.424c 0.003 0.002 0.003 0.003a 3 0.0006 0.0001Cg 76.7 1 74.780 3 0.847 0.938A3 0.158 0.170'Region 1 (Stockholm) is a singlecommune region; hence there exists no intraregional schedule for it,leaving 8' - 1 = 6 3 schedules, of which 6 were deleted.

It is also important to note the erratic behavior of the retirement peak, apparentlydue to its extreme sensitivity to the loss of information arising out of the aggregation.Thus, although we shall continue to present results relating to the post-labor force ages,they will not be a part of our search for families of schedules.


23/160

Model migration schedules 17

TABLE 5 Mean values of parameters defining the reduced set of observed model migration schedules:Sweden, 8 regions, 1974 observed data by five years of age unti l 80 years and over.a

ParametersMales FemalesWithout retirementpeak (49 chedules)0.0280.1150.06820.610.1050.3960.002

With retirementpeak (8 chedules)0.0260.0880.05220.260.0840.3900.0010.001777.470.6030.148

Without retirementpeak (54 chedules)0.0260.1080.08019.520.1330.3740.002

With retirementpeak (3 chedules)0.0260.0770.04419.180.0890.3410.0020.0036

77.720.3750.134a ~ e g i o n (Stockholm) is a single-commune region; hence there exists no intraregional schedule for it,leaving 8' -- 1 = 63 schedules, of which 6 were deleted.

3.2 National ContrastsTables 4 and 5 of the preceding subsection summarized average parameter values

for 57 male an d 5 7 female Swedish mode l migration schedules. In this subsection we shallexpand our analysis to include a mu ch larger data base, adding to the 11 4 Swedish modelschedules another 16 4 schedules from the United Kingdom (Table 6), 11 4 from Jap an, 20from the Netherlands (Table 7) , 58 from the Soviet Union, 8 from the U nited S tates, and32 from H ungary (Table 8). Sum mary statistics for these 51 0 schedules are set out in

TABLE 6 Mean values of parameters defining the reduced set of observed model migration schedules:the United Kingdom, 10 egions, 1970.~Males FemalesWithout retirement With retirement Without retirement With retirementParameters peak (59 chedules) peak (23 chedules) peak (61 chedules) peak (21 schedules)

a1 0.021 0.016 0.021 0.018a1 0.099 0.080 0.097 0.089a2 0.059 0.053 0.063 0.048Pa 22.00 20.42 21.35 21.56a1 0.127 0.120 0.151 0.153A2 0.259 0.301 0.327 0.333c 0.003 0.004 0.003 0.004a 3 0.007 0.002P3 71.11 71.84a3 0.692 0.583As 0.309 0.403a ~ ontraregional migration data were included in the United Kingdom data; hence 10' - 10 = 90schedules were analyzed, of which 8 were deleted.


24/160

18 A. Rogers, L.J. CnstroTABLE 7 Mean values of parameters defining the reduced set of observed model migration schedules:Japan, 8 regions, 1970; the Netherlands, 1 2 regions, 1 9 74 . ~

Japan NetherlandsMales Females Males FemalesWithout retirement Without retirement With retirement With retirement

Parameters peak (57 schedules) peak (57 schedules) slope (10 schedules) slope (10 schedules)

a ~ e g i o n in Japan (Hokkaido) is a single-prefecture region; hence there exists no intraregional schedulefor it, leaving 8' - = 63 schedules, of which 6 were deleted. The only migration schedules availablefor the Netherlands were the migration rates out of each region without regard to destinat ion; henceonly 12 schedules were used, of which 2 were deleted.TABLE 8 Mean values of parameters defining the reduced set of observed total (males plus females)model migration schedules: the Soviet Union, 8 regions, 1974; the United States, 4 regions, 1970-1971 Hungary, 6 regions, 1974.~

Soviet Union United States HungaryWithout retirement With retirement Without retirement With retirement

Parameters peak (58 schedules) peak (8 schedules) slope (7 schedules) slope (25 schedules)

aIntraregional migration was included in the Soviet Union and Hungarian data but not in the UnitedStates data; hence there were 8' = 64 schedules for the Soviet Union, of which 6 were deleted, 6' =36 schedules for Hungary, of which 4 were deleted, and 4' - 4 = 12 schedules for the United States,of which 2 were deleted because they lacked a retirement peak and another 2 were deleted because oftheir extreme values.

Appendix B ; 206 are male schedules, 206 are female schedules, and 98 are for the com-bination of bo th sexes (males plus females).**This total does not include the 56 schedules excluded as "extreme" schedules. During the process offitting the model schedule to these more than 500 interregional migration schedules, a frequentlyencountered problem was the occurrence of a negative value for the constant c. In all such instances


25/160

Model migration schedules 19A significant number of schedules exhibited a pattern of migration in the post-labor

force ages that differed from th at of th e 1 1 parameter model migration schedule definedin eq. ( 1 ) . Instead of a retirement peak, the age profile took on the form of an "upwardslope". In such instances the following 9-parameter modification of the basic model migra-tion was introduced

M ( x ) = a , exp ( - a , x ) 1

The right-hand side of Table 7, for example, sets ou t the m ean parameter estimatesof this modified form of th e model migration schedule for the Netherlands.Tables 4 through 8 present a wealth of information about national patterns of

migration by age. The parameters, given in columns, define a wide range of model migra-tion schedules. Four refer only to migration level: a l , a 2 ,a, , an d c . Their values are for aG M R of unity; to obtain corresponding values for other levels of migration, these fournumbers need to be multiplied by the desired level of GMR. For example, the observedG M R for female m igration out of th e Stockh olm region in 1 97 4 was 1.43. Multiplyinga, = 0.029 by 1.43 gives 0.041, the appropriate value of a , with which to generate themigration schedule having a G M R of 1.43.

The remaining model schedule param eters refer t o migration age profile: a,, p,, a, ,A,, p,, a,, an d A,. Their values remain con stant for all levels of th e GM R. Taken together,they define the age profile of migration from on e region to another . Schedules with out aretirement peak yield only the four profile parameters: a,, p,, a, , and A,, and scheduleswith a retirement slope have an additional profile parameter a,.

A detailed analysis of th e parameters defining the various classes of schedules isbeyond the scope of this rep ort. Nevertheless a few basic contrasts among national averageage profiles may be usefully highlighted.

Let us begin with an examination of the labor force component defined by th e fourparameters a,, p, , a, , and A,. The national average values for these parameters generallylie within t he following ranges:

the initial value of c was set equal to the lowest observed migration rate and the nonlinear estimationprocedure was started once again.


26/160

20 A . Rogers, L.J.G s t r oIn all but two instances, the female values for a , , a , , an d h, are larger than thosefor males. The reverse is the case for p ,, with two exceptions, the most imp ortan t of which

is extubited by Japan's females, who consistently show a high peak tha t is older than thatof males. This apparently is a consequence of the tradition in Japan that girls leave thefamily hom e at a later age than boys.The two parameters defming the pre-labor force compo nen t, a , and a , , generally liewithin the ranges of 0.01 to 0.03 and 0.08 to 0.12, respectively. The exceptions are theSoviet Union and Hungary, which exhibit unusually high values for a , . Unlike the case ofthe labor force co mp onen t, consistent sex differentials are difficult t o identify.Average national migration age profi les, like most aggregations, hide m ore th an th eyreveal. Some insight into the ranges of variations that are averaged out may be found byconsulting t he lower and upper bounds and standard-deviation-to-m ean ratios listed inAppend ix B for each set of national schedules. Additional details are set out in AppendixC. Finally, Table 9 illustrates how para me ters vary in severalunaveraged national schedules,by way of example. T he model schedules presented there describe migration flows out ofand int o th e capital regions of each of six countries: Helsinki, Finland; Budapest, Hungary;Tok yo, Japan; Amsterdam, the Netherlands; Stockh olm, Sweden; and Lo ndon, the UnitedKingdom . All are illustrated in Figure 8 .The most apparent difference between the age profiles of the outflow and inflowmigration schedules of the six national capitals is the dominance of young labor forcemigrants in the inflow , tha t is, proportionately m orem igran ts in the young labor force agesappear in the inflow schedules. The larger values of the prod uct a , h , in the inflow sched-ules and of the ratio A,, = a , / a , in the ou tflow schedules indicate this labor dominance.A second profile attrib ute is the degree of asymm etry in the labor force comp onen tof t he migration schedule, i.e., the ratio of the rate of ascent h, to th e rate of descent a ,defined as o, in section 2. In all bu t the Japanese case, the labor force curves of the capital-region outm igration profiles are more asymmetric th an those of the corresponding inmigra-tion profiles. We refer to this characteristic as labor asymmetry.Examining the observed rates of descent of the labor and pre-labor force curves, a ,and a , , respectively, we find , fo r exam ple, tha t they are close to being equal in the outflow

TABLE 9 Parameters defining observed total (males plus females) model migration schedules for flows1974; the United Kingdom, 1970.

Finland Hungary JapanParameters From Helsinki To Helsinki From Budapest To Budapest From Tok yo To Tok yo"I 0.037 0.024 0.015 0.008 0.019 0.008Q I 0.127 0 .170 0.239 0.262 0.157 0.149"1 0.081 0.130 0.082 0.094 0.064 0.096P l 21.42 22.1 3 17.10 17.69 20.70 15.74a1 0.124 0.198 0.130 0.152 0.111 0.134A 1 0.231 0.231 0.355 0.305 0.204 0.577c 0.000 0.003 0.003 0.003 0.003 0.002"3 0.00027 0.0000 1 0.00005 0.00002 0.0013 1k 99.32Q3 0.204 0.072 0.059 0.061 0.000A, 0.042


27/160

Model migration schedules 21schedules of Helsinki and Stockholm and are highly unequal in the cases of Budapest,Tokyo, and Amsterdam. In four of the six capital-region inflow profiles a, > a,. Profileswith significantly different values for a, an d a, re said to be irregular.

In conclusion, t he empirical migration data of six industrialized nations suggest thefollowing hypothes is. The age profile o f a typ ical capital-region inmigration schedule is, ingeneral, mo re labor domin ant and more labor symmetric than the age profile o f the corre-sponding capital-region outmigration schedule. No comparable hypothesis can be maderegarding its anticipated degree of irregularity.

3.3 Families of SchedulesThree sets of model migration schedules have been defined in this report: the 11

parameter scheduIe with a retirement peak, the alternative 9-parameter schedule with aretirement slope, and the simple 7-parameter schedule with neither a peak nor a slope.Thus we have a t least three broa d families of schedules.

Additional dimensions for classifying schedules into families are suggested by theabove comparative analysis of national migration age profiles and the basic measures andderived variables defined in section 2. These dimensions reflect different locations on thehorizontal and vertical axes of the schedu le, a s well as different ratios of slopes and heights.Of the 524 model migration schedules studied in this section, 412 are sex-specificand, of these, only 336 exhibit neither a retirement peak nor a retirement slope. Becausethe parameter estimates describing the age profile of post-labor force migration behaveerratically, we shall restrict our search for families of schedules t o these 16 4 male and 17 2female mode l schedu les, summary statistics for which are set ou t in Tables 1 0 and 1 I .

An ex amina tion of t he p aram etric values exhibited by the 33 6 migration schedulessummarized in Tables 1 0 and 11 suggests tha t a large fraction of the variation shown bythese schedules is a consequence of changes in the values of the following fou r parametersand derived variables: p , , 6,, ,o ,and P I , .

from and to capital cities: Finland, 19 74 ; Hungary, 1 97 4; Japan, 19 70 ; the Netherlands, 19 74 ; Swed en,

Netherlands Swed en United KingdomFrom Amsterdam T o Amsterdam From Stockh olm To Stockh olm From London To Lond on


28/160

A. Rogers, L.J. Cnstro


29/160



30/160



31/160

5.TABLE 10 Estimated summary statistics of parameters and variables associated with reduced sets of observed model migration schedules for Sweden, the gUnited Kingdom, and Japan: males, 1 64 schedulesa Esumm ary statist ics 9-Parameters Standa rd deviation1 "and variables Lowest value Highest value Mean value Median Mode Standa rd deviation mean

GMR (observed)GMR (model)Ea 1'4a2

a2A2C-n%(O-14)%(IS-64)% (6 5 + )&, c61 2P n0 29XhXAB'A list of defin itions for the param eters and variables appear s in Appendix B.


32/160

TABLE 11 Estimated summary statis tics of parameters and variables associated with reduced sets of observed model migration schedules for Sweden, theUnited Kingdom, and Japan: females, 172 schedulesaSummary statistics

Parameters Standard deviation/and variables Lowest value Highest value Mean value Median Mode Standard deviation meanGMR (observed) 0.00388 1.59564 0.19909GMR (model) 1.00000 1.00000 1.00000E 4.17964 60.83579 15.42092a1 0.00526 0.04496 0.02259(I1 0.01585 0.41038 0.1069801 0.02207 0.18944 0.07426fil 15.06610 37.76019 20.63237(12 0.05467 0.33556 0.14355A, 0.08367 1.49869 0.40032c 0.00012 0.00685 0.00347FT 24.51402 37.86541 30.65265%(O-14) 9.37675 3 1A7480 20.93872%(IS-64) 60.55278 81.17286 68.65491%(65+) 1.46164 19.56255 10.4063861c 0.89359 192.60318 9.39987611 0.02828 0.90435 0.34847012 0.09121 2.48385 0.814720 2 0.38917 12.23371 3.26434Xl 10.32012 21.79038 14.5 1330Xh 17.03028 30.92059 22.49959X 2.89007 15.09035 7.98629A 23.73040 37.24700 28.50972B 0.00831 0.09111 0.03118'A list of definitions for the parameters and variables appears in Appendix B.


33/160

Model migration schedules 27Migration schedules may be early or late peaking, depending o n the location of p,on the horizontal (age) axis. Although this parameter generally takes on a value close t o

2 0, roughly thre e ou t of fou r observations fall within t he range 17 -- 25 . We shall call thosebelow age 1 9 early peaking schedules and those above 22 late peaking schedules.The ratio of t he t wo basic vertical param eters, a , an d a,, is a measure of the relativeimportanc e of t he migration of children in a model migration schedule. The ind exo f childdependency, S,, = a,/a ,, tends to exhibit a mean value of abou t one-third with 8 0 per-

cent of the values falling between one-fifth and four-fifths. Schedules with an index ofone-fifth or less will be said to be labor dominant; those above two-fifths will be calledchild depen dent.Migration schedules with lab or force co mp onen ts that t ake the form of a relativelysymmetrical bell shape will be said to be labor symmetrical. These schedules will tend toexhibit an index of labor asymmetry (a, = A,/ol,) th at is less th an 2. Labo r asy mm etr icschedules, on the oth er h an d, will usually assume values for a, of 5 or m ore. The averagemigration schedule will tend to show a a, value of about 4, with approximately five outof six schedules exhib iting a a, within th e range 1-8.Finally, the index of parental-shift regularity in many schedules is close to unity,with approxim ately 7 0 percent of the values lying between one-third and four-thirds.Values of P I , = a,/% that are lower than four-fifths o r higher than six-fifths will be calledirregular.

We may imagine a 3 X 4 cross-classification of migration schedules that defines adozen "average families" (Table 12). Introducing a low and a high value for each param-eter gives rise to 1 6 additional families for each of the three classes of schedules. Thus wemay conceive of a m inimum set of 6 0 families, equally divided amon g schedules with aretirement peak, schedules with a retirement slope, and schedules with neither a retire-ment peak nor a retirement slope (a reduced form ).

TABL E 1 2 A cross-classification o f migration schedules.Measures (av eraee values)Peaking Dominance Asymmetry RegularitySchedule (r,= 20) (h,, = 1/31 (a , = 4) (s,, = 1)

Retirement peak + + + +Retirement slope + + + +Reduced fonn + + + +

T o comp lement the above discussion w ith a few visual illustrations, in Figure 9(a)we present six labor dominant profiles, with SICfixed at 22. The tallest three exhibit asteep rate of descent a, = 0.3; the shortest three show a m uch m ore moderate slope ofa, = 0.06. W ithin each family of three curves, one finds variations in p, an d in the rateof ascent A,. Increasing p, shifts the curve to th e right along the horizo ntal axis; increas-ing A, raises th e relative height of the high pea k.

The six schedules in Figure 9(b) depict the corresponding two families of childdependent profiles. The results are generally similar to those in Figure 9(a), with the


34/160



35/160


I I t I 1# . " "0 0 0

2 2 2 2 a" ,1 11 11 I 1 11 11


36/160

30 A. Rogers,L.J. Castroexception that the relative importance of migration in the pre-labor force age groups isincreased considerably. The principal effects of the change in 6,, are: ( I) a raising of theintercept a , + c along the vertical axis, and (2) a sim ultaneous reduction in the height ofthe labor force component in order to maintain a constant area of unity under each curve.

Finally, the dozen schedulcs in Figures 9(c) and 9(d) describe similar families ofmigration curves, but in these profiles t he relative c ontri butio n of the constan t componerltto the unit GMR has been increased significantly (i.e., 6,, = 2.6). It is importan t to notethat such "pure" measures of profiles a s x l, x,, ,X, and A remain unaffected by this change,whereas "impure" profile measures, such as the mean age of migration i i , now take o n adifferent set of values.

3.4 Sensitivity Analysis

The preceding subsections have focused on a compariso n of the funda mental param-eters defining the model migration age profiles of a number of nations. The comparisonyielded ranges of values within which each parameter may be expe cted t o fall and suggesteda classification of schedules int o families. We now tur n t o an analytic examination of howchanges in several of the mor e imp ortan t parameters become manifested in the age profileof the model schedule. For analytical convenience we begin by focusing on th e propertiesof the dou ble exponential curve tha t describes the labor force comp onen t:

We begin by observing that if a , is set equal t o A, in the above expression, then thelabor force component assumes the shape of a well-known extreme value distributionused in the stud y of flood flows (Gumbel 19 41 , Kimball 1946). In such a case x h = p,and th e function f,(x) achieves its maximum y h at that point. To analyze the more gen-eral case where a , # A,, we may derive analytical expressions for bo th of these variablesby differentiating eq. (4) with respect to x , setting the result equal to zero, and then solv-ing to find

an expression that does not involve a , , and

an expression th at does not involve p, .Note that if A, > a , , which is almost always the case, then xh > p,. And observethat if a , = A,, then t he above tw o equations simplify to

and


37/160

Model migration schedules 3 1Since p, affects xh only as a displacement, we may focus o n the variation of xh as afunction of a, and A, . A plot of xh against a,, for a fixed A,, shows th at increases in a,lead to decreases in xh . Analogously, increases in A,, for a fixed a,, produce increases in

xh bu t a t a rate t hat decreases rapidly as the latter variable approaches its asym ptote .The behavior of yh is independent of p, and varies proportionately with a,. Henceits variation also depends fundamentally only on the two variables a, and A,. A plot of

yh against a,, for a fixed A,, gives rise to a U-shaped curve that reaches its minimum ata, = A,. Increasing A, widens the shape of the U.

The influence of a, and A, on t he labor force com pon ent may be assessed by exam-ining the propo rtional rate of change of the fu nctio n f , x):

Equation (7) defines this rate of change as the sum of two components: -a, and theexponential A, exp[-A,(x - p , ) ] . To demonstrate how the actual rates of ascent anddescent are related to A, and a, we may t ake, for exam ple, a typical set of parameter val-ues such as a, = 0.1, A, = 0.4, and p, = 20 and then proceed t o calculate the quantitiespresented in Table 13. The calculations indicate tha t, at ages above 30 , the actual rate ofdescent is almost identical to -a,. The actual rates of ascent are very different from theA, value, exce pt for ages close to x = p,.*

TABLE 13 Impacts of A , and a, on the actual rates of ascent and descent o f the lab01force component: A , = 0.4 , a, = 0.1, and p, = 20.Actual rates of ascent and descent

Range of age Age (x ) dx) = A , exp [-A, (x - , ) ] -a , + g(x)1192 1192In this range the impact 161 161o f a, can be ignored 2 2 2 2

15 3 3

In this range the impact 0 . 0 0 7 4 . 0 9 3of A , can be ignored 0 . 0 0 1 4 . 1 0 0

*We are grateful to Kao-Lee Liaw for sug gesting the exam ination o f eq. (7 ) and for pointing out thatthe parameters A, an d a, are not truly rates o f ascent and descent, respectively.


38/160

32 A. Rogers, L.J. Castro

The introduction of the pre-labor force compo nent in to the profile generally movesx h to a slightly younger age and raises y h by a bou t a , ex p(- a,x h), usually a neghgiblequantity. The addition of the constant term c , of cou rse, affects only y h , raising it b y theamo unt of the co nstan t. Thus the migration rate a t age xh may be expressed as

A variable that interrelates the pre-labor force and labor force com pon ents is theparental shift A . T o simplify o ur analysis of its dependence on th e fundamen tal param-eters, it is convenient to assume tha t a , an d a, are approxim ately equal. In such instances,for ages immediately fol lowing the h g h peak x h , he labor force componen t of the modelmigration schedule is closely approximated by the function a, exp[-a,(x, - p,)] . Recall-ing tha t the pre-labor force curve is given by a , exp (-a, x,) when a , = a, , we may equatethe tw o functio ns to solve for the difference in ages th at we have called the parenta l shift:

This equation shows that the parental shift will increase with increasing values ofp2 and will decrease with increasing values of a, and ti,,. Table 1 4 com pares the values ofthis analytically defined "theoretical" parental shift with the corresponding observed paren -tal shifts presented earlier in Table 1 for Swedish males an d females. The two definitionsappear t o produc e similar numerical values, but th e analytical definition has the advantageof being simpler t o calculate and analyze.

Consider th e rural-to-urban migration age profile defined by the p arameters in Table15 . In this profile the values of a, an d A, are almost eq ual, making it a suitable illustrationof several points raised in the above discussion.

First, calculating xh with eq. (5) gives

as against x h = 21.59 set ou t in Table 15. Deriving y h using eq . (6) gives

where a,/A, = 0.23710.270 = 0.87 8. Thus M(21.59) is approximately e qual to y h + c =0 . 069 + 0.004 = 0.0 73 . The value given by the m odel migration schedule eq uation is also0.073.Since a, # a,, we cannot adequately test the accuracy of eq. (8) as an estimator ofA . Nevertheless, it can be used t o help account for the unusually large value of the paren -tal shift. Sub stituting the values for p,, a, , and ti,, in to eq. (8) , we find

And although this is an underestimate of 45.13, it does suggest that the principal causefor t he unusu ally high value o f A is the unusually low value of ti,,. If this latter paramete r


39/160

dTABLE 14 Observed and theoretical values of the p arental shift: Sw eden, 8 regions, 1 974 . 8

Regions of SwedenParental shift 1. Stockholm 2. East Middle 3. Sou th Middle 4. Sout h 5. West 6. North Middle 7. Lower North 8. Upper North~ b s e r v e d , ~ales 27.87 29.99 29.93 29.90 29.57 29.92 30.15 31.61Theoretical, males 25.14 29.24 30.01 29.65 28.97 29.43 26.6 1 29.89observed: females 25.49 27.32 27.27 27.87 27.42 27.01 26.94 28.30Theoretical, females 24.6 8 26.85 28.16 28.91 27.51 28.54 28.19 28.95a ~ o u r c e : ab le 1 .


40/160

34 A . Rogers, L.J.CastroT A B L E 15 Parameters and variables defining observed total (males plusfemales) model migration schedules for urban-to-rural and rural-to-urbanflows: the Soviet Union, 1974.Parameters and variablesa Urban-to-rural Rural-to-urbanGMR 0.74 3.41a 0.005 0.002&I 0.313 0.431a2 0.1 27 0.18 7Pz 19.26 21.10a2 0.177 0.237A, 0.286 0.270C- 0.005 0.00411 33.66 31.24%(O-14) 8.63 5.59%1(15-64) 78.30 84.60%(65+) 13.07 9.816 ~ c 0.977 0.5486 ~ z 0.038 0.011P12 1.77 1.8201 1.61 1.14X1 11.09 11.38Xh 20.94 21.59X 9.85 10.21A 42.30 45.13B 0.045 0.063a~ list of definitio ns for t he parameters and variables appears in App endix A .

had the value fo und for Stockholm's m ales, for example, the pare ntal shift would exhibitthe muc h lower value of 22.52.

4 ESTIMATED MODEL MIGRATION SCH EDULESAn estimated model schedule is a collection of age-specific rates derived from pat-

terns observed in various populations othe r than the one being studied plus some incom -plete da ta on the pop ulation under exam ination. The justification for such an approa ch isthat age profiles of fertility, mortality, and geographical mobility vary within predeter-mined limits for most human populations. B irth, de ath , and m igration rates for one agegroup are lughly correlated with the corresponding rates for othe r age groups, and expres-sions of such interrelationships form the basis of model schedule construction. T he use ofthese regularities to develop hypothe tical schedules that are deemed t o be close approxim a-tions of the unobserved schedules of populations lacking accurate vital and mobility regis-tration statistics has been a rapidly growing area of contempo rary demographic research.

4.1 Introd uctio n: Alternative PerspectivesThe earliest efforts in the development of model schedules were based on only one

parameter an d hence had very little flexibility (United Nations 1955). Demogra pherssoon


41/160

Model migration schedules 35discovered that variations in the mortality and fertility regimes of different populationsrequired more com plex formulations. In mortality studiesgreater flexibility was introducedby providing families of schedules (Coale and Demeny 1966) or by enlarging the numberof parameters used to describe the age pattern (Brass 1975). The latter strategy was alsoadop ted in the creation of im proved model fertility schedules and was augmented by theuse of analytical d escriptions of age profiles (Coale an d Trussell 1974).Since the age patterns of migration normally exhibit a greater degree of variabilityacross regions than do m ortality an d fertility schedules, it is to be expected tha t the devel-opment of an adequate set of model migration schedules will require a greater numberbo th of families and of parameters. Although many alternative m etho ds could be devisedto summarize regularities in the form of families of model schedules defined by severalparameters, three have received the widest popularity and dissemination:

1. The regression approach of the Co de-Dem eny model life tables (Coale andDemeny 1966)2. The logit system o f Brass (Brass 1 971 )3 . The double expone ntial graduation of Coale, McNeil, and Trussell (Coale 19 77 ,Coale and McNeil 19 72 , Coale and Trussell 197 4)The regression approach embo dies a correlational perspective t ha t associates rates at

different ages to an index of level, where the particular associations may differ from one"family" of schedules to ano ther. For exam ple, in the Code-Demeny model life tables,the index of level is the expectation of remaining life at age 10 , and a different set ofregression equ ation s is established f or each of four "regions" of the wo rld. Each of thefour regions (North, South, East, and West) defines a collection of similar mortality sched-ules that are more uniform in pattern than t he to tality of observed life tables.

Brass's logit system reflects a relational perspective in which rates at different agesare given by a standard schedule whose shape and level may be suitably modified to beappropriate for a particular p opulation.Th e Code-Trussell model fertility schedules are relational in perspective (using aSwedish stand ard first-marriage schedule), but they also introduce an analytic descriptionof the age profile by adopting a double exponential curve that defines the shape of theage-specific first-marriage func tion.In this stu dy we m ix the above three approaches to define two alternative perspec-tives for estimating model migration schedules in situations where only inadequate ordefective data on inte rnal (origin-destination) migration flows are available. Both perspec-

tives rely on the a nalytic (double plus single exponential) graduation defined by the basicmodel migration schedule set out earlier in this study. Both ultimately depend on theavailability of some limited data t o ob tain t he approp riate m odel schedule, for example,at least two age-specific rates, such as M(0-4) an d M(20-24), and inform ed guesses regard-ing the values of a few key variables, such as the lo w and high points of th e schedule. The ydiffer only in the m etho d by which a schedule is identified as being approp riate for a par-ticular population.

The first perspective, the regression approach, associatesvariations in the parametersand derived variables of the model schedule t o each other and th en to age-specific migra-tion rates. The second, the logit approach, embodies different relationships between themodel schedule parameters in several standard schedules and then associates the logits ofthe migration rates in a standard to those of th e population in question.


42/160

36 A. Rogers, L.J. Costro4.2 The Correlational Perspective: The Regression Migration System

A straightforward way of obtaining an estimated model migration schedule fromlimited observed data is to associate such data with the basic model schedule's parametersby means of regression equations. For example, given estimates of the migration rates ofinfants and young adults, M(0-4) and M(20-24) say, we may use equations of the form

t o estimate the set of parameters Qi hat define the model schedule. The parameters of thefitted mo del scl~ edu les re no t independent of each o the r, however. Higher than averagevalues of A,, for example, tend t o be associated with lower tha n average values of a , . Theincorporation of such dependencies int o the regression app roach w ould surely improve theaccuracy and consistency of the estimation procedure. An examination of empirical asso-ciations amo ng model schedule parameters a nd variables, therefor e, is a necessary first ste p.

Regularities in the covariations of the model schedule's parameters suggest astrat egyof model schedule construction that builds on regression equations embodying these co-variations. Given the values for 6 ,, ,x , , an d xh , or example, one can proceed to derive p, ,A,, o,, and PI , . Since a , = A,/a, we obtai n, at the same time, an estimate for a , , whichwe then can use to fmd a , . With a , established, a , may be ob tained by drawing on thedefinitional equation ti,, = a , / a , , an d a , may be found with the similar equation P , , =a , / a , . An initial value for c is obtained by setting c = a , / 6 , , , where 6 , , is estimated byregressing it o n 6 , , , a n d a , , a , , a n d c are scaled t o give a GMR of unity .

Conceptually, this approach to model schedule construction begins with the laborforce component and then appends to it the pre-labor force part of the curve. The valuegiven for 6 , , reflects the relative weights of these two co mp on ent s, with low values defin -ing a labor dominant curve and high values pointing to a family dominant curve. (Thebehavior of the post-labor force curve is assumed here to be treated exogenously.)

We begin the calculations with p, to establish the location of th e curve on the ageaxis; is it an early or late peaking curve? Next, we turn to the determination of its twoslope parameters A, and a, by resolving whether or no t it is a labor symmetric curve. Val-ues of a , between 1 and 2 generally characterize a labor sy mm etric cu rve; higher valuesdescribe an asymm etric age profile. T he regression of a , on a, produces the fourth param-eter needed to define the labor force component. With values for p,, A,, a , , and a , th econstruction procedure turns to the estimation of the pre-labor force curve, which isdefined by the two parameters a, an d a , . Its relative share of the total unit area underthe model migration schedule is set by the value given to 6 , , . The retirement peak and theupward slope are introduced exogenously by setting their parameters equal to those ofthe "observed" mo del migration schedule.

The collection of regression equations given in Table 16 exemplifies a regressionsystem that may be defined to represent the "child dependency " set, inasmuch as theircentral independent variable ti,, is the index of child dependency. It is also possible toreplace this independent variable with others, such as o, or P , , for ex ample, to create a"labor asym metr y" or a "parental-shift regularity" set. The regression coefficients wereobtained using the age-specific interregional migration schedules (scaled to unit GMR) ofSweden , the United Kingdom, and J apan . Of the three variants, the child depen dency setgave the best fits in abou t half of t he female schedules tes ted , whereas the parental-shift


43/160

Model migration schedulesTABLE 16 A basic set of regression equations.

Regression co efficients of independent variablesDepe ndent variables Intercep t 6,, X1 Xh a 2P, (males) -3.26 3.28 -4 .67 1 .39(females) - 7.69 -2 .1 4 -0.53 1.63A, (males) 1.31 0.15 0 .08 4 . 0 9(females) 1.19 0.13 0.08 4 . 0 9o, (males) 16.43 5.59 0.89 -1.17

(females) 10.97 6.05 0.63 -0.85P , , (males) 1.90 1.33 4 . 0 3 4 . 0 4(females) 1.82 1.42 0 . 0 4 4 . 0 4a, (males) 0.03

(females) 0.046,, (males) 9.41 13.83(females) 0.19 26.4 3

regularity set was overwhelmingly the best fitting variant for the male schedules (see Rogersand Castro 1981).To use th e basic regression equat ion s presented in T able 1 6, on e first needs to obtain

estimates of til2, x l , and xh.Values for these three variables may be selected to reflectinformed guesses, historical data, or empirical regularities between such model schedulevariables and observed m igration d ata . Fo r examp le, suppose th at a fertility survey hasproduced a crude es timate of the ratio of infant to parent migration rates: M = M(0--4)lM(20-24), say. A linear association betw een S12 and this M ra tio , with the regressionequa tion forced through the origin, gives

for females, and

for males.Figure 1 0 illustrates examples of the goodness-of-fit provided by the estimatedschedules to the observed model migration data. Two sets of estimated schedules are

shown: those obtained w ith the observed index of child dependency (S12) nd those foundwith the estimated index (i12), both calculated using the above regressions. In each casex1an d xh were set equal to the values given by the observed model migration schedules.

4.3 The Relational Perspective: The Logit Migration SystemAmong the most popular method s for estimating mortality from inadequate or defec-

tive da ta , is t he so-called logit system developed by William Brass abo ut t we nty years ago


44/160

A . Rogers, L.J. Castro


45/160

Model migration schedules 39and now widely applied by demographers all over the world (Brass 197 1, Brass and Coale196 8, Carrier and Hobcraft 197 1 , Hill and Trussell 1 97 7, Zaba 197 9). The logit approachto model schedules is founded on the assumption that different mortality schedules canbe related t o each ot he r by a linear tra nsfo rmati on of the logits of their respective survivor-ship probab~lities.That is, given an observed series of survivorship probabilities l(x) forages x = 1,2, . w, it is possible t o associate these observed series with a "standard" seriesl,(x) by m eans of the linear relationship

where, say,

The inverse of this fu nctio n is

The principal result of this mathematical transformation of the nonlinear l(x) func-tion is a mo re nearly linear functi on in x , with a range of minus and plus infinity ratherthan unity and zero.

Given a standard schedule, such as the set of standard logits, Y,(x), proposed byBrass, a life table can be created by selecting appropriate values for y and p . In the Brasssystem y reflects the level of mortality and p defines the relationship between child andadult m ortality. The closer y is to zero and p t o unity, the more the estimated life table islike the standard.The logit perspective c an be readily applied to migration schedules. Let .M(x) den otethe age-specific migration rates of a schedule scaled to a unit GMR, and let .M,(x) den ote

the corresponding standard schedule. Taking logits of both sets of rates gives the logitmigration system

an d

where, for example,

Th e selection of a particular migration schedule as a standard reflects the belief th atit is broadly representative of th e age pattern of m igration in th e multiregional pop ulation


46/160

40 A. Rogers, L.J. Gzstrosystem under consideration. (Our standard schedules will always have a unit GM R; henceth e left subscript o n .Y,(x) will be dropped.) T o illustr ate a num ber of calculations carriedou t with several sets of multiregional da ta, we shall ad op t the national age profile as thestandard in e ach case an d strive to estimate r.egiona1outmigration age profiles by relatingthem t o the national one. Specifically, given an m X m table of interregional migrationflows for any age x , we divide each origin-destination-specific flow Oii(x) by the pop ula-tion in the origin region Ki(x) to define the associated age-specific migration rate Mii(x).Summ ing these over all origins and de stinations gives the correspon ding national rate M. .(x),an d scaling all sche dules t o un it GMR gives .Mii(x) an d .M..(x), respectively.

Figure 1 1 presents national male standards for Sw eden, the United Kingdom, Jap an ,and the Netherlands. (We shall deal only with g raduated fits inasmuch as all of our non-Swedish data are for five-year age intervals and therefore need to be graduated first inorder to provide single-year profiles by means of interpolation.) The differences in ageprofiles are marked. Only the Sw edish an d the United K ingdom standards exhibit a retire-ment peak. Japan's profile is described without such a peak because the age distributionof migrants given by the census data e nds with th e ope n interval of 65 years and over. Thedata for the Netherlands, on the other han d, show a definite upward slope at the post-laborforce ages and therefore have been graduated with the 9-parameter model schedule withan upward slope.

Regressing th e logits of the age-specific outmigration rates o f eac h region on thoseof its na tional standard (the GMRs of bot h first being scaled t o unity) gives estimated val-ues for 7 nd p. Reversing the p rocedure and combining selected values of 7 nd p with anational standard of logit values, identifies the following important regularity: whenever7 = 2 (p - 1 ) then the GMR of the esti ma ted model schedu le is approxim ately unity(Rogers and Ca stro 1981). Linear regressions of th e form

fitted to our data f or Sw eden, the U nited Kingdom, Japa n, and the Netherlands, consis-tently produce estimates for do and dl that are approximately equal t o 2 in magnitudeand t ha t differ only in sign, i.e., 4 = -2, and 4 = +2. Thus

Differences in the national standa rd schedules illustrated in Figure 11 suggest tha t asingle standa rd schedule may be a more restrictive assump tion in migration analysis thanin mortality studies. It therefo re may be necessary t o follow the Code-De meny strategyof developing families of approp riate schedules (C od e and D emeny 1966).The comparative analysis of national and interregional migration patterns carriedout in section 3 identified a t least thre e distinct fam ilies of age profiles. First, there wasthe 1 1-param eter basic m odel migration schedule with a retiremen t peak tha t adequa telydescribed a number of interregional flows, for example, the age profiles of outmigrantsleaving capital regions such as Stockholm and London. The elimination of the retirementpeak gave rise t o the 7-param eter reduced form of this basic schedule, a form that wasused to describe a large numb er of labor dom inan t profiles and the age patter ns of migra-tion schedules with a single openended age interval for the post-labor force population,


47/160



48/160

4 2 A. Rogers, L.J . Castrofor exam ple, Japan's migration schedules. Finally, the existence of a mon otonically risingtail in m igration schedules such as those exhibited by t he Dutch da ta led to the definitionof a third profile: the 9-parameter model migration schedu le with art upward slope.

Within each family of schedules, a number of key parameters or variables may beput forward in order to further classify different categories of migration profiles. Forexam ple, in section 3 we identified the special importa nce of the following aspe cts ofshape and location along the age axis:

1. Peaking: early peaking versus late peaking (p , )2. Dom inance: child dependence versus labor dominan ce (6, , )3. Asymmetry: labor symmetry versus labor asymmetry ( 0 , )4. Regularity: parental-shift regu larity versus parental-shift irregularity (/3,,)These fundamental families and four key parameters give rise to a large variety of

standard schedules. For example, even if the four key parameters are restricted to onlydichotomous values, one already needs Z4 = 16 standard schedules. If, in addition, thesexes are to be differentiated, then 32 standa rd schedules are a minimum . A large num berof standard schedules would make the logit approach a less desirable alternative. Hencewe shall examine the feasibility of adopting only a single standard for both sexes andassume that the shape of the post-labor force part of the schedule may be determinedexogenously. In tests of our logit migration sy stem, there fore, we shall always set the pos t-labor force retirement peak or upward slope equal to observed m odel schedule values.

The similarity of the male and female median parameter values set ou t in Tables 10and 11 (for Sweden , the United Kingdom, and Japan), suggests that one could use theaverage of the values for the two sexes to define a unisexual stand ard. A rough roundingof these averages would simplify matters even more. Table 17 presents the simplified basicstand ard param eters o btaine d in this way. The values of a, , a,, and c are initial values onlyand need t o be scaled proportionately t o ensure a unit GMR. Figure 12 illustrates the ageprofile of this simplified basic standard migration schedule.

TABLE 17 Th e simplified basic standard migrationschedule. - - - -Fundamental parameters Fundamen tal ratios

We have noted before that when 7 = 0 and p = 1 , the estimated model schedule isidentical t o the stand ard. Moreover since the GMR of the s tandard is always unity , values of7 nd p tha t satisfy the equality 7= 2(p - 1) guarantee a GMR ofu nit y for the estimatedschedule. What are the effec ts of oth er combinations of values for these two parameters?


49/160


FIGURE 12 Simplified ba sic standard migration schedule.

Figure 1 3 illustrates how the sim plified basic stand ard schedule is transf orm ed when-y and p are assigned particular pairs of values. Figure 13(a) shows that fixing -y = 0 andincreasing p from 0.75 to 1.25 lowers the schedule, giving migration rates th at are smallerin value than those of the standard. On the other hand, fixing p = 0.7 5, and increasing -yfrom -1 to 0 raises the sched ule, according t o Figure 13(b). Finally, fixing GMR = 1 b yselecting values of -y and p that satisfy the equality -y = 2(p - 1) shows that as -y and pboth increase, so does the degree of labor dominance exhibited by the estimated sched-ule. For e xam ple, moving from an estimated schedule with -y = --0.5 and p = 0.75 t o onewith -y = 0.5 and p = 1.25 does no t alter the area under the curve (GMR = l ) , but i t doesincrease its labor d ominanc e (Figure 13(c)).Given a standard schedule and a few observed rates, such as M(0-4) and M(20-24),for example, how can one find estimates for -y an d p , and with those estimates go on toobtain th e entire estimated schedule?Fir st, taking logits of the two observed migration rates gives Y(0-4) an d Y(20-24)and associating these two logits with the pair of corresponding logits for th e standard gives

Solving these two equa tions in two unknow ns givescrude estimates for -y and p , and apply-ing them to the stand ard schedule's full set of logits results in a set of logits for the esti-mated schedule. From these one can ob tain the migration rates, as shown earlier. Tests ofsuch a procedure with the migration data for Sweden, the United Kingdom, Japan, andthe Netherlands, however, indicate that the me thod is very erratic in the goodness-of-fitsthat it produces and, therefore, more refined procedures are necessary. Such procedures(for tlie case of mortality) are described in the literature on the Brass logit system (forexam ple, in Brass 19 75 , Carrier and Go h 1972).

A reasonable first approxim ation t o an improved estimation method for the case ofmigration is suggested by the regression approach described in subsection 4.2. Imagine a


50/160

A. Rogers, I..J. Olstro


51/160

Model migration schedules 45regression of p on the M ratio, M(0-4) /M(20 -24) .Starting with the simplified basic stan -dard migration schedule and varying p within the range of observed values, one may obtaina corresponding set of M ratios. Associating p and the M ratio in this way, one may pro-ceed further and use the relational equation to estimate ? from p ^ :

A further simplification can be made by forcing the regression line to pass throughthe origin. Since the resulting regression coefficient has a negative sign and the interceptexhibits roughly the same absolute value, but with a positive sign, the regressionequationstake on the form

where M = M ( 0 - 4 ) / M ( 2 0 - 2 4 ) .Given a standard schedule and estimates for y and p , one can proceed to comp utethe associated estimated model migration schedule. Figure 14 illustrates representativeexamples of the goodness-of-fit obtained using this procedure. Two estimated schedulesare illustrated with each observed model migration schedule: those calculated with theinterpolated 85 single-year-of-age observations and t he resulting least-squares estimates ofy an d p , and those computed using the above regression equations of p on the M ratio.Although the fits are moderately successful, it is clear that furthe r stud y of this problemis necessary.

5 CONCLUSIONThis report began with the observation that empirical regularities characterize ob-

served migration schedules in ways that are no less imp ortan t tha n th e corresponding well-established regularities in observed fertility or mortality schedules. Section 2 was devotedto defining mathematically such regularities in observed migration schedules in order toexploit the notational, computational, and analytical advantages that such a formulationprovides. Section 3 reported on the results of an examination of over 500 migration sched-ules th at underscored the broad generality of the model m igration schedule proposed andhelped t o identify a number of families of such schedules.

Regularities in age profiles lead naturally to the development of hypothetical modelmigration schedules that might be suitable for studies of populations with inadequate ordefective data. Drawing on techniques used in the corresponding literature in fertility andmortality, section 4 develops procedures for inferring migration patte rns in the absence ofaccurate migration data .Of w hat use, th en, is the model migration schedule defined in this stud y? What aresome of its concrete practical applications?The model migration schedule may be used to graduate observed data, therebysmoo thing ou t irregularities and ascribing t o the data summ ary measures that can be usedfor comparative analysis. It may be used to interpolate to single years of age, observedmigration schedules that are reported for wider age intervals. Assessments of the reliability


52/160

Stockholm London %

O-" 1 Tokyo 0 . ~ ~ 1 Amsterdam

FIGURE 14 The fits o f the relational approach when using the estimated parameters from 85 observations (- - ) and the parameter from th e observedM ratio (. . . ) compared with the observed (-) data for the female populat ions of Stock holm, Lon don, Tokyo , and Amsterdam. 3


53/160

Model migration schedules 4 7of empirical migration data and indications of appropriate strategies for their correctionare aided by the availability of standard families of migration schedules. Finally, suchschedules also m ay be used to help resolve problems caused by missing data.

The analysis of national m igration age patterns reported in this stud y seeks to dem-onstrate the utility of examining the regularities in age profile exhibited by empiricalschedules of interregional migration. Although data limitations have restricted some ofthe findings to conjectures, a modest start has been made. It is hoped that the resultsreported here will induce others to devote more attention t o this topic.

ACKNOWLEDGMENTSThe autho rs are grateful t o t he many national collaborating scholars who have par-

ticipated in IIASA's Comparative Migration a nd S ettlem ent Stud y. This report could nothave been written without the da ta bank produced by their collective efforts. Thanks alsogo to Richard Raquillet for his contri butio ns to the early phases of this stud y and to WalterKogler for his untiring efforts on our behalf in front of a console in IIASA's computercent er. Kao-Lee Liaw, Philip Rees, W arren Sanderson, and an anon ym ous reviewer offeredseveral useful com men ts on a n earlier draft.

REFERENCESAndersson, A. and I. Holmberg (1980) Migration and Settlement: 3. Sweden. RR-80-5. Laxenburg,Austria: International Institute for Applied Systems Analysis.Bard, Y. (1974) Nonlinear Parameter Estimation. New York: Academic Press.Benson, M. (1979) Parameter Fitting in Dynamic Models. Ecological Modelling 6:97-115.Brass, W. (1971) On the scale of mortality. Pages 69-1 10 in Biological Aspects of Demography, editedby W. Brass. London: Taylor and Francis, Ltd.Brass, W. (1975) Methods for Estimating Fertility and Mortality from Limited and Defective Data.Chapel Hill, North Carolina: Laboratories for Population Statistics, University of North Carolinaat Chapel Hill.Brass, W. and A. Cod e (1968) Methods of analysis and estimation. Pages 88--150 in The Demographyof Tropical Africa, edited by W. Brass et al. Princeton, New Jersey: Princeton University Press.Brown, K.M. and J.E. Dennis (1972) Derivative free analogues of the Levenberg-Marquardt and Gaussalgorithms for nonlinear least squares approximations. Numerische Mathematik 18:289-297.Carrier, N. and J. Hobcraft (1971) Demographic Estimation for Developing Societies. London: Popula-tion Investigation Committee, London School of Economics.Carrier, N. and T. Goh (1972) The validation of Brass's model life table system. Population Studies 26(1):29-5 1.Coale, A. (1977) The development of new models of nuptiality and fertility. Population, special issue:131-154.Coale, A. and P. Demeny (1966) Regional Model Life Tables and Stable Populations. Princeton, NewJersey: Princeton University Press.Coale, A.J. and D.R. McNeil (1972) The distribution by age of the frequency of first marriage in afemale cohort. Journal of the American Statistical Association 67: 743 749.Coale, A. and J. Trussell (1974) Model fertility schedules: variations in the age structure of childbear-ing in human populations. Population Index 40(2):185-206.Fiacco, A. and G. McCormick (1968) Nonlinear Programming: Sequential Unconstrained MinimizationTechniques. New York: Wiley.


54/160

4 8 A. Rogers , L.J. G s t r oGumbel, E.J. (1941) The return period of flood flows. Annals of Mathematical Statistics 12:163-190.Hill. K. and J. Trussell (1977) Fu rther developments in indirect m ortality estimation. Population Stud -ies 31(2):313--33 4.KimbaU, B.F. (1946) Sufficient statistical estima tion functions for the parameters of the distribution

of maximum values. Annals of Mathematical Statistics 17:299--309.Levenberg, K. (1944) A method for the solution of certain nonlinear problems in least squares. Quar-terly of Applied Mathematics 2:16

Model Migration Schedules (1981)

Documents

Transcript of Model Migration Schedules (1981)