Models of migration. Frans Willekens. Colorado Conference on the Estimation of Migration September 2004

Transcription

1 Models of mgrato Fras Wllekes Colorado Coferece o the Estmato of Mgrato 4 6 Setember 004

2 Itroducto Mgrato : chage of resdece (relocato Mgrato s stuated tme ad sace Cocetual ssues Sace: admstratve boudares Tme: durato of resdece or teto to stay Measuremet of mgrato Evet: mgrato Perso: mgrat

3 Data tyes aled to lfe-hstory data: the case of mgrato

4 Data tyes aled to mgrato Mcro-data: data o dvduals or households Status data: Curret status (state occued mgrat status (eg ever mgrated / ever mgrated gve erod curret lace (rego of resdece Place of resdece at two ots tme: trasto data (mgrat data Tme terval of fxed legth: eg cesus ad 5 years ror Where dd you lve 5 years ago? Tme terval varable: eg cesus ad lace of brth Place of brth Place of resdece at 3 or more ots tme

5 Data tyes aled to mgrato Mcro-data: Evet data : mgrato data (movemet data Mgrato durg gve erod (yes/o: mgrat status Ever mgrated? Number of mgratos (quatum Tmg of mgrato (temo Tme scale: caledar tme, age, rocess tme (tme sce evetorg Measuremet of tme: exact tme, tme terval (dscrete tme, eg moth, year Tmg of all mgratos vs tmg of last mgrato

6 Data tyes aled to mgrato Groued data: data o grous of dvduals or households (actors Status data: Curret status: umber of actors (subjects gve status Number of actors by lace of resdece at two ots tme: trasto data (mgrat data CENSUS Number of actors by lace of resdece at 3 or more ots tme Evet data: Number of evets durg gve erod POP REGISTER

7 The Lexs dagram Dscrete age/tme

8 Idvdual as carrer of attrbutes Lfe course as a sequece of attrbutes Persoal attrbute at age x or tme t Attrbutes chage the course of lfe: evets Descrbe lfe course By attrbutes at each age: status-based aroach By tal attrbutes ad chages attrbutes: evetbased aroach Chages (ad evets occur cotuous tme but they are ofte measured dscrete tme

9 State-sace aroach to lfe hstores Attrbute state Set of ossble attrbutes state sace Attrbute at age x state occued at age x State occuacy Chage attrbute state trasto Drect trasto Dscrete-tme trasto Attrbutes of trastos: state of org, state of destato, reaso Tmg of state trastos Age structure of mgrato (age/durato deedece Deedece of destato o org Satal structure of mgrato

10 Probablty models of mgrato Rsk dcators: rsk of a trasto Number of trastos durg ut terval: couts How lkely s a trasto ut terval: robablty Tmg of trasto: trasto rate Uderlyg radom mechasm Cout data: Posso models Probablty (or roorto: logt model ad logstc regresso Rate: trasto rate model Rate occurreces / exosure

11 Model : state occuacy Y k State occued at x k π Pr{Y k } State robablty Idetcal dvduals: k π π for all k Idvduals dffer some attrbutes: k π π (x,z, Z covarates Prob of resdg rego by rego of brth Statstcal ferece: MLE of π Multomal dstrbuto Pr{ N, N,} m! I I! π

12 Model : state occuacy Statstcal ferece: MLE of state robablty π Multomal dstrbuto Pr{ N Lkelhood fucto L Log-lkelhood fucto I, N π l l( L,} m! I I! I π l ( π MLE πˆ m Exected umber of dvduals : E[N ]π m

13 Model : State occuacy wth covarates ( Z ( Z π log t[ π ( Z ] l η β + β Z + βz + β3z3 π 0 + π ex( η ex( η ex( η + ex( η I ex( η j j multomal logstc regresso model

14 Model : Trasto robabltes State robablty k π (x,z Pr{Y k (x,z Z} Trasto robablty Pr{Y(x+ j Y,Y(x-, ; Z} Pr{Y(x + j Y; Z} Pr {Y(x + j Y } Trasto robablty as a logt model ex[ β + β Y ( x ] log t[ π j ( x + ] β j0 + β jy ( x j dscrete-tme trasto robablty Mgrat data; Oto j ( x I r j0 ex[ β j0 j + β Y j r ( x]

15 Model : Trasto robabltes Trasto robablty as a logt model log t[ π j ( x + ] β j0( x + β j( x Y ( x j ( x I r ex [ β ( x + β ( x Y ( x ] j0 ex[ β j0 j ( x + β ( x Y j wth β jo logt of resdg j at x+ for referece category (ot resdg at x ad β j0 +β j logt of resdg j at x+ for resdet of at x r ( x]

16 Model : Trasto robabltes wth covarates Illustrato j - Mcro-data - Covarate: rego of brth ( x I r ex [ η ( x ] j ex[ η ( x] j wth η x β ( x + β ( x Z + β ( x Z + β ( x Z j ( j0 j j j3 3 + eg Z k f k s rego of brth (k ; 0 otherwse β j0 s logt of resdg j at x+ for someoe who resdes at x ad was bor multomal logstc regresso model

17 Model : Trasto robabltes wth covarates Illustrato j - Macro-data - Covarate: uderfve (or fat mgrato robablty ( x I ex [ η ( x ] j r j ( x β0( x + β( x j ( j ex[ η ( x] wth η 5 Rogers, Muhd, Jorda, Lea (004, 8: lear model wth regresso coeffcets deedet of x j ( x a + b ( 5 j

18 Trasto rates x y y x x j x y j, ( lm ( 0 ( for j s defed such that j j x 0 ( Hece x y x x x j x y j j ( lm ( ( 0 ( Force of reteto

19 Trasto rates: matrx of testes II I I I I ( ( ( x x dx x d P P (x (x (x (x (x (x (x (x (x NN N N N N,,,,,,,,, (x, P Dscrete-tme trasto robabltes:

20 Trasto rates: ecewse costat trasto testes (rates [ ( y x M( x, ] P( x, ex y 3 ex( A I + A + A + A! 3! + ex[ ( y x M( x, I - (y - x M( x, + (y - x! (y - x 3! 3 [ M( x, ] - [ M( x, ] + 3 [ I + M(x, ] [ I M (x,] P(x,

21 Trasto rates: geerato ad dstrbuto j + ( x ( x ξ ( x where ξ j s the robablty that a dvdual who leaves selects j as the destato It s the codtoal robablty of a drect trasto from to j j Cometg rsk model - - I - - I - - I I II ξ -ξ -ξ I -ξ ξ -ξ I -ξ -ξ ξ I I II I+

22 Trasto rates: geerato ad dstrbuto wth covarates Log-lear model m [ β + β Z + β ] ex Z 0 + l m β + β Z + β Z + 0 Cox model m [ β + β Z + β ] ( x mo ( xex Z 0 +

23 From trasto robabltes to trasto rates The verse method (Sger ad Slerma [ I + M(x, ] [ I M (x,] P(x, M( x, [ I P( x, ][ I + P( x, ] y y x From 5-year robablty to -year robablty: [ M( x, ] P( x, x + ex x +

24 Cout data Posso model: Pr{ N } λ! ex[-λ ] Covarates: E [ N ] λ [ β + β Z + β Z ] ex 0 + l λ β + β Z + βz 0 + The log-rate model s a log-lear model wth a offset: N E PY E λ PY [ β + β Z + β Z ] ex 0 + [ N ] λ PY [ β + β Z + β Z ] ex 0 +

25 Posso model: Data avalablty: Icomlete data Pr{ N E j j j λj }! ex[-λ ] [ N ] j λ j α β j The maxmzato of the robablty s equvalet to l l[α β ] maxmzg the log-lkelhood [ α β ] ˆ α + j ˆ β j ˆ β j The EM algorthm results the well-kow exresso λ j + j ˆ α j j j j j j j

26 Cocluso Ufed ersectve o modelg of mgrato: robablty models of couts, robabltes (roortos or rates (rsk dcators State occuaces ad state trastos Trasto rate ext rate * destato robabltes Tmg of evet Drecto of chage