Impact of duration of stay definition on migration

Impact of duration of stay definition on migration levels Frans Willekens DGINS Budapest 2017, 21 September 2017 Outline Theory: migration = relocation with duration of stay criterion Numerical illustration Migration from Poland to Sweden Conclusion Modelling of count data: introduction Relocation is repeatable (recurrent) event

Relocation rate (occurrence-exposure rate) (occurrence-exposure rate) Probability of at least one relocation during interval (0,t): ( ) =1 exp ( ) Prob of relocation in interval (t,t+dt): Expected number of relocations in interval (0,t): = = (occurrence-exposure rate)t Probability of n relocations during interval Pr { ()=| }= [ (0,t): ] ! = [ ( ) ] = ; ( )= Migration:

relocation followed by minimum duration of stay Minimum duration of stay of dm years dm ={0, 0.25 (3 months), 0.5 (6 months), 1, 10} dead time (blocked time) (dm) = time after each event during which the system does not record another event -> count loss Relocation (event) that occur during the blocked time are lost In renewal theory, the counter is called Type I counter (e.g. Geiger counter to count radioactive impulses) (Pyke, 1958) Literature

Cox, D., Isham, V. (1980). Point Processes. London: Chapman and Hall, p. 102 Blocked time Pyke, R. (1958) On renewal processes related to type I and type II counter models. Annals of Mathematical Statistics, 29(3):737-754 Distinction: and event has happened and an event has been registered Type I counter: counter in which deadtime is produced only after event has been registered (or detected) He, S., G. Yang, K.T.Fang and J.F. Widmann (2005) Estimation of Poisson Intensity in the Presence of Dead Time. Journal of the American Statistical Association, 100(470):669-679 Migration: relocation followed by minimum duration of stay

Probability of measuring a relocation at time t (t > dm) ( ) ( ) = exp ( )

= [ ( ) ] exp ( ) = exponential distribution shifted by dm Expected time between measurements: 1/ + d + dm True number of relocations during observation 1 period (0,t): / with Nm the observed number of relocations during interval (d m,t) and + d the event (relocation) rate and + dm the detection rate

Migration = relocation with duration threshold Probability of n migrations in (0,t)-interval if the duration threshold is dm ( ) Pr { ( )=| , }= [ ] ! where is the probability of no relocation within dm years Migration rate: z Expected number of migrations during the interval of length t

Migration: relocation with duration threshold Limited number of duration thresholds Let dm=1 year be reference category Migration count when dm is duration threshold, relative to count when UN (+Eurostat) recommendation is ( ) followed: = = [ ( 12 ) ] ( ) [

12 ] Overestimation is % If >0, it measures undercount is independent of length of observation period Migration: relocation with duration threshold Relocation intensity =0.2 Overestimation 10%: Migration count is 10% higher than with a duration threshold of 1 year (UN recom.) Overestimation by duration threshold

relocation rate Threshold 0 0.25 (3 months) 0.5 (6 months) 1 (12 months) 5 (60 months) 10 (120 months) 0.2 1.22 1.16 1.1 1 0.44 0.17

0.1 1.11 1.08 1.05 1 0.67 0.41 Migration from Poland to Sweden, average 20022007 Emigrations reported by Poland: o Total to EU18+EFTA: 22,306 o To Sweden: 303 Immigrations from Poland reported by o EU18+EFTA: 217,977 o Sweden: 3,718

Population Poland: 38 million Poisson model of migration from Poland to Sweden o Emigration rate: 303/38million = 7.97e-06 o Overestimation: = 0.9999282 Hypothetical case in which Poisson model results in correct over(under)estimation o Emigration rate 0.2 o Duration threshold 13.5 years (=permanent) o Overestimation: = 0.082085 o Polish data underreport underreport the true migration flow to Sweden by 92 percent. o

Mover-stayer model of Polands emigration to EU18+EFTA Polands emigration rate to EU18+EFTA: 22306/38million = 0.0006 (0.6 per thousand) Suppose 2.5 per thousand of the residents of Poland considers emigration to EU18+EFTA within a year. Their emigration rate is hence Assume that, on average, EU18+EFTA countries use dm of 0.5 years and Poland uses dm of 10 years (permanent) Mover-stayer model of Polands emigration to

EU18+EFTA Polands reporting of emigration (22,306) is about 10 percent of immigrants count reported by EU18+EFTA (217,977) Mover-stayer model of Polands emigration to Sweden During the observation period (2002-2007), 1.7 percent of the emigrants from Poland emigrated to Sweden. Suppose residents of Poland have a slight preference for Sweden -> emigration rate to Sweden is 0.27 (instead of 0.24). 8.8% of migration from Poland to Sweden is reported by Poland Close to observation: 303/3718=0.082 Mover-stayer model should replace Poisson model

Winiowski (2017, p. 193) considers the maximal fraction of the population that can emigrate in a given year (considering emigration data of sending country and immigration data from receiving country) and sets is to 0.02. Expert judgment vs mover-stayer model Mover-stayer model Proportion movers (mobile): 6% Observation period: 1 year Emigration rate for movers: 1.8 (move every 6 months) Emigration rate for stayers: 0.1

E[Ndm] = 0.94*0.1*exp(-0.1*dm)+0.06*1.8*exp(1.8*dm) Proportion of migration with duration threshold of 1 Model and expert judgment: comparison Table 1. True migration flow (UN definition) as fraction of recorded flow. Expert judgments, Poisson model and mixture model Duration Experts Poisson Mixture threshold judgment model model (=0.24)

No time limit 0.51 0.79 0.51 3 months 0.61 0.84 0.64 6 months 0.81 0.89 0.77 12 months 1.00 1.00 1.00 Permanent 1.64

(p)) 2.61 1.80 5 years 8.67 2.98 Conclusion Duration of stay criterion in definition of migration has large impact on migration counts Poisson process with duration threshold = blocked Poisson process (blocked time or dead time) Blocked Poisson model cannot describe the differences in migration counts Mover-stayer model can describe the differences in migration counts Mover-stayer model also describes outcome of

expert judgments thank you [email protected] NIDI is an institute of the Royal Netherlands Academy of Arts and Sciences KNAW and is affiliated to the University of Groningen www.nidi.nl