Models of migration Observations and judgments

1. Models of migrationObservations and judgments In: Raymer and Willekens, 2008, International migration in Europe, Wiley

2. Introduction:models • To interpret the world, we use models (mental schemes; mental structures) • Models are representations of portions of the real world • Explanation, understanding, prediction, policy guidance • Models of migration

3. Introduction: migration • Migration : change of residence (relocation) • Migration is situated in time and space • Conceptual issues • Space: administrative boundaries • Time: duration of residence or intention to stay • Lifetime (Poland); one year (UN); 8 days (Germany) • Measurement issues • Event: ‘migration’ • Event-based approach; movement approach • Person: ‘migrant’ • Status-based approach; transition approach => Data types and conversion

4. Introduction: migration • Multistate approach • Place of residence at x = state (state occupancy) • Life course is sequence of state occupancies • Change in place of residence = state transition • Continuous vs discrete time • Migration takes place in continuous time • Migration is recorded in continuous time or discrete time • Continuous time: direct transition or event (Rajulton) • Discrete time: discrete-time transition

5. Introduction: migration • Level of measurement or analysis • Micro: individual • Age at migration, direction of migration, reason for migration, characteristic of migrant • Macro: population (or cohort) • Age structure, spatial structure, motivational structure, covariate structure • Structure is represented by models • Structures exhibit continuity and change

6. Probability models • Models include • Structure (systematic factors) • Chance (random factors) • Variate  random variable • Not able to predict its value because of chance • Types of data (observations) => models • Counts: Poisson variate => Poisson models • Proportions: binomial variate => logit models (logistic) • Rates: counts / exposure => Poisson variate with offset

7. Model 1: state occupancy • Yk State occupied by individual k • ki = Pr{Yk=i} State probability • Identical individuals: ki = i for all k • Individuals differ in some attributes: ki = i(Z), Z = covariates • Prob. of residing in i region by region of birth • Statistical inference: MLE of i • Multinomial distribution

8. Model 1: state occupancy • Statistical inference: MLE of state probability i • Multinomial distribution • Likelihood function • Log-likelihood function • MLE • Expected number of individuals in i: E[Ni]=i m

9. Model 1: State occupancy with covariates multinomial logistic regression model

10. Count data Poisson model: Covariates: The log-rate model is a log-linear model with an offset:

11. Model 2: Transition probabilitiesAge x • State probability ki(x,Z) = Pr{Yk(x,Z)=i | Z} • Transition probability discrete-time transition probability Migrant data; Option 2

12. Model 2: Transition probabilities • Transition probability as a logit model with jo(x) = logit of residing in j at x+1 for reference category (not residing in i at x)and j0(x) +j1(x) = logit of residing in j at x+1 for resident of i at x.

13. Model 2: Transition probabilities with covariates with e.g. Zk = 1 if k is region of birth (ki); 0 otherwise. ij0 (x) is logit of residing in j at x+1 for someone who resides in i at x and was born in i. multinomial logistic regression model

14. Model 3: Transition rates for i  j ii(x) is defined such that Hence Force of retention

15. Transition rates: matrix of intensities Discrete-time transition probabilities:

16. Transition rates: piecewise constant transition intensities (rates) Exponential model: Linear approximation:

17. Transition rates: generation and distribution where ij(x) is the probability that an individual who leaves i selects j as the destination. It is the conditional probability of a direct transition from i to j. Competing risk model

18. Transition rates: generation and distribution with covariates Let ij be constant during interval => ij = mi Log-linear model Cox model

19. From transition probabilities to transition ratesThe inverse method (Singer and Spilerman) From 5-year probability to 1-year probability:

20. Incomplete data Expectation (E) Poisson model: Data availability: The maximization(m) of the probability is equivalent to maximizing the log-likelihood The EM algorithm results in the well-known expression

21. Incomplete data: Prior information Gravity model Log-linear model Model with offset

22. 1845 / 1269 = 1.454 1800 / 753 = 2.390 2.390 / 1.454 = 1.644 ODDS ODDS Ratio [1614/632] / [1977/1272] = 1.644 Interaction effect is ‘borrowed’ Source: Rogers et al. (2003a)

23. Adding judgmental data • Techniques developed in judgmental forecasting: expert opinions • Expert opinion viewed as data, e.g. as covariate in regression model with known coefficient (Knudsen, 1992) • Introduce expert knowledge on age structure or spatial structure through model parameters that represent these structures

24. Adding judgmental data • US interregional migration • 1975-80 matrix + migration survey in West • Judgments • Attractiveness of West diminished in early 1980s • Increased propensity to leave Northeast and Midwest • Quantify judgments • Odds that migrant select South rather than West increases by 20% • Odds that migrant into the West originates from the Northeast (rather than the West) is 9 % higher. For Northeast it is 20% higher.

26. Conclusion • Unified perspective on modeling of migration: probability models of counts, probabilities (proportions) or rates (risk indicators) • State occupancies and state transitions • Transition rate = exit rate * destination probabilities • Judgments Timing of event Direction of change