1. Models of migration Observations 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 Y k State occupied by individual k k  i = Pr{Y k =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[N i ]=  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 probabilities Age x State probability k  i (x,Z) = Pr{Y k (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. Z k = 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) Linear approximation: Exponential model:

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 Cox model Log-linear model Let  ij be constant during interval =>  ij = m i

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

20. Incomplete data 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 Expectation (E)

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 eventDirection of change