Title: Models of migration Observations and judgments
1Models of migrationObservations and judgments
In Raymer and Willekens, 2008, International
migration in Europe, Wiley
2Introductionmodels
- 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
3Introduction 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
- gt Data types and conversion
4Introduction 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
5Introduction 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
6Probability 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) gt models
- Counts Poisson variate gt Poisson models
- Proportions binomial variate gt logit models
(logistic) - Rates counts / exposure gt Poisson variate with
offset
7Model 1 state occupancy
- Yk State occupied by individual k
- k?i PrYki 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
8Model 1 state occupancy
- Statistical inference MLE of state probability
?i - Multinomial distribution
- Likelihood function
- Log-likelihood function
- MLE
- Expected number of individuals in i ENi?i m
9Model 1 State occupancy with covariates
multinomial logistic regression model
10Count data
Poisson model
Covariates
The log-rate model is a log-linear model with an
offset
11Model 2 Transition probabilitiesAge x
- State probability k?i(x,Z) PrYk(x,Z)i Z
- Transition probability
discrete-time transition probability Migrant
data Option 2
12Model 2 Transition probabilities
- Transition probability as a logit model
-
- with ?jo(x) logit of residing in j at x1 for
reference category (not residing in i at x) and
?j0(x) ?j1(x) logit of residing in j at x1
for resident of i at x.
13Model 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
x1 for someone who resides in i at x and was
born in i.
multinomial logistic regression model
14Model 3 Transition rates
for i ? j
?ii(x) is defined such that
Hence
Force of retention
15Transition rates matrix of intensities
Discrete-time transition probabilities
16Transition rates piecewise constant transition
intensities (rates)
Exponential model
Linear approximation
17Transition 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
18Transition rates generation and distribution
with covariates
Let ?ij be constant during interval gt ?ij mi
Log-linear model
Cox model
19From transition probabilities to transition
ratesThe inverse method (Singer and Spilerman)
From 5-year probability to 1-year probability
20Incomplete 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
21Incomplete 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)
23Adding 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
24Adding 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.
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26Conclusion
- 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