Models of migration Observations and judgments

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Models of migrationObservations and judgments
In Raymer and Willekens, 2008, International
migration in Europe, Wiley
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Introductionmodels

• 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

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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
• gt Data types and conversion

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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

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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

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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) gt models
• Counts Poisson variate gt Poisson models
• Proportions binomial variate gt logit models
  (logistic)
• Rates counts / exposure gt Poisson variate with
  offset

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Model 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

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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 ENi?i m

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Model 1 State occupancy with covariates
multinomial logistic regression model
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Count data
Poisson model
Covariates
The log-rate model is a log-linear model with an
offset
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Model 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
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Model 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.

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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
x1 for someone who resides in i at x and was
born in i.
multinomial logistic regression model
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Model 3 Transition rates
for i ? j
?ii(x) is defined such that
Hence
Force of retention
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Transition rates matrix of intensities
Discrete-time transition probabilities
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Transition rates piecewise constant transition
intensities (rates)
Exponential model
Linear approximation
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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
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Transition rates generation and distribution
with covariates
Let ?ij be constant during interval gt ?ij mi
Log-linear model
Cox model
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From transition probabilities to transition
ratesThe inverse method (Singer and Spilerman)
From 5-year probability to 1-year probability
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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
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Incomplete data Prior information
Gravity model
Log-linear model
Model with offset
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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)
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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

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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.

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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