Models of migration
Frans Willekens
Colorado Conference on the Estimation of
Migration
24 – 26 September 2004
Introduction
• 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
•
Measurement of migration
−
−
Event: ‘migration’
Person: ‘migrant’
Data types applied to life-history data:
the case of migration
Data types applied to migration
• Micro-data: data on individuals or households
– Status data:
• Current status (state occupied)
– migrant status (e.g. ever migrated / never migrated in given
period)
– current place (region) of residence
• Place of residence at two points in time: transition data
(migrant data)
– Time interval of fixed length: e.g. census and 5 years prior
⇒ “Where did you live 5 years ago?”
– Time interval variable: e.g. census and place of birth
⇒ “Place of birth”
• Place of residence at 3 or more points in time
Data types applied to migration
• Micro-data:
– Event data : migration data (movement data)
•
•
•
•
Migration during given period (yes/no): migrant status
Ever migrated?
Number of migrations (quantum)
Timing of migration (tempo)
– Time scale: calendar time, age, process time (time since eventorigin)
– Measurement of time: exact time, time interval (discrete time,
e.g. month, year)
– Timing of all migrations vs timing of last migration
Data types applied to migration
• Grouped data: data on groups of individuals or
households (actors)
– Status data:
• Current status: number of actors (subjects) in given status
• Number of actors by place of residence at two points in
time: transition data (migrant data)
CENSUS
• Number of actors by place of residence at 3 or more
points in time
– Event data:
• Number of events during given period POP. REGISTER
The Lexis diagram
Discrete age/time
Individual as ‘carrier’ of attributes
Life course as a sequence of attributes
• Personal attribute at age x or time t
• Attributes change in the course of life: events
• Describe life course
– By attributes at each age: status-based approach
– By initial attributes and changes in attributes: eventbased approach
• Changes (and events) occur in continuous time but
they are often measured in discrete time
State-space approach to life histories
• Attribute = state
• Set of possible attributes = state space
• Attribute at age x = state occupied at age x
– State occupancy
• Change in attribute = state transition
– Direct transition
– Discrete-time transition
– Attributes of transitions: state of origin, state of
destination, reason
• Timing of state transitions
– Age structure of migration (age/duration dependence)
• Dependence of destination on origin
– Spatial structure of migration
Probability models of migration
• Risk indicators: risk of a transition
– Number of transitions during unit interval: counts
– How likely is a transition in unit interval: probability
– Timing of transition: transition rate
• Underlying random mechanism
– Count data: Poisson models
– Probability (or proportion): logit model and logistic
regression
– Rate: transition rate model
• Rate = occurrences / exposure
Model 1: state occupancy
• Yk(x) State occupied at x
• kπi(x) = Pr{Yk(x)=i} State probability
– Identical individuals: kπi(x) = πi(x) for all k
– Individuals differ in some attributes:
kπi(x)= πi(x,Z), Z = covariates
• Prob. of residing in i region by region of birth
• Statistical inference: MLE of πi(x)
– Multinomial distribution
Pr{N1 = n 1 , N 2 = n 2 , ...} =
m!
I
∏ ni!
i =1
I
∏π
i =1
ni
i
Model 1: state occupancy
• Statistical inference: MLE of state probability πi
– Multinomial distribution
m! I
ni
Pr{N1 = n1 , N 2 = n 2 , ...} = I
∏
π
i
∏ n i ! i=1
i =1
I
– Likelihood function
L = ∏π ini
– Log-likelihood function
i =1
l = ln( L) = ∑i =1 ni ln(π i )
I
ni
– MLE πˆ i =
m
– Expected number of individuals in i: E[Ni]=πi m
Model 1: State occupancy with covariates
π i (Z )
log it [π i (Z )] = ln
= η i = β i 0 + β i1Z1 + β i 2 Z 2 + β i 3 Z 3 + ...
1 − π i (Z )
πi =
exp(η i )
exp(η i )
= I
exp(η1 ) + exp(η 2 ) + ... + 1 + ...
∑ exp(η j )
j =1
multinomial logistic regression model
Model 2: Transition probabilities
• State probability kπi(x,Z) = Pr{Yk(x,Z)=i | Z}
• Transition probability
Pr{Y(x + 1 ) = j | Y(x),Y(x - 1 ), ...; Z} = Pr{ Y(x + 1 ) = j | Y(x); Z}
Pr {Y(x + 1 ) = j | Y(x) = i} = p ij (x)
discrete-time transition probability
Migrant data; Option 2
• Transition probability as a logit model
exp[β + β
p ( x) =
log it[π j ( x + 1)] = β j 0 + β j1Yi ( x)
j0
ij
I
∑ exp[β
r =1
j0
Y ( x)
j1 i
]
+ β j1Yr ( x)]
Model 2: Transition probabilities
• Transition probability as a logit model
log it[π j ( x + 1)] = β j 0 ( x) + β j1 ( x )Yi ( x )
pij ( x) =
[
exp β j 0 ( x) + β j1 ( x)Yi ( x)
I
∑exp[β
r =1
j0
]
( x) + β j1 ( x)Yr ( x)]
with βjo(x) = logit of residing in j at x+1 for reference category
(not residing in i at x) and βj0(x) +βj1(x) = logit of residing in
j at x+1 for resident of i at x.
Model 2: Transition probabilities with covariates
Illustration 1 - Micro-data
- Covariate: region of birth
pij ( x) =
[
I
∑exp[η (x)]
r =1
with
]
expηij ( x)
ij
η ij ( x) = β ij 0 ( x) + β ij1 ( x) Z1 + β ij 2 ( x) Z 2 + β ij 3 ( x) Z 3 + ...
e.g. Zk = 1 if k is region of birth (k≠i); 0
otherwise.
βij0 (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
Model 2: Transition probabilities with covariates
Illustration 1 - Macro-data
- Covariate: underfive (or infant)
migration probability
pij ( x) =
[
I
∑exp[η (x)]
r =1
with
]
expηij ( x)
ij
η ij ( x) = β 0 ( x) + β1 ( x) pij (−5)
Rogers, Muhidin, Jordan, Lea (2004, p. 8):
linear model with regression coefficients
independent of x
pij ( x) = a + b pij (−5)
Transition rates
µ ij ( x) = lim
( y − x )→0
pij ( x, y )
for i ≠ j
y−x
µii(x) is defined such that
∑µ
ij
j
Hence µ ii ( x) = ∑ µ ij ( x) = lim
j ≠i
( x) = 0
( y − x ) →0
1 − pij ( x)
y−x
Force of retention
Transition rates: matrix of intensities
 µ (x)
 11
- µ 12 (x)
µ(x) = 
.


.

 - µ 1I (x)
-µ
µ
21
(x) . .
(x) .
.
.
.
.
-µ (x) .
22
2I
.
.
.
.
-µ
-µ
(x) 

(x)
I2

.


.

(x)
µ II 
I1
Discrete-time transition probabilities:
 p (x, y)
 11
 p12 (x, y)
P(x, y) = 
.


.

 p1N (x, y)
p
p
p
21
22
2N
(x, y)
. .
(x, y) . .
.
. .
.
. .
(x, y) . .
p
p
(x, y) 

(x, y)
N2

.


.
p NN (x, y)
N1
dP ( x )
= −µ ( x ) P ( x )
dx
Transition rates: piecewise constant
transition intensities (rates)
P( x, y ) = exp[− ( y − x)M( x, y )]
exp( A ) = I + A +
1 2 1 3
A + A + ...
2!
3!
3
(y - x) 2
2 (y - x)
exp[−( y − x)M ( x, y ) = I - (y - x)M ( x, y ) +
[M( x, y )] [M( x, y )]3 + . . .
2!
3!
P(x, y) = [I + 12 M(x, y)] [I − 12 M(x, y)]
−1
Transition rates: generation and distribution
µ ij ( x) = µ i + ( x) ξ ij ( x)
where ξij(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
 µ (x)
 11
- µ 12 (x)

.

.


 - µ 1I (x)
-µ
µ
-µ
21
22
(x) . .
(x) . .
.
. .
.
. .
(x) . .
2I
-µ
-µ
(x)   ξ (x)
  11
(x) -ξ (x)
I2
12


=
.
.
 
.
  .
µ II (x)  -ξ 1I (x)
I1
-ξ
ξ
-ξ
21
22
(x) . .
(x) . .
.
. .
.
. .
(x) . .
2I
-ξ
-ξ
(x) 

(x)
I2
. 

. 
ξ II (x) 
I1








µ
1+
0
.
.
0
(x)
0
µ
2+
. .
(x) . .
.
. .
.
0
. .
. .


0 

.

.

µ I+ (x)
0
Transition rates: generation and distribution
with covariates
Log-linear model
mi = exp[β i 0 + β i1 Z1 + β i 2 Z 2 + ...]
ln mi = β i 0 + β i1 Z1 + β i 2 Z 2 + ...
Cox model
mi ( x) = mio ( x) exp[β i 0 + β i1 Z1 + β i 2 Z 2 + ...]
From transition probabilities to
transition rates
The inverse method (Singer and Spilerman)
P(x, y) = [I + 12 M(x, y)] [I − 12 M(x, y)]
−1
M ( x, y ) =
y−x
2
[I − P( x, y)][I + P( x, y)]−1
From 5-year probability to 1-year probability:
P( x, x + 1) = exp[− M( x, x + 1)]
Count data
Poisson model:
Covariates:
Pr{N i = n i } =
λin
i
ni !
exp[-λi ]
E[N i ] = λi = exp[β i 0 + β i1Z1 + β i 2 Z 2 + ...]
ln λi = β i 0 + β i1 Z1 + β i 2 Z 2 + ...
The log-rate model is a log-linear model with an offset:
 N i  λi
E
= exp[β i 0 + β i1 Z1 + β i 2 Z 2 + ...]
=
 PYi  PYi
E [N i ] = λi = PYi exp[β i 0 + β i1 Z1 + β i 2 Z 2 + ...]
Incomplete data
nij
Poisson model:
Data availability:
Pr{ N ij = n ij } =
[ ]
λij
nij !
exp[-λij ]
E N ij = λij = α i β j
The maximization of the probability is equivalent to
maximizing the log-likelihood l = ∑ nij ln[α i β j ] − α i β j
[
αˆ i =
ni +
βˆ
∑
j
j
n+ j
ˆ
βj =
∑ αˆ i
ij
i
The EM algorithm results in the well-known expression
ni +
λij =
n+ j
n+ +
]
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
Timing of event
Direction of change