# Models of migration

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```					          Models of migration
Observations and judgments

In: Raymer and Willekens, 2008, International migration
in Europe, Wiley
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
Introduction: migration
• Migration : change of residence (relocation)
• Migration is situated in time and space
– Conceptual issues
• 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
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
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
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
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
I
m!
Pr{ N1  n 1 , N 2  n 2 , ...}     I              ni
i

n !
i 1
i
i 1
Model 1: state occupancy
• Statistical inference: MLE of state probability i
– Multinomial distribution
m! I
Pr{ N1  n1 , N 2  n 2 , ...}  I      i ni
 n i ! i1
i 1
I

– Likelihood function      L    ini
i 1

l  ln( L)  i 1 ni ln  i 
I
– Log-likelihood function
ni
– MLE  i 
ˆ
m

– Expected number of individuals in i: E[Ni]=i m
Model 1: State occupancy with covariates
 i Z 
logit  i Z   ln           i   i 0   i1Z1   i 2 Z 2   i 3 Z 3  ...
 n Z 

exp(i )                    exp(i )
i                                         I
exp(1 )  exp( 2 )  ... 1  ...
 exp( j )
j 1

multinomial logistic regression model
Count data
in
i
Poisson model:      Pr{N i  n i }           exp[-i ]
ni !
Covariates:      EN i   i  exp  i 0   i1Z1   i 2 Z 2  ...

ln i   i 0   i1 Z1   i 2 Z 2  ...

The log-rate model is a log-linear model with an offset:
 N i  i
E          exp i 0   i1Z1   i 2 Z 2  ...
 PYi  PYi
EN i   i  PYi exp  i 0   i1Z1   i 2 Z 2  ...
Model 2: Transition probabilities
Age x
• State probability ki(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} = pij (x)
discrete-time transition probability
Migrant data; Option 2
Model 2: Transition probabilities
• Transition probability as a logit model

log it[ j ( x  1)]   j 0 ( x)   j1 ( x)Yi ( x)

pij ( x) 

exp  j 0 ( x)   j1 ( x)Yi ( x)   
I

 exp[
r 1
j0   ( x)   j1 ( x)Yr ( x)]

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.
Model 2: Transition probabilities with covariates

pij ( x) 

exp ij ( x)   
I

 exp[ ( x)]
r 1
ij

with     ij ( x)   ij 0 ( x)   ij1 ( x) Z1   ij 2 ( x) Z 2   ij 3 ( x) Z 3  ...

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
Model 3: Transition rates
pij ( x, y )
 ij ( x)  lim                                    for i  j
( y  x )0     yx

ii(x) is defined such that             
j
ij   ( x)  0

1  pij ( x)
Hence ii ( x)   ij ( x)  lim
j i
( y  x )0      yx
Force of retention
Transition rates: matrix of intensities

  (x)              -            (x) . .   -   (x)
 11                          21                  I1

- 12 (x)                        (x) . .   -   (x)
μ(x)                                                       
22                   I2
.                        .        . .     .
                                                     
    .                        .        . .     .      

 - 1I (x)
                    -       2I
(x) . .    II (x) 



Discrete-time transition probabilities:
 p (x, y)     p        (x, y)   . .   p    (x, y) 
 11               21                      N1      
 p12 (x, y)   p        (x, y) . .     p    (x, y)
P(x, y)  
22                    N2
           dP( x)
     .                 .        . .        .                         μ( x)P( x)
     .                 .        . .        .                  dx
                                                  
 p1N (x, y)
              p   2N
(x, y) . .     p NN 
(x, y)
Transition rates: piecewise constant
transition intensities (rates)
Exponential model:
P( x, y )  exp  ( y  x)M( x, y )

1 2 1 3
exp( A) = I + A +          A + A + ...
2!    3!

(y - x) 2                     3
exp[ ( y  x)M( x, y ) = I - (y - x) M( x, y ) +           M( x, y) -
2 (y - x)
M( x, y)3 + . . .
2!                    3!

Linear approximation:
P(x, y)  I  1 M(x, y) I  1 M(x, y)
1
2               2
Transition rates: generation and distribution
 ij ( x)   i  ( x)  ij ( x)

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

  (x)       -        (x) . .   -   (x)   (x)       -        (x) . .   -   (x)               (x)       0        . .     0     
 11               21                  I1
  11              21                  I1
          1

- 12 (x)             (x) . .   -   (x) - (x)                (x) . .   -   (x)           0             2
(x) . .     0 
                                           .                                .                                                 . 
22               I2           12            22               I2
.             .        . .     .                          .        . .                        .              .        . .
                                                                                                                                    
    .             .        . .     .        .               .        . .     .                .              .        . .      . 

 - 1I (x)
             -   2I
(x) . .    II (x)  - 1I (x)
 
             -   2I
(x) . .    II (x) 





0              0        . .    I (x)


Transition rates: generation and distribution
with covariates
Let ij be constant during interval => ij = mi

Log-linear model    mi  exp  i 0   i1 Z1   i 2 Z 2  ...

ln mi   i 0   i1 Z1   i 2 Z 2  ...

Cox model      mi ( x)  mio ( x) exp  i 0   i1 Z1   i 2 Z 2  ...
From transition probabilities to
transition rates
The inverse method (Singer and Spilerman)
P(x, y)  I  1 M(x, y) I  1 M(x, y)
1
2               2

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)
Incomplete data
Expectation (E)
ij
nij

Poisson model:            Pr{N ij  n ij }              exp[-ij ]
nij !

Data availability:              
E N ij  ij   i  j
The maximization (m) of the probability is equivalent to

maximizing the log-likelihood l   nij ln[ i  j ]   i  j        
ij
ni 
i 
ˆ                  ˆ  n j
j

j
 ˆ
j      i ˆ
i

The EM algorithm results in the well-known expression
ni 
ij       n j
n 
Incomplete data: Prior information

 
E N ij  ij  k ai  j exp  cij     Gravity model

 
ln ij  u  uiA  u B  uijAB
j                 Log-linear model

 
E Nij  ij   i  m  *    *
j
0
ij     Model with
offset
A.               Region of       Region of destination
Time period      origin          Northeast   Midwest     South    West     Total
1975–1980        Northeast       43,123      462         1,800    753      46,138
Data             Midwest         350         51,136      1,845    1,269    54,600
South           695         1,082       67,095   1,141    70,013       1845 / 1269 = 1.454
ODDS
West            287         677         1,120    37,902   39,986
1800 / 753 = 2.390
Total           44,455      53,357      71,860   41,065   210,737
ODDS
2.390 / 1.454 = 1.644    Ratio
B.               Origin          Northeast   Midwest     South    West     Total
1980–1985        Northeast       44,845      379         1,387    473      47,084
Data             Midwest         326         52,311      1,954    1,144    55,735
South           651         855         68,742   1,024    71,272
West            237         669         1,085    40,028   42,019
Total           46,059      54,214      73,168   42,669   216,110

C.             Flows predicted based on marginal totals and 1975-80 matrix
Origin        Northeast     Midwest       South    West               Total
1980-1985      Northeast     44,445        393           1,614    632                47,084
Predicted      Midwest       431           52,055        1,977    1,272              55,735
South         814           1,047         68,324   1,087              71,272
West          369           719           1,253    39,678             42,019
Total         46,059        54,214        73,168   42,669             216,110

[1614/632] / [1977/1272] = 1.644

Interaction effect is ‘borrowed’
Source: Rogers et al. (2003a)
• 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
• 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.
A.            Region of   Region of destination
Time period   origin      Northeast      Midwest    South     West     Total
1975–1980     Northeast   43,123         462        1,800     753      46,138
Data          Midwest     350            51,136     1,845     1,269    54,600
South       695            1,082      67,095    1,141    70,013
West        287            677        1,120     37,902   39,986
Total       44,455         53,357     71,860    41,065   210,737

D.        Flows predicted based on West survey and 1975-80 matrix
Origin        Northeast     Midwest      South    West       Total
1980-1985 Northeast     21,181        272          1,037    473        22,963
Predicted Midwest       247           43,135       1,526    1,144      46,052
South         488           909          55,235   1,024      57,656
West          237           669          1,085    40,028     42,019
Total         22,153        44,985       58,883   42,669     168,690

E.        Flows predicted based on West survey, 1975-80 matrix and judgmental data
Origin        Northeast     Midwest      South      West     Total
1980-1985 Northeast     40,243        516          2,365      899      44,023
Predicted Midwest       296           51,762       2,197      1,373    55,628
South         488           909          66,282     1,024    68,703
West          237           669          1,302      40,028   42,236
Total         41,264        53,856       72,146     43,324   210,590
Source: Rogers et al. (2003a)
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
• Judgments

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