Models of migrationObservations 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• 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

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

•ki = Pr{Yk=i} State probability– Identical individuals: ki = i for all k – Individuals differ in some attributes:

ki = i(Z), Z = covariates• Prob. of residing in i region by region of birth

• Statistical inference: MLE of i – Multinomial distribution

I

1i

iI

1i

2211n

!n

m! }... ,n N ,nNPr{ i

i

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[Ni]=i m

I

1i

iI

1i

2211n

!n

m! }... ,n N ,nNPr{ i

i

I

1i

iniL

I

i iinLl1

ln)ln(

m

nii ̂

Model 1: State occupancy with covariates

...lnlogit 3322110 ZZZZ

ZZ iiiii

n

ii

I

jj

iii

1

21 )exp(

)exp(

...1...)exp()exp(

)exp(

multinomial logistic regression model

Count data

]exp[- !

}n Pr{ iii

i

ni

nN

i

Poisson model:

...exp 22110 ZZNE iiiii Covariates:

...ln 22110 ZZ iiii

The log-rate model is a log-linear model with an offset:

...exp 22110

ZZ

PYPY

NE iii

i

i

i

i

...exp 22110 ZZPYNE iiiiii

Model 2: Transition probabilitiesAge x

• State probability ki(x,Z) = Pr{Yk(x,Z)=i | Z}

• Transition probability

Z}Y(x |j = )Y(xPr{ = Z}... ),-Y(x Y(x), |j = )Pr{Y(x ;)1;11

(x)p = i} = Y(x) |j = ){Y(x ij1Pr discrete-time transition probability

Migrant data; Option 2

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.

)()()()]1([log 10 xYxxxit ijjj

I

rrjj

ijjij

xYxx

xYxxxp

110

10

)]()()(exp[

)()()(exp)(

Model 2: Transition probabilities with covariates

I

rij

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1

)](exp[

)(exp)(

with ...)()()()()( 3322110 ZxZxZxxx ijijijijij

e.g. Zk = 1 if k is region of birth (ki); 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

xy

yxpx ij

xyij

),(lim)(

0)( for i j

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Hencexy

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Force of retention

Transition rates: matrix of intensities

(x)..(x)(x)

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.....

(x)..(x)(x)

(x)..(x)(x)

(x)

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- ---

II2I1I

I22212

I12111

μ

)()()(

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y)(x..y)(xy)(x

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y)(x..y)(xy)(x

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NNNN

N

N

,,,

,,,

,,,

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21

22212

12111

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Discrete-time transition probabilities:

Transition rates: piecewise constant transition intensities (rates)

),()(exp),( yxxyyx MP

. . . + 3!

1 +

2!

1 + + = )exp 32 AAAI A(

. . . + ),(3!

x)-(y - ),(

2!

x)-(y + ),(x)-(y - = ),()(exp[ 3

32

2

yxyxyxyxxy MMMI M

y)(x,y)(x, y)(x, 211

21 MIMIP

Linear approximation:

Exponential model:

Transition rates: generation and distribution

)()()( xxx ijiij

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)..00

.....

.....

0..(x)0

0..0(x)

(x)..(x)(x)

.....

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(x)..(x)(x)

(x)..(x)(x)

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(x)..(x)(x)

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--

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1

II2I1I

I22212

I12111

II2I1I

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I12111

Transition rates: generation and distribution with covariates

...exp 22110 ZZm iiii

...ln 22110 ZZm iiii

...exp)()( 22110 ZZxmxm iiiioi Cox model

Log-linear model

Let ij be constant during interval => ij = mi

From transition probabilities to transition rates

The inverse method (Singer and Spilerman)

y)(x,y)(x, y)(x, 211

21 MIMIP

12 ),(),(),( yxyxyx xy PIPIM

)1,(exp)1,( xxxx MP

From 5-year probability to 1-year probability:

Incomplete data

jiijijNE

]exp[- !

}n Pr{ ijijij

ij

nij

nN

ij

Poisson model:

Data availability:

The maximization (m) of the probability is equivalent to maximizing the log-likelihood

ijjijiijnl ]ln[

jj

ii

n

ˆˆ

ii

jj

n

ˆˆ

ji

ij nn

n

The EM algorithm results in the well-known expression

Expectation (E)

Incomplete data: Prior information

0**ijjijij mNE

i

ijjiijij cakNE exp

ABij

Bj

Aiij uuuu ln

Gravity model

Log-linear model

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 West 287 677 1,120 37,902 39,986 Total 44,455 53,357 71,860 41,065 210,737 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

1845 / 1269 = 1.454

1800 / 753 = 2.390

2.390 / 1.454 = 1.644

ODDS

ODDS Ratio

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)

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

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.

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

• Judgments Timing of event Direction of change