analysis of the salary trajectories in luxembourg...binary data )binary logit distribution censored...
TRANSCRIPT
![Page 1: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/1.jpg)
includegraphics[height=0.5cm]lsf.eps
Analysis of the salary trajectories in LuxembourgA finite mixture model approach
Jang SCHILTZ (University of Luxembourg)joint work with Jean-Daniel GUIGOU (University of Luxembourg)
& Bruno LOVAT (University Nancy II)
January 19, 2010
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 1 / 125
![Page 2: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/2.jpg)
includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 2 / 125
![Page 3: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/3.jpg)
includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 2 / 125
![Page 4: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/4.jpg)
includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 2 / 125
![Page 5: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/5.jpg)
includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 2 / 125
![Page 6: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/6.jpg)
includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 3 / 125
![Page 7: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/7.jpg)
includegraphics[height=0.5cm]lsf.eps
General description of Nagin’s model
We have a collection of individual trajectories.
We try to divide the population into a number of homogenoussubpopulations and to estimate a mean trajectory for each subpopulation.
This is still an inter-individual model, but unlike other classical modelssuch as standard growth curve models, it allows the existence ofsubpolulations with completely different behaviors.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 4 / 125
![Page 8: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/8.jpg)
includegraphics[height=0.5cm]lsf.eps
General description of Nagin’s model
We have a collection of individual trajectories.
We try to divide the population into a number of homogenoussubpopulations and to estimate a mean trajectory for each subpopulation.
This is still an inter-individual model, but unlike other classical modelssuch as standard growth curve models, it allows the existence ofsubpolulations with completely different behaviors.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 4 / 125
![Page 9: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/9.jpg)
includegraphics[height=0.5cm]lsf.eps
General description of Nagin’s model
We have a collection of individual trajectories.
We try to divide the population into a number of homogenoussubpopulations and to estimate a mean trajectory for each subpopulation.
This is still an inter-individual model, but unlike other classical modelssuch as standard growth curve models, it allows the existence ofsubpolulations with completely different behaviors.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 4 / 125
![Page 10: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/10.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 11: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/11.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 12: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/12.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 13: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/13.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 14: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/14.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 15: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/15.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 16: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/16.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (1)
Consider a population of size N and a variable of interest Y .
Let Yi = yi1 , yi2 , ..., yiT be T measures of the variable, taken at timest1, ...tT for subject number i .
P(Yi ) denotes the probability of Yi
count data ⇒ Poisson distribution
binary data ⇒ Binary logit distribution
censored data ⇒ Censored normal distribution
Aim of the analysis: Find r groups of trajectories of a given kind (forinstance polynomials of degree 4, P(t) = β0 + β1t + β2t
2 + β3t3 + β4t
4.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 5 / 125
![Page 17: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/17.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 18: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/18.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 19: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/19.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 20: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/20.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 21: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/21.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 22: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/22.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)
finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 23: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/23.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 24: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/24.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (2)
πj : probability of a given subject to belong to group number j
⇒ πj is the size of group j .
We try to estimate a set of parameters Ω =βj
0, βj1, β
j2, β
j3, β
j4, πj
which
allow to maximize the probability of the measured data.
P j(Yi ) : probability of Yi if subject i belongs to group j
⇒ P(Yi ) =r∑
j=1
πjPj(Yi ). (1)
Finite mixture model(Daniel S. Nagin (Carnegie Mellon University)
)finite : sums across a finite number of groups
mixture : population composed of a mixture of unobserved groups
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 6 / 125
![Page 25: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/25.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (3)
Hypothesis: In a given group, conditional independence is assumed for thesequential realizations of the elements of Yi !!!
⇒ P j(Yi ) =T∏
t=1
pj(yit ), (2)
where pj(yit ) denotes the probability of yit given membership in group j .
Likelihood of the estimator:
L =N∏
i=1
P(Yi ) =N∏
i=1
r∑j=1
πj
T∏t=1
pj(yit ). (3)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 7 / 125
![Page 26: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/26.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (3)
Hypothesis: In a given group, conditional independence is assumed for thesequential realizations of the elements of Yi !!!
⇒ P j(Yi ) =T∏
t=1
pj(yit ), (2)
where pj(yit ) denotes the probability of yit given membership in group j .
Likelihood of the estimator:
L =N∏
i=1
P(Yi ) =N∏
i=1
r∑j=1
πj
T∏t=1
pj(yit ). (3)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 7 / 125
![Page 27: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/27.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (3)
Hypothesis: In a given group, conditional independence is assumed for thesequential realizations of the elements of Yi !!!
⇒ P j(Yi ) =T∏
t=1
pj(yit ), (2)
where pj(yit ) denotes the probability of yit given membership in group j .
Likelihood of the estimator:
L =N∏
i=1
P(Yi ) =N∏
i=1
r∑j=1
πj
T∏t=1
pj(yit ). (3)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 7 / 125
![Page 28: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/28.jpg)
includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (3)
Hypothesis: In a given group, conditional independence is assumed for thesequential realizations of the elements of Yi !!!
⇒ P j(Yi ) =T∏
t=1
pj(yit ), (2)
where pj(yit ) denotes the probability of yit given membership in group j .
Likelihood of the estimator:
L =N∏
i=1
P(Yi )
=N∏
i=1
r∑j=1
πj
T∏t=1
pj(yit ). (3)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 7 / 125
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includegraphics[height=0.5cm]lsf.eps
The Likelihood Function (3)
Hypothesis: In a given group, conditional independence is assumed for thesequential realizations of the elements of Yi !!!
⇒ P j(Yi ) =T∏
t=1
pj(yit ), (2)
where pj(yit ) denotes the probability of yit given membership in group j .
Likelihood of the estimator:
L =N∏
i=1
P(Yi ) =N∏
i=1
r∑j=1
πj
T∏t=1
pj(yit ). (3)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 7 / 125
![Page 30: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/30.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (1)
Y ∗: latent variable measured by Y .
y∗it = βj
0 + βj1Ageit + βj
2Age2it + βj
3Age3it + βj
4Age4it + εit , (4)
where εit ∼ N (0, σ), σ being a constant standard deviation.
Hence,
yit = Smin si y∗it < Smin,
yit = y∗it si Smin ≤ y∗
it ≤ Smax ,
yit = Smax si y∗it > Smax ,
where Smin and Smax dennote the minimum and maximum of the censorednormal distribution.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 8 / 125
![Page 31: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/31.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (1)
Y ∗: latent variable measured by Y .
y∗it = βj
0 + βj1Ageit + βj
2Age2it + βj
3Age3it + βj
4Age4it + εit , (4)
where εit ∼ N (0, σ), σ being a constant standard deviation.
Hence,
yit = Smin si y∗it < Smin,
yit = y∗it si Smin ≤ y∗
it ≤ Smax ,
yit = Smax si y∗it > Smax ,
where Smin and Smax dennote the minimum and maximum of the censorednormal distribution.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 8 / 125
![Page 32: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/32.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (1)
Y ∗: latent variable measured by Y .
y∗it = βj
0 + βj1Ageit + βj
2Age2it + βj
3Age3it + βj
4Age4it + εit , (4)
where εit ∼ N (0, σ), σ being a constant standard deviation.
Hence,
yit = Smin si y∗it < Smin,
yit = y∗it si Smin ≤ y∗
it ≤ Smax ,
yit = Smax si y∗it > Smax ,
where Smin and Smax dennote the minimum and maximum of the censorednormal distribution.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 8 / 125
![Page 33: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/33.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 34: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/34.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 35: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/35.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 36: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/36.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 37: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/37.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 38: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/38.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 39: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/39.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 40: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/40.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (2)
Notations :
βjxit = βj0 + βj
1Ageit + βj2Age2
it+ βj
3Age3it
+ βj4Age4
it.
φ: density of standard centered normal law.
Φ: cumulative distribution function of standard centered normal law.
Hence,
pj(yit = Smin) = Φ
(Smin − βjxit
σ
), (5)
pj(yit ) =1
σφ
(yit − βjxit
σ
)pour Smin ≤ yit ≤ Smax , (6)
pj(yit = Smax) = 1− Φ
(Smax − βjxit
σ
). (7)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 9 / 125
![Page 41: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/41.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (3)
If all the measures are in the interval [Smin,Smax ], we get
L =1
σ
N∏i=1
r∑j=1
πj
T∏t=1
φ
(yit − βjxit
σ
). (8)
It is too complicated to get closed-forms equations
⇒ quasi-Newton procedure maximum research routine
Software:
SAS-based Proc Traj procedureby Bobby L. Jones (Carnegie Mellon University).
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 10 / 125
![Page 42: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/42.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (3)
If all the measures are in the interval [Smin,Smax ], we get
L =1
σ
N∏i=1
r∑j=1
πj
T∏t=1
φ
(yit − βjxit
σ
). (8)
It is too complicated to get closed-forms equations
⇒ quasi-Newton procedure maximum research routine
Software:
SAS-based Proc Traj procedureby Bobby L. Jones (Carnegie Mellon University).
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 10 / 125
![Page 43: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/43.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (3)
If all the measures are in the interval [Smin,Smax ], we get
L =1
σ
N∏i=1
r∑j=1
πj
T∏t=1
φ
(yit − βjxit
σ
). (8)
It is too complicated to get closed-forms equations
⇒ quasi-Newton procedure maximum research routine
Software:
SAS-based Proc Traj procedureby Bobby L. Jones (Carnegie Mellon University).
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 10 / 125
![Page 44: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/44.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (3)
If all the measures are in the interval [Smin,Smax ], we get
L =1
σ
N∏i=1
r∑j=1
πj
T∏t=1
φ
(yit − βjxit
σ
). (8)
It is too complicated to get closed-forms equations
⇒ quasi-Newton procedure maximum research routine
Software:
SAS-based Proc Traj procedureby Bobby L. Jones (Carnegie Mellon University).
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 10 / 125
![Page 45: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/45.jpg)
includegraphics[height=0.5cm]lsf.eps
The case of a censored normal distribution (3)
If all the measures are in the interval [Smin,Smax ], we get
L =1
σ
N∏i=1
r∑j=1
πj
T∏t=1
φ
(yit − βjxit
σ
). (8)
It is too complicated to get closed-forms equations
⇒ quasi-Newton procedure maximum research routine
Software:
SAS-based Proc Traj procedureby Bobby L. Jones (Carnegie Mellon University).
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 10 / 125
![Page 46: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/46.jpg)
includegraphics[height=0.5cm]lsf.eps
A computational trick
The estimations of πj must be in [0, 1].
It is difficult to force this constraint in model estimation.
Instead, we estimate the real parameters θj such that
πj =eθjr∑
j=1
eθj
, (9)
Finally,
L =1
σ
N∏i=1
r∑j=1
eθjr∑
j=1
eθj
T∏t=1
φ
(yit − βjxit
σ
). (10)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 11 / 125
![Page 47: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/47.jpg)
includegraphics[height=0.5cm]lsf.eps
A computational trick
The estimations of πj must be in [0, 1].
It is difficult to force this constraint in model estimation.
Instead, we estimate the real parameters θj such that
πj =eθjr∑
j=1
eθj
, (9)
Finally,
L =1
σ
N∏i=1
r∑j=1
eθjr∑
j=1
eθj
T∏t=1
φ
(yit − βjxit
σ
). (10)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 11 / 125
![Page 48: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/48.jpg)
includegraphics[height=0.5cm]lsf.eps
A computational trick
The estimations of πj must be in [0, 1].
It is difficult to force this constraint in model estimation.
Instead, we estimate the real parameters θj such that
πj =eθjr∑
j=1
eθj
, (9)
Finally,
L =1
σ
N∏i=1
r∑j=1
eθjr∑
j=1
eθj
T∏t=1
φ
(yit − βjxit
σ
). (10)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 11 / 125
![Page 49: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/49.jpg)
includegraphics[height=0.5cm]lsf.eps
A computational trick
The estimations of πj must be in [0, 1].
It is difficult to force this constraint in model estimation.
Instead, we estimate the real parameters θj such that
πj =eθjr∑
j=1
eθj
, (9)
Finally,
L =1
σ
N∏i=1
r∑j=1
eθjr∑
j=1
eθj
T∏t=1
φ
(yit − βjxit
σ
). (10)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 11 / 125
![Page 50: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/50.jpg)
includegraphics[height=0.5cm]lsf.eps
A computational trick
The estimations of πj must be in [0, 1].
It is difficult to force this constraint in model estimation.
Instead, we estimate the real parameters θj such that
πj =eθjr∑
j=1
eθj
, (9)
Finally,
L =1
σ
N∏i=1
r∑j=1
eθjr∑
j=1
eθj
T∏t=1
φ
(yit − βjxit
σ
). (10)
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 11 / 125
![Page 51: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/51.jpg)
includegraphics[height=0.5cm]lsf.eps
Muthen’s model (1)
Muthen and Shedden (1999): Generalized growth curve model
Elegant and technically demanding extension of the uncensored normalmodel.
Adds random effects to the parameters βj that define a group’s meantrajectory.
Trajectories of individual group members can vary from the grouptrajectory.
Software:
Mplus package by L.K. Muthen and B.O Muthen.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 12 / 125
![Page 52: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/52.jpg)
includegraphics[height=0.5cm]lsf.eps
Muthen’s model (1)
Muthen and Shedden (1999): Generalized growth curve model
Elegant and technically demanding extension of the uncensored normalmodel.
Adds random effects to the parameters βj that define a group’s meantrajectory.
Trajectories of individual group members can vary from the grouptrajectory.
Software:
Mplus package by L.K. Muthen and B.O Muthen.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 12 / 125
![Page 53: Analysis of the salary trajectories in Luxembourg...binary data )Binary logit distribution censored data )Censored normal distribution Aim of the analysis: Find r groups of trajectories](https://reader036.vdocuments.site/reader036/viewer/2022062506/5fb7506c56cece166b73588e/html5/thumbnails/53.jpg)
includegraphics[height=0.5cm]lsf.eps
Muthen’s model (1)
Muthen and Shedden (1999): Generalized growth curve model
Elegant and technically demanding extension of the uncensored normalmodel.
Adds random effects to the parameters βj that define a group’s meantrajectory.
Trajectories of individual group members can vary from the grouptrajectory.
Software:
Mplus package by L.K. Muthen and B.O Muthen.
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includegraphics[height=0.5cm]lsf.eps
Muthen’s model (1)
Muthen and Shedden (1999): Generalized growth curve model
Elegant and technically demanding extension of the uncensored normalmodel.
Adds random effects to the parameters βj that define a group’s meantrajectory.
Trajectories of individual group members can vary from the grouptrajectory.
Software:
Mplus package by L.K. Muthen and B.O Muthen.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 12 / 125
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includegraphics[height=0.5cm]lsf.eps
Muthen’s model (1)
Muthen and Shedden (1999): Generalized growth curve model
Elegant and technically demanding extension of the uncensored normalmodel.
Adds random effects to the parameters βj that define a group’s meantrajectory.
Trajectories of individual group members can vary from the grouptrajectory.
Software:
Mplus package by L.K. Muthen and B.O Muthen.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 12 / 125
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includegraphics[height=0.5cm]lsf.eps
Muthen’s model (2)
Advantage of GGCM
Fewer groups are required to specify a satisfactory model.
Disadvantages of GGCM
1 Difficult to extend to other types of data.
2 Group cross-over effects.
3 can create the illusion of non-existing groups.
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includegraphics[height=0.5cm]lsf.eps
Muthen’s model (2)
Advantage of GGCM
Fewer groups are required to specify a satisfactory model.
Disadvantages of GGCM
1 Difficult to extend to other types of data.
2 Group cross-over effects.
3 can create the illusion of non-existing groups.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 13 / 125
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includegraphics[height=0.5cm]lsf.eps
Muthen’s model (2)
Advantage of GGCM
Fewer groups are required to specify a satisfactory model.
Disadvantages of GGCM
1 Difficult to extend to other types of data.
2 Group cross-over effects.
3 can create the illusion of non-existing groups.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 13 / 125
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includegraphics[height=0.5cm]lsf.eps
Muthen’s model (2)
Advantage of GGCM
Fewer groups are required to specify a satisfactory model.
Disadvantages of GGCM
1 Difficult to extend to other types of data.
2 Group cross-over effects.
3 can create the illusion of non-existing groups.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 13 / 125
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includegraphics[height=0.5cm]lsf.eps
Model Selection
Bayesian Information Criterion:
BIC = log(L)− 0, 5k log(N), (11)
where k denotes the number of parameters in the model.
Rule:
The bigger the BIC, the better the model!
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includegraphics[height=0.5cm]lsf.eps
Model Selection
Bayesian Information Criterion:
BIC = log(L)− 0, 5k log(N), (11)
where k denotes the number of parameters in the model.
Rule:
The bigger the BIC, the better the model!
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 14 / 125
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includegraphics[height=0.5cm]lsf.eps
Model Selection
Bayesian Information Criterion:
BIC = log(L)− 0, 5k log(N), (11)
where k denotes the number of parameters in the model.
Rule:
The bigger the BIC, the better the model!
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 14 / 125
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includegraphics[height=0.5cm]lsf.eps
Posterior Group-Membership Probabilities
Posterior probability of individual i ’s membership in group j : P(j/Yi ).
Bayes’s theorem
⇒ P(j/Yi ) =P(Yi/j)πjr∑
j=1
P(Yi/j)πj
. (12)
Bigger groups have on average larger probability estimates.
To be classified into a small group, an individual really needs to bestrongly consistent with it.
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includegraphics[height=0.5cm]lsf.eps
Posterior Group-Membership Probabilities
Posterior probability of individual i ’s membership in group j : P(j/Yi ).
Bayes’s theorem
⇒ P(j/Yi ) =P(Yi/j)πjr∑
j=1
P(Yi/j)πj
. (12)
Bigger groups have on average larger probability estimates.
To be classified into a small group, an individual really needs to bestrongly consistent with it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 15 / 125
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includegraphics[height=0.5cm]lsf.eps
Posterior Group-Membership Probabilities
Posterior probability of individual i ’s membership in group j : P(j/Yi ).
Bayes’s theorem
⇒ P(j/Yi ) =P(Yi/j)πjr∑
j=1
P(Yi/j)πj
. (12)
Bigger groups have on average larger probability estimates.
To be classified into a small group, an individual really needs to bestrongly consistent with it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 15 / 125
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includegraphics[height=0.5cm]lsf.eps
Posterior Group-Membership Probabilities
Posterior probability of individual i ’s membership in group j : P(j/Yi ).
Bayes’s theorem
⇒ P(j/Yi ) =P(Yi/j)πjr∑
j=1
P(Yi/j)πj
. (12)
Bigger groups have on average larger probability estimates.
To be classified into a small group, an individual really needs to bestrongly consistent with it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 15 / 125
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includegraphics[height=0.5cm]lsf.eps
Posterior Group-Membership Probabilities
Posterior probability of individual i ’s membership in group j : P(j/Yi ).
Bayes’s theorem
⇒ P(j/Yi ) =P(Yi/j)πjr∑
j=1
P(Yi/j)πj
. (12)
Bigger groups have on average larger probability estimates.
To be classified into a small group, an individual really needs to bestrongly consistent with it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 15 / 125
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includegraphics[height=0.5cm]lsf.eps
Use for Model diagnostics (2)
Diagnostic 1: Average Posterior Probability of Assignment
AvePP should be at least 0, 7 for all groups.
Diagonostic 2: Odds of Correct Classification
OCCj =AvePPj/1− AvePPj
πj/1− πj. (13)
OCCj should be greater than 5 for all groups.
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includegraphics[height=0.5cm]lsf.eps
Use for Model diagnostics (2)
Diagnostic 1: Average Posterior Probability of Assignment
AvePP should be at least 0, 7 for all groups.
Diagonostic 2: Odds of Correct Classification
OCCj =AvePPj/1− AvePPj
πj/1− πj. (13)
OCCj should be greater than 5 for all groups.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 16 / 125
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includegraphics[height=0.5cm]lsf.eps
Use for Model diagnostics (2)
Diagnostic 1: Average Posterior Probability of Assignment
AvePP should be at least 0, 7 for all groups.
Diagonostic 2: Odds of Correct Classification
OCCj =AvePPj/1− AvePPj
πj/1− πj. (13)
OCCj should be greater than 5 for all groups.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 16 / 125
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includegraphics[height=0.5cm]lsf.eps
Use for Model diagnostics (2)
Diagonostic 3: Comparing πj to the Proportion of the SampleAssigned to Group j
The ratio of the two should be close to 1.
Diagonostic 4: Confidence Intervals for Group MembershipProbabilities
The confidence intervals for group membership probabilities estimatesshould be narrow, i.e. standard deviation of πj should be small.
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includegraphics[height=0.5cm]lsf.eps
Use for Model diagnostics (2)
Diagonostic 3: Comparing πj to the Proportion of the SampleAssigned to Group j
The ratio of the two should be close to 1.
Diagonostic 4: Confidence Intervals for Group MembershipProbabilities
The confidence intervals for group membership probabilities estimatesshould be narrow, i.e. standard deviation of πj should be small.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 17 / 125
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includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
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The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
The data
Salaries of workers in the private sector in Luxembourg from 1940 to 2006.
About 7 million salary lines corresponding to 718.054 workers.
Some sociological variables:
gender (male, female)
nationality and residentship (luxemburgish residents, foreign residents,foreign non residents)
working status (white collar worker, blue collar worker)
year of birth
age in the first year of professional activity
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 19 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Dataset transformations
Mathematica programming
1 row per year → 1 row per worker
selection of the period we are interested in
taking out the years without years up to a maximum of five years
selection of the people who worked the number of years we areinterested in
Transformations in SPSS
elimination of all the workers who had monthly salaries above 15.000
transforming all the salaries above 7.577 to 7.577
creation of the time variables necessary for the Proc Traj procedure
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 20 / 125
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includegraphics[height=0.5cm]lsf.eps
Proc Traj procedure
Selection of the time period for macroeconomic reasons
(Crisis in the steelindustry and emergence of the financial market place of Luxembourg)
20 years of work for workers beginning their carrier between 1982 and 1987
Proc Traj Macro:
DATA TEST;INPUT ID O1-O20 T1-T20;CARDS;
dataRUN;
PROC TRAJ DATA=TEST OUTPLOT=OP OUTSTAT=OS OUT=OFOUTEST=OE ITDETAIL;
ID ID; VAR O1-O20; INDEP T1-T20;MODEL CNORM; MAX 8000; NGROUPS 6; ORDER 4 4 4 4 4 4;
RUN;
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includegraphics[height=0.5cm]lsf.eps
Proc Traj procedure
Selection of the time period for macroeconomic reasons (Crisis in the steelindustry and emergence of the financial market place of Luxembourg)
20 years of work for workers beginning their carrier between 1982 and 1987
Proc Traj Macro:
DATA TEST;INPUT ID O1-O20 T1-T20;CARDS;
dataRUN;
PROC TRAJ DATA=TEST OUTPLOT=OP OUTSTAT=OS OUT=OFOUTEST=OE ITDETAIL;
ID ID; VAR O1-O20; INDEP T1-T20;MODEL CNORM; MAX 8000; NGROUPS 6; ORDER 4 4 4 4 4 4;
RUN;
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includegraphics[height=0.5cm]lsf.eps
Proc Traj procedure
Selection of the time period for macroeconomic reasons (Crisis in the steelindustry and emergence of the financial market place of Luxembourg)
20 years of work for workers beginning their carrier between 1982 and 1987
Proc Traj Macro:
DATA TEST;INPUT ID O1-O20 T1-T20;CARDS;
dataRUN;
PROC TRAJ DATA=TEST OUTPLOT=OP OUTSTAT=OS OUT=OFOUTEST=OE ITDETAIL;
ID ID; VAR O1-O20; INDEP T1-T20;MODEL CNORM; MAX 8000; NGROUPS 6; ORDER 4 4 4 4 4 4;
RUN;
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includegraphics[height=0.5cm]lsf.eps
Proc Traj procedure
Selection of the time period for macroeconomic reasons (Crisis in the steelindustry and emergence of the financial market place of Luxembourg)
20 years of work for workers beginning their carrier between 1982 and 1987
Proc Traj Macro:
DATA TEST;INPUT ID O1-O20 T1-T20;CARDS;
dataRUN;
PROC TRAJ DATA=TEST OUTPLOT=OP OUTSTAT=OS OUT=OFOUTEST=OE ITDETAIL;
ID ID; VAR O1-O20; INDEP T1-T20;MODEL CNORM; MAX 8000; NGROUPS 6; ORDER 4 4 4 4 4 4;
RUN;
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includegraphics[height=0.5cm]lsf.eps
Proc Traj procedure
Selection of the time period for macroeconomic reasons (Crisis in the steelindustry and emergence of the financial market place of Luxembourg)
20 years of work for workers beginning their carrier between 1982 and 1987
Proc Traj Macro:
DATA TEST;INPUT ID O1-O20 T1-T20;CARDS;
dataRUN;
PROC TRAJ DATA=TEST OUTPLOT=OP OUTSTAT=OS OUT=OFOUTEST=OE ITDETAIL;
ID ID; VAR O1-O20; INDEP T1-T20;MODEL CNORM; MAX 8000; NGROUPS 6; ORDER 4 4 4 4 4 4;
RUN;
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includegraphics[height=0.5cm]lsf.eps
Proc Traj procedure
Selection of the time period for macroeconomic reasons (Crisis in the steelindustry and emergence of the financial market place of Luxembourg)
20 years of work for workers beginning their carrier between 1982 and 1987
Proc Traj Macro:
DATA TEST;INPUT ID O1-O20 T1-T20;CARDS;
dataRUN;
PROC TRAJ DATA=TEST OUTPLOT=OP OUTSTAT=OS OUT=OFOUTEST=OE ITDETAIL;
ID ID; VAR O1-O20; INDEP T1-T20;MODEL CNORM; MAX 8000; NGROUPS 6; ORDER 4 4 4 4 4 4;
RUN;
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includegraphics[height=0.5cm]lsf.eps
Results for 9 groups (1)
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includegraphics[height=0.5cm]lsf.eps
Results for 9 groups (1)
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Results for 9 groups (2)
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Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
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includegraphics[height=0.5cm]lsf.eps
1st group
13.4 % of the population
P(x) = 590 + 388t − 14t2 + 0.009t4
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includegraphics[height=0.5cm]lsf.eps
1st group
13.4 % of the population
P(x) = 590 + 388t − 14t2 + 0.009t4
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includegraphics[height=0.5cm]lsf.eps
1st group
13.4 % of the population
P(x) = 590 + 388t − 14t2 + 0.009t4
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includegraphics[height=0.5cm]lsf.eps
1st group
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includegraphics[height=0.5cm]lsf.eps
1st group
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includegraphics[height=0.5cm]lsf.eps
1st group
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includegraphics[height=0.5cm]lsf.eps
1st group
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1st group
Men:
Women:
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includegraphics[height=0.5cm]lsf.eps
1st group
Men:
Women:
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1st group
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includegraphics[height=0.5cm]lsf.eps
1st group
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1st group
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1st group
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1st group
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2nd group
16.9 % of the population
P(x) = 785 + 278t − 28t2 + 1.18t3 − 0.016t4
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includegraphics[height=0.5cm]lsf.eps
2nd group
16.9 % of the population
P(x) = 785 + 278t − 28t2 + 1.18t3 − 0.016t4
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includegraphics[height=0.5cm]lsf.eps
2nd group
16.9 % of the population
P(x) = 785 + 278t − 28t2 + 1.18t3 − 0.016t4
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2nd group
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2nd group
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2nd group
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2nd group
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2nd group
Men:
Women:
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2nd group
Men:
Women:
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2nd group
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2nd group
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2nd group
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2nd group
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2nd group
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3rd group
20.8 % of the population
P(x) = 709 + 318t − 21.5t2 + 0.62t3
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3rd group
20.8 % of the population
P(x) = 709 + 318t − 21.5t2 + 0.62t3
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3rd group
20.8 % of the population
P(x) = 709 + 318t − 21.5t2 + 0.62t3
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3rd group
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3rd group
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3rd group
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3rd group
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3rd group
Men:
Women:
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3rd group
Men:
Women:
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3rd group
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3rd group
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3rd group
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3rd group
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3rd group
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4th group
7.9 % of the population
P(x) = 976 + 474t − 29.6t2 − 0.029t4
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4th group
7.9 % of the population
P(x) = 976 + 474t − 29.6t2 − 0.029t4
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4th group
7.9 % of the population
P(x) = 976 + 474t − 29.6t2 − 0.029t4
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4th group
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4th group
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4th group
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4th group
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4th group
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4th group
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4th group
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4th group
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4th group
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5th group
14.9 % of the population
P(x) = 1452 + 490t − 29.6t2 + 1.38t3 − 0.028t4
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5th group
14.9 % of the population
P(x) = 1452 + 490t − 29.6t2 + 1.38t3 − 0.028t4
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5th group
14.9 % of the population
P(x) = 1452 + 490t − 29.6t2 + 1.38t3 − 0.028t4
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5th group
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5th group
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5th group
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5th group
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5th group
Men:
Women:
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5th group
Men:
Women:
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5th group
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5th group
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5th group
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5th group
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5th group
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6th group
4.8 % of the population
P(x) = 2089− 0.017t4
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6th group
4.8 % of the population
P(x) = 2089− 0.017t4
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6th group
4.8 % of the population
P(x) = 2089− 0.017t4
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6th group
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6th group
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6th group
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6th group
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6th group
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6th group
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6th group
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6th group
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6th group
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7th group
6.6 % of the population
P(x) = 2556 + 484t − 29.9t2 + 0.66t3
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7th group
6.6 % of the population
P(x) = 2556 + 484t − 29.9t2 + 0.66t3
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7th group
6.6 % of the population
P(x) = 2556 + 484t − 29.9t2 + 0.66t3
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7th group
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7th group
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7th group
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7th group
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7th group
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8th group
8.4 % of the population
P(x) = 1987 + 537t − 52.7t2 + 2.06t3 − 0.028t4
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8th group
8.4 % of the population
P(x) = 1987 + 537t − 52.7t2 + 2.06t3 − 0.028t4
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8th group
8.4 % of the population
P(x) = 1987 + 537t − 52.7t2 + 2.06t3 − 0.028t4
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8th group
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8th group
Men:
Women:
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8th group
Men:
Women:
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8th group
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8th group
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8th group
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8th group
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8th group
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9th group
6.4 % of the population
P(x) = 3873 + 206t + 30t2− 2.89t3 + 0.06t4
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9th group
6.4 % of the population
P(x) = 3873 + 206t + 30t2− 2.89t3 + 0.06t4
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9th group
6.4 % of the population
P(x) = 3873 + 206t + 30t2− 2.89t3 + 0.06t4
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9th group
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9th group
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9th group
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includegraphics[height=0.5cm]lsf.eps
9th group
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 114 / 125
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includegraphics[height=0.5cm]lsf.eps
Salary Dynamics
Mean annual salary growth:
Group 1 Group 2 Group 3 Group 4 Group 5
λi = 3.07% λ2 = 0.96% λ3 = 1.45% λ4 = 2.82% λ5 = 0.19%
Group 6 Group 7 Group 8 Group 9λ6 = 2.58% λ7 = 1.28% λ8 = 0.48% λ9 = 1.09%
Flat curves : groups 2,5 and 8.
Normal salary growth: groups 3,7 and 9.
Dynamic trajectories: groups 1,4 and 6.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 115 / 125
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includegraphics[height=0.5cm]lsf.eps
Salary Dynamics
Mean annual salary growth:
Group 1 Group 2 Group 3 Group 4 Group 5
λi = 3.07% λ2 = 0.96% λ3 = 1.45% λ4 = 2.82% λ5 = 0.19%
Group 6 Group 7 Group 8 Group 9λ6 = 2.58% λ7 = 1.28% λ8 = 0.48% λ9 = 1.09%
Flat curves : groups 2,5 and 8.
Normal salary growth: groups 3,7 and 9.
Dynamic trajectories: groups 1,4 and 6.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 115 / 125
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includegraphics[height=0.5cm]lsf.eps
Salary Dynamics
Mean annual salary growth:
Group 1 Group 2 Group 3 Group 4 Group 5
λi = 3.07% λ2 = 0.96% λ3 = 1.45% λ4 = 2.82% λ5 = 0.19%
Group 6 Group 7 Group 8 Group 9λ6 = 2.58% λ7 = 1.28% λ8 = 0.48% λ9 = 1.09%
Flat curves : groups 2,5 and 8.
Normal salary growth: groups 3,7 and 9.
Dynamic trajectories: groups 1,4 and 6.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 115 / 125
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includegraphics[height=0.5cm]lsf.eps
Salary Dynamics
Mean annual salary growth:
Group 1 Group 2 Group 3 Group 4 Group 5
λi = 3.07% λ2 = 0.96% λ3 = 1.45% λ4 = 2.82% λ5 = 0.19%
Group 6 Group 7 Group 8 Group 9λ6 = 2.58% λ7 = 1.28% λ8 = 0.48% λ9 = 1.09%
Flat curves : groups 2,5 and 8.
Normal salary growth: groups 3,7 and 9.
Dynamic trajectories: groups 1,4 and 6.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 115 / 125
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includegraphics[height=0.5cm]lsf.eps
Outline
1 Nagin’s Finite Mixture Model
2 The Luxemburgish salary trajectories
3 Description of the groups
4 Economic Modeling
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 116 / 125
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includegraphics[height=0.5cm]lsf.eps
Dummy example
2 trajectories S1 and S2 with group size 60% and 40% of the population.
Length of the professional life: T = 40 years.
Additional life expectancy: T ∗ = 20 years.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 117 / 125
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includegraphics[height=0.5cm]lsf.eps
Dummy example
2 trajectories S1 and S2 with group size 60% and 40% of the population.
Length of the professional life: T = 40 years.
Additional life expectancy: T ∗ = 20 years.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 117 / 125
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includegraphics[height=0.5cm]lsf.eps
Dummy example
2 trajectories S1 and S2 with group size 60% and 40% of the population.
Length of the professional life: T = 40 years.
Additional life expectancy: T ∗ = 20 years.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 117 / 125
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includegraphics[height=0.5cm]lsf.eps
Dummy example
2 trajectories S1 and S2 with group size 60% and 40% of the population.
Length of the professional life: T = 40 years.
Additional life expectancy: T ∗ = 20 years.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 117 / 125
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includegraphics[height=0.5cm]lsf.eps
Hypotheses
Salaries grow linearly, S1 with a starting value of 1500 and a growthcoefficient of 3 %, S2 with a starting value of 1000 and a growthcoefficient of 2 %.
Pensions grow also linearly, S1 with a starting value of 2718 and a growthcoefficient of 2%, S2 with a starting value of 1104 and a growthcoefficient of 1%.
Luxembourg adopts a repartition model, which means that the currentpensions are paid with the tax incomes from the current workers. Eachgeneration hence pays the pension for the generation before it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 118 / 125
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includegraphics[height=0.5cm]lsf.eps
Hypotheses
Salaries grow linearly, S1 with a starting value of 1500 and a growthcoefficient of 3 %, S2 with a starting value of 1000 and a growthcoefficient of 2 %.
Pensions grow also linearly, S1 with a starting value of 2718 and a growthcoefficient of 2%, S2 with a starting value of 1104 and a growthcoefficient of 1%.
Luxembourg adopts a repartition model, which means that the currentpensions are paid with the tax incomes from the current workers. Eachgeneration hence pays the pension for the generation before it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 118 / 125
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includegraphics[height=0.5cm]lsf.eps
Hypotheses
Salaries grow linearly, S1 with a starting value of 1500 and a growthcoefficient of 3 %, S2 with a starting value of 1000 and a growthcoefficient of 2 %.
Pensions grow also linearly, S1 with a starting value of 2718 and a growthcoefficient of 2%, S2 with a starting value of 1104 and a growthcoefficient of 1%.
Luxembourg adopts a repartition model, which means that the currentpensions are paid with the tax incomes from the current workers. Eachgeneration hence pays the pension for the generation before it.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 118 / 125
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includegraphics[height=0.5cm]lsf.eps
Replacement rate in the repartition model
Replacement rate = first pension / last salary
For S1, trep = 27181500(1+0.03)39 ' 57%.
For S2, trep = 11041000(1+0.02)39 = 50%.
A worker who’s trajectory is S1 with a probability of 75 % and S2 with aprobability of 25 % has a replacement rate of
trep =0.75× 2718 + 0.25× 1104
0.75× 1500(1 + 0.03)39 + 0.25× 1000(1 + 0.02)39' 56%.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 119 / 125
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includegraphics[height=0.5cm]lsf.eps
Replacement rate in the repartition model
Replacement rate = first pension / last salary
For S1, trep = 27181500(1+0.03)39 ' 57%.
For S2, trep = 11041000(1+0.02)39 = 50%.
A worker who’s trajectory is S1 with a probability of 75 % and S2 with aprobability of 25 % has a replacement rate of
trep =0.75× 2718 + 0.25× 1104
0.75× 1500(1 + 0.03)39 + 0.25× 1000(1 + 0.02)39' 56%.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 119 / 125
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includegraphics[height=0.5cm]lsf.eps
Replacement rate in the repartition model
Replacement rate = first pension / last salary
For S1, trep = 27181500(1+0.03)39 ' 57%.
For S2, trep = 11041000(1+0.02)39 = 50%.
A worker who’s trajectory is S1 with a probability of 75 % and S2 with aprobability of 25 % has a replacement rate of
trep =0.75× 2718 + 0.25× 1104
0.75× 1500(1 + 0.03)39 + 0.25× 1000(1 + 0.02)39' 56%.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 119 / 125
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includegraphics[height=0.5cm]lsf.eps
Replacement rate in the repartition model
Replacement rate = first pension / last salary
For S1, trep = 27181500(1+0.03)39 ' 57%.
For S2, trep = 11041000(1+0.02)39 = 50%.
A worker who’s trajectory is S1 with a probability of 75 % and S2 with aprobability of 25 % has a replacement rate of
trep =0.75× 2718 + 0.25× 1104
0.75× 1500(1 + 0.03)39 + 0.25× 1000(1 + 0.02)39' 56%.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 119 / 125
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includegraphics[height=0.5cm]lsf.eps
Coverage potential in a repartition & capitalization modelWe want to know the sum a that we have to put every year in a savingaccount to get a desired replacement rate taim.
a of course depends on the account’s interest rate i .
If i ∼ U(2%; 7%), a varies between 46 euros and 252 euros with a mean of124 euros.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 120 / 125
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includegraphics[height=0.5cm]lsf.eps
Coverage potential in a repartition & capitalization modelWe want to know the sum a that we have to put every year in a savingaccount to get a desired replacement rate taim.
a of course depends on the account’s interest rate i .
If i ∼ U(2%; 7%), a varies between 46 euros and 252 euros with a mean of124 euros.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 120 / 125
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includegraphics[height=0.5cm]lsf.eps
Coverage potential in a repartition & capitalization modelWe want to know the sum a that we have to put every year in a savingaccount to get a desired replacement rate taim.
a of course depends on the account’s interest rate i .
If i ∼ U(2%; 7%), a varies between 46 euros and 252 euros with a mean of124 euros.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 120 / 125
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includegraphics[height=0.5cm]lsf.eps
Capitalization effort coefficient
τ2 = Sum of the salaries on the salary trajectory / sum of the investmentreturns = 16.5 on average.
That means that you need 16.5 euros from the salary to get 1 euro bycapitalization for the pension.
In fact, if i ∼ U(2%; 7%), τ2 varies between 18 euros and 15 euros.
τ2 =Sj
aj(i − λj)i(1 + i)T − (1 + λj)
T
(1 + i)T − 1.
τ2 depends on a, hence a not only allows to get the desired replacementrate, but a also serves to control the variability of the capitalization effortcoefficient.
We need a compromise between a high replacement rate and a smallcapitalization effort coefficient.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 121 / 125
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includegraphics[height=0.5cm]lsf.eps
Capitalization effort coefficient
τ2 = Sum of the salaries on the salary trajectory / sum of the investmentreturns = 16.5 on average.
That means that you need 16.5 euros from the salary to get 1 euro bycapitalization for the pension.
In fact, if i ∼ U(2%; 7%), τ2 varies between 18 euros and 15 euros.
τ2 =Sj
aj(i − λj)i(1 + i)T − (1 + λj)
T
(1 + i)T − 1.
τ2 depends on a, hence a not only allows to get the desired replacementrate, but a also serves to control the variability of the capitalization effortcoefficient.
We need a compromise between a high replacement rate and a smallcapitalization effort coefficient.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 121 / 125
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includegraphics[height=0.5cm]lsf.eps
Capitalization effort coefficient
τ2 = Sum of the salaries on the salary trajectory / sum of the investmentreturns = 16.5 on average.
That means that you need 16.5 euros from the salary to get 1 euro bycapitalization for the pension.
In fact, if i ∼ U(2%; 7%), τ2 varies between 18 euros and 15 euros.
τ2 =Sj
aj(i − λj)i(1 + i)T − (1 + λj)
T
(1 + i)T − 1.
τ2 depends on a, hence a not only allows to get the desired replacementrate, but a also serves to control the variability of the capitalization effortcoefficient.
We need a compromise between a high replacement rate and a smallcapitalization effort coefficient.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 121 / 125
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includegraphics[height=0.5cm]lsf.eps
Capitalization effort coefficient
τ2 = Sum of the salaries on the salary trajectory / sum of the investmentreturns = 16.5 on average.
That means that you need 16.5 euros from the salary to get 1 euro bycapitalization for the pension.
In fact, if i ∼ U(2%; 7%), τ2 varies between 18 euros and 15 euros.
τ2 =Sj
aj(i − λj)i(1 + i)T − (1 + λj)
T
(1 + i)T − 1.
τ2 depends on a, hence a not only allows to get the desired replacementrate, but a also serves to control the variability of the capitalization effortcoefficient.
We need a compromise between a high replacement rate and a smallcapitalization effort coefficient.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 121 / 125
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includegraphics[height=0.5cm]lsf.eps
Capitalization effort coefficient
τ2 = Sum of the salaries on the salary trajectory / sum of the investmentreturns = 16.5 on average.
That means that you need 16.5 euros from the salary to get 1 euro bycapitalization for the pension.
In fact, if i ∼ U(2%; 7%), τ2 varies between 18 euros and 15 euros.
τ2 =Sj
aj(i − λj)i(1 + i)T − (1 + λj)
T
(1 + i)T − 1.
τ2 depends on a, hence a not only allows to get the desired replacementrate, but a also serves to control the variability of the capitalization effortcoefficient.
We need a compromise between a high replacement rate and a smallcapitalization effort coefficient.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 121 / 125
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includegraphics[height=0.5cm]lsf.eps
Repartition effort coefficient
τ1 = weighted mean of the salaries on the salary trajectory / weightedmean of the pensions in the repartition model on the pension trajectory =2.7 on average.
That means that the active worker have to earn 2.7 euros to pay 1 euro ofpension by repartition. pause
τ1 =
k(1+d)T+1 PT+1 + ...+ k
(1+d)T+T∗ PT+T∗
S0 + ...+ ST
(1+d)T
.
τ1 depends on the demographic rate d . In fact, if d ∼ U(0%; 5%), τ1varies between 6.7 euros and 1.6 euros.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 122 / 125
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includegraphics[height=0.5cm]lsf.eps
Repartition effort coefficient
τ1 = weighted mean of the salaries on the salary trajectory / weightedmean of the pensions in the repartition model on the pension trajectory =2.7 on average.
That means that the active worker have to earn 2.7 euros to pay 1 euro ofpension by repartition. pause
τ1 =
k(1+d)T+1 PT+1 + ...+ k
(1+d)T+T∗ PT+T∗
S0 + ...+ ST
(1+d)T
.
τ1 depends on the demographic rate d .
In fact, if d ∼ U(0%; 5%), τ1varies between 6.7 euros and 1.6 euros.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 122 / 125
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includegraphics[height=0.5cm]lsf.eps
Repartition effort coefficient
τ1 = weighted mean of the salaries on the salary trajectory / weightedmean of the pensions in the repartition model on the pension trajectory =2.7 on average.
That means that the active worker have to earn 2.7 euros to pay 1 euro ofpension by repartition. pause
τ1 =
k(1+d)T+1 PT+1 + ...+ k
(1+d)T+T∗ PT+T∗
S0 + ...+ ST
(1+d)T
.
τ1 depends on the demographic rate d . In fact, if d ∼ U(0%; 5%), τ1varies between 6.7 euros and 1.6 euros.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 122 / 125
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includegraphics[height=0.5cm]lsf.eps
Systemic risk
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includegraphics[height=0.5cm]lsf.eps
Global effort coefficient
τ = xτ1 + (1− x)τ2
is the number of euros necessary to pay 1 euro for the pension.
Here x euros come from repartition and 1− x euros from capitalization.
We want to limit the risk of the hybrid system without reducing thepension and in the same time minimize the capitalization effort.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 124 / 125
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includegraphics[height=0.5cm]lsf.eps
Global effort coefficient
τ = xτ1 + (1− x)τ2
is the number of euros necessary to pay 1 euro for the pension.
Here x euros come from repartition and 1− x euros from capitalization.
We want to limit the risk of the hybrid system without reducing thepension and in the same time minimize the capitalization effort.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 124 / 125
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includegraphics[height=0.5cm]lsf.eps
Aim and solution
Aim : volatility of τ = (volatility of τ1)/m.
Solution:
x =1
m2.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 125 / 125
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includegraphics[height=0.5cm]lsf.eps
Aim and solution
Aim : volatility of τ = (volatility of τ1)/m.
Solution:
x =1
m2.
Jang SCHILTZ (University of Luxembourg) joint work with Jean-Daniel GUIGOU (University of Luxembourg) & Bruno LOVAT (University Nancy II) ()Analysis of the salary trajectories in Luxembourg January 19, 2010 125 / 125