Estimating Dynamic Models and their use for evaluations
Costas Meghir
February 2009
Costas Meghir (UCL)
Dynamic Models
February 2009
1 / 31
Dynamic Models and Policy Evaluation
In many economic settings, policy reform can have as much impact
on current actions as on future ones
Costas Meghir (UCL)
Dynamic Models
February 2009
2 / 31
Dynamic Models and Policy Evaluation
In many economic settings, policy reform can have as much impact
on current actions as on future ones
Think of taxation, where taxes can a¤ect the incentive to work and
the incentive to save
Costas Meghir (UCL)
Dynamic Models
February 2009
2 / 31
Dynamic Models and Policy Evaluation
In many economic settings, policy reform can have as much impact
on current actions as on future ones
Think of taxation, where taxes can a¤ect the incentive to work and
the incentive to save
Think of minimum wages or work subsidies: they can a¤ect work
incentive and savings of those currently working and educational
choices of those still in education
Costas Meghir (UCL)
Dynamic Models
February 2009
2 / 31
Dynamic Models and Policy Evaluation
In many economic settings, policy reform can have as much impact
on current actions as on future ones
Think of taxation, where taxes can a¤ect the incentive to work and
the incentive to save
Think of minimum wages or work subsidies: they can a¤ect work
incentive and savings of those currently working and educational
choices of those still in education
Think of public pensions, which can crowd out private savings
Costas Meghir (UCL)
Dynamic Models
February 2009
2 / 31
Dynamic Models and Policy Evaluation
In many of the cases above, both standard treatment/control
evaluations and models may have something to say.
Costas Meghir (UCL)
Dynamic Models
February 2009
3 / 31
Dynamic Models and Policy Evaluation
In many of the cases above, both standard treatment/control
evaluations and models may have something to say.
For example we can use as outcome variables both labour supply and
savings when we consider tax reform.
Costas Meghir (UCL)
Dynamic Models
February 2009
3 / 31
Dynamic Models and Policy Evaluation
In many of the cases above, both standard treatment/control
evaluations and models may have something to say.
For example we can use as outcome variables both labour supply and
savings when we consider tax reform.
However, in many cases the standard treatment/control evaluations
have little or nothing to say about key longer term e¤ects of policy.
Costas Meghir (UCL)
Dynamic Models
February 2009
3 / 31
Dynamic Models and Policy Evaluation
In many of the cases above, both standard treatment/control
evaluations and models may have something to say.
For example we can use as outcome variables both labour supply and
savings when we consider tax reform.
However, in many cases the standard treatment/control evaluations
have little or nothing to say about key longer term e¤ects of policy.
This point goes beyond the issue of external validity we discussed in
lecture 7.
Costas Meghir (UCL)
Dynamic Models
February 2009
3 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
They are supposed to improve the incentive to work. However they
can have e¤ects on human capital accumulation through many
di¤erent channels:
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
They are supposed to improve the incentive to work. However they
can have e¤ects on human capital accumulation through many
di¤erent channels:
1
If they encourage work and experience is non-rivalrous to work then
they also encourage human capital accumulation
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
They are supposed to improve the incentive to work. However they
can have e¤ects on human capital accumulation through many
di¤erent channels:
1
2
If they encourage work and experience is non-rivalrous to work then
they also encourage human capital accumulation
If however, on-the-job training requires a reduction in work the
incentive to train will be reduced
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
They are supposed to improve the incentive to work. However they
can have e¤ects on human capital accumulation through many
di¤erent channels:
1
2
3
If they encourage work and experience is non-rivalrous to work then
they also encourage human capital accumulation
If however, on-the-job training requires a reduction in work the
incentive to train will be reduced
There is also an intertemporal incentive problem: Individuals planning
pre-labour market education, may be discouraged from more education
as a result of compressed returns.
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
They are supposed to improve the incentive to work. However they
can have e¤ects on human capital accumulation through many
di¤erent channels:
1
2
3
If they encourage work and experience is non-rivalrous to work then
they also encourage human capital accumulation
If however, on-the-job training requires a reduction in work the
incentive to train will be reduced
There is also an intertemporal incentive problem: Individuals planning
pre-labour market education, may be discouraged from more education
as a result of compressed returns.
To understand such e¤ects we need to specify and estimate dynamic
models.
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models and Policy Evaluation
One of the best examples are the e¤ects of tax credits
They are supposed to improve the incentive to work. However they
can have e¤ects on human capital accumulation through many
di¤erent channels:
1
2
3
If they encourage work and experience is non-rivalrous to work then
they also encourage human capital accumulation
If however, on-the-job training requires a reduction in work the
incentive to train will be reduced
There is also an intertemporal incentive problem: Individuals planning
pre-labour market education, may be discouraged from more education
as a result of compressed returns.
To understand such e¤ects we need to specify and estimate dynamic
models.
These come in various shapes and forms.
Costas Meghir (UCL)
Dynamic Models
February 2009
4 / 31
Dynamic Models - Setting the scene
We will present …rst a very simple model inspired by one of the …rst in
its kind - Eckstein and Wolpin (1989)
Costas Meghir (UCL)
Dynamic Models
February 2009
5 / 31
Dynamic Models - Setting the scene
We will present …rst a very simple model inspired by one of the …rst in
its kind - Eckstein and Wolpin (1989)
Our aim is to use this simple model to illustrate some key points. We
will then present a richer model to provide an idea of how more
complex models are built.
Costas Meghir (UCL)
Dynamic Models
February 2009
5 / 31
Dynamic Models - Setting the scene
We will present …rst a very simple model inspired by one of the …rst in
its kind - Eckstein and Wolpin (1989)
Our aim is to use this simple model to illustrate some key points. We
will then present a richer model to provide an idea of how more
complex models are built.
I will present a simpler model than Eckstein and Wolpin: they also
include the fertility decision.
Costas Meghir (UCL)
Dynamic Models
February 2009
5 / 31
Dynamic Models - Setting the scene
We will present …rst a very simple model inspired by one of the …rst in
its kind - Eckstein and Wolpin (1989)
Our aim is to use this simple model to illustrate some key points. We
will then present a richer model to provide an idea of how more
complex models are built.
I will present a simpler model than Eckstein and Wolpin: they also
include the fertility decision.
This is particularly interesting because it shows that policy can have
many di¤erent e¤ects on many di¤erent margins.
Costas Meghir (UCL)
Dynamic Models
February 2009
5 / 31
Dynamic Models - Setting the scene
We will present …rst a very simple model inspired by one of the …rst in
its kind - Eckstein and Wolpin (1989)
Our aim is to use this simple model to illustrate some key points. We
will then present a richer model to provide an idea of how more
complex models are built.
I will present a simpler model than Eckstein and Wolpin: they also
include the fertility decision.
This is particularly interesting because it shows that policy can have
many di¤erent e¤ects on many di¤erent margins.
However, here I am presenting a simple model building exercise.
Costas Meghir (UCL)
Dynamic Models
February 2009
5 / 31
A Simple Dynamic Model of Female Labour Supply and
Human Capital Accumulation
Individuals have a utility of work
UitP = (a0 + a1 edi ) Wit + Fi
Costas Meghir (UCL)
Dynamic Models
εit
February 2009
6 / 31
A Simple Dynamic Model of Female Labour Supply and
Human Capital Accumulation
Individuals have a utility of work
UitP = (a0 + a1 edi ) Wit + Fi
εit
In the above Wit is the wage, Fi is a …xed cost of working (P = 1)
and εit is a random preference shock. In our simple model we will
take this as independently and identically distributed. The role of this
is to explain why there is no perfect persistence from period to period
and why seemingly identical individuals take di¤erent decisions.
Costas Meghir (UCL)
Dynamic Models
February 2009
6 / 31
A Simple Dynamic Model of Female Labour Supply and
Human Capital Accumulation
Individuals have a utility of work
UitP = (a0 + a1 edi ) Wit + Fi
εit
In the above Wit is the wage, Fi is a …xed cost of working (P = 1)
and εit is a random preference shock. In our simple model we will
take this as independently and identically distributed. The role of this
is to explain why there is no perfect persistence from period to period
and why seemingly identical individuals take di¤erent decisions.
The marginal utility of income depends on education.
Costas Meghir (UCL)
Dynamic Models
February 2009
6 / 31
A Simple Dynamic Model of Female Labour Supply and
Human Capital Accumulation
Individuals have a utility of work
UitP = (a0 + a1 edi ) Wit + Fi
εit
In the above Wit is the wage, Fi is a …xed cost of working (P = 1)
and εit is a random preference shock. In our simple model we will
take this as independently and identically distributed. The role of this
is to explain why there is no perfect persistence from period to period
and why seemingly identical individuals take di¤erent decisions.
The marginal utility of income depends on education.
Non-work time (home production, leisure) is valued as
UitNP = λedi
Note that we can only identify the di¤erence in the utilities not the
absolute level.
Costas Meghir (UCL)
Dynamic Models
February 2009
6 / 31
A Simple Dynamic Model of Female Labour Supply
Wages are determined by education (exogenous for now) and by
accumulated experience (this is the human capital accumulation
aspect). So we write
Wit = δ0 + δ1 edi + δ2 Xit + δ3 Xit2 + δ4 (Xit
edi ) + γZi + ζFi + vit
where Xit is the number of periods the person has worked up to this
point. This is the source of intertemporal non-separability. Otherwise
this would have been a completely static model.
Costas Meghir (UCL)
Dynamic Models
February 2009
7 / 31
A Simple Dynamic Model of Female Labour Supply
Wages are determined by education (exogenous for now) and by
accumulated experience (this is the human capital accumulation
aspect). So we write
Wit = δ0 + δ1 edi + δ2 Xit + δ3 Xit2 + δ4 (Xit
edi ) + γZi + ζFi + vit
where Xit is the number of periods the person has worked up to this
point. This is the source of intertemporal non-separability. Otherwise
this would have been a completely static model.
The variable Zi enters wages but not preferences and will be the
source of identi…cation. What this is, is important and needs to be
discussed separately. We are going to take it as …xed over time for
simplicity.
Costas Meghir (UCL)
Dynamic Models
February 2009
7 / 31
A Simple Dynamic Model of Female Labour Supply
Wages are determined by education (exogenous for now) and by
accumulated experience (this is the human capital accumulation
aspect). So we write
Wit = δ0 + δ1 edi + δ2 Xit + δ3 Xit2 + δ4 (Xit
edi ) + γZi + ζFi + vit
where Xit is the number of periods the person has worked up to this
point. This is the source of intertemporal non-separability. Otherwise
this would have been a completely static model.
The variable Zi enters wages but not preferences and will be the
source of identi…cation. What this is, is important and needs to be
discussed separately. We are going to take it as …xed over time for
simplicity.
Finally, note two important features:
Costas Meghir (UCL)
Dynamic Models
February 2009
7 / 31
A Simple Dynamic Model of Female Labour Supply
Wages are determined by education (exogenous for now) and by
accumulated experience (this is the human capital accumulation
aspect). So we write
Wit = δ0 + δ1 edi + δ2 Xit + δ3 Xit2 + δ4 (Xit
edi ) + γZi + ζFi + vit
where Xit is the number of periods the person has worked up to this
point. This is the source of intertemporal non-separability. Otherwise
this would have been a completely static model.
The variable Zi enters wages but not preferences and will be the
source of identi…cation. What this is, is important and needs to be
discussed separately. We are going to take it as …xed over time for
simplicity.
Finally, note two important features:
1
The unobserved factor Fi a¤ects wages. This makes wages endogenous
for labour supply
Costas Meghir (UCL)
Dynamic Models
February 2009
7 / 31
A Simple Dynamic Model of Female Labour Supply
Wages are determined by education (exogenous for now) and by
accumulated experience (this is the human capital accumulation
aspect). So we write
Wit = δ0 + δ1 edi + δ2 Xit + δ3 Xit2 + δ4 (Xit
edi ) + γZi + ζFi + vit
where Xit is the number of periods the person has worked up to this
point. This is the source of intertemporal non-separability. Otherwise
this would have been a completely static model.
The variable Zi enters wages but not preferences and will be the
source of identi…cation. What this is, is important and needs to be
discussed separately. We are going to take it as …xed over time for
simplicity.
Finally, note two important features:
1
2
The unobserved factor Fi a¤ects wages. This makes wages endogenous
for labour supply
Xit is the accumulation of past labour supply decisions, also
endogenous because of the persistent unobservable F
Costas Meghir (UCL)
Dynamic Models
February 2009
7 / 31
A Simple Dynamic Model of Female Labour Supply
We now need to state the objective of the individual. In each period
t, the individual maximises expected utility
T
V0t = max
∑ βτ
P t ,...,P T τ =t
t
h
Pi τ UiPτ + (1
Pi τ ) UiNP
τ
i
subject to the law of motion
Xit = Xit
Costas Meghir (UCL)
1
+ Pit
Dynamic Models
1
February 2009
8 / 31
A Simple Dynamic Model of Female Labour Supply
We now need to state the objective of the individual. In each period
t, the individual maximises expected utility
T
V0t = max
∑ βτ
P t ,...,P T τ =t
t
h
Pi τ UiPτ + (1
Pi τ ) UiNP
τ
i
subject to the law of motion
Xit = Xit
The discount factor β =
Costas Meghir (UCL)
1
1 +ρ
1
+ Pit
1
ρ being the rate of time preference
Dynamic Models
February 2009
8 / 31
A Simple Dynamic Model of Female Labour Supply
We now need to state the objective of the individual. In each period
t, the individual maximises expected utility
T
V0t = max
∑ βτ
P t ,...,P T τ =t
t
h
Pi τ UiPτ + (1
Pi τ ) UiNP
τ
i
subject to the law of motion
Xit = Xit
The discount factor β =
1
1 +ρ
1
+ Pit
1
ρ being the rate of time preference
However we still need to close the model by saying what happens at
the end. With utility that is linear income we can say that T is the
retirement date.
Costas Meghir (UCL)
Dynamic Models
February 2009
8 / 31
Based on the Bellman principle we can write the value function as
n
o
Vt (Xt ) = max Pi τ UiPτ + (1 Pi τ ) UiNP
+
βEV
(
X
+
P
)
t +1
t
t
τ
Pt
where the expectation is taken over the shocks to wages vit and the
shocks to preferences εit .
Costas Meghir (UCL)
Dynamic Models
February 2009
9 / 31
Based on the Bellman principle we can write the value function as
n
o
Vt (Xt ) = max Pi τ UiPτ + (1 Pi τ ) UiNP
+
βEV
(
X
+
P
)
t +1
t
t
τ
Pt
where the expectation is taken over the shocks to wages vit and the
shocks to preferences εit .
We also specify the terminal condition as VT = 0. There are other
ways of doing this.
Costas Meghir (UCL)
Dynamic Models
February 2009
9 / 31
Based on the Bellman principle we can write the value function as
n
o
Vt (Xt ) = max Pi τ UiPτ + (1 Pi τ ) UiNP
+
βEV
(
X
+
P
)
t +1
t
t
τ
Pt
where the expectation is taken over the shocks to wages vit and the
shocks to preferences εit .
We also specify the terminal condition as VT = 0. There are other
ways of doing this.
Finally, we need to specify how the unobservables are distributed. So
lets say that vit and εit are each independent Normal with variance σ2v ,
σ2ε . We will also need to deal with the unobserved …xed cost of work.
Costas Meghir (UCL)
Dynamic Models
February 2009
9 / 31
Based on the Bellman principle we can write the value function as
n
o
Vt (Xt ) = max Pi τ UiPτ + (1 Pi τ ) UiNP
+
βEV
(
X
+
P
)
t +1
t
t
τ
Pt
where the expectation is taken over the shocks to wages vit and the
shocks to preferences εit .
We also specify the terminal condition as VT = 0. There are other
ways of doing this.
Finally, we need to specify how the unobservables are distributed. So
lets say that vit and εit are each independent Normal with variance σ2v ,
σ2ε . We will also need to deal with the unobserved …xed cost of work.
The variables that need to be remembered from the past constitute
what is known as the State space. Here the State space is Xit , edi ,
Zit and Fi . Once these are known no other variables need to be
remembered
Costas Meghir (UCL)
Dynamic Models
February 2009
9 / 31
A Simple Dynamic Model of Female Labour Supply
It is also possible to de…ne the value of work and the value of
unemployment to use these as the basis of comparison for
constructing a probability of work.
Costas Meghir (UCL)
Dynamic Models
February 2009
10 / 31
A Simple Dynamic Model of Female Labour Supply
It is also possible to de…ne the value of work and the value of
unemployment to use these as the basis of comparison for
constructing a probability of work.
Given our speci…cation these are
V P = (a0 + a1 edi ) Wit + Fi εit + βEVt +1 (Xt + Pt )
Work
V NP = λedi + βEVt +1 (Xt )
Unemployment
Work if V P > V NP
Costas Meghir (UCL)
Dynamic Models
February 2009
10 / 31
A Simple Dynamic Model of Female Labour Supply
It is also possible to de…ne the value of work and the value of
unemployment to use these as the basis of comparison for
constructing a probability of work.
Given our speci…cation these are
V P = (a0 + a1 edi ) Wit + Fi εit + βEVt +1 (Xt + Pt )
Work
V NP = λedi + βEVt +1 (Xt )
Unemployment
Work if V P > V NP
This would be a standard probit model if it were not for the following
complications:
Costas Meghir (UCL)
Dynamic Models
February 2009
10 / 31
A Simple Dynamic Model of Female Labour Supply
It is also possible to de…ne the value of work and the value of
unemployment to use these as the basis of comparison for
constructing a probability of work.
Given our speci…cation these are
V P = (a0 + a1 edi ) Wit + Fi εit + βEVt +1 (Xt + Pt )
Work
V NP = λedi + βEVt +1 (Xt )
Unemployment
Work if V P > V NP
This would be a standard probit model if it were not for the following
complications:
1
We do not know EVt +1 (Xt ). This will be obtained by solving the
model backwards from the terminal point
Costas Meghir (UCL)
Dynamic Models
February 2009
10 / 31
A Simple Dynamic Model of Female Labour Supply
It is also possible to de…ne the value of work and the value of
unemployment to use these as the basis of comparison for
constructing a probability of work.
Given our speci…cation these are
V P = (a0 + a1 edi ) Wit + Fi εit + βEVt +1 (Xt + Pt )
Work
V NP = λedi + βEVt +1 (Xt )
Unemployment
Work if V P > V NP
This would be a standard probit model if it were not for the following
complications:
1
2
We do not know EVt +1 (Xt ). This will be obtained by solving the
model backwards from the terminal point
We do not know wages for non workers. They have to be integrated out
Costas Meghir (UCL)
Dynamic Models
February 2009
10 / 31
A Simple Dynamic Model of Female Labour Supply
It is also possible to de…ne the value of work and the value of
unemployment to use these as the basis of comparison for
constructing a probability of work.
Given our speci…cation these are
V P = (a0 + a1 edi ) Wit + Fi εit + βEVt +1 (Xt + Pt )
Work
V NP = λedi + βEVt +1 (Xt )
Unemployment
Work if V P > V NP
This would be a standard probit model if it were not for the following
complications:
1
2
3
We do not know EVt +1 (Xt ). This will be obtained by solving the
model backwards from the terminal point
We do not know wages for non workers. They have to be integrated out
Fi is unobserved and needs to be integrated out
Costas Meghir (UCL)
Dynamic Models
February 2009
10 / 31
A Simple Dynamic Model of Female Labour Supply
Before we proceed note the contrast with the treatment e¤ects models
Costas Meghir (UCL)
Dynamic Models
February 2009
11 / 31
A Simple Dynamic Model of Female Labour Supply
Before we proceed note the contrast with the treatment e¤ects models
Despite the fact the model is really conceptually simple we have had
to specify everything relevant in the environment
Costas Meghir (UCL)
Dynamic Models
February 2009
11 / 31
A Simple Dynamic Model of Female Labour Supply
Before we proceed note the contrast with the treatment e¤ects models
Despite the fact the model is really conceptually simple we have had
to specify everything relevant in the environment
The great attraction of treatment e¤ect models is that they seem to
avoid to say much about the peripheral aspects of the model
Costas Meghir (UCL)
Dynamic Models
February 2009
11 / 31
A Simple Dynamic Model of Female Labour Supply
Before we proceed note the contrast with the treatment e¤ects models
Despite the fact the model is really conceptually simple we have had
to specify everything relevant in the environment
The great attraction of treatment e¤ect models is that they seem to
avoid to say much about the peripheral aspects of the model
This comes at the cost of not being able to have a framework to
think of longer term e¤ects of policy and not being able to deal with
external validity.
Costas Meghir (UCL)
Dynamic Models
February 2009
11 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Now lets see how these models are solved and estimated.
Costas Meghir (UCL)
Dynamic Models
February 2009
12 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Now lets see how these models are solved and estimated.
Remembering that εit is iid normal, the probability of work given
wages is
Pr(work jFi , Wi , edi , Xit ) = Pr(V P V NP > 0)
Φ ((a0 + a1 edi ) Wit + Fi λedi + β [EVt +1 (Xt + Pt ) EVt +1 (Xt )])
where Φ is the standard normal probability distribution.
Costas Meghir (UCL)
Dynamic Models
February 2009
12 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
The likelihood contribution for someone working and earning a wage
rate W with experience X is
LPi (Xit , Fi ) = Pr(work jFi , Wi , edi , Xit )
φ
Costas Meghir (UCL)
W it
(δ0 +δ1 edi +δ2 Xit +δ3 Xit2 +δ4 (Xit
ed i )+γZ i +ζF i )
σv
Dynamic Models
February 2009
13 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
If someone is not working in a period the wage is not observed and it
needs to be integrated out. The likelihood contribution in this case is
R
LNP
Pr(work jFi , Wi , edi , Xit )]
i ( X t , Fi ) = W [ 1
φ
W it
Costas Meghir (UCL)
(δ0 +δ1 edi +δ2 Xit +δ3 Xit2 +δ4 (Xit
σv
Dynamic Models
ed i )+γZ it +ζF i )
dW
February 2009
14 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Our …nal statistical di¢ culty is that Fi is not observed; moreover its
distribution will depend on Xit and Wit
Costas Meghir (UCL)
Dynamic Models
February 2009
15 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Our …nal statistical di¢ culty is that Fi is not observed; moreover its
distribution will depend on Xit and Wit
To deal with this we need to unwind this dependence by modelling
the complete sequence of work non-work spells from the start of one’s
career.
Costas Meghir (UCL)
Dynamic Models
February 2009
15 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Our …nal statistical di¢ culty is that Fi is not observed; moreover its
distribution will depend on Xit and Wit
To deal with this we need to unwind this dependence by modelling
the complete sequence of work non-work spells from the start of one’s
career.
Suppose we observe someone for 10 years. Then for them the
likelihood contribution will be ( note Pit = 1 or 0)
Ti
Li (Fi ) =
∏ Li (Xit , Fi )P
it
Li (Xit , Fi )(1
P it )
t =1
where Ti is the number of periods where we observe individual i from
the start of her career on and Xi 1 = 0.
Costas Meghir (UCL)
Dynamic Models
February 2009
15 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Now we need to integrate out the unobservable Fi . we assume this is
an independent random variable with a discrete distribution with
some number of points of support. How many can be determined
empirically. Say three
Costas Meghir (UCL)
Dynamic Models
February 2009
16 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Now we need to integrate out the unobservable Fi . we assume this is
an independent random variable with a discrete distribution with
some number of points of support. How many can be determined
empirically. Say three
Thus Fi can take one of three values f1 , f2 and f3 with respective
probabilities p1 , p2 and p3 = (1 p1 p2 ) respectively. These are
additional unknown parameters that need to be estimated.
Costas Meghir (UCL)
Dynamic Models
February 2009
16 / 31
A Simple Dynamic Model of Female Labour Supply
Solution and Estimation
Now we need to integrate out the unobservable Fi . we assume this is
an independent random variable with a discrete distribution with
some number of points of support. How many can be determined
empirically. Say three
Thus Fi can take one of three values f1 , f2 and f3 with respective
probabilities p1 , p2 and p3 = (1 p1 p2 ) respectively. These are
additional unknown parameters that need to be estimated.
Thus …nally the likelihood function contribution for an individual i is
3
Li =
∑ ps Li (Fi = fs )
s =1
Costas Meghir (UCL)
Dynamic Models
February 2009
16 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
All the above depends on the unknown function EV (X ) which
re‡ects future expectations. So now we need to construct it.
Costas Meghir (UCL)
Dynamic Models
February 2009
17 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
All the above depends on the unknown function EV (X ) which
re‡ects future expectations. So now we need to construct it.
First note how a likelihood is maximised by the computer: A set of
parameters is guessed (the initial values) and then derivatives of the
likelihood with respect to the parameters is computed to …nd the
direction in which to improve the parameters.
Costas Meghir (UCL)
Dynamic Models
February 2009
17 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
All the above depends on the unknown function EV (X ) which
re‡ects future expectations. So now we need to construct it.
First note how a likelihood is maximised by the computer: A set of
parameters is guessed (the initial values) and then derivatives of the
likelihood with respect to the parameters is computed to …nd the
direction in which to improve the parameters.
So we can start by taking these parameters at any iteration as given.
Here is what we do
Costas Meghir (UCL)
Dynamic Models
February 2009
17 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
All the above depends on the unknown function EV (X ) which
re‡ects future expectations. So now we need to construct it.
First note how a likelihood is maximised by the computer: A set of
parameters is guessed (the initial values) and then derivatives of the
likelihood with respect to the parameters is computed to …nd the
direction in which to improve the parameters.
So we can start by taking these parameters at any iteration as given.
Here is what we do
Simplify the notation by writing Wit = W (Xit , Zi ) + vit
Costas Meghir (UCL)
Dynamic Models
February 2009
17 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Say we assume (part of the model) that the total time in the labour
market is 40 years at which point we have that V41 = 0.
Costas Meghir (UCL)
Dynamic Models
February 2009
18 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Say we assume (part of the model) that the total time in the labour
market is 40 years at which point we have that V41 = 0.
P = a + a ed W (X , Z ) + F
If this is the case V40
( 0
1 i)
it
i
i
NP
V
= λedi .
Costas Meghir (UCL)
Dynamic Models
εit + vit and
February 2009
18 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Say we assume (part of the model) that the total time in the labour
market is 40 years at which point we have that V41 = 0.
P = a + a ed W (X , Z ) + F
If this is the case V40
( 0
1 i)
it
i
i
NP
V
= λedi .
εit + vit and
Say education can take two values. Experience can take a total of 39
di¤erent values, F three values and say Z 10 values. So the size of
the state space is 39 2 3 10 = 2340.
Costas Meghir (UCL)
Dynamic Models
February 2009
18 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Say we assume (part of the model) that the total time in the labour
market is 40 years at which point we have that V41 = 0.
P = a + a ed W (X , Z ) + F
If this is the case V40
( 0
1 i)
it
i
i
NP
V
= λedi .
εit + vit and
Say education can take two values. Experience can take a total of 39
di¤erent values, F three values and say Z 10 values. So the size of
the state space is 39 2 3 10 = 2340.
The size of the state space can become huge very fast - this is known
as the curse of dimensionality because we will have to compute the
value functions for the entire state space,making some problems
computationally intractable.
Costas Meghir (UCL)
Dynamic Models
February 2009
18 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Say we assume (part of the model) that the total time in the labour
market is 40 years at which point we have that V41 = 0.
P = a + a ed W (X , Z ) + F
If this is the case V40
( 0
1 i)
it
i
i
NP
V
= λedi .
εit + vit and
Say education can take two values. Experience can take a total of 39
di¤erent values, F three values and say Z 10 values. So the size of
the state space is 39 2 3 10 = 2340.
The size of the state space can become huge very fast - this is known
as the curse of dimensionality because we will have to compute the
value functions for the entire state space,making some problems
computationally intractable.
Costas Meghir (UCL)
Dynamic Models
February 2009
18 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
So now we need to compute the expected value from the point of
view of the previous period (39), allowing for the fact that in period
40 an optimal decision will be taken
P , V NP
EV40 (Xi 40 ) = E max V40
=
(Probability Work)X(Utilitygiven she works)
+(Probability NOT Work)X(Utility given she does not work)
Costas Meghir (UCL)
Dynamic Models
February 2009
19 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Formally: Whether the individual with experience X will work or not
will be determined by whether
εit
Costas Meghir (UCL)
vit < (a0 + a1 edi ) W (Xi 39 , Zi ) + Fi
Dynamic Models
λedi
February 2009
20 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Formally: Whether the individual with experience X will work or not
will be determined by whether
εit
vit < (a0 + a1 edi ) W (Xi 39 , Zi ) + Fi
λedi
Given this what is the expected value function in period 39?
Costas Meghir (UCL)
Dynamic Models
February 2009
20 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Formally: Whether the individual with experience X will work or not
will be determined by whether
εit
vit < (a0 + a1 edi ) W (Xi 39 , Zi ) + Fi
λedi
Given this what is the expected value function in period 39?
Denote for shorthand U40 = (a0 + a1 edi ) W (Xi 39 , Zi ) + Fi λedi .
This is given by
P , V NP
EV40 (Xi 40 ) = E max V40
=
Pr (εi 40 vi 40 < U40 )
[(a0 + a1 edi ) W (Xi 40 , Zi ) + Fi E (εi 40 vi 40 jεi 40
(1
Costas Meghir (UCL)
Pr (εi 40
vi 40 < U40 )] +
vi 40 < U40 )) λedi
Dynamic Models
February 2009
20 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Given the parameters this needs to be computed at all 2340
combinations of values above
Costas Meghir (UCL)
Dynamic Models
February 2009
21 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Given the parameters this needs to be computed at all 2340
combinations of values above
Now lets go back one more period and see how we compute V39. given
that we know what V40 is at all possible values of the state variables.
Costas Meghir (UCL)
Dynamic Models
February 2009
21 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Given the parameters this needs to be computed at all 2340
combinations of values above
Now lets go back one more period and see how we compute V39. given
that we know what V40 is at all possible values of the state variables.
We know that
P = a + a ed W (X , Z ) + F
V39
( 0
1 i)
i 39
i
i
εit + vit + βEV40 (Xi 39 + 1)
NP = λed + βEV (X )
V39
40
i
i 39
Costas Meghir (UCL)
Dynamic Models
February 2009
21 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Given the parameters this needs to be computed at all 2340
combinations of values above
Now lets go back one more period and see how we compute V39. given
that we know what V40 is at all possible values of the state variables.
We know that
P = a + a ed W (X , Z ) + F
V39
( 0
1 i)
i 39
i
i
εit + vit + βEV40 (Xi 39 + 1)
NP = λed + βEV (X )
V39
40
i
i 39
Experience now will take up to 38 possible values
Costas Meghir (UCL)
Dynamic Models
February 2009
21 / 31
A Simple Dynamic Model of Female Labour Supply
Solving the model
Now we apply the same idea as before
P , V NP
EV39 (Xi 39 ) = E max V39
=
vi 39 < U39 + βEV40 (Xi 39 + 1))
U39 + βEV40 (Xi 39 + 1)
vi 40 jεi 40 vi 40 < U39 + βEV40 (Xi 39 + 1))
Pr (εi 39
E (εi 40
(1
Costas Meghir (UCL)
Pr (εi 39
+
vi 39 < U39 + βEV40 (Xi 39 + 1))) λedi
Dynamic Models
February 2009
22 / 31
The likelihood and the END
By continuing to work backwards we will construct all possible values
of EVt everywhere on the state space, for the current values of the
parameters.
Costas Meghir (UCL)
Dynamic Models
February 2009
23 / 31
The likelihood and the END
By continuing to work backwards we will construct all possible values
of EVt everywhere on the state space, for the current values of the
parameters.
This is the missing piece for the likelihood function.
Costas Meghir (UCL)
Dynamic Models
February 2009
23 / 31
The likelihood and the END
By continuing to work backwards we will construct all possible values
of EVt everywhere on the state space, for the current values of the
parameters.
This is the missing piece for the likelihood function.
These computations need to be repeated at every iteration when the
parameters change.
Costas Meghir (UCL)
Dynamic Models
February 2009
23 / 31
The likelihood and the END
By continuing to work backwards we will construct all possible values
of EVt everywhere on the state space, for the current values of the
parameters.
This is the missing piece for the likelihood function.
These computations need to be repeated at every iteration when the
parameters change.
The overall sample likelihood is
N
L=
∏ Li
i =1
Costas Meghir (UCL)
Dynamic Models
February 2009
23 / 31
What sort of results?
Why all this trouble?
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
What sort of results?
Why all this trouble?
Suppose one wishes to simulate the e¤ects of a wage subsidy.
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
What sort of results?
Why all this trouble?
Suppose one wishes to simulate the e¤ects of a wage subsidy.
A standard treatment e¤ects model will measure the immediate
e¤ects.
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
What sort of results?
Why all this trouble?
Suppose one wishes to simulate the e¤ects of a wage subsidy.
A standard treatment e¤ects model will measure the immediate
e¤ects.
This sort of model is designed to measure the overall e¤ect.
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
What sort of results?
Why all this trouble?
Suppose one wishes to simulate the e¤ects of a wage subsidy.
A standard treatment e¤ects model will measure the immediate
e¤ects.
This sort of model is designed to measure the overall e¤ect.
For example a programme making work more attractive, will lead to
more people working.
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
What sort of results?
Why all this trouble?
Suppose one wishes to simulate the e¤ects of a wage subsidy.
A standard treatment e¤ects model will measure the immediate
e¤ects.
This sort of model is designed to measure the overall e¤ect.
For example a programme making work more attractive, will lead to
more people working.
They will accumulate more human capital and will become as a result
more attached to the labour market
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
What sort of results?
Why all this trouble?
Suppose one wishes to simulate the e¤ects of a wage subsidy.
A standard treatment e¤ects model will measure the immediate
e¤ects.
This sort of model is designed to measure the overall e¤ect.
For example a programme making work more attractive, will lead to
more people working.
They will accumulate more human capital and will become as a result
more attached to the labour market
We are thus able to estimate how many individuals will take up work
say with a temporary as opposed to a permanent subsidy and what
will be the long term e¤ects.
Costas Meghir (UCL)
Dynamic Models
February 2009
24 / 31
The Career Decisions of Young Men by Keane and Wolpin
This is a model of Schooling, work and occupational choice
Costas Meghir (UCL)
Dynamic Models
February 2009
25 / 31
The Career Decisions of Young Men by Keane and Wolpin
This is a model of Schooling, work and occupational choice
As such it is well suited to consider the dynamic e¤ects of policy
Costas Meghir (UCL)
Dynamic Models
February 2009
25 / 31
The Career Decisions of Young Men by Keane and Wolpin
This is a model of Schooling, work and occupational choice
As such it is well suited to consider the dynamic e¤ects of policy
It is estimated on 11 years of data from the NLSY 1979 cohort, which
follows young people from age 16 to 26.
Costas Meghir (UCL)
Dynamic Models
February 2009
25 / 31
The Career Decisions of Young Men by Keane and Wolpin
This is a model of Schooling, work and occupational choice
As such it is well suited to consider the dynamic e¤ects of policy
It is estimated on 11 years of data from the NLSY 1979 cohort, which
follows young people from age 16 to 26.
The mutually exclusive choices that individuals face are:
Costas Meghir (UCL)
Dynamic Models
February 2009
25 / 31
The Career Decisions of Young Men by Keane and Wolpin
This is a model of Schooling, work and occupational choice
As such it is well suited to consider the dynamic e¤ects of policy
It is estimated on 11 years of data from the NLSY 1979 cohort, which
follows young people from age 16 to 26.
The mutually exclusive choices that individuals face are:
Attending School, Being a White Collar Worker, Being a Blue Collar
Worker, Working in the Military and Home production.
Costas Meghir (UCL)
Dynamic Models
February 2009
25 / 31
The Career Decisions of Young Men by Keane and Wolpin
In what sense is this a Dynamic Problem?
Costas Meghir (UCL)
Dynamic Models
February 2009
26 / 31
The Career Decisions of Young Men by Keane and Wolpin
In what sense is this a Dynamic Problem?
We call it dynamic because current actions a¤ect future outcomes and
Costas Meghir (UCL)
Dynamic Models
February 2009
26 / 31
The Career Decisions of Young Men by Keane and Wolpin
In what sense is this a Dynamic Problem?
We call it dynamic because current actions a¤ect future outcomes and
Expectations about the future environment can a¤ect current choices.
Costas Meghir (UCL)
Dynamic Models
February 2009
26 / 31
The Career Decisions of Young Men by Keane and Wolpin
In what sense is this a Dynamic Problem?
We call it dynamic because current actions a¤ect future outcomes and
Expectations about the future environment can a¤ect current choices.
One source of Dynamics here is experience: As one works in a
particular occupation one becomes more experienced and can earn
more, making it less likely that one would gain from switching.
Costas Meghir (UCL)
Dynamic Models
February 2009
26 / 31
The Career Decisions of Young Men by Keane and Wolpin
In what sense is this a Dynamic Problem?
We call it dynamic because current actions a¤ect future outcomes and
Expectations about the future environment can a¤ect current choices.
One source of Dynamics here is experience: As one works in a
particular occupation one becomes more experienced and can earn
more, making it less likely that one would gain from switching.
Education is also a source of dynamics: Education choices now have
future bene…ts, but current costs.
Costas Meghir (UCL)
Dynamic Models
February 2009
26 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The utility in each period is written as
5
R (a ) =
∑
Rm (a)dm (a)
m =1
where dm (a) = 1 if action m is chosen at age a.
Costas Meghir (UCL)
Dynamic Models
February 2009
27 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The utility in each period is written as
5
R (a ) =
∑
Rm (a)dm (a)
m =1
where dm (a) = 1 if action m is chosen at age a.
In general we can write wages in an occupation as rm em (a), where rm
is the price of human capital (HC) in occupation m and em (a) is HC.
Costas Meghir (UCL)
Dynamic Models
February 2009
27 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The utility in each period is written as
5
R (a ) =
∑
Rm (a)dm (a)
m =1
where dm (a) = 1 if action m is chosen at age a.
In general we can write wages in an occupation as rm em (a), where rm
is the price of human capital (HC) in occupation m and em (a) is HC.
For the three work alternatives K&W specify
em (a) = exp(em (16) + em1 g (a) + em2 xm (a)
em3 xm2 (a) + εm (a)
where em (16) is HC accumulated until 16, g (a) is accumulated
schooling in years up to age a, xm (a) is years of experience in
occupation m and εm (a) is a shock.
Costas Meghir (UCL)
Dynamic Models
February 2009
27 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The non work alternatives are schooling which has an e¤ort cost
e4 (16) and direct costs if it involves College (tc1 ) and Graduate
school (tc2 ).
Costas Meghir (UCL)
Dynamic Models
February 2009
28 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The non work alternatives are schooling which has an e¤ort cost
e4 (16) and direct costs if it involves College (tc1 ) and Graduate
school (tc2 ).
Thus we have
Rm (a) = wm (a)
rm em (a)
Work (m = 1,2,3)
R5 (a) = e4 (16) 1(g (a) 12) tc1
1(g (a) 16) tc2 + ε4 (a)
Education
R6 (a) = e5 (16) + ε5 (a)
Home
Costas Meghir (UCL)
Dynamic Models
February 2009
28 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
Now de…ne the State vector. This is the complete set of relevant
information at age a.
S (a) = fe(16),g (a), x(a), ε(a)g
where each of the elements is a vector. For example
x(a) = fx1 (a), x2 (a), x3 (a)g
Costas Meghir (UCL)
Dynamic Models
February 2009
29 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
Now de…ne the State vector. This is the complete set of relevant
information at age a.
S (a) = fe(16),g (a), x(a), ε(a)g
where each of the elements is a vector. For example
x(a) = fx1 (a), x2 (a), x3 (a)g
Thus all decisions depend on the value of the state vector S.
Costas Meghir (UCL)
Dynamic Models
February 2009
29 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
Now de…ne the State vector. This is the complete set of relevant
information at age a.
S (a) = fe(16),g (a), x(a), ε(a)g
where each of the elements is a vector. For example
x(a) = fx1 (a), x2 (a), x3 (a)g
Thus all decisions depend on the value of the state vector S.
Here one can see how the dynamics matter: If one works as a blue
collar worker his experience in that sector will grow, changing the
future value of the state vector.
Costas Meghir (UCL)
Dynamic Models
February 2009
29 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The individual at every age maximises his remaining expected lifetime
value given the State S (a) he …nds himself in, by choosing which of
the discrete mutually exclusive actions dm (a) he will undertake:
(
)
A
V (S (a), a) = max E
d m (a )
Costas Meghir (UCL)
∑
δτ
a
τ =a
Dynamic Models
5
∑
Rm (a)dm (a)jS (a)
m =1
February 2009
30 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
The individual at every age maximises his remaining expected lifetime
value given the State S (a) he …nds himself in, by choosing which of
the discrete mutually exclusive actions dm (a) he will undertake:
(
)
A
V (S (a), a) = max E
d m (a )
∑
δτ
a
τ =a
5
∑
Rm (a)dm (a)jS (a)
m =1
The expectation is taken over the future realisations of the shocks
ε(a) which here are the source of uncertainty.
Costas Meghir (UCL)
Dynamic Models
February 2009
30 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
To solve this sort of problem we can now think of the problem as
follows: What would be the lifetime value of a particular action in this
period (say going to work) assuming that all future actions will be
optimal
Costas Meghir (UCL)
Dynamic Models
February 2009
31 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
To solve this sort of problem we can now think of the problem as
follows: What would be the lifetime value of a particular action in this
period (say going to work) assuming that all future actions will be
optimal
Call this alternative speci…c value Vm (S (a), a)
Costas Meghir (UCL)
Dynamic Models
February 2009
31 / 31
The Career Decisions of Young Men by Keane and Wolpin
Model Speci…cation
To solve this sort of problem we can now think of the problem as
follows: What would be the lifetime value of a particular action in this
period (say going to work) assuming that all future actions will be
optimal
Call this alternative speci…c value Vm (S (a), a)
The optimisation problem now is
V (S (a), a) = max fVm (S (a), a)g
m
where
Vm (S (a), a) = Rm (S (a), a) + δE [V (S (a + 1), a + 1)jS (a), dm (a) =
Costas Meghir (UCL)
Dynamic Models
February 2009
31 / 31