Estimating and Testing a Quantile Regression Model with
Interactive Effects
Matthew Harding1 and Carlos Lamarche2
1
Stanford University 2 University of Oklahoma
California Econometrics Conference, Sept 24, 2010
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Limitation
They assume that latent heterogeneity has the classical
additively separable, time-invariant structure.
Estimating and Testing a Quantile Regression Model with Interactive Effects
2 / 50
Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Limitation
They assume that latent heterogeneity has the classical
additively separable, time-invariant structure.
Estimating and Testing a Quantile Regression Model with Interactive Effects
2 / 50
Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Limitation
They assume that latent heterogeneity has the classical
additively separable, time-invariant structure.
Estimating and Testing a Quantile Regression Model with Interactive Effects
2 / 50
Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Limitation
They assume that latent heterogeneity has the classical
additively separable, time-invariant structure.
Estimating and Testing a Quantile Regression Model with Interactive Effects
2 / 50
Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Limitation
The estimation of N nuisance parameters is computationally
demanding.
Estimating and Testing a Quantile Regression Model with Interactive Effects
3 / 50
Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Contribution
This paper offers a simple procedure that allows estimation of
distributional effects, under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Contribution
This paper offers a simple procedure that allows estimation of
distributional effects, under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
4 / 50
Motivation
Motivation
Classical least squares methods for panel data are often
inadequate for empirical analysis.
They deal with individual heterogeneity, but fail to estimate
effects other than the mean.
Koenker (2004), Lamarche (2010), Harding and Lamarche
(2009), Abrevaya and Dahl (2008) suggest approaches but their
use is limited under general conditions.
Contribution
This paper offers a simple procedure that allows estimation of
distributional effects, under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Background
In the last half a century, understanding the drivers of students’
academic performance has been a major focus in the economics
of education.
A number of studies have focused on class size and peer effects
(e.g., Coleman 1966, Krueger 1999, Hoxby 2000, Hanushek et
al. 2003).
The empirical evidence on the effect of class size and class
composition on achievement remains mixed.
The literature offers a number of studies on the mean effect, but
few studies investigate its distributional effect. One exception is
Ma and Koenker (2006).
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Background
In the last half a century, understanding the drivers of students’
academic performance has been a major focus in the economics
of education.
A number of studies have focused on class size and peer effects
(e.g., Coleman 1966, Krueger 1999, Hoxby 2000, Hanushek et
al. 2003).
The empirical evidence on the effect of class size and class
composition on achievement remains mixed.
The literature offers a number of studies on the mean effect, but
few studies investigate its distributional effect. One exception is
Ma and Koenker (2006).
Estimating and Testing a Quantile Regression Model with Interactive Effects
5 / 50
Background
In the last half a century, understanding the drivers of students’
academic performance has been a major focus in the economics
of education.
A number of studies have focused on class size and peer effects
(e.g., Coleman 1966, Krueger 1999, Hoxby 2000, Hanushek et
al. 2003).
The empirical evidence on the effect of class size and class
composition on achievement remains mixed.
The literature offers a number of studies on the mean effect, but
few studies investigate its distributional effect. One exception is
Ma and Koenker (2006).
Estimating and Testing a Quantile Regression Model with Interactive Effects
5 / 50
Background
In the last half a century, understanding the drivers of students’
academic performance has been a major focus in the economics
of education.
A number of studies have focused on class size and peer effects
(e.g., Coleman 1966, Krueger 1999, Hoxby 2000, Hanushek et
al. 2003).
The empirical evidence on the effect of class size and class
composition on achievement remains mixed.
The literature offers a number of studies on the mean effect, but
few studies investigate its distributional effect. One exception is
Ma and Koenker (2006).
Estimating and Testing a Quantile Regression Model with Interactive Effects
5 / 50
20
22
24
test scores
26
28
30
Background
−0.5
0.0
0.5
1.0
1.5
large class
Estimating and Testing a Quantile Regression Model with Interactive Effects
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test scores
0.10
0.05
35
30
25
20
0.00
f(y|x)
0.15
0.20
Background
−0.5
15
0.0
0.5
1.0
1.5
large class
Estimating and Testing a Quantile Regression Model with Interactive Effects
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test scores
26
28
30
Background
20
22
24
^
!1=−0.16
−0.5
0.0
0.5
1.0
1.5
large class
Estimating and Testing a Quantile Regression Model with Interactive Effects
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test scores
26
28
30
Background
22
24
^
!1=−0.16
20
^
!1(0.1)=−0.30
−0.5
0.0
0.5
1.0
1.5
large class
Estimating and Testing a Quantile Regression Model with Interactive Effects
9 / 50
30
Background
test scores
26
28
^
!1(0.9)=−0.08
22
24
^
!1=−0.16
20
^
!1(0.1)=−0.30
−0.5
0.0
0.5
1.0
1.5
large class
Estimating and Testing a Quantile Regression Model with Interactive Effects
10 / 50
Background
It is standard to consider (e.g., Hanushek et al. 2003):
yict = d 0ct α + x 0i β + λi + Fct + uict
The λi ’s are associated with motivation and ability, and the Fct ’s
measure teaching quality.
Note that we are imposing λi + Fct .
High teaching quality may have a modest effect on performance
among unmotivated students, while it may dramatically affect
strong, motivated students.
Can we estimate this model?
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Background
It is standard to consider (e.g., Hanushek et al. 2003):
yict = d 0ct α + x 0i β + λi + Fct + uict
The λi ’s are associated with motivation and ability, and the Fct ’s
measure teaching quality.
Note that we are imposing λi + Fct .
High teaching quality may have a modest effect on performance
among unmotivated students, while it may dramatically affect
strong, motivated students.
Can we estimate this model?
Estimating and Testing a Quantile Regression Model with Interactive Effects
11 / 50
Background
It is standard to consider (e.g., Hanushek et al. 2003):
yict = d 0ct α + x 0i β + λi + Fct + uict
The λi ’s are associated with motivation and ability, and the Fct ’s
measure teaching quality.
Note that we are imposing λi + Fct .
High teaching quality may have a modest effect on performance
among unmotivated students, while it may dramatically affect
strong, motivated students.
Can we estimate this model?
Estimating and Testing a Quantile Regression Model with Interactive Effects
11 / 50
Background
It is standard to consider (e.g., Hanushek et al. 2003):
yict = d 0ct α + x 0i β + λi + Fct + uict
The λi ’s are associated with motivation and ability, and the Fct ’s
measure teaching quality.
Note that we are imposing λi + Fct .
High teaching quality may have a modest effect on performance
among unmotivated students, while it may dramatically affect
strong, motivated students.
Can we estimate this model?
Estimating and Testing a Quantile Regression Model with Interactive Effects
11 / 50
Background
It is standard to consider (e.g., Hanushek et al. 2003):
yict = d 0ct α + x 0i β + λi + Fct + uict
The λi ’s are associated with motivation and ability, and the Fct ’s
measure teaching quality.
Note that we are imposing λi + Fct .
High teaching quality may have a modest effect on performance
among unmotivated students, while it may dramatically affect
strong, motivated students.
Can we estimate this model?
Estimating and Testing a Quantile Regression Model with Interactive Effects
11 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Outline
Estimating and Testing a Quantile Regression Model with Interactive Effects
12 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
If r = Ft = 1, model with individual effects: λi .
If r = λi = 1, model with time effects: Ft .
If r = 2 and λi2 = Ft1 = 1, model with additive individual and
time effects: λi + Ft .
Estimating and Testing a Quantile Regression Model with Interactive Effects
13 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
If r = Ft = 1, model with individual effects: λi .
If r = λi = 1, model with time effects: Ft .
If r = 2 and λi2 = Ft1 = 1, model with additive individual and
time effects: λi + Ft .
Estimating and Testing a Quantile Regression Model with Interactive Effects
13 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
If r = Ft = 1, model with individual effects: λi .
If r = λi = 1, model with time effects: Ft .
If r = 2 and λi2 = Ft1 = 1, model with additive individual and
time effects: λi + Ft .
Estimating and Testing a Quantile Regression Model with Interactive Effects
13 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
If r = Ft = 1, model with individual effects: λi .
If r = λi = 1, model with time effects: Ft .
If r = 2 and λi2 = Ft1 = 1, model with additive individual and
time effects: λi + Ft .
Estimating and Testing a Quantile Regression Model with Interactive Effects
13 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
If r = Ft = 1, model with individual effects: λi .
If r = λi = 1, model with time effects: Ft .
If r = 2 and λi2 = Ft1 = 1, model with additive individual and
time effects: λi + Ft .
Estimating and Testing a Quantile Regression Model with Interactive Effects
13 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
The time effect Ft is stochastically dependent on d it .
The effects (Ft , λi ) are stochastically dependent on d it .
The variables (Ft , λi , uit ) and d it are stochastically dependent.
Estimating and Testing a Quantile Regression Model with Interactive Effects
14 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
The time effect Ft is stochastically dependent on d it .
The effects (Ft , λi ) are stochastically dependent on d it .
The variables (Ft , λi , uit ) and d it are stochastically dependent.
Estimating and Testing a Quantile Regression Model with Interactive Effects
14 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
The time effect Ft is stochastically dependent on d it .
The effects (Ft , λi ) are stochastically dependent on d it .
The variables (Ft , λi , uit ) and d it are stochastically dependent.
Estimating and Testing a Quantile Regression Model with Interactive Effects
14 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
The time effect Ft is stochastically dependent on d it .
The effects (Ft , λi ) are stochastically dependent on d it .
The variables (Ft , λi , uit ) and d it are stochastically dependent.
Estimating and Testing a Quantile Regression Model with Interactive Effects
14 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
A Panel Data Model
Recently, Pesaran (2006) and Bai (2009) write,
yit = α0 d it + β 0 x it + λ0i F t + uit .
A panel data model with r factors,
λ0i F t = λi1 Ft1 + λi2 Ft2 + . . . + λir Ftr .
The time effect Ft is stochastically dependent on d it .
The effects (Ft , λi ) are stochastically dependent on d it .
The variables (Ft , λi , uit ) and d it are stochastically dependent.
Estimating and Testing a Quantile Regression Model with Interactive Effects
14 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Least Squares Estimation of a Panel Data
Model
Lemma
Under regularity conditions, α can be estimated by
α̂ = (D 0 P̄ M̄W D)−1 (D 0 P̄ M̄W y) where P̄ M̄W is a projection matrix
that uses instruments W and cross-sectional averages.
Remark
The method extends Pesaran (2006) analysis accommodating
to issues associated with dependence between d and (λ0 , u)0 .
Remark
It is possible to obtain a feasible estimator in a model with
interactive effects and endogenous covariates.
Estimating and Testing a Quantile Regression Model with Interactive Effects
15 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Least Squares Estimation of a Panel Data
Model
Lemma
Under regularity conditions, α can be estimated by
α̂ = (D 0 P̄ M̄W D)−1 (D 0 P̄ M̄W y) where P̄ M̄W is a projection matrix
that uses instruments W and cross-sectional averages.
Remark
The method extends Pesaran (2006) analysis accommodating
to issues associated with dependence between d and (λ0 , u)0 .
Remark
It is possible to obtain a feasible estimator in a model with
interactive effects and endogenous covariates.
Estimating and Testing a Quantile Regression Model with Interactive Effects
15 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Least Squares Estimation of a Panel Data
Model
Lemma
Under regularity conditions, α can be estimated by
α̂ = (D 0 P̄ M̄W D)−1 (D 0 P̄ M̄W y) where P̄ M̄W is a projection matrix
that uses instruments W and cross-sectional averages.
Remark
The method extends Pesaran (2006) analysis accommodating
to issues associated with dependence between d and (λ0 , u)0 .
Remark
It is possible to obtain a feasible estimator in a model with
interactive effects and endogenous covariates.
Estimating and Testing a Quantile Regression Model with Interactive Effects
15 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
The Quantile Regression Model
The paper considers conditional quantile functions of the form
QYit (τ |d it , x it , λi , F t ) = α(τ )0 d it + β(τ )0 x it + λi (τ )0 F t (τ )
where τj ∈ (0, 1) is the quantile of interest, and
QYit (τj |d it , x it , λi , F t ) ≡ inf{yit : FYit (yit |d it , x it , λi , F t ) ≥ τj }
The covariate’s effect is to shift the location, scale and possibly
shape of the conditional distribution of the response.
The model also allows for individual and time specific
distributional shifts.
Estimating and Testing a Quantile Regression Model with Interactive Effects
16 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
The Quantile Regression Model
The paper considers conditional quantile functions of the form
QYit (τ |d it , x it , λi , F t ) = α(τ )0 d it + β(τ )0 x it + λi (τ )0 F t (τ )
where τj ∈ (0, 1) is the quantile of interest, and
QYit (τj |d it , x it , λi , F t ) ≡ inf{yit : FYit (yit |d it , x it , λi , F t ) ≥ τj }
The covariate’s effect is to shift the location, scale and possibly
shape of the conditional distribution of the response.
The model also allows for individual and time specific
distributional shifts.
Estimating and Testing a Quantile Regression Model with Interactive Effects
16 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
The Quantile Regression Model
The paper considers conditional quantile functions of the form
QYit (τ |d it , x it , λi , F t ) = α(τ )0 d it + β(τ )0 x it + λi (τ )0 F t (τ )
where τj ∈ (0, 1) is the quantile of interest, and
QYit (τj |d it , x it , λi , F t ) ≡ inf{yit : FYit (yit |d it , x it , λi , F t ) ≥ τj }
The covariate’s effect is to shift the location, scale and possibly
shape of the conditional distribution of the response.
The model also allows for individual and time specific
distributional shifts.
Estimating and Testing a Quantile Regression Model with Interactive Effects
16 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
An Estimator for Panel Data
The estimator θ̂(τ ) ≡ α̂(τ ), β̂(α̂(τ ), τ )), δ̂(α̂(τ ), τ )) is
arg min
β,γ,δ
T X
N
X
ρτ (yit − d 0it α − x 0it β − Ψ̂0t (τ )δ − Φ̂0it (τ )γ),
t=1 i=1
where ρτ is the quantile regression check function and,
α̂(τ ) = arg min γ̂(τ, α)0 Aγ̂(τ, α)
α
The first term provides an asymptotic (consistent) approximation
for the interactive effect.
The second term is a vector of transformations of instruments,
as in the classical IV case.
Estimating and Testing a Quantile Regression Model with Interactive Effects
17 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
An Estimator for Panel Data
The estimator θ̂(τ ) ≡ α̂(τ ), β̂(α̂(τ ), τ )), δ̂(α̂(τ ), τ )) is
arg min
β,γ,δ
T X
N
X
ρτ (yit − d 0it α − x 0it β − Ψ̂0t (τ )δ − Φ̂0it (τ )γ),
t=1 i=1
where ρτ is the quantile regression check function and,
α̂(τ ) = arg min γ̂(τ, α)0 Aγ̂(τ, α)
α
The first term provides an asymptotic (consistent) approximation
for the interactive effect.
The second term is a vector of transformations of instruments,
as in the classical IV case.
Estimating and Testing a Quantile Regression Model with Interactive Effects
17 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
An Estimator for Panel Data
The estimator θ̂(τ ) ≡ α̂(τ ), β̂(α̂(τ ), τ )), δ̂(α̂(τ ), τ )) is
arg min
β,γ,δ
T X
N
X
ρτ (yit − d 0it α − x 0it β − Ψ̂0t (τ )δ − Φ̂0it (τ )γ),
t=1 i=1
where ρτ is the quantile regression check function and,
α̂(τ ) = arg min γ̂(τ, α)0 Aγ̂(τ, α)
α
The first term provides an asymptotic (consistent) approximation
for the interactive effect.
The second term is a vector of transformations of instruments,
as in the classical IV case.
Estimating and Testing a Quantile Regression Model with Interactive Effects
17 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Existing Fixed Effects Methods
The estimator considered in Koenker (2004),
arg min
α,β,λ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi )
t=1 i=1
Our method is similar to Harding and Lamarche (2009), but
arg min
β,γ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi − Φ̂0it (τ )γ)
t=1 i=1
Estimation of N nuisance parameters could be, in some
applications, computationally demanding.
They can produce biased results under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
18 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Existing Fixed Effects Methods
The estimator considered in Koenker (2004),
arg min
α,β,λ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi )
t=1 i=1
Our method is similar to Harding and Lamarche (2009), but
arg min
β,γ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi − Φ̂0it (τ )γ)
t=1 i=1
Estimation of N nuisance parameters could be, in some
applications, computationally demanding.
They can produce biased results under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
18 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Existing Fixed Effects Methods
The estimator considered in Koenker (2004),
arg min
α,β,λ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi )
t=1 i=1
Our method is similar to Harding and Lamarche (2009), but
arg min
β,γ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi − Φ̂0it (τ )γ)
t=1 i=1
Estimation of N nuisance parameters could be, in some
applications, computationally demanding.
They can produce biased results under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
18 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Existing Fixed Effects Methods
The estimator considered in Koenker (2004),
arg min
α,β,λ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi )
t=1 i=1
Our method is similar to Harding and Lamarche (2009), but
arg min
β,γ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi − Φ̂0it (τ )γ)
t=1 i=1
Estimation of N nuisance parameters could be, in some
applications, computationally demanding.
They can produce biased results under mild conditions.
Estimating and Testing a Quantile Regression Model with Interactive Effects
18 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Useful variation on these models
Combine fixed effects with interactive effects specification
arg min
β,γ
T X
N
X
ρτ (yit − d 0it α − x 0it β − λi − Ψ̂0it (τ )δ − Φ̂0it (τ )γ)
t=1 i=1
Can be used to test whether the interactive effects are useful
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Regularity Conditions
Regularity Conditions
1
Yit has a conditional distribution Fit , and continuous densities fit
bounded away from 0 and ∞ at ξit (τj ).
2
(α(τ ), β(τ ), δ(τ )) ∈ int of a compact and convex set.
3
√
max kz it k/ NT → 0, for z = {d, x, w }.
4
The Jacobian matrices have full rank and are continuous.
5
There exist limiting positive definite matrices S(τ ) and J(τ ).
Estimating and Testing a Quantile Regression Model with Interactive Effects
20 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Regularity Conditions
Regularity Conditions
1
Yit has a conditional distribution Fit , and continuous densities fit
bounded away from 0 and ∞ at ξit (τj ).
2
(α(τ ), β(τ ), δ(τ )) ∈ int of a compact and convex set.
3
√
max kz it k/ NT → 0, for z = {d, x, w }.
4
The Jacobian matrices have full rank and are continuous.
5
There exist limiting positive definite matrices S(τ ) and J(τ ).
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Conclusions
Regularity Conditions
Regularity Conditions
1
Yit has a conditional distribution Fit , and continuous densities fit
bounded away from 0 and ∞ at ξit (τj ).
2
(α(τ ), β(τ ), δ(τ )) ∈ int of a compact and convex set.
3
√
max kz it k/ NT → 0, for z = {d, x, w }.
4
The Jacobian matrices have full rank and are continuous.
5
There exist limiting positive definite matrices S(τ ) and J(τ ).
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Regularity Conditions
Regularity Conditions
1
Yit has a conditional distribution Fit , and continuous densities fit
bounded away from 0 and ∞ at ξit (τj ).
2
(α(τ ), β(τ ), δ(τ )) ∈ int of a compact and convex set.
3
√
max kz it k/ NT → 0, for z = {d, x, w }.
4
The Jacobian matrices have full rank and are continuous.
5
There exist limiting positive definite matrices S(τ ) and J(τ ).
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Regularity Conditions
Regularity Conditions
1
Yit has a conditional distribution Fit , and continuous densities fit
bounded away from 0 and ∞ at ξit (τj ).
2
(α(τ ), β(τ ), δ(τ )) ∈ int of a compact and convex set.
3
√
max kz it k/ NT → 0, for z = {d, x, w }.
4
The Jacobian matrices have full rank and are continuous.
5
There exist limiting positive definite matrices S(τ ) and J(τ ).
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Asymptotic Theory
Application
Conclusions
Theoretical Results
Theorem
Under the regularity conditions, the estimator (α̂(τ )0 , β̂(τ )0 )0 is
consistent and asymptotically normally distributed with mean
(α(τ )0 , β(τ )0 )0 and covariance matrix J 0 (τ )S(τ )J(τ ).
Remark
The paper suggests ways of doing inference. The standard
errors are obtained by estimating the asymptotic covariance
matrices.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Theoretical Results
Theorem
Under the regularity conditions, the estimator (α̂(τ )0 , β̂(τ )0 )0 is
consistent and asymptotically normally distributed with mean
(α(τ )0 , β(τ )0 )0 and covariance matrix J 0 (τ )S(τ )J(τ ).
Remark
The paper suggests ways of doing inference. The standard
errors are obtained by estimating the asymptotic covariance
matrices.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Simulation design:
yit
=
β0 + β1 dit + γxt + λ1i f1t + λ2i f2t + (1 + hdit )uit
dit
=
π0 + π1 wit + π2 xt + π3 f1t + π3 f2t + π4 λ1i f1t + π4 λ2i f2t + i + vit
fjt
=
ρf fjt−1 + ηjt
ηjt
=
ρη ηjt−1 + ejt
for j = {1, 2}, . . . t = −49, . . . 0, . . . T in the last two equations. The
random variables are xt ∼ N (0, 1), λi1 , λi2 ∼ N (1, 0.2), and e, and
w are Gaussian independent random variables. The error terms are
(uit , vit )0 ∼ (0, Ω), distributed either as Gaussian or t-student
distribution with two degrees of freedom. The parameters are
assumed to be: β0 = π3 = 2, β1 = γ = π0 = π1 = π2 = 1, ρf = 0.90,
ρη = 0.25, and Ω11 = Ω22 = 1.
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Conclusions
Design 1 The endogenous variable d is not correlated with
the λ’s, and the variables u and v are independent
Gaussian variables. Although d is not correlated
with the individual effects and the error term, it is
correlated with the F ’s. We assume π4 = 0 and
Ω12 = Ω21 = 0.
Design 2 The variable d is correlated with F ’s and λ’s, and
the error terms in equations 2.1 and 2.1 are not
correlated. We assume π4 = 2 and Ω12 = Ω21 = 0.
Design 3 The error terms in equations 2.1 and 2.1 are now
correlated, assuming that Ω12 = Ω21 = 0.5. The
variable d is also correlated with the F ’s and λ’s as
in the experiment carried out in Design 2.
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Asymptotic Theory
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Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Asymptotic Theory
Application
Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Models and Estimators
Asymptotic Theory
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Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Monte Carlo Evidence (cont.)
We consider the same model expanding the design to include
different sample sizes N = {50, 100} and T = {4, 8}.
We compare estimators:
(1) quantile regression estimator (QR); (2) Koenker’s (2004) fixed
effects estimator (QRFE);
(3) Harding and Lamarche’s (2009) instrumental variable
estimator (QRIVFE); (4) the proposed approach (QRIE).
We report bias and root mean square error (RMSE) on 24
different monte carlo experiments.
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
QR: D1, N=50, T=4, N(0,1)
0.1
Bias
0.2
QRFE: D1, N=50, T=4, N(0,1)
0.0
QRIVFE: D1, N=50, T=4, N(0,1)
−0.1
QRIE: D1, N=50, T=4, N(0,1)
5
10
15
20
Monte Carlo experiment
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Models and Estimators
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Application
Conclusions
0.4
Comparing Methods
QR
QRFE
QRIVFE
QRIE
QR: D1, N=50, T=8, N(0,1)
QRIVFE: D1, N=50, T=8, N(0,1)
QRIE: D1, N=50, T=8, N(0,1)
−0.1
0.0
0.1
Bias
0.2
0.3
QRFE: D1, N=50, T=8, N(0,1)
5
10
15
20
Monte Carlo experiment
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Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
Bias
0.2
QRFE: D1, N=100, T=4, N(0,1)
QR: D1, N=100, T=4, N(0,1)
0.0
0.1
QRIVFE: D1, N=100, T=4, N(0,1)
−0.1
QRIE: D1, N=100, T=4, N(0,1)
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
QR
QRFE
QRIVFE
QRIE
QRFE: D1, N=100, T=8, N(0,1)
0.1
Bias
0.2
0.3
QR: D1, N=100, T=8, N(0,1)
0.0
QRIVFE: D1, N=100, T=8, N(0,1)
−0.1
QRIE: D1, N=100, T=8, N(0,1)
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
QRFE: D1, N=50, T=4, t_2
0.1
Bias
0.2
QR: D1, N=50, T=4, t_2
0.0
QRIE: D1, N=50, T=4, t_2
−0.1
QRIVFE: D1, N=50, T=4, t_2
5
10
15
20
Monte Carlo experiment
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Conclusions
0.4
Comparing Methods
QR
QRFE
QRIVFE
QRIE
QRFE: D1, N=50, T=8, t_2
0.1
Bias
0.2
0.3
QR: D1, N=50, T=8, t_2
0.0
QRIE: D1, N=50, T=8, t_2
−0.1
QRIVFE: D1, N=50, T=8, t_2
5
10
15
20
Monte Carlo experiment
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Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
0.2
QRFE: D1, N=100, T=4, t_2
0.1
Bias
QR: D1, N=100, T=4, t_2
0.0
QRIE: D1, N=100, T=4, t_2
−0.1
QRIVFE: D1, N=100, T=4, t_2
5
10
15
20
Monte Carlo experiment
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Conclusions
0.4
Comparing Methods
QR
QRFE
QRIVFE
QRIE
QRFE: D1, N=100, T=8, t_2
0.1
Bias
0.2
0.3
QR: D1, N=100, T=8, t_2
0.0
QRIE: D1, N=100, T=8, t_2
−0.1
QRIVFE: D1, N=100, T=8, t_2
5
10
15
20
Monte Carlo experiment
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Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
0.2
QR: D2, N=50, T=4, N(0,1)
0.1
Bias
QRIVFE: D2, N=50, T=4, N(0,1)
−0.1
0.0
QRIE: D2, N=50, T=4, N(0,1)
5
10
15
20
Monte Carlo experiment
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Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
QR: D2, N=50, T=8, N(0,1)
0.1
Bias
0.2
QRFE: D2, N=50, T=8, N(0,1)
0.0
QRIVFE: D2, N=50, T=8, N(0,1)
−0.1
QRIE: D2, N=50, T=8, N(0,1)
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
0.2
QR: D2, N=100, T=4, N(0,1)
0.1
Bias
QRFE: D2, N=100, T=4, N(0,1)
0.0
QRIVFE: D2, N=100, T=4, N(0,1)
−0.1
QRIE: D2, N=100, T=4, N(0,1)
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
QR: D2, N=100, T=8, N(0,1)
0.1
Bias
0.2
QRFE: D2, N=100, T=8, N(0,1)
0.0
QRIVFE: D2, N=100, T=8, N(0,1)
−0.1
QRIE: D2, N=100, T=8, N(0,1)
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
0.2
QR: D2, N=50, T=4, t_2
0.1
Bias
QRFE: D2, N=50, T=4, t_2
0.0
QRIVFE: D2, N=50, T=4, t_2
−0.1
QRIE: D2, N=50, T=4, t_2
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
QRFE: D2, N=50, T=8, t_2
0.1
Bias
0.2
QR: D2, N=50, T=8, t_2
−0.1
0.0
QRIE: D2, N=50, T=8, t_2
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
0.2
QR: D2, N=100, T=4, t_2
0.1
Bias
QRFE: D2, N=100, T=4, t_2
0.0
QRIVFE: D2, N=100, T=4, t
−0.1
QRIE: D2, N=100, T=4, t_2
5
10
15
20
Monte Carlo experiment
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Models and Estimators
Asymptotic Theory
Application
Conclusions
0.4
Comparing Methods
0.3
QR
QRFE
QRIVFE
QRIE
QRFE: D2, N=100, T=8,
0.1
Bias
0.2
QR: D2, N=100, T=8, t_2
0.0
QRIE: D2, N=100, T=8,
−0.1
QRIVFE: D2, N=100, T=
5
10
15
20
Monte Carlo experiment
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Conclusions
0.4
Comparing Methods
−0.1
0.0
0.1
Bias
0.2
0.3
QR
QRFE
QRIVFE
QRIE
5
10
15
20
Monte Carlo experiment
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Models and Estimators
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Application
Conclusions
QR
QRFE
QRIVFE
QRIE
0.0
0.1
0.2
RMSE
0.3
0.4
0.5
0.6
Comparing Methods
5
10
15
20
Monte Carlo experiment
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Models and Estimators
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Application
Conclusions
Bocconi University
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Data
We employ data from the administrative records at Bocconi
University.
The data set includes: educational attainment (mean 25.455 ≈
B+ in US), class size, class composition, demographic and
socioeconomic characteristics.
Class size and class composition may be endogenous.
We use instruments generated from a random assignment of
students into classes. Class size instrumented with number of
students generated by lotteries.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Data
We employ data from the administrative records at Bocconi
University.
The data set includes: educational attainment (mean 25.455 ≈
B+ in US), class size, class composition, demographic and
socioeconomic characteristics.
Class size and class composition may be endogenous.
We use instruments generated from a random assignment of
students into classes. Class size instrumented with number of
students generated by lotteries.
Estimating and Testing a Quantile Regression Model with Interactive Effects
44 / 50
Models and Estimators
Asymptotic Theory
Application
Conclusions
Data
We employ data from the administrative records at Bocconi
University.
The data set includes: educational attainment (mean 25.455 ≈
B+ in US), class size, class composition, demographic and
socioeconomic characteristics.
Class size and class composition may be endogenous.
We use instruments generated from a random assignment of
students into classes. Class size instrumented with number of
students generated by lotteries.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Model specification
We estimate the following model:
yict
= d 0ct α + x 0i β + f 0ct λi + uict
d ct
= w 0ct π1 + x 0i π2 + f 0ct λi + v ict ,
The λi ’s are associated with motivation and ability, and the fct ’s
measure teaching quality.
It is tempting to impose λi + fct as in Hanushek et al. (2003).
However, high teaching quality may have a modest effect on
performance among unmotivated students, while it may
dramatically affect strong, motivated students.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Model specification
We estimate the following model:
yict
= d 0ct α + x 0i β + f 0ct λi + uict
d ct
= w 0ct π1 + x 0i π2 + f 0ct λi + v ict ,
The λi ’s are associated with motivation and ability, and the fct ’s
measure teaching quality.
It is tempting to impose λi + fct as in Hanushek et al. (2003).
However, high teaching quality may have a modest effect on
performance among unmotivated students, while it may
dramatically affect strong, motivated students.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Model specification
We estimate the following model:
yict
= d 0ct α + x 0i β + f 0ct λi + uict
d ct
= w 0ct π1 + x 0i π2 + f 0ct λi + v ict ,
The λi ’s are associated with motivation and ability, and the fct ’s
measure teaching quality.
It is tempting to impose λi + fct as in Hanushek et al. (2003).
However, high teaching quality may have a modest effect on
performance among unmotivated students, while it may
dramatically affect strong, motivated students.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Model specification
We estimate the following model:
yict
= d 0ct α + x 0i β + f 0ct λi + uict
d ct
= w 0ct π1 + x 0i π2 + f 0ct λi + v ict ,
The λi ’s are associated with motivation and ability, and the fct ’s
measure teaching quality.
It is tempting to impose λi + fct as in Hanushek et al. (2003).
However, high teaching quality may have a modest effect on
performance among unmotivated students, while it may
dramatically affect strong, motivated students.
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Estimating and Testing a Quantile Regression Model with Interactive Effects
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Models and Estimators
Asymptotic Theory
Application
Conclusions
Specification tests: p-values
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Conclusions
Brief summary and extensions
1
We propose a quantile approach for panel data models with
interative effects and endogenous independent variables.
2
The approach is simple and seems to provide satisfactory large
sample and finite sample performance measured in terms of
quadratic loss.
3
Next stop: forecasting in panel data models. Preliminary
evidence shows that QRIE models perform very well.
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