Estimating and Testing a Quantile Regression Model with Interactive - - PowerPoint PPT Presentation

Estimating and Testing a Quantile Regression Model with Interactive Effects

Matthew Harding1 and Carlos Lamarche2

1Stanford University 2University of Oklahoma

California Econometrics Conference, Sept 24, 2010

1 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

2 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

2 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

2 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

2 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

3 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

4 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

4 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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.

4 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

In the last half a century, understanding the drivers of students’ academic performance has been a major focus in the economics

• f 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).

5 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

In the last half a century, understanding the drivers of students’ academic performance has been a major focus in the economics

• f 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).

5 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

In the last half a century, understanding the drivers of students’ academic performance has been a major focus in the economics

• f 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).

5 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

In the last half a century, understanding the drivers of students’ academic performance has been a major focus in the economics

• f 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).

5 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

−0.5 0.0 0.5 1.0 1.5 20 22 24 26 28 30 large class test scores

6 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

−0.5 0.0 0.5 1.0 1.5 0.00 0.05 0.10 0.15 0.20 15 20 25 30 35

large class test scores f(y|x)

7 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

−0.5 0.0 0.5 1.0 1.5 20 22 24 26 28 30 large class test scores !1 ^ =−0.16

8 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

−0.5 0.0 0.5 1.0 1.5 20 22 24 26 28 30 large class test scores !1 ^ =−0.16 !1 ^ (0.1)=−0.30

9 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

−0.5 0.0 0.5 1.0 1.5 20 22 24 26 28 30 large class test scores !1 ^ =−0.16 !1 ^ (0.9)=−0.08 !1 ^ (0.1)=−0.30

10 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

It is standard to consider (e.g., Hanushek et al. 2003): yict = d′

ctα + x′ iβ + 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?

11 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

It is standard to consider (e.g., Hanushek et al. 2003): yict = d′

ctα + x′ iβ + 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?

11 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

It is standard to consider (e.g., Hanushek et al. 2003): yict = d′

ctα + x′ iβ + 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?

11 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

It is standard to consider (e.g., Hanushek et al. 2003): yict = d′

ctα + x′ iβ + 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?

11 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Background

It is standard to consider (e.g., Hanushek et al. 2003): yict = d′

ctα + x′ iβ + 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?

11 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Outline

12 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

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.

13 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

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.

13 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

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.

13 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

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.

13 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

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.

13 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

The time effect Ft is stochastically dependent on dit. The effects (Ft, λi) are stochastically dependent on dit. The variables (Ft, λi, uit) and dit are stochastically dependent.

14 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

The time effect Ft is stochastically dependent on dit. The effects (Ft, λi) are stochastically dependent on dit. The variables (Ft, λi, uit) and dit are stochastically dependent.

14 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

The time effect Ft is stochastically dependent on dit. The effects (Ft, λi) are stochastically dependent on dit. The variables (Ft, λi, uit) and dit are stochastically dependent.

14 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

The time effect Ft is stochastically dependent on dit. The effects (Ft, λi) are stochastically dependent on dit. The variables (Ft, λi, uit) and dit are stochastically dependent.

14 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

A Panel Data Model

Recently, Pesaran (2006) and Bai (2009) write, yit = α′dit + β′xit + λ′

iF t + uit.

A panel data model with r factors, λ′

iF t = λi1Ft1 + λi2Ft2 + . . . + λirFtr.

The time effect Ft is stochastically dependent on dit. The effects (Ft, λi) are stochastically dependent on dit. The variables (Ft, λi, uit) and dit are stochastically dependent.

14 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Least Squares Estimation of a Panel Data Model

Lemma Under regularity conditions, α can be estimated by ˆ α = (D′ ¯ P ¯

MWD)−1(D′ ¯

P ¯

MWy) where ¯

P ¯

MW 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 (λ′, u)′. Remark It is possible to obtain a feasible estimator in a model with interactive effects and endogenous covariates.

15 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Least Squares Estimation of a Panel Data Model

Lemma Under regularity conditions, α can be estimated by ˆ α = (D′ ¯ P ¯

MWD)−1(D′ ¯

P ¯

MWy) where ¯

P ¯

MW 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 (λ′, u)′. Remark It is possible to obtain a feasible estimator in a model with interactive effects and endogenous covariates.

15 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Least Squares Estimation of a Panel Data Model

Lemma Under regularity conditions, α can be estimated by ˆ α = (D′ ¯ P ¯

MWD)−1(D′ ¯

P ¯

MWy) where ¯

P ¯

MW 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 (λ′, u)′. Remark It is possible to obtain a feasible estimator in a model with interactive effects and endogenous covariates.

15 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

The Quantile Regression Model

The paper considers conditional quantile functions of the form QYit(τ|dit, xit, λi, F t) = α(τ)′dit + β(τ)′xit + λi(τ)′F t(τ) where τj ∈ (0, 1) is the quantile of interest, and QYit(τj|dit, xit, λi, F t) ≡ inf{yit : FYit(yit|dit, xit, 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.

16 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

The Quantile Regression Model

The paper considers conditional quantile functions of the form QYit(τ|dit, xit, λi, F t) = α(τ)′dit + β(τ)′xit + λi(τ)′F t(τ) where τj ∈ (0, 1) is the quantile of interest, and QYit(τj|dit, xit, λi, F t) ≡ inf{yit : FYit(yit|dit, xit, 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.

16 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

The Quantile Regression Model

The paper considers conditional quantile functions of the form QYit(τ|dit, xit, λi, F t) = α(τ)′dit + β(τ)′xit + λi(τ)′F t(τ) where τj ∈ (0, 1) is the quantile of interest, and QYit(τj|dit, xit, λi, F t) ≡ inf{yit : FYit(yit|dit, xit, 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.

16 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

An Estimator for Panel Data

The estimator ˆ θ(τ) ≡

• ˆ

α(τ), ˆ β( ˆ α(τ), τ)), ˆ δ( ˆ α(τ), τ))

• is

arg min

β,γ,δ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − ˆ

Ψ′

t(τ)δ − ˆ

Φ′

it(τ)γ),

where ρτ is the quantile regression check function and, ˆ α(τ) = arg min

α ˆ

γ(τ, α)′Aˆ γ(τ, 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.

17 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

An Estimator for Panel Data

The estimator ˆ θ(τ) ≡

• ˆ

α(τ), ˆ β( ˆ α(τ), τ)), ˆ δ( ˆ α(τ), τ))

• is

arg min

β,γ,δ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − ˆ

Ψ′

t(τ)δ − ˆ

Φ′

it(τ)γ),

where ρτ is the quantile regression check function and, ˆ α(τ) = arg min

α ˆ

γ(τ, α)′Aˆ γ(τ, 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.

17 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

An Estimator for Panel Data

The estimator ˆ θ(τ) ≡

• ˆ

α(τ), ˆ β( ˆ α(τ), τ)), ˆ δ( ˆ α(τ), τ))

• is

arg min

β,γ,δ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − ˆ

Ψ′

t(τ)δ − ˆ

Φ′

it(τ)γ),

where ρτ is the quantile regression check function and, ˆ α(τ) = arg min

α ˆ

γ(τ, α)′Aˆ γ(τ, 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.

17 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Existing Fixed Effects Methods

The estimator considered in Koenker (2004), arg min

α,β,λ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi)

Our method is similar to Harding and Lamarche (2009), but arg min

β,γ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi − ˆ

Φ′

it(τ)γ)

Estimation of N nuisance parameters could be, in some applications, computationally demanding. They can produce biased results under mild conditions.

18 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Existing Fixed Effects Methods

The estimator considered in Koenker (2004), arg min

α,β,λ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi)

Our method is similar to Harding and Lamarche (2009), but arg min

β,γ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi − ˆ

Φ′

it(τ)γ)

Estimation of N nuisance parameters could be, in some applications, computationally demanding. They can produce biased results under mild conditions.

18 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Existing Fixed Effects Methods

The estimator considered in Koenker (2004), arg min

α,β,λ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi)

Our method is similar to Harding and Lamarche (2009), but arg min

β,γ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi − ˆ

Φ′

it(τ)γ)

Estimation of N nuisance parameters could be, in some applications, computationally demanding. They can produce biased results under mild conditions.

18 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Existing Fixed Effects Methods

The estimator considered in Koenker (2004), arg min

α,β,λ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi)

Our method is similar to Harding and Lamarche (2009), but arg min

β,γ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi − ˆ

Φ′

it(τ)γ)

Estimation of N nuisance parameters could be, in some applications, computationally demanding. They can produce biased results under mild conditions.

18 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Useful variation on these models

Combine fixed effects with interactive effects specification arg min

β,γ T

• t=1

N

• i=1

ρτ(yit − d′

itα − x′ itβ − λi − ˆ

Ψ′

it(τ)δ − ˆ

Φ′

it(τ)γ)

Can be used to test whether the interactive effects are useful

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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 zit/ √ 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).

20 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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 zit/ √ 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).

20 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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 zit/ √ 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).

20 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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 zit/ √ 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).

20 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

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 zit/ √ 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).

20 / 50 Estimating and Testing a Quantile Regression Model with Interactive Effects

Models and Estimators Asymptotic Theory Application Conclusions

Theoretical Results

Theorem Under the regularity conditions, the estimator ( ˆ α(τ)′, ˆ β(τ)′)′ is consistent and asymptotically normally distributed with mean (α(τ)′, β(τ)′)′ and covariance matrix J′(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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Theoretical Results

Theorem Under the regularity conditions, the estimator ( ˆ α(τ)′, ˆ β(τ)′)′ is consistent and asymptotically normally distributed with mean (α(τ)′, β(τ)′)′ and covariance matrix J′(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 + β1dit + γxt + λ1if1t + λ2if2t + (1 + hdit)uit dit = π0 + π1wit + π2xt + π3f1t + π3f2t + π4λ1if1t + π4λ2if2t + ǫi + vit fjt = ρffjt−1 + ηjt ηjt = ρηηjt−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, ǫ and w are Gaussian independent random variables. The error terms are (uit, vit)′ ∼ (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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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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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

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

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=50, T=4, N(0,1) QRFE: D1, N=50, T=4, N(0,1) QRIVFE: D1, N=50, T=4, N(0,1) QRIE: D1, N=50, T=4, N(0,1)

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Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=50, T=8, N(0,1) QRFE: 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)

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Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=100, T=4, N(0,1) QRFE: D1, N=100, T=4, N(0,1) QRIVFE: D1, N=100, T=4, N(0,1) QRIE: D1, N=100, T=4, N(0,1)

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=100, T=8, N(0,1) QRFE: D1, N=100, T=8, N(0,1) QRIVFE: D1, N=100, T=8, N(0,1) QRIE: D1, N=100, T=8, N(0,1)

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=50, T=4, t_2 QRFE: D1, N=50, T=4, t_2 QRIVFE: D1, N=50, T=4, t_2 QRIE: D1, N=50, T=4, t_2

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=50, T=8, t_2 QRFE: D1, N=50, T=8, t_2 QRIVFE: D1, N=50, T=8, t_2 QRIE: D1, N=50, T=8, t_2

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=100, T=4, t_2 QRFE: D1, N=100, T=4, t_2 QRIVFE: D1, N=100, T=4, t_2 QRIE: D1, N=100, T=4, t_2

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D1, N=100, T=8, t_2 QRFE: D1, N=100, T=8, t_2 QRIVFE: D1, N=100, T=8, t_2 QRIE: D1, N=100, T=8, t_2

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Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=50, T=4, N(0,1) QRIVFE: D2, N=50, T=4, N(0,1) QRIE: D2, N=50, T=4, N(0,1)

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=50, T=8, N(0,1) QRFE: D2, N=50, T=8, N(0,1) QRIVFE: D2, N=50, T=8, N(0,1) QRIE: D2, N=50, T=8, N(0,1)

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=100, T=4, N(0,1) QRFE: D2, N=100, T=4, N(0,1) QRIVFE: D2, N=100, T=4, N(0,1) QRIE: D2, N=100, T=4, N(0,1)

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=100, T=8, N(0,1) QRFE: D2, N=100, T=8, N(0,1) QRIVFE: D2, N=100, T=8, N(0,1) QRIE: D2, N=100, T=8, N(0,1)

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=50, T=4, t_2 QRFE: D2, N=50, T=4, t_2 QRIVFE: D2, N=50, T=4, t_2 QRIE: D2, N=50, T=4, t_2

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=50, T=8, t_2 QRFE: D2, N=50, T=8, t_2 QRIE: D2, N=50, T=8, t_2

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=100, T=4, t_2 QRFE: D2, N=100, T=4, t_2 QRIVFE: D2, N=100, T=4, t QRIE: D2, N=100, T=4, t_2

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE QR: D2, N=100, T=8, t_2 QRFE: D2, N=100, T=8, QRIVFE: D2, N=100, T= QRIE: D2, N=100, T=8,

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 −0.1 0.0 0.1 0.2 0.3 0.4 Monte Carlo experiment Bias QR QRFE QRIVFE QRIE

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Models and Estimators Asymptotic Theory Application Conclusions

Comparing Methods

5 10 15 20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Monte Carlo experiment RMSE QR QRFE QRIVFE QRIE

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Models and Estimators Asymptotic Theory Application Conclusions

Bocconi University

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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.

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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.

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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.

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Models and Estimators Asymptotic Theory Application Conclusions

Model specification

We estimate the following model: yict = d′

ctα + x′ iβ + f ′ ctλi + uict

dct = w′

ctπ1 + x′ iπ2 + f ′ ctλi + vict,

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.

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Models and Estimators Asymptotic Theory Application Conclusions

Model specification

We estimate the following model: yict = d′

ctα + x′ iβ + f ′ ctλi + uict

dct = w′

ctπ1 + x′ iπ2 + f ′ ctλi + vict,

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.

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Models and Estimators Asymptotic Theory Application Conclusions

Model specification

We estimate the following model: yict = d′

ctα + x′ iβ + f ′ ctλi + uict

dct = w′

ctπ1 + x′ iπ2 + f ′ ctλi + vict,

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.

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Models and Estimators Asymptotic Theory Application Conclusions

Model specification

We estimate the following model: yict = d′

ctα + x′ iβ + f ′ ctλi + uict

dct = w′

ctπ1 + x′ iπ2 + f ′ ctλi + vict,

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.

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Models and Estimators Asymptotic Theory Application Conclusions

Specification tests: p-values

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Models and Estimators Asymptotic Theory Application 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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