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

Estimating and Testing a Quantile Regression Model with Interactive Effects Matthew Harding 1 and Carlos Lamarche 2 1 Stanford University 2 University of Oklahoma California Econometrics Conference, Sept 24, 2010 Estimating and Testing a Quantile Regression Model with Interactive Effects 1 / 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 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 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 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 4 / 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

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 30 28 26 test scores 24 22 20 − 0.5 0.0 0.5 1.0 1.5 large class Estimating and Testing a Quantile Regression Model with Interactive Effects 6 / 50

Background 0.20 0.15 f(y|x) 0.10 test scores 35 0.05 30 25 20 0.00 15 − 0.5 0.0 0.5 1.0 1.5 large class Estimating and Testing a Quantile Regression Model with Interactive Effects 7 / 50

Background 30 28 26 test scores ^ = − 0.16 ! 1 24 22 20 − 0.5 0.0 0.5 1.0 1.5 large class Estimating and Testing a Quantile Regression Model with Interactive Effects 8 / 50

Background 30 28 26 test scores ^ = − 0.16 ! 1 24 22 ^ ( 0.1 ) = − 0.30 ! 1 20 − 0.5 0.0 0.5 1.0 1.5 large class Estimating and Testing a Quantile Regression Model with Interactive Effects 9 / 50

Background 30 ^ ( 0.9 ) = − 0.08 28 ! 1 26 test scores ^ = − 0.16 ! 1 24 22 ^ ( 0.1 ) = − 0.30 ! 1 20 − 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): y ict = d ′ ct α + x ′ i β + λ i + F ct + u ict The λ i ’s are associated with motivation and ability, and the F ct ’s measure teaching quality. Note that we are imposing λ i + F ct . 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): y ict = d ′ ct α + x ′ i β + λ i + F ct + u ict The λ i ’s are associated with motivation and ability, and the F ct ’s measure teaching quality. Note that we are imposing λ i + F ct . 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