Xiaozhou Wang Graduate Fellow Quantitative Social Science Initiative Penn State University ([email protected]) 1.

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Xiaozhou Wang

Graduate Fellow

Quantitative Social Science Initiative

Penn State University

([email protected])

An Introduction to Quantile Regression:

Modeling Group Disparity by Comparing Entire

Distributions

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MotivationGroup disparity

GenderRaceClassImmigration…

How to model group disparityDescriptive statisticsRegression model

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Standard approach to describe data: comparing the means

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“It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once”.

— Sir Francis Galton Natural Inheritance

(1889): 62-3.

Critiques of the mean-comparison approach

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Normal distribution

Are distributions identical when they have equal means and variances?

Gamma distribution

-4 -2 0 2 4 6

0.0

0.5

1.0

1.5

x

dn

orm

(x, m

ea

n =

mu

, sd

= s

igm

a)

fx1

22exp

x 2

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: mean

: standard deviation

Normal distribution: mean=1 and variance=2

0 2 4 6 8 10

0.0

0.5

1.0

1.5

x

dg

am

ma

(x, s

ha

pe

= a

, sca

le =

s)

fx1

saaxa 1

expx

s

a 0: shape parameter

s 0: scale parameter

Gamma distribution: shape=0.5 and scale=2

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The basic idea (Silverman 1986)

K(x) is the kernel function, which satisfies:

Very often, K will be a symmetric probability density function.

h is the smoothing parameter (bandwidth).

Kernel density estimation

Or,

where

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Let’s compare the distributions.

Both of the two earnings distributions changed from 1996 to 2006.

For native whites, there is a growing share of population who make about 80,000 or more.

Hispanic immigrants also made some progress – less people are found at the lower end of their earnings distribution.

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The OLS model

The conditional mean of the outcome variable is expressed as a product of characteristics and the "prices" of these characteristics.

The estimated model provides “a grand summary for the averages of the distributions corresponding to the set of Xs” (Mosteller and Tukey 1977).

Regression model

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Application with earnings data from CPSLet’s model the log

earnings of the foreign-born Hispanic immigrants as a linear combination of the following covariates.ExperienceExperience^2Year of educationLiving in the southLiving in a non-

metropolitan areaBeing married

Data: CPS March supplements in 2006A monthly survey of

about 50,000 householdsThe primary source of

detailed information on income and work experience in the United States

The best source to study labor market outcome of immigrants so far (Farley and Alba 2002)

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DataSample of analysis

Non-institutional population

Men of Hispanic origin

Aged at 25-64 by the time of survey interview

Worked for at least one week during the last calendar year

Coefficients:Estimate Std. Error t value Pr(>|t|)

(Intercept) 8.84 0.09 101.87 0.00 ***exper 0.02 0.01 3.84 0.00 ***exper2 0.00 0.00 -2.73 0.01 **yredu 0.06 0.00 16.20 0.00 ***south -0.02 0.03 -0.52 0.60nonmetro 0.09 0.06 1.57 0.12married 0.15 0.03 4.88 0.00 ***

Var. Mean sd. Median log earnings 9.89 1.00 9.97experience 23.03 10.49 22.00experience^2 640.40 555.00 484.00year of education 10.06 4.04 12.00south 0.34 0.34 0.00nonmetro 0.07 0.07 0.00married 0.65 0.65 1.00

Coefficients:Estimate Std. Error Pr(>|t|)

(Intercept) 8.84 0.09 0.00 ***experience 0.02 0.01 0.00 ***experience^2 -3.00E-04 0.00 0.01 **year of education 0.06 0.00 0.00 ***south -0.02 0.03 0.60nonmetro 0.09 0.06 0.12married 0.15 0.03 0.00 ***

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Can we generalize our findings to the entire distribution?

“Just as the mean gives an incomplete picture of a single distribution, so the regression curve gives a correspondingly incomplete picture for a set of distributions” (Mosteller and Tukey 1977).

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The solution – Quantile RegressionNow we model the θth quantile of the

outcome variable (logged earnings in this case) as a product of the characteristics and premiums to the characteristics. That is, . is a vector of coefficients (or premiums)X is a vector of the covariates (characteristics)For any given θ in (0,1), can be estimated

by minimizing in β (Koenker and Bassett 1978) ,

where .

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Foreign-born Hispanics

Quantile regression (QR): application

Native whites

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PDFTo what extent is the

observed group disparity due to: A. difference in

covariates?B. difference in

coefficients ?

The observed group disparity

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Use QR to estimate the conditional distribution given the covariates

In Bassett and Koenker (1982; 1986), it is established that under regularity conditions, the estimated conditional quantile function is a strongly consistent estimator of the population quantile function.

Probability Integral Transformation Theorem: if U is a uniform random variable on [0,1], then F-1(U) has distribution F.

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• We estimate the quantile of earnings distribution given the observed characteristics of the respondents, where are drawn from a uniform (0, 1) distribution.

• By the Probability Integral Transformation Theorem,

constitute a random sample from F(Y|X), the (estimated) conditional distribution of earnings given X (Albrecht et al 2009: 384).

Use QR to estimate the conditional distribution given the covariates

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According to Machado & Mata (2005), to ‘integrate X out’ and get a sample from the marginal earnings distribution, we can generate a random sample of size m with replacement from the rows of X, denoted by

. Now form a random sample of size m from the marginal earnings distribution for the respondents.

From F(Y|X) to F(Y)

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Foreign-born Hispanics

Marginal distributions

Native whites

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PDF

Comparing the marginal distributions

Quantile function

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To what extent the observed group disparity is due to: A. difference in covariates?B. difference in coefficients?

Counterfactual analysisA. when group A have same coefficients as

group BB. when group A have same covariates as

group B

The questions yet to answer

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Once we have the marginal earnings distributions for each group, we can obtain the counterfactual densities by weighting the marginal density of wages with appropriate weights. The counterfactual earnings distribution for the

Hispanic immigrants would be fully characterized by

if the returns to characteristics are identical to the “prices” for native whites.

The counterfactual earnings distribution for the Hispanic immigrants given they share the same characteristics as the native do in the same year can be characterized by .

The counterfactual analysis

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With the covariates of the native whites

The counterfactual earnings distribution for the foreign-born HispanicsWith the

coefficients of the native whites

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With the covariates of the native whites

Quantile functions for the counterfactuals (foreign-born Hispanics)With the

coefficients of the native whites

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With the covariates of the native whites

Quantile functions for the counterfactuals (foreign-born Hispanics)With the

coefficients of the native whites

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Pessimistic view Segmented assimilation

model (López and Stanton-Salazar 2001; Portes and Rumbaut 2001; Zhou 1997)

Racial disadvantaged model (Glazer and Moynihan 1963)

Declining quality of immigrants (Borjas 1995, 1999; Borjas and Friedberg 2007)

Classical assimilation theory (Robert Park (1926); Irving Child (1943); Oscar Handlin (1951); Milton Gordon (1964))

Institutional perspective (Alba and Nee; Alba and Farley)

Earnings assimilation of Hispanic immigrants in the U.S.

Optimistic view

Perspectives about economic assimilation of the Hispanic immigrants

Pessimistic view Segmented assimilation model (López and

Stanton-Salazar 2001; Portes and Rumbaut 2001; Zhou 1997)Declining employment in manufacturingRacial discrimination against ethnic minoritiesDeteriorated urban infrastructure – in

particular ,the public education system may not provide advanced training for immigrant students to move upward

Transnationalism – it is easier to maintain strong tie to immigrants’ own community

Less-educated second generation may look down upon the job their parents take and then join the “underclass” – that is, a “downward assimilation” trajectory

Perspectives about economic assimilation of the Hispanic immigrants

Pessimistic view Racial disadvantaged model (Glazer and

Moynihan 1963)The host society is highly stratified according to

race and ethnicity and thus the earning differentials associated with nativity is just a reflection of the ethnic inequality in earnings

Discrimination may undermine immigrants’ motivation to move upward (Bean & Stevens: 99)

Declining quality of immigrants (Borjas 1995, 1999; Borjas and Friedberg 2007)The education gap between immigrants and

native whites has been widen by the 1990sDeclining quality of immigrant cohorts

Perspectives about economic assimilation of the Hispanic immigrants

Optimistic viewClassical assimilation theory (Robert Park

(1926); Irving Child (1943); Oscar Handlin (1951); Milton Gordon (1964))Immigrants experience steady improvement in

socioeconomic outcomes over time, within and across generations

Institutional perspective (Alba and Nee; Alba and Farley)Legal, social and economic changes may now

guarantee the second generation protections and opportunities

Labor markets have been extremely tight throughout the 1990s, which “should have benefited the bottom of the labor queue”(Alba and Farley: 677)

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Assimilation as a product of assimilation in characteristics and assimilation in returns to characteristics

Earnings assimilation of Hispanic immigrants in the U.S.

  Assimilation in characteristics  Low High

Assimilation in returns to characteristics

Low Downward assimilation;Declining quality of immigrants

Racial disadvantage model

High New mode of assimilation

Classical assimilation; Upward assimilation;Institutional perspective

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With the covariates of the native whites

Quantile functions for the counterfactuals (foreign-born Hispanics)With the

coefficients of the native whites

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With the covariates of the native whites

Quantile functions for the counterfactuals (2nd gen. Hispanics)With the

coefficients of the native whites

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Foreign-born HispanicsAssimilation in covariates

will increase the earnings of immigrants across the whole distribution.

Assimilation in coefficients will decrease the earnings of those at the bottom of the distribution, while benefit those who have higher earnings.

Assimilation in covariates and in coefficients will increase of the earnings of the immigrants. However, it is the assimilation in covariates that plays a more important role in reducing the earnings gap across the whole distribution.

Summary of findings

2nd gen. Hispanics

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Foreign-born HispanicsRacial disadvantage

model and the declining quality of immigrants argument do not hold for the immigrants at the lower end of the earnings distribution.

Downward assimilation does not hold for the immigrants since their difference with the native whites is mainly due to difference in covariates.

Conclusion

2nd gen. Hispanics

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Albrecht, J., A. Van Vuuren, et al. (2009). "Counterfactual distributions with sample selection adjustments: Econometric theory and an application to the Netherlands." Labour Economics 16(4): 383-396.

Bassett, G. W. and R. W. Koenker (1986). "Strong consistency of regression quantiles and related empirical processes." Econometric Theory 2(2): 191-201.

Bassett Jr, G. and R. Koenker (1982). "An empirical quantile function for linear models with iid errors." Journal of the American Statistical Association 77(378): 407-415.

Machado, J. and J. Mata (2005). "Counterfactual decomposition of changes in wage distributions using quantile regression." Journal of Applied Econometrics 20(4): 445-466.

Mosteller, F. and J. W. Tukey (1977). "Data analysis and regression. A second course in statistics."

Silverman, B (1986). "Density estimation for statistics and data analysis. " Monographs on Statistics and Applied Probability. Chapman and Hall, New York.

Some References on Quantile Regression

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R: Package ‘quantreg’ by Roger Koenker

Software

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Thank you!