Lecture 8: Decomposition Method · 2020. 12. 30. · – Machado and Mata(2005): MM – DiNardo, Fortin and Lemieux(1996): DFL – Firpo, Fortin and Lemieux(2007,2010): FFL Zhaopeng

Lecture 8: Decomposition MethodIntroduction ot Econometrics,Fall 2019

Zhaopeng Qu

Nanjing University

11/12/2020

Zhaopeng Qu (Nanjing University) Lecture 8: Decomposition Method 11/12/2020 1 / 70

1 Review Previous Lectures

2 Oaxaca-Blinder Decomposition

3 Reference group problem

4 Detailed Decomposition

5 Standard Errors

6 Introduction to bootstrap(Maybe Skipped)

7 Representative Applications


Review Previous Lectures

Section 1




Topics covered

Main ContentBuild a framework of Causal InferenceReview Basic Probability and StatisticsSimple OLS: Estimation and InferenceMultiple OLS: Estimation and InferenceFunction forms: Nonlinear in independent variablesNonlinear Regression model: Dummy dependent variableComprehensive Evaluations to OLS regression Model



Two explicite Assumptions

So far, all models we learned have to be satisfied two stronghypotheses:

1 No Hetergeneity: If the sample could be divied by 𝑚 heterogenousgroups, then we assume that the estimate coeffient 𝛽𝑗 for the jthindependent variable, 𝑋𝑗 are the same among all groups of the sample.Thus

𝛽𝑗,1 = 𝛽𝑗,2 = ... = 𝛽𝑗,𝑀

for any group 𝐺𝑚 ∶ 𝑚 = 1, 2, ..., 𝑀2 No Endogeneity(Internal Valid): there is no endogeneity in these

estimating models. Essentially,the 1st Assumption of indentificationin OLS model is satisfied.Thus

𝐸(𝑢𝑖|𝑋1, 𝑋2, ..., 𝑋𝑘) = 0



An Simple Extension: Decomposition Method

1 Heterogeneity: Gap between two groups(more than interactions)2 Exogenous conditional on controlling variables

Ignoriable or Conditional Independence Assumption(CIA), thusassume that

𝐸(𝑢𝑖|𝑋1, 𝑋2, ..., 𝑋𝑘) = 0, the first assumption of OLS regression.



Wage Decomposition: Introduction

Wage Decomposition methods are used to analyze distributionaldifferences in an outcome variable between groups or time points.

In particular, the methods decompose the observed differencebetween two groups(or across two time points) into a componentsthat is due to compositional differences between the groups, and acomponent that is due to differential mechanisms.



Wage Decomposition Methods: IntroductionA Classical Case: Gender Wage Gap

How can the difference in average wages between men and women beexplained?

Is the gap due to1 group differences in wage determinants (i.e. in characteristics that are

relevant for wages, such as education)? (compositional differences)2 differential compensation for these determinants (e.g. different returns

to education for men an women, or wage discrimination againstwomen)? (differential mechanisms)

The typical question is needed to answer is “what the pay(or otheroutcomes) would be if women had the same characteristics as men?”

It will help us construct a counterfactual state by usingdecomposition method to recovery the causal effect(sort of causal) ofa certain factor or a groups of determinants.



Decomposition Methods: Introduction

The method can be tracked back from the seminal work bySolow(1957) for “growth accounting”.

Seminal Work: Oaxaca (1973) and Blinder (1973), who analyzedmean wage differences between groups (males vs. females, whitesvs. blacks).

Once widely used in the area of labor economics, especially in thetopics of earnings inequality since 1990s. Now it is popular used inmany topics in many fields of economics.

These more recent developments focus on topics such asdistributional measures other than the meannon-linear models for categorical variablestaking into selection bias and other type endogenous.combine with other quasi-experimental methodsextending into spatial econometric model



Decomposition Methods to Gaps: Two Categories

1 In Mean

– Oaxaca-Blinder(1974):OB

– Brown(1980): Brown

– Fairlie(1999): Fairlie2 In Distribution(Skipped)

– Juhn, Murphy and Pierce(1993): JMP

– Machado and Mata(2005): MM

– DiNardo, Fortin and Lemieux(1996): DFL

– Firpo, Fortin and Lemieux(2007,2010): FFL



Decomposition Methods: Pros and Cons

ProsIt is a naturally way to distengle cause and effect based on OLS orother Linear regressions.

ConsIn particular, decomposition methods inherently follow a partialequilibrium.The results cannot be fully explained as causal inference.



Decomposition Methods to Gaps

Although some of methods listed above is quite sophisicated andfrontier in the filed, the OB is so fundemental that all other methodscan explained by it.

Therefore,in our lecture, we will only cover OB and its extensionversions.


Oaxaca-Blinder Decomposition

Section 2




A naive way to identification gender gap

Use a dummy variable in a regression function

𝑌 = 𝛽0 + 𝛽1𝐷 + Γ𝑋′ + 𝑢

𝐷 = 1 denotes that the gender of the sample is male, and 𝐷 = 0denotes female.𝑋′ denotes a series control variables, thus personal characteristicssuch as education,working experience,etc.So if 𝛽1 is large enough and significant statistically,then the result can only answer to that question: “is there a wage gapbetween men and women in the labor market when other thingsequal(X)?”



Oaxaca-Blinder DecompositionAssume that a multiple OLS regression equation is

𝑌𝑖 = 𝛽0 + 𝛽1𝑋1𝑖 + ... + 𝛽𝑘𝑋𝑘𝑖 + 𝑢𝑖

where 𝑌𝑖 is dependent variable,𝑋𝑖𝑠 are a seriesindependent(controlling) variables which affect 𝑌𝑖.And 𝑢𝑖 are errorterms which satisfied by 𝐸(𝑢𝑖|𝑋1, ..., 𝑋𝑘) = 0The means of 𝑌𝑖

𝐸(𝑌 ) = 𝛽0 + 𝛽1𝐸(𝑋1) + ... + 𝛽𝑘𝐸(𝑋𝑘) + 𝐸(𝑢𝑖)

Using the sample estimator to replace the population parameters andconsidering the definition of error term, thus ∑ 𝑢𝑖 = 0, then

𝑌 = 𝛽0 + 𝛽1��1 + ... + 𝛽𝑘��𝑘



Oaxaca-Blinder Decomposition: Two groups

If we assume that whole sample can be divided into 2 groups: A andB,then we could regress the similar regression using A and Bsubsamples, repectively. Thus,

𝑌𝐴𝑖 = 𝛽𝐴0 + 𝛽𝐴1𝑋1𝑖 + ... + 𝛽𝐴𝑘𝑋𝑘𝑖 + 𝑢𝐴𝑖𝑌𝐵𝑖 = 𝛽𝐵0 + 𝛽𝐵1𝑋1𝑖 + ... + 𝛽𝐵𝑘𝑋𝑘𝑖 + 𝑢𝐵𝑖

Accordingly, we can obtain the means of outcome 𝑌 for group A andgroup B are

𝑌𝐴 = 𝛽𝐴0 + 𝛽𝐴

1 ��𝐴1 + ... + 𝛽𝐴𝑘 ��𝐴𝑘

𝑌𝐵 = 𝛽𝐵0 + 𝛽𝐵

1 ��𝐵1 + ... + 𝛽𝐵𝑘 ��𝐵𝑘



Oaxaca-Blinder Decomposition: Two groups

Denote��𝐴 = (1, ��𝐴1, ��𝐴2, ..., ��𝐴𝑘)

And𝛽𝐴 = ( 𝛽𝐴

0 , 𝛽𝐴1 , 𝛽𝐴

2 , ..., 𝛽𝐴𝑘 )

Then𝑌 𝐴 = 𝛽𝐴��′

𝐴

Denote as the same way

𝑌 𝐵 = 𝛽𝐵��′𝐵



Oaxaca-Blinder Decomposition: difference in meanThe difference in mean of 𝑌𝑖 of group A and B is

𝑌𝐴 − 𝑌𝐵 = 𝛽𝐴��′𝐴 − 𝛽𝐵��′

𝐵

A small trick: plus and minus a term 𝛽𝐵��′𝐴,then

𝑌𝐴 − 𝑌𝐵 = 𝛽𝐴��′𝐴 − 𝛽𝐵��′

𝐵

= 𝛽𝐴��′𝐴 − 𝛽𝐵��′

𝐴 + 𝛽𝐵��′𝐴 − 𝛽𝐵��′

𝐵

= ( 𝛽𝐴 − 𝛽𝐵)��′𝐴 + 𝛽𝐵(��′

𝐴 − ��′𝐵)

Then the second term is characteristics effect which describes howmuch the difference of outcome, 𝑌 , in mean is due to differences inthe levels of explanatory variables(characteristics).

the first term is coefficients effect which describes how much thedifference of outcome, 𝑌 , in mean is due to differences in themagnitude of regression coefficients.



A Classical Case: Gender Wage Gap

Male-female average wage gap can be attributed into two parts:1 Explained Part: due to differences in the levels of explanatory variables:

such as schooling years,experience,tenure,industry,occupation,etc

characteristics effectendownment effectcomposition effect

In the literature of labor economics, we think that the wage gap dueto this part is reasonable…



A Classical Case: Gender Wage Gap

Male-female average wage gap can be attributed into two parts:2 Unexplained Part:due to differences in the coefficients to explanatory

variables: such as returns to schooling years, experience and tenureand premium in industry and occupation,etc

coefficients effectreturns effectstructure effect

In the literature of labor economics, we think that the wage gap dueto this part is unreasonable,often it is called discrimination part…



Gustafusson and Li(2000): Gender gaps in China



Decomposition Methods to Gaps

OB Decomposition is a tool for separating the influences ofquantities and prices on an observed mean difference.

The aim of the OB decomposition is to explain

how much of the difference in mean outcomes across two groups isdue to group differences in the levels of explanatory variables,andhow much is due to differences in the magnitude of regressioncoefficients(Oaxaca 1973; Blinder 1973).

Although most applications of the technique can be found in thelabor market and discrimination literature, it can also be useful inother fields.

In general, the technique can be employed to study group differencesin any (continuous or categorical) outcome variable.


Reference group problem

Section 3




A different reference groupwhat if we use a different reference group: plus and minus a term

𝛽𝐴��′𝐵,then

𝑌𝐴 − 𝑌𝐵 = 𝛽𝐴��′𝐴 − 𝛽𝐵��′

𝐵

= 𝛽𝐴��′𝐴 − 𝛽𝐴��′

𝐵 + 𝛽𝐴��′𝐵 − 𝛽𝐵��′

𝐵

= ( 𝛽𝐴 − 𝛽𝐵)��′𝐴 + 𝛽𝐵(��′

𝐴 − ��′𝐵)

Then again the first term is characteristics effect or endownmenteffect as the amount of 𝑋𝑗 can be seen as an endownment for groupA or B.

The second term is coefficients effect or price(returns) effect asthe estimate coefficents 𝛽𝑗 can be seen as the market price of or thereturns to a certain 𝑋𝑗.

Question: is the result as same as the first decomposition?Zhaopeng Qu (Nanjing University) Lecture 8: Decomposition Method 11/12/2020 24 / 70



What is the ture coefficient or characteristics effect ?



Oaxaca-Blinder Decomposition: a general framework

Let 𝑌 ∗ be a nondiscriminatory potential outcome, so 𝛽∗ is such anondiscriminatory coefficient vector, and 𝑋 is still a vector of many𝑥(characteristics). Then they satisfy as a following equation

𝑌 ∗ = 𝑋𝛽∗ + 𝜖

where 𝜖 is the error term and satisfies 𝐸(𝜖|𝑋) = 0




Then the difference of the potential outcomes between two groupscan then be decomposed as follows

𝑌𝐴 − 𝑌𝐵 = ��′𝐴 𝛽𝐴 − ��′

𝐵 𝛽𝐵

= ��′𝐴 𝛽𝐴 − ��′

𝐴 𝛽∗ + ��′𝐴 𝛽∗−��′

𝐵 𝛽∗ + ��′𝐵 𝛽∗−��′

𝐵 𝛽𝐵

= (��′𝐴 − ��′

𝐵) 𝛽∗ + [��′𝐴( 𝛽𝐴 − 𝛽∗) + ��′

𝐵( 𝛽∗− 𝛽𝐵)]




The first term, (��′𝐴 − ��′

𝐵) 𝛽∗ is the explained part as usualcharacteristics effectendownment effectcomposition effect




The second term, the unexplained part can further be subdividedinto

1 “discrimination” in favor of group A(such as Men)

��′𝐴( 𝛽𝐴 − 𝛽∗)

2 “discrimination” against group B(such as Women)

��′𝐵( 𝛽∗ − 𝛽𝐵)

All variables are known but the nondiscriminatory coefficients 𝛽∗. Sohow to determine it?



Oaxaca-Blinder Decomposition: One referencegroup

Assume that discrimination is directed toward only one group.

Recall

𝑌𝐴 − 𝑌𝐵 = (��′𝐴 − ��′

𝐵) 𝛽∗ + [��′𝐴( 𝛽𝐴 − 𝛽∗) + ��′

𝐵( 𝛽∗− 𝛽𝐵)]

Assume that wage discrimination is directed only againstwomen(denoted as group B) and there is no (positive)discrimination(favor) of men(denoted as group A).

Then 𝛽∗ = 𝛽𝐴 and the wage gap can be decomposed into as

𝑌𝐴 − 𝑌𝐵 = (��′𝐴 − ��′

𝐵) 𝛽𝐴 + ��′𝐵( 𝛽𝐴 − 𝛽𝐵)



Oaxaca-Blinder Decomposition: One referencegroup

Similarly, if there is only (positive) discrimination(favor) of men butno discrimination of women, Then 𝛽∗ = 𝛽𝐵, and the decomposition is

𝑌𝐴 − 𝑌𝐵 = (��′𝐴 − ��′

𝐵) 𝛽𝐵 + ��′𝐴( 𝛽𝐴 − 𝛽𝐵)



Oaxaca-Blinder Decomposition: Weighted referencegroup

However, there is no specific reason to assume that the coefficients ofone or the other group are nondiscriminating.

So the value of 𝛽∗ should be a math combination of 𝛽𝐴 and 𝛽𝐵,Reimers(1983)therefore proposes using the average coefficients overboth groups as an estimate for the nondiscriminatory parameter vector;that is,

𝛽∗ = 0.5 𝛽𝐴 + 0.5 𝛽𝐵

Cotton (1988) suggests to weight the coefficients by the group sizes,𝑛𝐴 and 𝑛𝐵,

𝛽∗ = 𝑛𝐴𝑛𝐴 + 𝑛𝐵

𝛽𝐴 + 𝑛𝐵𝑛𝐴 + 𝑛𝐵

𝛽𝐵



Oaxaca-Blinder Decomposition: Weighted Matrix

More general,let 𝑊 be a 𝑘 + 1 diagonal matrix of weights, such that

𝛽∗ = 𝑊 𝛽𝐴 + (1 − 𝑊) 𝛽𝐵

Note here𝛽𝐴,𝐵 = ( 𝛽𝐴,𝐵

0 , 𝛽𝐴,𝐵1 , 𝛽𝐴,𝐵

2 , ..., 𝛽𝐴,𝐵𝑘 )



Oaxaca-Blinder Decomposition: Weighted MatrixThen the difference between two groups can be expressed as

𝑌𝐴 − 𝑌𝐵 = (��′𝐴 − ��′

𝐵)[𝑊 𝛽𝐴 + (𝐼 − 𝑊) 𝛽𝐵][(𝐼 − 𝑊)′��𝐴 + 𝑊��𝐵]( 𝛽𝐴 − 𝛽𝐵)]

𝑊 is a matrix of relative weights given to the coefficients of group A,and 𝐼 is the identity matrix.

e.g. If we choose 𝑊 = 𝐼 , then it is equivalent to setting

𝛽∗ = 𝛽𝐴

.e.g. If we choose 𝑊 = 0.5𝐼 , then it is equivalent to setting

𝛽∗ = 0.5𝛽𝐴 + 0.5𝛽𝐵

.Zhaopeng Qu (Nanjing University) Lecture 8: Decomposition Method 11/12/2020 34 / 70


Oaxaca-Blinder Decomposition: Weighted Matrix

Oaxaca and Ransom (1994) show that �� can be use followingeqation to estimate

�� = Ω = (𝑋′𝑋)−1(𝑋′𝐴𝑋𝐴)

Neumark(1988) also use the coeffients from a pooled model over bothgroups as the reference coefficients,thus

We pool all data and run a regression

𝑌 = 𝑋𝛽

Then 𝛽∗ can be obtained by

𝛽∗ = (𝑋′𝑋)−1(𝑋′𝑌 )



Oaxaca-Blinder Decomposition:Weighted(Continued)

However, Oaxaca and Ransom(1994) and Neumark(1998) caninappropriately transfer some of the unexplained parts of thedifferential into the explained component.Assume a simple OLS equation: 𝑌𝑖 on a single regressor 𝑋𝑖 and agroup specific intercepts 𝛼𝐴 and 𝛼𝐵

𝑌𝐴𝑖 = 𝛼𝐴 + 𝛾𝐴𝑋𝐴𝑖 + 𝑢𝐴𝑖𝑌𝐵𝑖 = 𝛼𝐵 + 𝛾𝐵𝑋𝐴𝑖 + 𝑢𝐵𝑖

Let 𝛼𝐴 = 𝛼 and 𝛼𝐵 = 𝛼 + 𝛿, where 𝛿 is the discriminationparameter. Then the model can also be expressed as

𝑌 = 𝛼 + 𝛾𝑋 + 𝛿𝐷 + 𝑢where D as an indicator for group B, such as “female” in gender wagegap case



Oaxaca-Blinder Decomposition: OVB and Weighted

Assume that 𝛾 > 0 (positive relation between X and Y) and 𝛿 < 0(discrimination against women).

The true model is𝑌 = 𝛼 + 𝛾𝑋 + 𝛿𝐷 + 𝑢

But if as Oaxaca and Ransom (1994) suggested,we only estimate

𝑌 = 𝛼 + 𝛾𝑋 + 𝑒

Then following the Omitted Variable Bias formula, we can obtain

𝛾∗ = 𝛾 + 𝛿 𝐶𝑜𝑣(𝑋, 𝐷)𝑉 𝑎𝑟(𝑋)



Oaxaca-Blinder Decomposition: OVB and Weighted

Then the explained part of the differential is

(��𝐴 − ��𝐵)𝛾∗ = (��𝐴 − ��𝐵)[𝛾 + 𝛿 𝐶𝑜𝑣(𝑋, 𝐷)𝑉 𝑎𝑟(𝑋) ]

Note: 𝛿,the discrimination parameter,which belongs to theunexplained parts of the gap,now attributes to the explained part ofthe gap.



Oaxaca-Blinder Decomposition:Weighted(Continued)

To address the OVB problem in decomposition, Jann(2008) suggestedestimate a pooled regression over both groups but controlling groupmembership(a dummy variable D), that is

𝑌 = 𝛽∗𝑋 + 𝛿𝐷 + 𝜀

In this case,𝛽∗ = ((𝑋, 𝐷)′(𝑋, 𝐷))−1(𝑋, 𝐷)′𝑌

And the coefficient effect or unexplained part of the difference is𝛿, which is the the coefficient of D in the pooled regression now.

The most widely used weighted method for OB decomposition rightnow.



Reference group: Wrap up

On the different circumstances, the value could be quite different.One reference group: A or B

Weighted reference groupsimple weight:weighted in matrix: Omega and Pooled

In practice,1 We could use one reference group method but both A and B. If the

two result are similar, then it is OK.2 If the simple method does not work(not similar), then we have to

adjusted the weight.


Detailed Decomposition

Section 4




Introduction

The detailed contributions of the single predictors or sets ofpredictors are subject to investigation.

For example, one might want to evaluate how much of the genderwage gap is due to differences in education and how much is due todifferences in work experience.

Similarly, it might be informative to determine how much of theunexplained gap is related to differing returns to education and howmuch is related to differing returns to work experience.



Detailed Decomposition: the explained part

Identifying the contributions of the individual predictors to theexplained part of the differential is relative easy.

Because the total component is a simple sum over the individualcontributions.Thus

(��𝐴 − ��𝐵)′ 𝛽𝐴 = (��1𝐴 − ��1𝐵) 𝛽1𝐴 + (��2𝐴 − ��2𝐵) 𝛽2𝐴 + ...

The first summand reflects the contribution of the group differencesin 𝑋1; the second, of differences in 𝑋2; and so on.



Detailed Decomposition: the unexplained part

the individual contributions to the unexplained part are thesummands in

��′𝐵( 𝛽𝐴 − 𝛽𝐵) = ( 𝛽0𝐴 − 𝛽0𝐵) + ( 𝛽1𝐴 − 𝛽1𝐵)��1𝐵 + ( 𝛽2𝐴 − 𝛽2𝐵)��2𝐵...



Detailed Decomposition: sets of covariates

Furthermore, it is easy to subsum the detailed decomposition by setsof covariates

the explained part of every set

(��𝐴 −��𝐵)′ 𝛽𝐴 =𝑎

∑𝑘=1

𝛽𝑘𝐴(��𝑘𝐴 −��𝑘𝐵)+𝑏

∑𝑗=𝑎+1

𝛽𝑗𝐴(��𝑗𝐴 −��𝑗𝐵)+...

the unexplained part of every set

��′𝐵( 𝛽𝐴− 𝛽𝐵) = ( 𝛽0𝐴− 𝛽0𝐵)+

𝑎∑𝑘=1

( 𝛽𝑘𝐴− 𝛽𝑘𝐵)��𝑘𝐵+𝑏

∑𝑗=𝑎+1

( 𝛽𝑗𝐴− 𝛽𝑗𝐵)��𝑗𝐵...


Standard Errors

Section 5

Standard Errors


Standard Errors

Introduction

The computation of the decomposition components is straightforward:

Estimate OLS models and insert the coefficients and the means of theregressors into the formulas.

For a long time, results from OB decompositions were reportedwithout information on statistical inference (standard errors,confidence intervals).

Without reporting s.e. or C.I is problematicbecause it is hard to evaluate the significance of reporteddecomposition results without knowing anything about their samplingdistribution.


Standard Errors

Standard Errors: Jan(2005)

Think of a term such as �� 𝛽,where �� is a row vector of samplemeans and 𝛽 is a column vector of regression coefficients (the resultis a scalar).

How can its sampling variance,𝑉 (�� 𝛽) be estimated?

Following Jann(2005), the sampling variance is

𝑉 𝑎𝑟(�� 𝛽) = ��𝑉 𝑎𝑟( 𝛽)��′ + 𝛽′𝑉 𝑎𝑟(��) 𝛽 + 𝑡𝑟𝑎𝑐𝑒[𝑉 𝑎𝑟(��)𝑉 𝑎𝑟( 𝛽)]


Standard Errors

Standard Errors: Jan(2005)

The last term, 𝑡𝑟𝑎𝑐𝑒[𝑉 𝑎𝑟(��)𝑉 𝑎𝑟( 𝛽)],will be asymptoticallyvanishing and can be ignored when 𝑛 is enough large.

To estimate 𝑉 𝑎𝑟(�� 𝛽), plug in estimates for 𝑉 𝑎𝑟( 𝛽) (thevariance-covariance matrix of the regression coefficients) and 𝑉 𝑎𝑟(��)(the variance-covariance matrix of the means), which are readilyavailable.


Standard Errors

Standard Errors: Jan(2005)Recall OB decomposition:

𝑌𝐴 − 𝑌𝐵 = ��𝐴( 𝛽𝐴 − 𝛽𝐵) + (��𝐴 − ��𝐵) 𝛽𝐵

So corresponding the first term’s variance is as follows𝑉 𝑎𝑟[��𝐴( 𝛽𝐴 − 𝛽𝐵)] = 𝑉 𝑎𝑟[��𝐴 𝛽𝐴 − ��𝐴 𝛽𝐵]

= 𝑉 𝑎𝑟[��𝐴 𝛽𝐴] − 𝑉 𝑎𝑟[��𝐴 𝛽𝐵]≈ ��𝐴[𝑉 ( 𝛽𝐴) + 𝑉 ( 𝛽𝐵)]��′

𝐴+( 𝛽𝐴 − 𝛽𝐵)′𝑉 ( 𝑋𝐴)( 𝛽𝐴 − 𝛽𝐵)

Similarly,𝑉 𝑎𝑟[(��𝐴 − ��𝐵) 𝛽𝐵] ≈ (��𝐴 − ��𝐵)𝑉 ( 𝛽𝐵)(��𝐴 − ��𝐵)′+

𝛽′𝐵𝑉 ( 𝑋𝐴 + 𝑋𝐵) 𝛽𝐵

Equations for other variants of the decomposition, for elements of thedetailed decomposition, and for the covariances among componentscan be derived similarly.


Introduction to bootstrap(Maybe Skipped)

Section 6




Introduction

In many cases where formulas for standard errors are hard to obtainmathematically, or where they are thought not to be very goodapproximations to the true sampling variation of an estimator, we canrely on a resampling method.

The general idea is to treat the observed data as a population that wecan draw samples from. The most common resampling method is thebootstrap.



Introduction

In short, the bootstrap takes the sample (the values of theindependent and dependent variables) as the population and theestimates of the sample as true values.

Instead of drawing from a specified distribution (such as the normal)by a random number generator, the bootstrap draws withreplacement from the sample.

It therefore takes the empirical distribution function as the truedistribution function.

The great advantage is that we neither make assumption about thedistributions nor about the true values of the parameters.



The Method: Nonparametric Bootstrap

actually there are several bootstrap method.

A very simple approach is to use the quantiles of the bootstrapsampling distribution of the estimator to establish the end points of aconfidence interval nonparametrically.



Bootstrap Standard Errors

The empirical standard deviation of a series of bootstrap replicationsof 𝛽 can be used to approximate the standard error 𝑠𝑒( 𝛽)

1 Draw B independent bootstrap samples (𝑌 ∗𝑖 , 𝑋∗

𝑖 ) of size N fromoriginal sample (𝑌𝑖, 𝑋𝑖).Usually 𝐵 = 100 replications are sufficient.

2 Estimate the parameter 𝛽 of interest for each bootstrap sample:

𝛽∗𝑏 𝑓𝑜𝑟 𝑏 = 1, 2, ..., 𝐵



Bootstrap Standard Errors3 Estimate 𝑠𝑒( 𝛽) by

𝑠𝑒𝐵𝑜𝑜𝑡( 𝛽) =√√√⎷

1𝐵 − 1

𝐵∑𝑏=1

( 𝛽∗𝑏 − 𝛽∗)2

where 𝛽∗ = 1𝐵 ∑𝐵

𝑏=1𝛽∗

𝑏, thus the average of the B bootstrapestimates.

When the estimator 𝛽 is consistent and asymptotically normallydistributed, bootstrap standard errors can be used to constructapproximate confidence intervals and to perform asymptotic testsbased on the normal distribution.

4 Then we can construct 95% confidence interval for 𝛽

[ 𝛽 − 1.96 × 𝑠𝑒𝐵𝑜𝑜𝑡( 𝛽), 𝛽 + 1.96 × 𝑠𝑒𝐵𝑜𝑜𝑡( 𝛽)]Zhaopeng Qu (Nanjing University) Lecture 8: Decomposition Method 11/12/2020 56 / 70


Bootstrap: Alternative way to construct ConfidenceIntervals

We can construct a two-sided equal-tailed 1 − 𝛼 confidence intervalfor an estimate 𝛽 from the empirical distribution function of a seriesof bootstrap replications.The 𝛼

2 and the 1 − 𝛼2 empirical percentiles of the bootstrap

replications are used as lower and upper confidence bounds. Thisprocedure is called percentile bootstrap.

1 Draw B independent bootstrap samples (𝑌 ∗𝑖 , 𝑋∗

𝑖 ) of size N fromoriginal sample (𝑌𝑖, 𝑋𝑖).Usually 𝐵 = 1000 replications are sufficient.

2 Estimate the parameter 𝛽 of interest for each bootstrap sample:

𝛽∗𝑏 𝑓𝑜𝑟 𝑏 = 1, 2, ..., 𝐵



Bootstrap: Confidence Intervals

3 Order the bootstrap replications of 𝛽 such that 𝛽∗1 ≤ ... ≤ 𝛽∗

𝐵.

The lower and upper confidence bounds are the 𝐵 × 𝛼2 𝑡ℎ and

𝐵 × (1 − 𝛼2 ) − 𝑡ℎ ordered elements, respectively.

For example, 𝐵 = 1000 and 𝛼 = 0.05, then these are the 25𝑡ℎ and975𝑡ℎ ordered elements.

The estimated 1 − 𝛼 confidence interval of 𝛽 is

[ 𝛽∗𝐵 𝛼

2, 𝛽∗

𝐵(1− 𝛼2 )]



Bootstrap: t-statistic

Review: Assume that we have consistent estimates of 𝛽 and 𝑠𝑒( 𝛽) athand and that the asymptotic distribution of the t-statistic is thestandard normal,thus

𝑡 =𝛽 − 𝛽0𝑠𝑒( 𝛽)

𝑑−→ 𝑁(0, 1)

Then we can calculate approximate critical values from percentiles ofthe empirical distribution of a series of bootstrap replications for thet- statistic.

1 Consistently estimate 𝛽 and 𝑠𝑒(𝛽) using the originally observedsample:

𝛽, 𝑠𝑒( 𝛽)



Bootstrap: t-statistic2 Draw B independent bootstrap samples (𝑌 ∗

𝑖 , 𝑋∗𝑖 ) of size N from

original sample (𝑌𝑖, 𝑋𝑖). Usually 𝐵 = 1000 replications are sufficient.

3 Estimate the t-value assuming 𝛽0 = 𝛽 for each bootstrap sample:

𝑡∗𝑏 =

𝛽∗𝑏 − 𝛽𝑠𝑒∗𝑏( 𝛽)

𝑓𝑜𝑟 𝑏 = 1, 2, ..., 𝐵

4 Order the bootstrap replications of t such that 𝑡∗1 ≤ ... ≤ 𝑡∗

𝐵.

The lower and upper confidence bounds are the 𝐵 × 𝛼2 − 𝑡ℎ and

𝐵 × (1 − 𝛼2 ) − 𝑡ℎ ordered elements, respectively.

For example, 𝐵 = 1000 and 𝛼 = 0.05, then these are the 25𝑡ℎ and975𝑡ℎ ordered elements.

So the critical values are𝑡 𝛼

2= 𝑡∗

𝐵 𝛼2, 𝑡1− 𝛼

2= 𝑡∗

𝐵(1− 𝛼2 )



Bootstrap: Standard Errors to Decomposition

What are the estimators in decomposition?

the unexplained part:��′

𝐵( 𝛽𝐴 − 𝛽𝐵)

the explained part:(��𝐴 − ��𝐵)′ 𝛽𝐴



Concluding Remarks: Bootstrap

If the bootstrap is so simple and of such broad application, why isn’tit used more in the social sciences?

Because the bootstrap is computationally intensive. This barrier tobootstrapping is more apparent than real.

When the outcome of one of many small steps immediately affectsthe next, rapid results are important.


Representative Applications

Section 7




Examples

1 Labor Economics: Wage or Income Gaps

Gender: Male-FemaleUrban-Rural(or Urban-Migrant)Minority-Majority(Racial Gaps)Poor-NonpoorPublic-Private Sectors gapsUnion-NonUnion gaps



章莉等 (2014): 中国劳动力市场上工资收入的户籍歧视



章莉等 (2014): 中国劳动力市场上工资收入的户籍歧视



Fortin and Lemiux(2011)



Examples

2 Other Fields:

Educational Performance: Fortin, Oreopoulos and Phipps(2017)Marketing: Liu et al(2016) “Movie Stars Effects”Family Origins: Li,Ling and Qu(2018)House Price:Health Status:



Concluding Remarks and Discussions:

OB decomposition can be easily extended in some nonlinearregression models.

But OB method decompose the gap only on the mean.

The result may depends on the choice of counterfactual fact if youneglect the reference group problem.

Intrinsically,a partial equilibrium approach to analyze a generalequilibrium question.

Question: how is extent to trust that the result have a causalexplanation in the decomposition?



Reference王美艳 (2005),“城市劳动力市场上的就业机会与工资差异——外来劳动力就业与报酬研究”，《中国社会科学》第 5 期，第 36-46 页。

章莉，李实，William Darity, Rhonda Sharpe(2014),“中国劳动力市场上工资收入的户籍歧视” 《管理世界》第 11 期，第 35-46 页。

Fortin,Nicole M.,Philip Oreopoulos and Shelley Phipps(2017),“Leaving Boys Behind Gender Disparities in High AcademicAchievement”,Journal of Human Resources,vol. 50(3),pp.549-579

Gustafsson,Björn & Shi Li(2000), “Economic transformation and thegender earnings gap in urban China”,Journal of PopulationEconomics, 13, pp.305–329.

Li, Shi, Xiaoguang Ling and Zhaopeng Qu(2018), “The Red and TheBlack in the 21st Century: The Long-Term Effects of ParentalSocioeconomic Status on Children’s Well-Beings in China”, Workingpaper

Liu,(Xia)Angela, Tridib Mazumdar, Bo Li(2014), “CounterfactualDecomposition of Movie Star Effects with Star Selection”,Management Science,61(7),pp.1473-1740


Documents

Lecture 8: Decomposition Method · 2020. 12. 30. · – Machado and Mata(2005): MM – DiNardo, Fortin and Lemieux(1996): DFL – Firpo, Fortin and Lemieux(2007,2010): FFL Zhaopeng