decomposition methods: the case of mean and beyond · 2015. 5. 6. · lecture on decomposition...
TRANSCRIPT
Decomposition Methods:
The Case of Mean and Beyond
Nicole M Fortin
Vancouver School of Economics
University of British Columbia
Lecture on Decomposition Methods
Relies heavily on the chapter “Decomposition Methods in Economics”
(joint with Thomas Lemieux and Sergio Firpo) in the recent Handbook of
Labor Economics (Volume 4A, 2011)
Uses the classic work of Oaxaca (1973) and Blinder (1973) for the mean
as its point of departure
Focuses on recent developments (last 15 years) on how to go beyond
the mean
Connection with the treatment effect literature
Procedures specific to the case of decompositions
Provides some examples to motivate the use of decomposition methods
Provides empirical illustrations and discusses applications
Suggests a “user guide” of best practices
Plan for the first part
Introduction
A few motivating examples
Oaxaca decompositions: a refresher
Issues involved when going “beyond the mean”
A formal theory of decompositions
Clarify the question being asked
Decompositions as counterfactual exercises
Formal connection with the treatment effect literature:
Identification: the central role of the ignorability (or
unconfoundedness) assumption
Estimators for the “aggregate” decomposition: propensity score
matching, reweighting, etc.
Introduction: motivating examples
1. Gender wage gap (Oaxaca-Blinder)
Construction of counterfactuals
2. Union wage gap
Link to the treatment effects literature
3. Understanding changes in wage inequality
The case for going beyond the mean
Gender Wage Gap
Oaxaca (1973) was the first study (along with Blinder, 1973) aimed
at understanding the sources of the large gender gap in wage.
Gender gap (difference between male and female average wages)
of 43 percent in the 1967 Survey of Economics Opportunity
Question: how much of the gender gap can be “explained” by male-
female differences in human capital (education and labor market
experience), occupational choices, etc?
The “unexplained” part of the gender gap is often interpreted as representing
labor market discrimination, though other interpretations(unobserved skills) are
possible too
Estimate OLS regressions of (log) wages on
covariates/characteristics/endowments
Use the estimates to construct a counterfactual wage such as
“what would be the average wage of women if they had the same
characteristics as men?”
Glass ceiling effects give a case for of going beyond the case
•If the proportion of men across ranks was identical to women, the overall
counterfactual average male salary would be: 31/100×152493.4 +
43.9/100×121483.4 + 25.1/100×106805.6 =127426.43, and the overall ratio would
be 120623.1/127426.43 (*100)= 94.66%.
• If the proportion of women across rank was identical to men, the overall
counterfactual average female salary would be: 51.8/100×146047.5 +
30.7/100×114594.9 + 17.6/100×99708.87 =128259.3, and the overall ratio would be
134955.3/128259.3(*100)=95.03%.
•Either way, more than 50% of the gap is accounted for by the gender differences in
the proportion of faculty members across rank.
Table 1. Average Professorial Salaries at UBC in 2010
Gender Rank Numbers % of All
% of women
Average Salary
Female/ Male Ratio
Men All 968 100.0 134955.3 0.89
Women All 419 100.0 30.2 120623.1
Men Full 501 51.8 152493.4 0.96
Women Full 130 31.0 20.6 146047.5
Men Associate 297 30.7 121483.4 0.94
Women Associate 184 43.9 38.3 114594.9
Men Assistant 170 17.6 106805.6 0.93
Women Assistant 105 25.1 38.2 99708.87
Union Wage Gap
Workers covered by collective bargaining agreement (minority of
workers in Canada, the U.S. or U.K.), or “union workers”, earn more
on average than other workers.
One can do a Oaxaca-type decomposition to separate the “true”
union effect from differences in characteristics between union and
nonunion workers. The “unexplained” part of the decomposition is
now the union effect.
Unlike the case of the gender wage gap, we can think of the union
effect as a treatment effect since union status is manipulable (can
in principle be switched on and off for a given worker)
Useful since tools and results from the treatment effect literature can
be used in the context of this decomposition.
Results from the treatment effect can also be used in cases where
the group indicator is not manipulable (e.g. men vs. women)
Classic example of variance decomposition!
Source: Fortin and Lemieux (1997)
Changes in Wage Inequality
Wage/earnings inequality has increased in many countries over the
last 30 years; in the 1990s we saw a polarization of earnings
Decomposition methods have been used to look for explanations for
these changes, such as:
Decline in unions and in the minimum wage
Increase in the rate of return to education
Technological change, international competition, etc.
We need to go “beyond the mean” which is more difficult than
performing a standard Oaxaca decomposition for the mean.
Active area of research over the last 20 years. A number of
procedures are now available, a few examples:
Juhn, Murphy, and Pierce (1993): Imputation method
DiNardo, Fortin, and Lemieux (1996): Reweighting method
Machado and Mata (2005): Quantile regression-based method
-.2
-.1
0.1
.2
Log
Wag
e D
iffe
ren
tial
0 .2 .4 .6 .8 1Quantile
Observed 1988/90-1976/78
Smoothed
Observed 2000/02-1988/90
Smoothed
Figure 1. Changes in Real ($1979) Log Male Wages by Percentiles –
MORG-CPS
Refresher on Oaxaca decomposition We want to decompose the difference in the mean of an
outcome variable Y between two groups A and B
Groups could also be periods, regions, etc.
Postulate linear model for Y, with conditionally independent
errors (E(υ|X)=0):
The estimated gap
Can be decomposed as
Technical note (algebra)
To get the Oaxaca decomposition start with (in simplified notation,
where subscripted letters stand for means: YA=∑A 1/NA Yi )
ΔO=YB-YA
Now use the regression models to get:
ΔO= XBβB - XAβA
and then add and subtract XBβA :
ΔO = (XBβB - XBβA) + (XBβA - XAβA)
This yields
ΔO = XB(βB - βA) + (XB - XA) βA
= ΔS + ΔX
Some terminology
The lecture focuses on wage decompositions
The “explained” part of the decomposition will be called the
composition effect since it reflects differences in the distribution of
the X’s between the two groups
The “unexplained” part of the decomposition will be called the wage
structure effect as it reflects differences in the β’s, i.e. in the way
the X’s are “priced” (or valued) in the labor market.
Studies sometimes refer to “price” (β’s) and “quantity” (X’s) effects
instead.
In the “aggregate” decomposition, we only divide Δ into its two components ΔS (wage structure effect) and ΔX (composition effect).
In the “detailed” decomposition we also look at the contribution of each individual covariate (or corresponding β)
A few remarks We focus on this particular decomposition, but we could also change
the reference wage structure
Or add an interaction term:
Does not affect the substance of the argument in most cases.
The “intercept” component of ΔS, βB0- βA0, is the wage structure effect for the base group. Unless the other β’s are the same in group A and B, βB0- βA0 will arbitrarily depend on the base group chosen.
Well known problem, e.g. Oaxaca and Ransom (1999).
Oaxaca decompositions in Stata
Main command is “oaxaca.ado” (due to Ben Jann, U of
Zurich, has to be downloaded)
Runs the regressions, computes the means, and
computes the elements of the decomposition along with
standard errors (see A Stata implementation of the
Blinder-Oaxaca decomposition)
We provide all the programs used to produce the tables
in the Handbook chapter at:
http://faculty.arts.ubc.ca/nfortin/datahead.html
Here is an example of how this works for a standard male-female
decomposition (Table 3 in our Handbook chapter from O’Neill
and O’Neill (2006) using data from the NLSY79 in 2000)
*** Table 3, Column 1;
oaxaca lropc00 age00 msa ctrlcity north_central south00 west
hispanic black sch_10 diploma_hs ged_hs smcol bachelor_col
master_col doctor_col afqtp89 famrspb wkswk_18
yrsmil78_00 pcntpt_22 manuf eduheal othind,
by(female) weight(1)
/*weight(1) uses the group 1 coefficients as the reference
coefficients, and weight(0) uses the group 2 coefficients */
detail(groupdem:age00 msa ctrlcity north_central south00
west hispanic black,
groupaf:afqtp89,
grouped:sch_10 diploma_hs ged_hs smcol bachelor_col
master_col doctor_col ,
groupfam:famrspb,
groupex:wkswk_18 yrsmil78_00 pcntpt_22 ,
groupind: manuf eduheal othind) ;
Here the base group is white, sch10-12, primary sector,
Blinder-Oaxaca decomposition Number of obs = 5309
1: female = 0
2: female = 1
------------------------------------------------------------------------------
lropc00 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Differential |
Prediction_1 | 2.762557 .0106598 259.16 0.000 2.741664 2.78345
Prediction_2 | 2.529257 .0100367 252.00 0.000 2.509585 2.548928
Difference | .2333003 .0146413 15.93 0.000 .2046039 .2619967
-------------+----------------------------------------------------------------
Explained |
groupdem | .0115371 .0032919 3.50 0.000 .0050851 .0179891
grouped | -.0124049 .0055175 -2.25 0.025 -.023219 -.0015907
groupaf | .0108035 .0034414 3.14 0.002 .0040584 .0175486
groupfam | .0328186 .0106373 3.09 0.002 .0119698 .0536674
groupex | .137095 .0112599 12.18 0.000 .1150259 .159164
groupind | .0174583 .0061707 2.83 0.005 .005364 .0295526
Total | .1973076 .0180079 10.96 0.000 .1620128 .2326024
-------------+----------------------------------------------------------------
Unexplained |
groupdem | -.0978872 .2338861 -0.42 0.676 -.5562956 .3605212
grouped | .0454348 .0344576 1.32 0.187 -.0221009 .1129705
groupaf | .0026284 .023485 0.11 0.911 -.0434014 .0486582
groupfam | .0025869 .0174562 0.15 0.882 -.0316266 .0368005
groupex | .0475104 .0616535 0.77 0.441 -.0733281 .168349
groupind | -.0920156 .03294 -2.79 0.005 -.1565769 -.0274543
_cons | .1277349 .2127739 0.60 0.548 -.2892944 .5447641
Total | .0359927 .0185897 1.94 0.053 -.0004425 .0724279
------------------------------------------------------------------------------
Problems when going beyond the mean
The case of the mean is simple because we can use the law of
conditional expectations, and then plug-in sample estimates
In non-linear models, things are also
more complicated
Problems when going beyond the mean
Things get more complicated in cases such as the variance
Using the analysis of variance formula, we can write the
unconditional variance as
So that the decomposition, where ,
has to include interactions terms between the covariates,
And could become even more complicated for more general
distributional statistics such as quantiles, the Gini coefficient, etc.
Back to identification: what can we
estimate using decompositions? Computing a Oaxaca decomposition is easy enough
Run OLS and compute mean values of X for each of the two groups
In Stata, “oaxaca.ado” does that automatically and also yields standard
errors that reflect the fact both the β’s and the mean values of the X’s
are being estimated.
But practitioners often wonder what we are really estimating for the
following reasons (to list a few):
What if some the X variables (e.g. years of education) are endogenous?
What if there is some selection between (e.g. choice of being unionized or not) or
within (participation rates for men and women) the two groups under
consideration?
What about general equilibrium effects? For example if we increase the level of
human capital (say experience) of women to the level of men, this may depress
the return to experience and invalidate the decomposition
Key question: Can we construct a valid counterfactual?
Recall that decompositions are based on
counterfactuals Recall that we can write:
ΔO=YB-YA=(YB-YC) + (YC-YA),
where YC is the counterfactual wage members of group B (say
women) would earn if they were paid like men (group A). Under the
linearity assumption (of the regression model), we get:
YA= XAβA, YB= XBβB, and YC= XBβA.
We then obtain the Oaxaca decomposition as follows:
ΔO = (XBβB - XBβA) + (XBβA - XAβA)
= XB(βB - βA) + βA(XB - XA)
= ΔS + ΔX
So with an estimate of the counterfactual YC one can compute the
decomposition. This is a general point that holds for all
decompositions, and not only for the mean
Counterfactual and treatment effects Generally speaking, one can think of treatment effects as the
difference between how people are actually paid, and how they
would be paid under a different “treatment”.
Take the case of unions.
YA : Average wage of non-union workers
YB : Average wage of union workers
YC : Counterfactual wage union members would earn if the were not
unionized
ΔS = YB-YC = “union effect” on union workers is an average treatment
effect on the treated (ATET)
This means that the identification/estimation issue involved in the
case of decomposition/counterfactual are the same as in the case of
treatment effects.
This is very useful since we can then use results from the treatment
effect literature.
The wage structure effect (ΔS) can be
interpreted as a treatment effect
The conditional independence assumption (E(ε|X)=0) usually invoked in Oaxaca decompositions can be replaced by the weaker ignorability assumption (Dg ╨ ε|X) to compute the aggregate decomposition
For example, ability (ε) can be correlated with education (X) as long as the correlation is the same in groups A and B.
This is the standard assumption used in “selection on observables” models where matching methods are typically used to estimate the treatment effect.
Main result: If we have YG=mG(X, ε) and ignorability, then:
ΔS solely reflects changes in the m(.) functions (ATET)
ΔX solely reflects changes in the distribution of X and ε (ignorability key for this last result).
The wage structure effect (ΔS) can be
interpreted as a treatment effect A number of estimators for ATET= ΔS have been proposed in the
treatment effect literature
Inverse probability weighting (IPW), matching, etc.
Formal results exist, e.g. IPW is efficient for
ATET (Hirano, Imbens, and Ridder, 2003)
Quantile treatment effects (Firpo, 2007)
This has been widely used in the decomposition literature since
DiNardo, Fortin, and Lemieux (1996).
Formal derivation of the identification
result (Handbook chapter)
Formal derivation (2)
Formal derivation (3)
Some intuition about Proposition 1
The wage setting model we use, Yg =mg (X,ε) is very general
Includes the linear model yg = xβg + ε as a special case
There are three reasons why wages can be different between
groups g=A and g=B:
Differences in the wage setting equations ma (.) and mb (.)
Differences in the distribution of X for the two groups
Differences in the distribution of ε for the two groups
The ignorability assumption states that the distribution of ε given X is
the same for the two groups, though this does not mean that
E(ε|X)=0.
So once we control for differences between the X’s in the two
groups, we also implicitly control for differences in the ε’s.
Only source of difference left is, thus, differences in the wage
structures ma (.) and mb (.).
A few caveats discussed in the chapter
This general result (Proposition 1) only works for the aggregate
decomposition. More assumptions have to be imposed to get at the
detailed decomposition.
We are implicitly ruling out general equilibrium effects under
assumption (simple counterfactual treatment).
For instance, in the absence of unions, wages in the nonunion
sector may change as firms are no longer confronted with the threat
of unionization
The wage structure observed among nonunion workers, ma (.), is no
longer a valid counterfactual for union workers.
This is closely related to the difference between union wage gap (no
general equilibrium effects) and union wage gain (possible general
equilibrium effects) discussed by H. Gregg Lewis decades ago
Plan for the second part: Going beyond the mean
Examples of distributional questions
Quantile regressions: analogy with standard regressions
doesn’t work
Constructing counterfactual distributions
Aggregate decomposition: reweighting procedure
Detailed decomposition:
Conditional reweighting
Quantile regression based decomposition (Machado-Mata)
Distributional regressions (Chernozhukov et al.)
RIF-regression (Firpo, Fortin, and Lemieux)
Empirical application and implementation in Stata
Example of distributional questions
Glass ceiling: Is the gender gap larger in the upper part of the
distribution (e.g. Albrecht and Vroman, 2004)?
What explains the growth in earnings inequality in the United States
(large literature) and other countries?
Changes over time in the world distribution of income (twin peaks,
etc.)
Changes or differences in the distribution of firm size.
A popular way to address these issues has been to look log wage
differentials at different percentiles
0.1
.2.3
.4
Dif
fere
nti
al i
n L
og T
ota
l C
om
pen
sati
on
0 .2 .4 .6 .8 1Quantile
Raw Gap
Mean Gap
Gender glass ceiling effect in executive
compensation (ExecuComp data)
The analogy with quantile regressions
does not work Tempting to run quantile regressions (say for the median) and
perform a decomposition as in the case of the mean (Oaxaca)
Does not work because there are two interpretations to β for the mean
Conditional mean: E(Y|X) = Xβ
Uncond. mean (LIE): E(Y) = EX(E(Y|X)) = EX(X)β
But the LIE does not work for quantiles
Conditional quantile: Qτ(Y|X) = Xβτ
Uncond. quantile: Qτ ≠ EX(Qτ(Y|X)) = EX(X)βτ
Only the first interpretation works for βτ, which is not useful for decomposing unconditional quantiles (unless one estimates quantiles regressions for all quantiles as in Machado-Mata)
Constructing counterfactual distributions
We can construct the counterfactual distribution as follows:
Or in simpler notation:
The idea is to integrate group A’s conditional distribution of Y given
X over group B’s distribution of X
Constructing counterfactual distributions
There are two general ways of estimating the counterfactual
distribution
First approach is to start with group B and replace the conditional
distribution FYB|XB(y|X) with FYA|XA
(y|X)
This requires estimating the whole conditional distribution.
Machado and Mata (2005) do so by estimating quantile
regressions for all quantiles, and inverting back
Second approach is the opposite. Start with group A but replace the
distribution of X for group A (FXA(X)) by the distribution of X for group
B (FXB(X)).
This is simpler since this only depends on X, while the conditional
distribution depends on both X and Y.
Even better, all we need to do is to compute a reweighting factor
Constructing counterfactual distributions
This is the approach suggested by DiNardo, Fortin and Lemieux
(1996)
The reweighting factor can be estimated using a simple logit (or
probit) model since after some manipulations we get
To estimate Prob(DB=1|X), pool the two groups and estimate a logit
for the probability of belonging to group B as a function of X
•The reweighting procedure is identical the weights substitution done earlier
•The average wage of women can be written as
31/100×146047.5 + 43.9/100×114594.9 + 25.1/100×99708.87 =120623.1
The reweighting formula would be:
31*(51.8/31)/100×146047.5 + 43.9*(30.7/43.9)/100×114594.9 +
25.1*(17.6/25.1)/100×99708.87 =128259.3
which gives use overall counterfactual average female salary, if the proportion of
women across rank was identical to men:
51.8/100×146047.5 + 30.7/100×114594.9 + 17.6/100×99708.87 =128259.3
Table 1. Average Professorial Salaries at UBC in 2010
Gender Rank Numbers % of All
% of women
Average Salary
Female/ Male Ratio
Men All 968 100.0 134955.3 0.89
Women All 419 100.0 30.2 120623.1
Men Full 501 51.8 152493.4 0.96
Women Full 130 31.0 20.6 146047.5
Men Associate 297 30.7 121483.4 0.94
Women Associate 184 43.9 38.3 114594.9
Men Assistant 170 17.6 106805.6 0.93
Women Assistant 105 25.1 38.2 99708.87
Reweighting in DFL for change in unionization rate
where group A is 1988 and group B is 1979
Reweighting in DFL for all X’s:
Reweighting procedure
Reweighting procedure:
a few lines of code *include a lot of non-linear interactions;
local edvar "sch_10 diploma_hs ged_hs smcol bachelor_col master_col
doctor_col";
foreach kv of local edvar {;
gen `kv'afqt=`kv‘*afqtp89; gen `kv'exp=`kv'*wkswk_18;
}; gen expafqt=afqtp89*wkswk_18; gen expsq=wkswk_18^2;
gen yrsmilsq=yrsmil78_00^2;
***probit for male;
probit male age00 msa ctrlcity north_central south00 west hispanic black
schl00 sch_10* diploma_hs* ged_hs* smcol* bachelor_col* master_col*
doctor_col* afqtp89 expafqt wkswk_18 expsq yrsmil78_00 yrsmilsq
pcntpt_22 manuf eduheal othind if female==0 | female==1 ;
predict pmale, p;
*/ top code weights if necessary */
summ pmale if male~=1, detail;
replace pmale=0.99 if pmale>0.99 & male~=1;
gen phix=(pmale)/(1-pmale)*((1-pbar)/pbar) if female==2;
Reweighting procedure
Very easy to use regardless of the distribution statistics being
considered (variance, Gini coefficient, inter-quartile range, etc.)
Just compute the statistic for group A, group B, and group A
reweighted to have the distribution of X of group B.
Example: in the case of the Gini coefficient, this yields estimates of
GiniA, GiniB, and GiniAC
We can then estimate the composition effect as:
ΔX = GiniAC - GiniA
And the wage structure effect as
ΔS = GiniB – GiniAC
Standard errors can be computed using a bootstrap procedure
(bootstrap the whole procedure starting with the logit or probit)
Going beyond the mean is a “solved”
problem for the aggregate decomposition Can directly apply non-parametric methods (reweighting, matching,
etc.) from the treatment effect literature.
Ignorability is crucial, but mG(X, ε) does not need to be linear
Inference by bootstrap or analytical standard errors in the case of reweighting (“generated regressor” correction required)
Reweighting very easy to use with large and well behaved (no support problem) data sets.
Occasionally top-coding weight may be needed
Good idea to check the fit!
Densities of male, female wages and female
reweighted as male (NLSY79)
In this small sample, reweighting is not so successful!
Going beyond the mean is more difficult
for the detailed decomposition Until recently, there were only a few partial (and not always fully
satisfactory) ways of performing a detailed decomposition for general distributional measures (quantiles in particular):
DFL conditional reweighting for the components of ΔX linked to dummy covariates (e.g. unions)
Machado-Mata quantile regressions for components of ΔS.
Sequential DFL-type reweighting (Altonji, Bharadwaj, and Lange, 2008) and adding one covariate at a time. Sensitive to order used as in a simple regression.
Can use Gelbach (2009) last-in type of approach, but may not add up.
A more promising approach is to estimate for proportions, and invert back to quantiles.
RIF-regressions of Firpo, Fortin and Lemieux (2009) is one easier possible way of doing so
Decomposing proportions is easier than
decomposing quantiles Easy to do a decomposition by running LP models for the
probability of earning less (or more) than say $17/hrs, and compute a Oaxaca counterfactual on the proportions:
We may find 50% of men earn more than $17/hrs, while only 35% of women do, but if women “were paid as men”, perhaps 45% of women would earn more than $17/hrs
By contrast, much less obvious how to decompose the difference between the 50th quantile for men ($17/hrs) and women (say $14/hrs)
But the function linking proportions and quantiles is the cumulative distribution.
Counterfactual proportions → Counterfactual cumulative → Counterfactual quantiles
Can be illustrated graphically
Figure 1: Relationship Between Proportions and Quantiles
0
1
0 4
Y (quantiles)
Pro
po
rtio
n (
cu
mu
l. p
rob
. (F
(Y))
Men (A)
Women (B)
Q.5
.5
.1
Q.9Q.1
.9
Counterfactual
proportions computed
using LP model
Figure 3: Inverting Globally
0
1
0 4
Y (quantiles)
Pro
po
rtio
n (
cu
mu
l. P
rob
. (F
(Y))
Men (A)
Women (B)
Q.5
.5
.1
Q.9Q.1
.9
Counterfactual
proportions computed
using LP model
Figure 2: RIF Regressions: Inverting Locally
0
1
0 4
Y (quantiles)
Pro
po
rtio
n (
cu
mu
l. p
rob
. (F
(Y))
Q.5
.5
Counterfactual
quantile (median)
.1
Q.9Q.1
.9
Counterfactual
proportion
Slope of the
cumulative
(density)
Decomposing proportions is easier than
decomposing quantiles Firpo, Fortin, and Lemieux (2009)
Estimate Recentered Influence Function (RIF) regressions
For the τ-th quantile of the distribution, qτ , we have :
RIF(y;)= qτ + IF(Y; qτ) = qτ + I [τ –1(Y≤ qτ) ]/f(qτ)
Run LP models (or logit/probit) for being below a given quantiles, and divide by density (slope of cumulative) to locally invert.
Dependent variable is dummy 1(Y<Qτ) divided by density → influence function for the quantile.
RIF approach works for other distributional measures (Gini, variance, etc.)
Chernozhukov, Fernandez-Val and Melly (CFVM) (2009)
Estimate “distributional regressions” (LP, logit or probit) for each value of Y (say at each percentile)
Invert back globally to recover counterfactual quantiles
Computing the RIF: a few lines of code
forvalues qt = 10(40)90 {; gen rif_`qt'=.; };
*get the quantile and the kernel density estimates
pctile eval1=lropc00 if female==1 , nq(100) ;
kdensity lropc00 if female==1, at(eval1) gen(evalf densf);
forvalues qt = 10(40)90 { ;
local qc = `qt'/100.0;
*compute qτ+[τ –1(Y≤ qτ) ]/f(qτ)
replace rif_`qt'=evalf[`qt']+`qc'/densf[`qt'] if
lropc00>=evalf[`qt'] & female==1;
replace rif_`qt'=evalf[`qt']-(1-`qc')/densf[`qt'] if
lropc00<evalf[`qt']& female==1;
};
Application to change in wage inequality
in the United States
Wage dispersion measured using “90-10” gap (difference between
90th and 10th percentile of log wage), Gini, etc…
Table 5: aggregate decomposition comparing DFL, CFVM, and RIF-
regressions
Table 6: detailed decomposition using RIF and FFL:DFL+RIF
Aggregate Decomposition
FFL Methodology: RIF-regressions+ Reweighting
Because ν(F) = EX [E[RIF(y; ν) |X = x]] = E[X|T = t] γν , by analogy with the Oaxaca decomposition, we can write
ΔνO = E [X|T = 1](γν1- γ
ν0)+ (E [X|T = 1] -E [X|T = 0]) γν0
If we are concerned that the linearity assumption may not hold, we can use the reweighted sample
ΔνO,R = E [X|T = 1](γν1- γ
ν01) + (E [X|T =1]- E [X0|T = 1]) γν01
ΔνS,P Δ
νS,e
+ (E [X0|T = 1] -E [X|T = 0]) γν0 + E [X0|T = 1](γν01 - γ ν0)
ΔνX ,P Δ
νX,e
It is like running two Oaxaca-Blinder decompositions on RIF(y; ν) OB1) with sample 1 and sample 01 to get the pure wage structure
effect ΔνS,P ,
OB2) with sample 0 and sample 01 to get the pure composition effect Δν
X ,P.
FFL Methodology: RIF-regressions+ Reweighting
The wage structure effect: ΔνS,R = E [X|T = 1](γν1- γ
ν01) + Δν
S,e
where γν01 are the period 1 coefficients estimated in sample 0 reweighted to look like period 1,
The reweighting error ΔνS,e =(E [X|T =1]- E [X0|T = 1]) γν01
goes to zero in a large sample, where E [X0|T = 1] is the mean of the reweighted sample. Nice way to check reweighting spec!
The composition effect: ΔνX ,R = (E [X0|T = 1] -E [X|T = 0]) γν0+ Δν
X,e
where γν01 are the period 1 coefficients estimated in sample 0 reweighted to look like period 1.
The specification error ΔνX,e =E [X0|T = 1](γν01 - γ ν0)
corresponds to the difference in the composition effects estimated by reweighting and RIF regressions, where E [X0|T = 1] is the mean of the reweighted sample.
Detailed Decomposition
Detailed Decomposition
Standard errors are obtained by bootstrapping the entire procedure,
typical of models involving quantiles, i.e. dependent variable is an
estimated variable
Example of bootstrapping procedures, as well as all programs and
data for Handbook chapter are downloadable at
http://faculty.arts.ubc.ca/nfortin/datahead.html
The “rifreg.ado” routine, that include RIF for variance and Gini, is
also available there