Empirical likelihood inference forOaxaca-Blinder decomposition

Taisuke Otsu and Shiori TanakaApril 13, 2022
KEO Discussion Paper No. 168

KEO Discussion Paper No. 168

EMPIRICAL LIKELIHOOD INFERENCE FOR OAXACA-BLINDERDECOMPOSITIONTAISUKE OTSU AND SHIORI TANAKA
Abstract. This paper proposes an empirical likelihood inference method for the Oaxaca-Blinderdecompositions. In contrast to the conventional Wald statistic using the delta method, our approachcircumvents the linearization errors and estimation of the variance terms. Furthermore, the shapeof the resulting empirical likelihood confidence set is determined flexibly by data. Simulation results
illustrate usefulness of the proposed inference method.

1. Introduction
Since the seminal works by Oaxaca (1973) and Blinder (1973), the Oaxaca-Blinder (hereafter,OB) decomposition has been widely used in empirical economic analyses to investigate differencesin mean outcomes between groups (see, For tin, Lemieux and Firpo, 2011, for an overview). The
OB decompositions are typically obtained as products of (differences of) the OLS estimators andaverages of regressors. Statistical inference on the OB decompositions is usually conducted basedon their standard errors, which are derived by the delta method; see, Oaxaca and Ransom (1998)and Jann (2008) for the cases of non-stochastic and stochastic regressors, respectively.
0 (β̂ − β̂ ), where β̂ andTo fix the idea, consider a typical form of the OB decomposition θ̂ = X̄BABAβ̂B are the OLS estimators of the groups A and B, respectively, and X̄B is the sample mean of theregressors of the group B. As shown in Jann (2008), the standard error of θ̂ is obtained by applying
the delta method and estimating the variance components Var(X̄B ), Var(β̂A ), and Var(β̂B ). As iswell known in the econometrics literature (e.g., Gregory and Veal, 1985), the t or Wald test basedon the delta method is not invariant to formulations of nonlinear parameter hypotheses and theapproximation errors by the delta method may not be negligible in finite samples. Intuitively, eventhough X̄B and β̂A − β̂B are well approximated by normal distributions, their product θ̂ may be far
from normal in finite samples.
To address this issue on the conventional inference method, we adapt the method of empirical likelihood by Owen (1988) to conduct inference on the OB decomposition parameters. In particular, weconstruct the empirical likelihood functions of two groups that involve their regression parameters,
and then combine these likelihood functions to impose the null hypothesis on the OB decomposition1

parameters of interest. Our empirical likelihood ratio statistic converges to the chi-squared distribution at the true parameter value so that we can conduct hypothesis testing and interval estimation byusing the chi-squared critical values. Although our empirical likelihood inference is computationallymore expensive than the conventional inference based on the t or Wald statistic, there are several
advantages for our method. First, the proposed empirical likelihood inference does not rely on thedelta method so that it is free from the linearization error and invariant to formulations of nonlinear
hypotheses. Second, since the empirical likelihood ratio is asymptotically pivotal, there is no needto estimate the variance components, such as Var(X̄B ), Var(β̂A ), and Var(β̂B ) for θ̂. Third, theempirical likelihood ratio-based confidence set is range preserving and transformation respecting.
Fourth, in contrast to the Wald-type confidence set whose shape is constrained to be symmetricaround the point estimate, the shape of the empirical likelihood ratio-based confidence set is naturally determined by data. Finally, the empirical likelihood inference can easily accommodate sideinformation on data, such as knowledge on some moments or additional instruments.
To illustrate advantages of the proposed method, we conduct a simulation study to investigate thesize and power properties. Our simulation results show that our empirical likelihood test for the OB
decomposition parameters shows similar size properties as the conventional t test across different
setups. On the other hand, the proposed empirical likelihood and conventional t tests exhibit rather
different power properties. In particular, when the error term follows an asymmetric distribution
such as log-normal, then the empirical likelihood test shows higher power for the negative sidewhich contains zero and is economically more important. This result illustrates that our empirical
likelihood inference can be a useful complement to the conventional one.

2. Main ResultA
B
Let {YAi , XAi }ni=1
and {YBi , XBi }ni=1
be two independent and identically distributed (iid) samples

for the groups A and B, where YA and YB are scalar outcome variables and XA and XB are kdimensional regressors. For each group, consider the regression modelY` = X`0 β` + ` ,for ` = A, B, where β` is a k-dimensional vector of parameters and the error term ` is independentof X` and has zero mean. One common way to decompose the difference of the means of the groupoutcomes is
E[YA ] − E[YB ] = (E[XA ] − E[XB ])0 βA + E[XB ]0 (βA − βB ),where the first term is the outcome difference explained by the difference in the regressors, and thesecond term is the unexplained part (often attributed to discrimination). Hereafter, we focus on
inference for the second term θ = E[XB ]0 (βA − βB ). However, our empirical likelihood approach canbe easily extended to conduct inference on the first term or other decompositions as far as they are
expressed by (nonlinear) functions of regression parameters and means of regressors.
2Let X̄` be the sample mean of X` , and β̂` be the OLS estimator for the regression of Y` on X` .
0 (β̂ − β̂ ). By applying the delta method, Jann
Then the point estimator of θ is given by θ̂ = X̄BAB(2008) suggested to construct the t-ratio by estimating the asymptotic variance of θ̂ as0 dd X̄B )(β̂A − β̂B ).
d θ̂) = X̄Bd β̂B )}X̄B + (β̂A − β̂B )0 Var(Var({Var(β̂A ) + Var(0 (β̂ − β̂ ) aroundWe note that this variance estimator is derived via linearization of θ̂ = X̄BABd β̂A ), Var(d β̂B ), and Var(d X̄B ).
(X̄B , β̂A , β̂B ) = (E[XB ], βA , βB ) and requires the variance estimators Var(This paper proposes an alternative inference method for θ = E[XB ]0 (βA − βB ) based on empirical
Alikelihood. For the sample {YAi , XAi }ni=1of the group A, by using the moment condition E[XA (YA −0 β )] = 0, the empirical likelihood function for β is obtained as
XAAA`A (βA ) = −2 maxnnAX

A{pAi }i=1i=1nA
X

s.t. pAi ≥ 0,log(nA pAi ),pAi = 1,i=1nA
X

0pAi XAi (YAi −XAi
βA ) = 0.

i=1

By the Lagrange multiplier method, its dual form is written as`A (βA ) = 2 supnAX

λA i=10log(1 + λ0A XAi (YAi − XAiβA )),and we employ this dual form to compute `A (βA ) in practice. We next construct the empirical
Blikelihood function for the sample {YBi , XBi }ni=1
of the group B. Since θ involves βB and µB = E[XB ],the empirical likelihood function for the group B is constructed as`B (βB , µB ) = −2 max
nnBXB{pBj }j=1
j=1

pBj ≥ 0,log(nB pBj ),nB
X

pBj = 1,j=1

nB
X"pBj0 β )
XBi (YBi − XBi
B

XBi − µB

j=1and its dual form is`B (βB , µB ) = 2 supnBX"log

λB j=1

1 + λ0B0 β )
XBi (YBi − XBi
B

XBi − µB#!
.

By combining the empirical likelihood functions `A (βA ) and `B (βB , µB ), we obtain the (profile)empirical likelihood function for θ as
L(θ) ={`A (βA ) + `B (βB , µB )}.

min(βA ,βB ,µB ):θ=µ0B (βA −βB )The asymptotic property of L(θ) is obtained as follows.
AB
Proposition. Suppose {YAi , XAi }ni=1
and {YBi , XBi }ni=1
are independent and identically distributedAB(iid) sequences with finite fourth moments. Also, {YAi , XAi }ni=1
and {YBi , XBi }ni=1
are independent.

ThendL(θ) → χ21 ,as nA , nB → ∞ with nA /nB → c ∈ (0, ∞).
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#= 0,

This proposition can be shown as follows. First, by adapting the proof of Jing (1995, The0 β )” andorem2) (i.e., replacing# “Xi − µx ” and “Yi − µy ” in Jing (1995) with “XAi (YAi − XAiA"0 β )
XBi (YBi − XBidB“”, respectively), we obtain `A (βA ) + `B (βB , µB ) → χ23k . Second, since
XBi − µB
θ = µ0B (βA − βB ) is a smooth function of (βA , βB , µB ), the argument in Hall and LaScala (1990,Theorem 2.1) applied to `A (βA ) + `B (βB , µB ) yields the conclusion.
This proposition shows that our empirical likelihood statistic L(θ) achieves asymptotic pivotalness
without estimating the variance components such as Var(X̄B ), Var(β̂A ), and Var(β̂B ). Based on thisproposition, the 100(1−α)% empirical likelihood confidence set of θ is given by ECLIθ = {θ : L(θ) ≤χ21,α }, where χ21,α is the (1 − α)-th quantile of the χ21 distribution.
Although we focus on the unexplained (discrimination) part of decomposition, our method canbe applied to the other term of the twofold decomposition (X̄A − X̄B )0 β̂A (explained or composition
effect). In addition, as Jann (2008) explained, there is a threefold decomposition where the overall
difference is decomposed of three components including an interaction term between the differenceof means and that of coefficients. Indeed, the twofold decomposition is the equivalent conversionof the threefold one with introducing the benchmark coefficient (β̂A here). Again, we can build ananalogous inference method for the threefold decomposition with the empirical likelihood approach.

3. Simulation
We conduct a simulation study to check the size and power properties of our empirical likelihood
method in comparison to Jann’s (2008) t ratio and the delta method. As a simulation setup, weconsider
YAi = 7XAi + Ai ,YBi = 5XBi + Bi ,where gi ∼ N (0, 1) and Xgi and gi are independent for g = A, B. For XAi and XBi , we consider thecases of normal, χ23q, and log-normal with mean 1 and variance 1. We also consider heteroskedastic
error terms gi =2v
0.5 + 0.5Xgi
gi with vgi ∼ N (0, 1). We are interested the OB decomposition

parameter θ = µ0B (βA − βB ) = 2. The sample size is set as n = 100. The number of Monte Carlo
replications is 500 for both the size and power analyses.
We first compare the size of three methods: Jann’s t ratio (Jann), delta method (Delta), andempirical likelihood (EL). The delta method included into comparison since previous research suchas Oaxaca and Ransom (1998) and Greene (2008) employs it to estimate the variance of the decomposition estimators. The rejection frequencies at 5% significance level of the three methods arepresented in Table 1. There are no significant differences in the size properties among these methods.
This is true even for asymmetric distributions, such as χ23 and Log-normal, and heteroskedasticity.
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Table 1. Simulation result: SizeJann DeltaNormal0.06 0.06
χ230.064 0.064
Log-normal
0.086 0.086
Normal heteroskedastic
0.054 0.054
χ23 heteroskedastic
0.066 0.066
Log-normal heteroskedastic 0.11 0.11

EL
0.062
0.064
0.072
0.05
0.068
0.096

We next compare the power of these methods by setting the null hypothesis H0 : θ = 2 + cwith c ∈ {−2.0, −1.9, . . . , 1.9, 2.0}. Since we are interested in the performance of our method (EL)in the cases where the data are asymmetrically distributed, we present the power curves for thecases of normal heteroskedastic and log-normal heteroskedastic errors in Figure 1. Although there
is no significant differences in the normal heteroskedastic case, EL performs better on the negativeside of the deviation in the log-normal heteroskedastic case. To illustrate this point, we draw theempirical likelihood function L(θ) in Figure 2, where the horizontal line is the .95-th quantile ofthe χ21 distribution used for the critical value. Due to the positive skewness of L(θ), the rejection
rate is higher on the left side of the true value θ = 2. We note that the left side contains zero (i.e.,no discrimination in the OB decomposition parameter). Thus in this example, EL exhibits better
power properties than other methods to detect presence of discrimination. Overall EL can be auseful complement to the conventional t ratio in this context.

Figure 1. Simulation result: Power5

Figure 2. Statistical Inference

References[1] Blinder, A. S. (1973) Wage discrimination: reduced form and structural estimates, Journal of Human Resources,
8, 436-455.
[2] Fortin, N., T. Lemieux and S. Firpo (2011) Decomposition methods in economics, in Ashenfelter, O. and D. Card(eds.), Handbook of Labor Economics, 4a, 1-102, North Holland, Amsterdam.
[3] Gregory, A. W. and M. R. Veall (1985) Formulating Wald tests of nonlinear restrictions, Econometrica, 53,1465-1468.
[4] Greene, W. H. (2008) Econometric Analysis, 6th ed., Upper Saddle River, New Jersey.
[5] Hall, P. and B. La Scala (1990) Methodology and Algorithms of Empirical Likelihood, International StatisticalReview, 58(2), 109-127.
[6] Jann, B. (2008) The Blinder–Oaxaca decomposition for linear regression models, Stata Journal, 8, 453-479.
[7] Jing, B. (1995) Two-sample empirical likelihood method, Statistics & Probability Letters, 24, 315-319.
[8] Oaxaca, R. L. (1973) Male-Female wage differentials in urban labor markets, International Economic Review, 14,693-709.
[9] Oaxaca, R. L. and L. Ransom (1998) Calculation of approximate variances for wage decomposition differentials,
Journal of Economic and Social Measurement, 24, 55-61.
[10] Owen, A. (1988) Empirical likelihood ratio confidence intervals for a single functional, Biometrika, 75, 237-249.
Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE,UK, and Keio Economic Observatory (KEO), 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan.
Email address: t.otsu@lse.ac.uk
Graduate School of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan.
Email address: tshio1997@keio.jp

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