RWI : Discussion Papers

Transcription

1 Thomas K. Bauer and Mathias Sinnin No. 49 RWI : Discussion Papers RWI ESSEN

2 Rheinisch-Westfälisches Institut für Wirtschaftsforschun Board of Directors: Prof. Dr. Christoph M. Schmidt, Ph.D. (President), Prof. Dr. Thomas K. Bauer Prof. Dr. Wim Kösters Governin Board: Dr. Eberhard Heinke (Chairman); Dr. Dietmar Kuhnt, Dr. Hennin Osthues-Albrecht, Reinhold Schulte (Vice Chairmen); Prof. Dr.-In. Dieter Amelin, Manfred Breuer, Christoph Dänzer-Vanotti, Dr. Hans Geor Fabritius, Prof. Dr. Harald B. Giesel, Dr. Thomas Köster, Heinz Krommen, Tillmann Neinhaus, Dr. Torsten Schmidt, Dr. Gerd Willamowski Advisory Board: Prof. David Card, Ph.D., Prof. Dr. Clemens Fuest, Prof. Dr. Walter Krämer, Prof. Dr. Michael Lechner, Prof. Dr. Till Requate, Prof. Nina Smith, Ph.D., Prof. Dr. Harald Uhli, Prof. Dr. Josef Zweimüller Honorary Members of RWI Essen Heinrich Frommknecht, Prof. Dr. Paul Klemmer RWI : Discussion Papers No. 49 Published by Rheinisch-Westfälisches Institut für Wirtschaftsforschun, Hohenzollernstrasse 1/3, D Essen, Phone +49 (0) 201/ All rihts reserved. Essen, Germany, 2006 Editor: Prof. Dr. Christoph M. Schmidt, Ph.D. ISSN ISBN ISBN The workin papers published in the Series constitute work in proress circulated to stimulate discussion and critical comments. Views expressed represent exclusively the authors own opinions and do not necessarily reflect those of the RWI Essen.

3 RWI : Discussion Papers No. 49 Thomas K. Bauer and Mathias Sinnin RWI ESSEN

4 Bibliorafische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliorafie; detaillierte bibliorafische Daten sind im Internet über abrufbar. ISSN ISBN ISBN

5 Thomas K. Bauer and Mathias Sinnin* An Extension of the Blinder-Oaxaca Decomposition to Non-Linear Models Abstract In this paper, a eneral Blinder-Oaxaca decomposition is derived that can also be applied to non-linear models, which allows the differences in a non-linear outcome variable between two roups to be decomposed into a part that is explained by differences in observed characteristics and a part attributable to differences in the estimated coeffcients. Departin from this eneral model, we show how it can be applied to different models with discrete and limited dependent variables. JEL Classification: C13, C20 Keywords: Blinder-Oaxaca decomposition, non-linear models October 2006 *Thomas K. Bauer, IZA-Bonn and CEPR London; Mathias Sinnin, RWI Essen. All correspondence to Mathias Sinnin, Rheinisch-Westfälisches Institut für Wirtschaftsforschun (RWI Essen), Hohenzollernstr. 1 3, Essen, Germany, Tel: / , Fax: , sinnin@rwi-essen.de.

6

7 1 Introduction The decomposition method developed by Blinder (1973) and Oaxaca (1973) and eneralized by Juhn, Murphy, and Pierce (1991), Neumark (1988) and Oaxaca and Ransom (1988, 1994) is a very popular descriptive tool in empirical economics, since it allows the decomposition of outcome variables between two roups into a part that is explained by differences in observed characteristics and a part attributable to differences in the returns to these characteristics. So far, these decomposition methods have mainly been applied in the context of linear reression models. In many cases, however, the outcome variable is non-linear, requirin the estimation of non-linear models because OLS yields inconsistent parameter estimates and in turn misleadin decomposition results. In particular, since the parameter estimates of non-linear models typically differ from the marinal effects of the latent outcome variable, they cannot be used to perform a standard Blinder-Oaxaca decomposition. A decomposition method for models with binary dependent variables has been developed by Fairlie (1999, 2003). In this paper, we eneralize the Blinder-Oaxaca decomposition method to other non-linear models. Based on this eneralized decomposition, we than demonstrate how the Blinder-Oaxaca decomposition can be applied to models with discrete and limited dependent variables. 2 An Extension of the Blinder-Oaxaca Decomposition to Non-linear Models Consider the followin linear reression model, which is estimated separately for the roups =(A, B), Y i = X i β + ε i, for i =1,..., N, and N = N. For these models, Blinder (1973) and Oaxaca (1973) propose the decomposition Y A Y B = OLS = (X A X B ) β A + X B ( β A β B ), (1) where Y = N 1 N i=1 Y i and X = N 1 N i=1 X i. The first term on the riht hand side of equation (1) displays the difference in the outcome variable between 1

8 the two roups due to differences in observable characteristics, whereas the second term shows the differential that is due to differences in coefficient estimates. A decomposition of the outcome variable similar to equation (1) is not appropriate in the non-linear (NL) case, because the conditional expectations E(Y i X i ) may differ from X β. For that reason, the decomposition of the mean difference of Y i between the two roups has to be considered: NL A =E βa (Y ia X ia ) E βa (Y ib X ib )+E βa (Y ib X ib ) E βb (Y ib X ib ), (2) where E β (Y i X i ) refers to the conditional expectation of Y i and E β (Y ih X ih ) to the conditional expectation of Y ih evaluated at the parameter vector β, with, h =(A, B) and h. Chanin the reference roup, an alternative expression for the decomposition is NL B =E βb (Y ia X ia ) E βb (Y ib X ib ) + E βa (Y ia X ia ) E βb (Y ia X ia ). (3) In both equations, the first term on the riht hand side aain displays the part of the differential in the outcome variable between the two roups that is due to differences in the covariates X i, and the second term the part of the differential in Y i that is due to differences in coefficients. To apply this decomposition to different nonlinear models, one just has to derive the respective sample counterparts S( ˆβ, X i ) and S( ˆβ h, X i ) of the conditional expectations E β (Y i X i ) and E βh (Y i X i ) for, h =(A, B) and h. The followin section illustrates the application of equation (2) for different models with discrete and limited dependent variables. An estimation of the correspondin components of equation (3) is straihtforward. Note that this decomposition shares all problems of the oriinal Blinder-Oaxaca decomposition, such as, e.., a potential sensitivity of the results with respect to the choice of the reference roup and the specification of the reression model. 3 Discrete Dependent Variable Models 3.1 Loit and Probit Models Discrete dependent variable models comprise binary and ordered Loit and Probit models as well as models for count data. Because binary Loit and Probit models 2

9 may be considered as a special case of ordered Loit and Probit models, the decomposition method for binary dependent variables proposed by Fairlie (1999, 2003) represents a special case of the Blinder-Oaxaca decomposition for ordered choice models. Ordered Loit and Probit models (O) are frequently used as a framework for analyzin outcomes of opinion surveys. These models are based on a latent reression of the form Yi = X i β O + ε O i, where Y i is unobserved. Instead of Y i, only the followin realizations are observed: Y i = 0 if Y i 0, = 1 if 0 <Yi µ 1, = 2 if µ 1 <Yi µ 2,... = J if µ J 1 Y i, where the µ s are unknown parameters to be estimated toether with the coefficients β O. The conditional expectation of Y i evaluated at the parameter vector β O can be written as E β O (Y i X i ) = F (µ 1 X i β O ) F ( X i β O ) + 2F (µ 2 X i β O ) F (µ 1 X i β O ) J1 F (µ J 1 X i β O ). Assumin that the error term ε O i is normally distributed across observations leads to the ordered Probit model, where F ( ) is defined as the cumulative standard normal distribution Φ( ). The Loit model is obtained when the error term ε O i is assumed to follow a loistic distribution, i.e. when F ( ) represents a cumulative loistic distribution Λ( ). Given the estimates of the parameter vector β O, the sample counterparts of the 3

10 sinle components of the decomposition equation can be calculated by N { S( ˆβ O, X i ) = N 1 F (ˆµ 1 X i ˆβO ) F ( X i ˆβO ) i=1 + 2F (ˆµ 2 X i ˆβO ) F (ˆµ 1 X i ˆβO ) +... } + J1 F (ˆµ J 1 X i ˆβO ). The sample counterpart of E β O h (Y i X i ), S( ˆβ h O, X i), is obtained by just replacin ˆβ O with ˆβ h O in the above equation.1 These sample counterparts can then be used to calculate the sinle parts of the decomposition equation (2) as ˆ O = S( ˆβ A, O X ia ) S( ˆβ A, O X ib ) + S( ˆβ A, O X ib ) S( ˆβ B, O X ib ). The Blinder-Oaxaca decomposition for ordered choice models reduces to the decomposition method for binary choice models if J = Count Data Models The Poisson reression model (P), which has been widely used to study count data, assumes that the dependent variable Y i conditional on the covariates X i is Poisson distributed with density f(y i X i ) = exp( µ i)µ Yi i, Y i =0, 1, 2,... Y i! and conditional expectation E(Y i X i )=µ i = exp(x i β P ). The sample counterpart of E β P (Y i X i ) which is necessary to estimate the decomposition equation is iven by S( ˆβ P, X i )=Y,ˆβP = 1 N N i=1 exp(x i ˆβP ). 1 Because the calculation of S( ˆβ h, X i ) is straihtforward, we will present just the calculation of S( ˆβ, X i ) in the remainder of the paper. 4

11 A well-known problem of the Poisson-model is the assumption that the dependent variable has the same mean and variance µ i = exp(x i β P ). If this assumption is violated, an alternative conditional distribution of the dependent variable needs to be specified that permits a more flexible specification of the variance of the dependent variable. The neative binomial (Nebin) reression model (NB) represents such an alternative. The Nebin reression model relaxes the assumption of equality of the conditional mean and the variance of the dependent variable while assumin the same form of the conditional mean as the Poisson-model. Hence, the sample counterpart of the conditional mean of the Nebin reression model is S( NB ˆβ, X i )=Y = 1,ˆβNB N N i=1 exp(x i ˆβNB ). Different to the Poisson model, the Nebin model assumes a quadratic relationship between the variance and the mean, i.e. V (Y i X i )=µ i + αµ 2 i. where α is a scalar parameter to be estimated toether with β NB. In addition to the Poisson and Nebin reression models, zero-inflated models are frequently used when analyzin count data. These models take into account that real-life data may contain excess zeros, causin a hiher probability of zero values than is consistent with the Poisson and neative binomial distribution. In this case it could be assumed that zeros and positive values do not come from the same data eneratin process (Winkelmann 2000). In order to investiate the probability of excess zeros, Lambert (1992) proposed a zero-inflated Poisson model, that allows for two different data eneratin reimes: the outcome of reime 1 (R1) is always zero, whereas the outcome of reime 2 (R2) is enerated by a poisson process. In this model, the unconditional expectation of the dependent variable consists of the conditional probability of observin reime 2 and the conditional expectation of the zero-truncated density: E(Y i X i )=(1 Pr(R1 X i ))E(Y i R2, X i ). (4) Lambert (1992) specifies the conditional probability of reime 1, that always leads 5

12 to a zero outcome, as a Loit model: Pr(R1 X i )= exp(z iγ ) 1 + exp(z i γ ), where Z i contains the covariates of the conditional probability of excess zeros and γ is the parameter vector to be estimated. The unconditional mean of the dependent variable specified by equation (4) can then be estimated for the zero-inflated Poisson and the zero-inflated Nebin model by N N S( ˆβ, j X i )= 1 (1 ( Pr(R1) X i ))ˆµ i = 1 exp(x i ˆβj ) N N i=1 1 + exp(z i=1 iˆγ ) j, for j = ZIP,ZINB. Hurdle models represent another modification of count data models. The hurdle model can be interpreted as a two-part model, where the first part is a binary outcome model, and the second part a truncated count data model. The unconditional mean of the dependent variable in these models is iven by: E(Y i X i )=Pr(Y i > 0 X i )E(Y i Y i > 0, X i ). Accordin to Cameron and Trivedi (1998) the conditional expected values of Y i of the hurdle Poisson (HP) and the hurdle Nebin (HNB) model are iven by and E(Y i Y i > 0, X i )= E(Y i Y i > 0, X i )= exp(x i β HP ) 1 exp( exp(x i β HP )) exp(x i β HNB ) 1 (1 + α exp(x i β HNB )) 1 α respectively. Assumin a loistic distribution for the underlyin zero eneratin process, the unconditional expected values can be estimated by and S( S( HP ˆβ, X i )= 1 N HNB ˆβ, X i )= 1 N N i=1 N i=1 exp(x i ˆβHP ) (1 exp( exp(x ˆβHP i )))(1 + exp(z iˆγ HP )) exp(x i ˆβHNB ) (1 (1 + α exp(x ˆβHNB i )) 1 α )(1 + exp(z iˆγ HNB ))., 6

13 4 Limited Dependent Variable Models 4.1 Tobit Models Limited dependent variable models comprise truncated reression models and models for censored and corner solution outcome variables. Technically, censored and corner solution outcome variables may be described appropriately by a Tobit model (Wooldride 2002). While censored outcome variables are not observable for a part of the population (such as top-coded wae information or preferred labor supply), corner solution outcome variables take on the value zero with positive probability but represent a continuous random variable over strictly positive values (such as actual labor supply). In the Tobit model (TB), the dependent variable takes on the values a 1 and a 2 with positive probability and represents a continuous random variable over values between a 1 and a 2, i.e. Yi = X i β TB + ε TB i, Y i = a 1 if Yi a 1 Y i = a 2 if Y i a 2 Y i = Yi = X i β TB + ε TB i if a 1 <Yi <a 2, ε i N(0, (σ TB ) 2 ). If one is interested in the marinal effects of a latent censored outcome variable, the stratey would be to use the Tobit estimator in the standard Blinder-Oaxaca decomposition depicted in equation (1). However, the conventional decomposition method leads to erroneous predictions of the components of the decomposition equation if we aim at analyzin the observable corner solution outcome variable Y i.in this case, an alternative decomposition method must be applied. Assumin homoscedastic and normal distributed error terms ε TB i, the unconditional expectation of Y i iven X i consists of the conditional expectations of Y i, weihted by the respective probabilities of observin a 1, a 2, or a value between a 1 7

14 and a 2, i.e. where E(Y i X i ) = a 1 Φ 1 (β TB, X,σ TB + Λ(β TB, X,σ TB ) )+a 2 Φ 2 (β TB, X,σ TB ) X i β TB + σ TB λ(β TB, X,σ TB Λ(β TB ), X,σ TB ). (5) Λ(β TB, X,σ TB )=1 Φ(σ TB ) 1 (a 1 X i β TB ) Φ(σ TB ) 1 (a 2 X i β TB ) and λ(β TB, X,σ TB )=φ(σ TB ) 1 (a 1 X i β TB ) φ(σ TB ) 1 (a 2 X i β TB ). φ( ) represents the standard normal density function. Equation (5) shows that a decomposition of the outcome variable similar to equation (2) is not appropriate for censored outcome variables, because the conditional expectations E(Y i X i ) in the Tobit model depend on the variance of the error term σ TB. Even thouh the ancillary parameter σ TB does not affect the sin of the marinal effects, it affects their manitudes and therefore becomes important for the decomposition. Dependin on which σ TB is used in the counterfactual parts of the decomposition equation, several possibilities of decomposin the mean difference of Y i between the two roups can be derived. Two possibilities are TB AB = E β TB A +,σtb A E β TB A (Y ia X ia ) E β TB (Y ib X A,σTB B ib ),σtb B (Y ib X ib ) E β TB (Y ib X B,σTB B ib ), (6) and TB AA = E β TB A +,σtb A E β TB A (Y ia X ia ) E β TB (Y ib X A,σTB A ib ),σtb A (Y ib X ib ) E β TB (Y ib X B,σTB B ib ), (7) where E β TB,σ TB (Y i X i ) now refers to the unconditional expectation of Y i evaluated at the parameter vector β TB and the error variance σ TB. In both equations, the first term on the riht hand side displays the part of the differential in the outcome variable between the two roups that is due to differences in the covariates X i, 8

15 and the second term the part of the differential in Y i that is due to differences in coefficients. The two versions of the decomposition equation differ from each other as soon as lare differences in the variance of the error term between the two roups exist. Note however, that the decomposition usin σb TB to calculate the counterfactual parts, as in equation (6), is more comparable to the OLS decomposition described in equation (1), since the counterfactual parts differ from E β TB B (Y ib X,σTB B ib ) only by usin the parameter vector for roup A, βa TB, rather than by usin the parameter vector and the error variance for roup A in the alternative decomposition described in equation (7). Usin the sample counterpart of equation (5), S( TB ˆβ, X i, ˆσ TB ) = N 1 equation (6) can be estimated by ˆ TB TB AB = S( ˆβ A + S( N a 1 Φ 1 ( i=1 ˆβ TB + Λ(ˆβ TB, X i, ˆσ TB ), X ia, ˆσ A TB ) S( ˆβ TB A, X ib, ˆσ B TB ) S(, X, ˆσ TB )+a 2 Φ 2 ( X i ˆβTB TB ˆβ A, X ib, ˆσ B TB ) TB ˆβ B, X ib, ˆσ B TB ) +ˆσ TB. λ( Λ( TB ˆβ, X, ˆσ TB ) ˆβ TB, X i, ˆσ TB ) ˆβ TB, X i, ˆσ TB ), (8) Similarly, equation (7) can be estimated by ˆ TB TB AA = S( ˆβ A, X ia, ˆσ A TB ) S( TB + S( ˆβ A, X ib, ˆσ A TB ) S( TB ˆβ A, X ib, ˆσ A TB ) TB ˆβ B, X ib, ˆσ B TB ) (9) If the dependent variable is not truncated, i.e. if a 1 and a 2, equations (6) and (7) reduce to the oriinal Blinder-Oaxaca decomposition described in equation (1). 4.2 Truncated Reression Models The results derived for the Tobit model can be easily transferred to a truncated reression model of the form Y i = X i β TR + ε TR i, 9

16 where the dependent variable is truncated at a lower limit a 1 and a hiher limit a 2. The error terms ε TR i are assumed to be homoscedastic and distributed normally with mean zero and variance (σ TR ) 2. Consequently, Y i X i N(X i β TR, (σ TR ) 2 ). In this model, the unconditional expectation of Y i iven X i consists of the conditional expectation of Y i weihted with the probability of observin a value between a 1 and a 2 : λ(β TR E β TR,σ TR (Y i X i ) = Λ(β TR, X,σ TR ) X i β TR + σ TR, X,σ TR )., X,σ TR Λ(β TR Consequently, similar to equations (8) and (9), the components of the decomposition equation of the truncated reression model can be estimated by usin the sample counterpart of the unconditional expectation: S( TR ˆβ, X i, ˆσ TR ) = N 1 5 Conclusion N Λ( i=1 X i ˆβTR TR ˆβ, X i, ˆσ TR ) +ˆσ TR λ( Λ( ˆβ TR, X i, ˆσ TR ) ˆβ TR, X i, ˆσ TR ) In this paper, the decomposition method proposed by Blinder (1973) and Oaxaca (1973) is extended to non-linear models. The extension of the conventional decomposition method permits a decomposition of differences in a non-linear outcome variable between two roups into a part that may be explained by differences in observed characteristics and a part that is attributable to differences in the estimated coefficients. The paper illustrates how the Blinder-Oaxaca decomposition can be applied to models with discrete and limited dependent variables. In particular, a Blinder- Oaxaca decomposition method for ordered Loit and Probit models is derived which represents a eneralization of the decomposition method for binary Loit and Probit models proposed by Fairlie (1999, 2003). Moreover, the Blinder-Oaxaca decomposition is applied to count data models, includin Poisson and Neative Binomial 10. )

17 models, zero-inflated Poisson and Neative Binomial models as well as Hurdle Poisson and Neative Binomial Models. An empirical application of the decomposition method for count data models is provided by Bauer, Göhlmann, and Sinnin (2006). Finally, the Blinder-Oaxaca decomposition is extended to truncated reression and Tobit models, where the latter has been used by Bauer and Sinnin (2005) to analyze differences in the savins behavior between natives and immirants in Germany. 11

18 References Bauer, T. K., S. Göhlmann, and M. Sinnin (2006): Gender Differences in Smokin Behavior, RWI Discussion Papers No. 44, pp Bauer, T. K., and M. Sinnin (2005): Blinder-Oaxaca Decomposition for Tobit Models, RWI Discussion Papers No. 32, pp Blinder, A. S. (1973): Wae Discrimination: Reduced Form and Structural Estimates, Journal of Human Resources, 8, Cameron, A. C., and P. K. Trivedi (1998): Reression Analysis of Count Data, Cambride University Press, Cambride. Fairlie, R. W. (1999): The Absence of the African-American Owned Business: An Analysis of the Dynamics of Self-Employment, Journal of Labor Economics, 17, (2003): An Extension of the Blinder-Oaxaca Decomposition Technique to Loit and Probit Models, Yale University Economic Growth Center Discussion Paper No. 873, pp Juhn, C., K. M. Murphy, and B. Pierce (1991): Accountin for the Slowdown in Black-White Wae Converence, in Workers and Their Waes: Chanin Patterns in the United States, ed. by M. H. Kosters. American Enterprise Institute, Washinton. Lambert, D. (1992): Zero-inflated Poisson Reression with an Application to Defects in Manufacturin, Technometrics, 34, Neumark, D. (1988): Employers Discriminatory Behavior and the Estimation of Wae Discrimination, Journal of Human Resources, 23, Oaxaca, R. L. (1973): Male-Female Wae Differentials in Urban Labor Markets, International Economic Review, 14, Oaxaca, R. L., and M. Ransom (1988): Searchin for the Effect of Unionism on the Waes of Union and Nonunion Workers, Journal of Labor Research, 9, (1994): On Discrimination and the Decomposition of Wae Differentials, Journal of Econometrics, 61, Winkelmann, R. (2000): Econometric Analysis of Count Data, Spriner Verla, Berlin, Heidelber. Wooldride, J. M. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambride, Massachusetts. 12