Box–Cox power transformation unconditional quantile regressions with an application on wage inequality

Abstract

This study proposes a semi-parametric estimation method, Box–Cox power transformation unconditional quantile regression, to estimate the impact of changes in the distribution of the explanatory variables on the unconditional quantile of the outcome variable. The proposed method consists of running a nonlinear regression of the recentered influence function (RIF) of the outcome variable on the explanatory variables. We also show the asymptotic properties of the proposed estimator and apply the estimation method to address an existing puzzle in labor economics–why the 50th/10th percentile wage gap has been falling in the USA since the late 1980s. Our results show that declining unionization can explain approximately 10% of the decline in the 50/10 wage gap in 1990–2000 and 23% in 2000–2010.

Keywords:

• Unconditional quantile regression
• extended Box–Cox power transformation unconditional quantile regression
• Yeo–Johnson RIF regression
• Unionization
• 50/10 and 90/50 percentile wage gaps

JEL Code:

• C14
• C21
• J31

Acknowledgments

I am thankful to Christoph Rothe, Sergio Firpo, Hugo Jales, Aureo De Paula, Thomas Lemieux, David Autor, Anil Bera, Badi Baltagi, Thomas Kniesner, Le Wang, Carlos Lamarche, and Jeffrey Kubik as well as seminar participants at the North American Econometric Society Meeting, Royal Economic Society, Canadian Economic Association, European Association of Labour Economists Annual Conference, Syracuse University and Oklahoma University for their helpful comments and suggestions. Any errors or omissions are my own.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The RIF is defined in Section 2.

2 $Q_{\tau }(\stackrel{.}{})$ is the conditional quantile function, ${\beta }_{\tau }$ is the regression parameter and τ denotes a quantile.

3 The unconditional quantile partial effect is defined in Section 2.

4 We use the extended Box–Cox power transformation proposed by Yeo and Johnson [54].

5 The support of X sufficiently rich implies that there exists a huge variation in X so that the choice of link function is not important to approximate the conditional distribution function $F_{Y|X}(y|x)$ arbitrarily well.

6 In this context implies that we refer to the situation where our dependent variables $RIF(\stackrel{.}{})$ are binary for a given value of the outcome variable when we consider the distributional statistic quantile. $RIF(\stackrel{.}{})$ for quantile is defined in Equation (1(1) $\int {\mathrm{\partial }}_{x}E(RIF(Y,q_{\tau })|X=x)\mathrm{d}F(x)$(1) ) and the detailed explanation are provided in Section 2.

7 As we mentioned earlier, the Firpo et al. [26] nonparametric model is a power approximation of logistic model specification.

8 $RIF(\stackrel{.}{})$ is defined in Equation (1(1) $\int {\mathrm{\partial }}_{x}E(RIF(Y,q_{\tau })|X=x)\mathrm{d}F(x)$(1) ). As shown, for a given value of Y = y, $RIF(\stackrel{.}{})$ takes value $q_{\tau }+(\tau -1)/f_Y$ for $y\le q_{\tau }$ and $q_{\tau }+\tau /f_Y$ otherwise.

9 Koenker and Park [42] develop an algorithm to implement the CBTS estimator. Fitzenberger et al. [28] suggest a modified version of the CBTS estimator.

10 Yeo and Johnson [54] show that the two transformation differ in their behavior the values of Y are close to zero, with Box–Cox transformation providing a much larger change for small values than does the Yeo–Johnson transformation.

11 Note that this model setup is similar to Firpo et al. [26] in which the authors also estimate the RIF regression model parameters using the strict exogeneity assumption.

12 Similarly, when ${\lambda }_{\tau }=1$ and $x^T{\beta }_{\tau }<0$, UQPE equals to -${\beta }_{\tau }$.

13 The assumptions A1–A6 are shown in the proof of Proposition 2.2.

14 In this context, skill refers to years of schooling.

15 Technological changes are measured by changes in the market price of cognitive and non-cognitive skills.

16 Unionization includes both union members and workers covered by unions.

17 The dark and light blue colors represent the high school dropouts and high school graduates; the green, yellow and purple colors represent union members who attended college, graduated from college and pursued postgraduate education.

18 During the period from 1990 to 2010, the unionization rate has declined 7.7$\mathrm{\%}$ (= 22.5$\mathrm{\%}$ – 14.8$\mathrm{\%}$), and the proportion of high school graduate union workers declined 5.47$\mathrm{\%}$ (= 10.84$\mathrm{\%}$ – 5.37$\mathrm{\%}$). Thus the contribution of high school graduate union workers is $\approx (5.47/7.7)\times 100$ = 71%.

19 Over the last five decades of unionization literature, we still do not have a valid instrument.

20 Fortin et al. [29] provide a detail discussion on the methodological development of decomposition method since [46] and [16].

21 In this empirical setup, one of those assumptions implies that the distribution of workers' unobserved skills remains the same over the two periods. Thus taking the difference in wages at the same quantile of the wage distribution in two time periods yields a consistent estimate of the impact of unionization.

22 We report the Yeo–Johnson RIF regression estimates of ${\beta }_{\tau }$ and ${\lambda }_{\tau }$ for years 1990, 2000 and 2010 in Appendix Table 2. Because ${\hat{\lambda }}_{\tau }$ do not vary much across those three years for a given τ, we could apply the same Yeo–Johnson RIF regression model to construct counterfactual distributions for different years.

23 See [20] for details.

24 Card [21] shows that the wages of nonunion workers fall because of the increased supply of workers squeezed out from union jobs.

25 See [12] for a detailed discussion.

26 The third row of the bottom panel shows the contribution of unionization in terms of percentages to explain the change in the wage gap as reported in the top panel.