Copula-based Semiparametric Modeling of. Intergenerational Earnings Mobility

Transcription

1 Copula-based Semiparametric Modeling of Intergenerational Earnings Mobility Shigeki Kano First draft: December 2008 This version: February 2009 Correspondence: Shigeki Kano, College of Economics, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka , Japan. address: Tel: This research is financially supported by KAKENHI ( ), JSPS. The author is grateful to Colin McKenzie and seminar participants at Keio University and Osaka City University for helpful comments. Remaining errors are due to the author. 1

2 Abstract In this paper, we estimate the joint distribution of son s and father s log-earnings in the United States for examining non-linear intergenerational earnings mobility. We model the joint distribution semiparameterically by a copula function. Conditional means and probabilities of son s log-earnings given father s log-earnings and their derivatives are derived from the estimated copula-based joint distribution. We further compare alternative methods of investigating the non-linear association, the OLS with higher order polynomials and the nonparametric local polynomial estimation, to the copula-based conditional mean estimation. Using the PSID data, we find that the earnings of son-father pairs exhibits a weak lower tail dependence. This result suggests that the US society is more mobile at the poorer population groups. JEL classification numbers: C14; D31; J62. Keywords: Copula; Intergenerational earnings mobility. 2

3 1 Introduction The association between a person s socio-economic status and family backgrounds has been a great concern for social scientists and policy makers since it provides important information on the equal-opportunity and anti-poverty actions. The influence of parental resources on one s economic success also have drawn a public attention: Whether being assigned wealthy parents or not is the first and largest natural experiment every person experienced over one s life. Solon (1992) argues that the estimates of son s income elasticity with respect to his father s income proposed by previous studies suffer from attenuation bias due to measurement errors. Using the Panel Study of Income Dynamics (PSID) data, he demonstrates that the error-free estimates of intergenerational income elasticity in the United Status is about 0.4, far above the previously cited estimates, less than 0.2, of Behrman and Taubman (1985). Subsequent studies estimating the intergenerational elasticities also report similar degree of son-father associations for the United States (See the review articles of Solon, 1999, 2002). In the literature, the authors often focus on a linear relationship between the earnings of two generations. However, some models of child investments predict non-linear associations of earnings between parents and offspring. Becker and Tomes (1979, 1986) show that, in the presence of credit constraints, the effect of wealth endowments is more important for poor families, yielding a concave intergenerational relationship. In contrast to Becker and Tomes, Bratsberg et al. (2007) propose a model where convex intergenerational relationship can arise. If the patterns of intergenerational mobility are different largely between the poor and nonpoor populations, it is misleading to cast anti-poverty policies based on an erroneously approximated linear relationship. So several authors examine the non-linearity of intergenerational mobility empirically. See Solon (1992, the US), Couch and Lillard (2004, the US and Germany), Bratsberg et al. (2007, the US, the UK and three nordic countries; Denmark, Finland, Norway), among others. Their OLS estimates of polynomial regressions suggest that the convex intergener- 3

4 ational relationship seems supported by the data. Corak and Heisz (1999) apply a nonparametric regression method to the Canadian data and find similar results obtained by the aforementioned parametric analysis. The purpose of this paper is to investigate the joint distribution of son s and father s logearnings in the United States, revisiting the non-linearity of the two variables. In general, a non-linear association between two random variables arises because they follow a non-elliptical joint distribution with asymmetric tail dependence. Therefore, if we capture the whole picture of the joint distribution of the son-father earnings pairs, it is possible to examine any stochastic non-linearities between the two variables, e.g., conditional means or conditional probabilities. For this purpose, we use a notion of copula which is detailed by Joe (1997), Nelsen (2007), and Trivedi and Zimmer (2007). In short, copula is a function coupling given marginal distributions to a joint distribution. There are several advantages of copula-based modeling of a joint distribution over the alternatives. First, among others, existing families of parametric copulas generate a wide variety of dependence patterns of random variables, e.g., asymmetric tail dependence. Second, the copula approach enables us to separate modeling of dependence structure of random variables (i.e., the choice of copula function) and marginal distributions each variable follows. It should be noted that different families of distributions are allowed to be used for each margin here. Consequently, the copula-based modeling describes dependence patterns of multivariate data in much more flexible manner than using conventional joint distributions, e.g., multivariate normal, multivariate Student t, or multivariate extreme value distributions. Our empirical analysis proceeds as follows. First, we apply the two-step semiparametric maximum likelihood method proposed by Genest et al. (1995) to the current data. In this method, the first step estimates the marginal distributions of log-earnings of both generations separately and nonparametrically, and then the second step joins the margins to the joint distribution via parametric copula. Next, from the estimated joint distribution, we derive (1) tail dependence parameters (defined later), (2) conditional means, and (3) conditional probabilities of son s log- 4

5 earnings given father s log-earnings. Further we estimate the derivatives of the conditional means (i.e., elasticities) and probabilities (i.e., marginal effects). Since the parameter values of a copula function per se usually have not direct information on the strength and patters of dependence, obtaining more interpretable quantities form the estimated copula parameters is an important task. Recently some authors demonstrate the usefulness of copula-based modeling in examining non-linear relationships of random variables. For univariate autoregressive time series data, Chen and Fan (2006) propose a copula-based semiparametric estimator of conditional means and quantiles. However, they do not mention on the estimation of the conditional probability nor the derivatives of population parameters. On the other hand, in the current study, estimating the derivative of conditional mean is quite useful since intergenerational earnings mobility is often measured by the elasticity estimates. In addition, since our analysis is based on the independent observations, much simpler asymptotic theory than Chen and Fan (2006) is allowed. Zimmer and Trivedi (2006) estimate the conditional mean from the copula-based joint distribution where margins are event-count and continuous variables. However, their analysis is fully parametric. Crane and Hoek (2008) derive the explicit formulas of conditional expectations for some copulabased models, but their approach requires strong distributional assumptions. They also do not discuss on statistical inferences. The remaining part of the paper is organized as follows. Section 2 reviews the properties of copula functions and estimation method. Section 3 proposes the estimator of conditional means and probabilities from the copula-based joint distribution. Section 4 describes the data used. Section 5 shows the empirical results. Section 6 concludes the paper. 5

6 2 Copula-based Semiparametric Joint Distribution 2.1 Copula: Overview Copula is a device for modeling a multivariate distribution. As mentioned earlier, there are several advantages of copula-based modeling over alternative methods. Its usefulness leads copula to become popular in the field of risk management, finance, and time series analysis (e.g., Junker and May, 2005; Patton, 2006; Hu, 2006). However, its application to microeconometric data is relatively few. Smith (2003) uses copula functions for modeling sample selection process in two empirical studies; female labor supply and hospitalization. Cameron et al. (2004) analyze the joint distribution of two count variables, two measurements of doctor visits, and their gap based on a copula function. Zimmer and Trivedi (2006) construct a three dimensional model where margins are continuous (a latent variable determining whether a couple takes out independent insurance policy or not) and count distributions (doctor visits of each couple). See Trivedi and Zimmer (2007) for further reference. The definition of copula function is given as follows. Definition 1 Function C :(u, v) [0, 1] 2 [0, 1] is a copula if and only if C(1, v) = v, C(u, 1) = u, (1) C(0, v) = C(u, 0) = 0, (2) C(u 2, v 2 ) C(u 2, v 1 ) C(u 1, v 2 ) + C(u 1, v 1 ) 0, u 1 u 2, v 1 v 2. (3) Let (X, Y) be a continuous random vector with marginal cdf s F X (x) = Pr(X x) and F Y (y) = Pr(Y y) and joint cdf F(x, y) = Pr(X x, Y y). In the current study, X and Y are father s and son s log-earnings, respectively. Our purpose is to estimate F(x, y) from the random samples of son-father pairs by means of a copula function. The relationship between a copula function and marginal and joint distributions is established by Sklar (1959): 6

7 Theorem 1 (Sklar) There exists copula C(, ) such that F(x, y) = C(u, v) = C [ F X (x), F y (y) ], x, y, R. (4) If F X ( ) and F Y ( ) are continuous, then C is unique. (See Nelsen, 2007, chapter 2.3, for the proof.) This uniqueness result is important in empirical applications since it says that the correctly specified set of margins and copula function lead to the correctly specified joint distribution a researcher want to know. 1 We assume that a true copula function is characterized by a vector of parameters, γ: F(x, y) = C(u, v; γ), u = F X (x), v = F Y (y). (5) By this assumption, the estimation of joint distribution F(x, y) is reduced to the estimation of γ given copula function C(, ; γ). On the other hand, the joint density of (X, Y) is given by f (x, y; γ) = F(x, y; γ) y x = f X (x) f Y (y)c [ F X (x), F Y (y); γ ], (6) where f X ( ) and f Y ( ) denote the marginal densities, and c(, ; γ) = C(u,v;γ) u v is the crossderivative of a copula function. Then, given N random draws from the joint distribution with typical draw being (X i, Y i ), we have the log-likelihood function of the model, L(γ) = N [ log fx (X i ) + log f Y (Y i ) ] + i=1 N log c [ F X (X i ), F Y (Y i ); γ ]. (7) i=1 For obtaining a consistent estimator of γ, say, ˆγ, further assumptions on the margins are needed. 1 Note that, in most applications of copula (including the current study), a vector of control variables, say, Z, enters each of margins. However, since the Sklar s theorem is not affected by the presence of covariates, we suppress Z. 7

8 2.2 Semiparametric estimation Once marginal distributions and a copula function are specified, conventional numerical optimization algorithm is readily applicable to equation (7) for obtaining the maximum likelihood estimator (See chapter 4 of Trivedi and Zimmer, 2007). However, this approach involves a risk of specification errors in marginal distributions and a copula function. So we apply the semiparametric two-step method proposed by Genest et al. (1995) where the marginal distributions are left unspecified. The point of the two-step estimation of Genest et al. is that dependence parameter γ enters only the second term of likelihood function (7) and the specification of the marginal densities, f X (x) and f Y (y), is not needed for constructing this second term. The two-step procedure by Genest et al. (1995) proceeds as follow. In the first step, we estimate F X (X i ) and F Y (X i ) nonparametrically by the empirical distribution functions: ˆF X (X i ) = ˆF Y (Y i ) = 1 N N + 1 N 1[X j X i ], (8) j=1 N 1[Y j Y i ], (9) j=1 where 1[ ] is an indicator function. 2 Then, in the second step, we maximize pseudo-likelihood function L(γ) = N log c [ ˆF X (X i ), ˆF Y (Y i ); γ ], (10) i=1 with respect to γ. Ignorance of the parts on the marginal distribution in equation (7) causes the loss of information but gives us robustness against specification errors. Genest et al. (1995) prove the N-consistency of the two-step estimator for independent observations and derive its 2 For some copula functions, evaluating them at u = 1orv = 1 or both causes numerical problems. As suggested by Genest et al. (1995), to avoid this problem, here the quantities are divided by not N but N + 1. By doing so, the estimated empirical distribution functions do not reach unity. This re-scaling does not affect the uniform convergences of ˆF X (x) tof X (x) and of ˆF Y (y) tof Y (y). 8

9 variance estimator. 2.3 Selection of parametric copula function There are numerous parametric alternatives for copula function C(, ; γ). For selecting a better approximation of the true joint distribution, we estimate several families of copulas and then compare the goodness-of-fit of them. The families we investigate are; the Gaussian, Frank, Clayton, and Gumble copulas, all of which are popular in data applications since they are computationally tractable and cover rich dependence patterns. Functional forms, cross-derivatives, and parameter spaces are listed by Table 1. The Gaussian and Frank exhibit a symmetric tail dependence, i.e., the strength of association is symmetric between lower and upper tail of a joint distribution. On the other hand, the Clayton copula has a stronger dependence at a lower tail. The Gumbel has an opposite dependence pattern to the Clayton. For a copula with asymmetric dependence (the Clayton and Gumbel here), a survival copula, defined by C S (u, v; γ) = u + v 1 + C(1 u, 1 v; γ), (11) is useful for modeling since it also serves as a copula and the structure of dependence is reversed to the original copula. (See Nelsen, 2007; Hu, 2006). For example, for Clayton copula C, survival Clayton C S exhibits upper tail dependence. Thus, in addition to the aforementioned popular copula families, we also take the survival Clayton and survival Gumbel to the data. The copula functions mentioned above are all one-parameter families and may explain limited aspects of dependence properties in a given data. Therefore some authors develop and apply to the data more general, multi-parameter copulas. Patton (2006) proposes symmetric Joe-Clayton copula which is a two-parameter generalization of the Clayton. The transformation method by Junker and May (2005) also incorporate additional parameters into a conventional 9

10 copula to pursuit flexible dependence structures. Hu (2006) utilizes the fact that the convex sum or mixture of any copula functions is also a copula (Nelsen, 2007, chapter 3.2.4). 3 We apply the mixture approach of Hu (2006) to the current analysis. Specifically, for J parametric copulas C j (u, v,γ j ), define mixture copula C M (u, v; α, γ) = α 1 C 1 (u, v; γ 1 ) + α 2 C 2 (u, v; γ 2 ) + + α J C 2 (u, v; γ J ), (12) where α = [α 1,α 2,...,α J ], γ = [γ 1,γ 2,...,γ J ], and J j=1 α j = 1. In the mixture copula, the parameters to be estimated are dependence parameters γ and mixture probabilities α. As a goodness of fit measure for the model choice, we mainly rely on the Akaike information criteria (AIC). Alternatively, we also compare an estimated parametric copula, C(u, v; ˆβ), to empirical copula, defined by C E (u, v) = 1 N N 1 [ u j u, v j v ], (13) j=1 where u j = F X (X j ) and u j = F Y (Y j ). Specifically, we use the following distance measure: D = 1 N N C E (u i, v i ) C(u i, v i ; ˆβ). (14) i=1 3 Dependence Measures for Copula-based Joint Distribution 3.1 Lower and upper tail dependence parameters Since dependence parameter γ itself do not measure strength nor patterns of dependence, its direct comparison among different data sets is subtle. So the estimation result on γ is often translated into other interpretable quantities measuring dependence. A popular dependence mea- 3 See Nelsen (2007, chapter 3) and Trivedi and Zimmer (2007, chapter 3) for further information on the construction and generalization of copulas. 10

11 sure is left and right tail dependence parameters (LTD and RTD, respectively). First, define the following conditional probabilities by means of copula: Pr(X x t, Y y t X x t ) = Pr(X x t, Y y t ) C(t, t) =, (15) Pr(X x t ) t Pr(X > x t, Y > y t X > x t ) = 1 Pr(X x t) Pr(Y y t ) + Pr(X x t, Y y t ) 1 Pr(X x t ) 1 t t + C(t, t) =, (16) 1 t where (x t, y t ) are quantiles of log-earnings such that t = Pr(X x t ) = Pr(Y y t ) for a given probability t. Then the LTD and RTD are given by C(t, t) LTD = lim, t 0 t (17) 1 2t C(t, t) RTD = lim. t 1 1 t (18) It is intuitive that the LTD or RTD capture the associations of X and Y at their extremely small or large values. In the empirical studies, the copula-based estimation of the LTD and RTD is useful since there are few observations at tail areas and so accurate statistical inference is difficult by conventional methods, e.g., kernel-based nonparametric regressions. Another attractive feature of the copula-based LTD and RTD is that explicit formulas are available for popular copula families with respect to γ. (For example, LTD = 2 1 γ and RTD = 0 for the Clayton.) We summarize the relationship between γ and tail dependence parameters for each copula in Table 1. It is also easily verified that, for a mixture copula, it follows that LTD M = RTD M = J α j LTD j, (19) j=1 J α j RTD j. (20) j=1 11

12 3.2 Conditional means Although the LTD and RTD are operative for inferring dependence patterns of random variables at extreme events, in the current literature, dependences at moderately small and moderately large observations, e.g., at the 10-percentile and 90-percentile of log-earnings, may be more interesting. For this purpose, we derive the non-linear conditional mean function of son s logearnings Y on father s log-earnings X from the copula-distribution. Given the derivative of parametric copula function, c(, ; γ), conditional distribution of Y given X is expressed as f Y (y x; γ) = f (x, y; γ) f X (x) = f Y (y)c [ F X (x), F Y (y); γ ]. (21) So the conditional mean function evaluated at X = x is given by m(x) E(Y X = x) = yc [ F X (x), F Y (y); γ ] f Y (y)dy, (22) a weighted mean of Y with weight c(, ; γ). Let us define the estimator of m(x)by ˆm(x) = 1 N N Y i w(x, Y i ;ˆγ), (23) i=1 where ˆγ is a consistent estimator of γ, and w(x, Y i ;ˆγ) = c [ ˆF X (x), ˆF Y (Y i ); ˆγ ]. (24) Then, by lemma 4.3 of Newey and McFadden (1994), ˆm(x) converges to m(x) in probability. 4 4 The result of Newey and McFadden (1994, lemma 4.3) is summarized as follows. For random variable Y with density f Y (y), define a continuous function g(y,θ), where θ denotes an unknown parameter vector. Let ˆθ is an estimator such that ˆθ p 1 θ. Then, given random sample Y i of size N from f Y (y), N N i=1 g(y p i, ˆθ) E [ g(y,θ) ] = g(y,θ) f y Y(y)dy. In the current study, an alternative method for obtaining an estimator of m(x) may be Monte Carlo integration with distribution f Y (y). 12

13 In the current study, the derivative of m(x), i.e., the elasticity, is of interest since previous studies investigating intergenerational mobility often report the results in the form of earnings elasticity. We estimate the derivative of m(x) by the following finite numerical approximation: ˆd m (x) = 1 [ ˆm(x + δ) ˆm(x δ)], (25) 2δ where δ denotes a small positive number disturbing ˆm( ) around x, a `la Pagan and Ullah (1999, chapter 4.3.1). As δ 0, ˆd m ( ) approaches to the derivative of ˆm( ). 3.3 Conditional probabilities In addition to conditional means and elasticities, conditional probabilities, e.g., the probability of son s log-earnings exceeding a threshold value given father s log-earnings, are of interest in the literature. In the existing studies, estimation of such quantities is often done by first estimating the conditional mean and then assuming normal errors (See, for example, Solon, 1992). We propose the method estimating conditional probabilities of various kinds semiparametrically by a copula. Let us define the conditional probability of son s log-earnings Y falling into interval [a Y, b Y ] when father s log-earnings X = x as π(a Y, b Y x) Pr(a Y Y b Y X = x) = F Y (b Y X = x) F Y (a Y X = x). (26) Let us further define the partial derivative of a copula function, c v (u, v; γ) = C(u, v; γ). (27) v 13

14 Then it follows from the result of Nelsen (2007, chapter 5.5) that Pr(Y y X = x) = c v [ FX (x), F Y (y); γ ]. (28) We thus have conditional probability function π(a Y, b Y x) = c v [ FX (x), F Y (b Y ); γ ] c v [ FX (x), F Y (a Y ); γ ]. (29) The sample counterpart of π(a Y, b Y x)isgivenby ˆπ(a Y, b Y x) = c v [ ˆF X (x), ˆF Y (b Y ); ˆγ ] c v [ ˆF X (x), ˆF Y (a Y ); ˆγ ]. (30) We estimate the derivative of π(a Y, b Y x) with respect to x, i.e., the marginal effect, by ˆd P (x) = 1 2δ [ˆπ(a Y, b Y x + δ) ˆπ(a Y, b Y x δ)], (31) where δ denoets, again, a arbitrary small positive number. Since ˆπ(a Y, b Y x) and ˆd P (x) are all continuous transformation of γ, the Slutsky s theorem suffices to their consistency given the consistent estimator of γ. Another conditional probabilities of interest may be intergenerational transition probabilities; the probabilities that son s log-earnings Y enters some inter-quantiles given father s log-earnings X. This probability is defined by π(a Y, b Y a X, b X ) Pr(a Y Y b Y a X X b X ) = F(b X, b Y ) F(a X, b Y ) F(b X, a Y ) + F(a X, a Y ). (32) F X (b X ) F X (a X ) 14

15 So, based on a copula, its sample counterpart is given by ˆπ(a Y, b Y a X, b X ) = D(a X, b X, a Y, b Y ) ˆF X (b X ) ˆF X (a X ), (33) where D(a X, b X, a Y, b Y ) =C[ ˆF X (b X ), ˆF Y (b Y ); ˆγ] C[ ˆF X (a X ), ˆF Y (b Y ); ˆγ] C[ ˆF X (b X ), ˆF Y (a Y ); ˆγ] + C[ ˆF X (a X ), ˆF Y (a Y ); ˆγ]. (34) 4 Data Description 4.1 Data source and summary statistics The data we use comes from the PSID. 5 As noted by Solon (1992), one-year observations of earnings are an error-ridden variable for the permanent component of one s income, so using such measurements for estimating an intergenerational earnings elasticity can cause attenuation biases. Solon demonstrates that using the time series averages of individual earnings circumvents the bias. Following Solon and subsequent studies, we employ this strategy. First, we average the logarithms of individual earnings (deflated by the CPI for every year) at age from 30 to 55 for male samples. Then we match a son and his father in the data set guided by Chiteji et al. (2008). Thus, the data, originally collected as panel, is cross-sectional in the current study. We drop the samples with missing or top-coded earnings. The resultant number of individuals is N = Table 2 presents the summary statistics of log-earnings of the both generations. Although we have imposed cut-off ages on the sample observations for calculating the earning averages, due to attritions and differences in the starting years of follow-ups, actually there are substantial variations in time-averaged ages of sample individuals. So we incorporate the 5 The data set is downloaded from: 15

16 son s and father s average ages and their square roots into the regressor vector for controlling for life-cycle variations in earnings. This is a common practice in the literature (Solon, 1999). In addition, since attritions (due to failure of follow-up or observing zero earnings) may not be at random, we use the years of observations of sons as a regressor for eliminating possible selectivity bias. Summary statistics of the regressors are shown in Table 3. Figure 1 depicts the scatter plot of regressor-adjusted log-earnings of sons and fathers. We made the regressor-adjustment by first regressing log-earnings of the both generations on the age variables and years of observations, then computing predicted values evaluated at the sample means of the regressors. As well as positive associations between the two variables, we know from the figure that the both variables are right-skewed, contrary to conventional cross-sectional income distribution. Interestingly, the shape of the scatter plot is similar to that from the data other than the PSID, e.g., Corak and Heisz (1999, Figure 2). The straight line passing through the figure denotes the linear regression. Although it is hard to inspect the properties of dense area of the scatter plot, the linear regression line does not seem to pass the ridge of the joint distribution, suggesting a non-linear intergenerational earnings mobility. 4.2 Preliminary analysis In order to reproduce the estimates of intergenerational earnings elasticities in the existing papers, we run the following linear regression. Y i = J j=0 β j X j i + Z i η + u i, (35) where Y i and X i denote son s and father s log-earnings, respectively, and Z i denotes the regressor vector. As the regressors, we use ages and its square roots of son and father and the number of years son s earnings are observed. J j=0 β 0X j i is the polynomial terms of father s earnings, which approximates the non-linearity in the conditional mean of Y i with respect to X i. 16

17 Table 4 exhibits regression results where the order of polynomials J = 1, 2, 3, 4. For the linear case (J = 1), the estimated intergenerational elasticity is about 0.34, falling into consensus ranges observed by the previous studies (See Solon, 1999). When we choose the quadratic functional form (J = 2), the coefficients of polynomial terms are statistically significant and, compared to the linear case, adding the square root term greatly improves the goodness-of-fit. The signs of estimated coefficients suggests convex relationship between son s and father s earnings abilities. This result is consistent with the statistical evidences of the existing studies (See Bratsberg et al., 2007, and reference therein). For the cubic and quartic cases (J = 3, 4), the coefficients of polynomials are imprecisely estimated. The additional polynomial terms do not seem to improve the fit of regression to the data. It is difficult to interpret the result of higher order polynomial models by the point estimates. So we will show the shape of the regression curve and its derivative for the quartic OLS in the later subsection. 5 Estimation Results 5.1 Parameter estimates and the choice of copula Before estimating the copula-based semiparametric model by Genest et al. (1995), we first generate the regressor-adjusted log-earnings values just as done when depicting scatter plot of Figure 1. Then we apply the maximum likelihood method to pseudo-likelihood function (10) with the pre-whitened earnings variables. 6 In Table 5, we present the estimates of parameters for the several families of copula functions, ˆγ, as well as estimated LTD and UTD parameters. Note that, except the Gaussian, the dependence parameters of copula do not have intuitive information for the strength of dependence. So comparison of parameter estimates of different copulas per se does not make sense. Only the Gaussian dependence parameter corresponds to the Peason s 6 The following results are obtained by using Ox version 4.1. See Doornik (2007) for more details on Ox. 17

18 product-moment correlation coefficient which, as well known, captures a linear association of random variables. Genest et al. (1995) propose the variance estimator for ˆγ which corrects the sampling errors of estimating F X (X i ) and F Y (Y i ) in the first step. However, since we made regressor-adjustments before the first step, further variations should be added to the final stage estimation. We compare two types of standard errors in Table 5. The first (denoted by S.E. in the table) is obtained by ignoring estimations before the final stage and using the inverse of the numerical hessian. The second (denoted by B.S.) is based on nonparametric bootstrap with 500 repetitions. The two standard errors are very similar for all the copulas we estimate. So generated regressor problem seems not to severe in this case. By comparing the goodness-of-fit measures, the AIC (or equivalently pseudo log-likelihood value here) and mean absolute deviations from the empirical copula, we find that the Frank copula outperforms others and the Gumbel and Gaussian follow the Frank. On the other hand, the Clayton and survival Gumbel copulas, both of which exhibit strong lower tail dependence, seem inappropriate for the data of the son-father earnings distribution. In order to improve the model-data match, we employ mixture copulas where the Frank copula is treated as the baseline. Specifically, we estimate the Frank-Gauss, Frank-Clayton, and Frank-Gumbel mixtures. Further we estimate the Clayton-Gumbel mixture which, in theory, covers both lower and upper tail dependences. The results are shown in Table 6. 7 We obtain the standard errors of the estimates by (1) ignoring generated regressors problem and (2) by the bootstrap with 500 repetitions. Compared to the single-copula cases in Table 5, there are non-negligible discrepancies between the two standard errors. For the Gumbel-Clayton mixture, bootstrap standard errors is not reported since its iterative maximizations fail to converge many 7 We examine mixtures of three different copulas but estimates are statistically insignificant, or convergence of iterations fails, or both. So we do note show the results here. Similar results are reported in Hu (2006), where the mixture of Gaussian, Gumbel and survival Gumbel is employed. Further, Based on our experience, not reported here again, the mixtures of the same copula family seem have difficulties in identification. Trivedi and Zimmer (2007) argue that inappropriately choosing copula, i.e, choosing copula not capture the dependence patterns of the data, often causes computational problems. 18

19 times during bootstrapping. The improvements of average absolute deviations over the single copulas are small and similar for all the mixture cases. However, by mixing the Gumbel with the Frank reduces the AIC greatly. The AIC of the Frank alone was , but that of the Frank-Gumbel mixture attains the value of Hence we use the Frank-Gumbel mixture for estimating conditional means, probabilities, and quantiles of the son-father earnings distribution. 5.2 Estimated conditional means and elasticities Figure 2 depicts the estimated conditional mean function of son s log-earnings on father s logearnings. As mentioned in the previous subsection, hereafter we use the Frank-Gumbel mixture as a preferable copula. The left panel of the table shows conditional mean estimates ˆm(x) in equation (23), while the right panel does the estimates of derivative ˆd m (x) in equation (24). These quantities are evaluated at 100 grid points of the X-axis. (Note that, since the variables are in logarithm, the derivatives correspond to elasticities.) For comparison, we estimate the conditional mean by alternative methods. The first is the nonparametric, local polynomial regression (LPR) method by Fan and Gijbels (1996). The second is parametric, quartic OLS given in equation (35) with J = 4 whose estimation result has been in Table 4. For both alternatives, conditional means and its derivatives are also estimated. 8 These results are reported in Figure 2 coupled with the copula-based regression curve. For the conditional mean function, the three different methods provide very similar estimation results. At the area below the 10-th quantile of father s log-earnings distribution, the LPR line is heaving due to, presumably, the presence of outliers. For the elasticities, all the different methods reveal the tendency that the intergenerational elasticity increases as the father s log- 8 The LPR is a kernel-based nonparametric regression and so requires bandwidth selections. We use the rule-ofthumb bandwidth selector of Fan and Gijbels (1996, Chapter 4.2). For facilitating comparison of the other methods, actually, we apply partially linear model (Fan and Gijbels, 1996, Chapter 7.3), where regressor vector Z enters into a regression equation but constant for all X. In the current analysis, the same regressors as in equation (35) is used. 19

20 earnings rising, ranging from 0.1 to 0.8. The consensus value of 0.4 (Solon, 1999) is found only at the inter-quantile of the domain. The elasticity estimates by the Frank-Gumbel copula are rickety due, presumably, to finite approximation of derivatives. Another concern is the sampling variations of competing estimation methods. We show in Figure 3 the 90% confidence intervals of the conditional mean and derivative estimates by the three alternatives. (The standard errors are obtained by the bootstrap technique with 500 resampling.) Overall, the regression line by the Frank-Gumbel copula seems very stable. It should be stressed that the width of the confidence interval for the copula-means do not inflate at the tail area of distribution where observations are scarce. Contrary to the copula result, the regression line of LPR is very noisy at the lower tail. The quartic OLS line is the middle of the two results. The copula-based method, however, do not outperform others in estimating the elasticities. Yet, this may reflect the poor finite approximation of the derivatives. 5.3 Estimated conditional probabilities and marginal effects Next, we report the estimation results of conditional probabilities of son s log-earnings Y given father s log-earnings X and its marginal effects. Specifically, we estimate the conditional probability that Y exceeds some quantile value q α given X = x, i.e., Pr(Y > q α X = x). (36) We choose 25-th, 50-th and 75-th quantiles of Y as q α. For estimating the conditional probability function, the derivative of copula function is needed. For the Frank-Gumbel mixture, the derivative with respect to the second argument is given by C M,v (u, v) = αc Frank,v (u, v) + (1 α)c Gumbel,v (u, v), (37) 20

21 where [ ] 1 1 exp( γ1 ) C Frank,v (u, v) = exp( γ 1 v) 1 exp( γ 1 u) 1 + exp( γ 1v) (38) and C Gumbel,v (u, v) = C Gumbel (u, v) [ ( log u) γ 2 + ( log v) γ 2 ] 1 γ 2 1 ( log v) γ 2 1 v 1. (39) The left panel of Figure 4 shows the estimates of the conditional probability functions. As before, we obtain 90% confidence intervals by the bootstrap. At the area below the 10-th quantile of X, each of conditional probability functions is almost flat. This result echoes the results observed in the previous subsection on the conditional mean estimation. The right panel of Figure 4 depicts the marginal effects. Contrary to the estimated elasticities reported in Figure 2 and 3, the marginal effects decay as X increasing. This is natural since here the outcome variable, Pr(Y > q α X = x), is bounded to unity. Finally, we show the transition probabilities estimated via the Frank-Gumbel mixture copula. By comparing them with empirical frequencies, we can examine the fit of the model to the data. Table 7 presents the result. For comparison, we also show the estimates based on the Gaussian copula. Note that the Gaussian copula distribution estimated here is more flexible than the bivariate normal because the margins are left unspecified for the Gaussian copula. Inspection of the table tells us that both the Gaussian and Frank-Gumbel mixture capture the patterns observed in the empirical transition matrix well. However, the latter copula seems more preferable: This result is just a reproduction of the AIC comparison done before. 21

22 6 Concluding Remarks This paper have investigated the dependence structures of log-earnings of sons and fathers in the United States. For this purpose, the copula-based semiparameteric modeling of a joint distribution has been applied. We propose the estimators of conditional mean and probability of son s log-earnings given father s earnings and their derivatives, all of which facilitate to interpret the dependence structure of the data brought by copula. Our main finding is that the intergenerational earnings mobility depends substantially on father s earnings in the United States. The copula which best fits the data is the Frank-Gumbel mixture, where dependence is weaker at the lower tail and vice versa. In terms of the conditional mean function, this asymmetric tail dependence means a convex curve. This result echoes empirical findings raised recently in the literature (Bratsberg et al., 2007). Linear elasticity estimates of the previous studies, e,g., Solon (1992), seems inappropriate in considering the population whose fathers earnings are far below or far above the average. The statistical evidence proposed by ours and by the previous studies may seem counterintuitive because they mean that father s economic status has a smaller impact on one s earnings at his/her adulthood for the poorer population groups. These results also do not support the theoretical prediction of Becker and Tomes (1979), where the relationship between parents and offspring s earnings ability should be concave. Why is the socio-economic status more mobile in the poor? In every industrialized country, including the United States, the government has made a large effort for anti-poverty actions. Therefore, the data sets of these countries used for studying intergenerational mobilities may reflect the effect of such policy interventions. 22

23 References Becker, G. S. and N. Tomes (1979). An equilibrium theory of the distribution of income and intergenerational mobility. Journal of Political Economy 87(6), Becker, G. S. and N. Tomes (1986). Human capital and the rise and fall of families. Journal of Labor Economics 4(3), S1 S39. Behrman, J. and P. Taubman (1985). Intergenerational earnings mobility in the united states: Some estimates and a test of becker s intergenerational endowments model. Review of Economics and Statistics 67(1), Bratsberg, B., K. Roed, O. Raaum, R. Naylor, M. Jantti, T. Eriksson, and E. Osterbacka (2007). Nonlinearities in intergenerational earnings mobility: Consequences for cross-country comparisons. Economic Journal 117(519), C72 C92. Cameron, A. C., T. Li, P. K. Trivedi, and D. M. Zimmer (2004). Modelling the differences in counted outcomes using bivariate copula models with application to mismeasured counts. Econometrics Journal 7(2), Chen, X. and Y. Fan (2006). Estimation of copula-based semiparametric time series models. Journal of Econometrics 130(2), Chiteji, N., E. Gouskova, and F. Stafford (2008). Intergenerational (ig) correlations in earnings. Mimeo. Corak, M. and A. Heisz (1999). The intergenerational earnings and income mobility of canadian men: Evidence from longitudinal income tax data. Journal of Human Resources 34(3), Couch, K. A. and D. R. Lillard (2004). Non-linear patterns of intergenerational mobility in 23

24 germany and united states. In M. Corak (Ed.), Generational Income Mobility in North America and Europe, Chapter 8, pp Cambridge, UK: Cambridge University Press. Crane, G. J. and J. v. Hoek (2008). Conditional expectations formulae for copulas. Australian & New Zealand Journal of Statistics 50(1), Doornik, J. A. (2007). Ox: An Object-Oriented Matrix Language. London, UK: Timberlake Consultants Press. Fan, J. and I. Gijbels (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability. Florida, USA: Chapman & Hall/CRC. Genest, C., K. Ghoudi, and L.-P. Rivest (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), Hu, L. (2006). Dependence patterns across financial markets: a mixed copula approach. Applied Financial Economics 16(10), Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. Monographs on Statistics and Applied Probability. Florida, USA: Chapman & Hall/CRC. Junker, M. and A. May (2005). Measurement of aggregate risk with copulas. Econometrics Journal 8(3), Nelsen, R. B. (2007). An Introduction to Copulas (2nd ed.). Springer Series in Statistics. Springer. Newey, W. K. and D. McFadden (1994). Large sample estimation and hypothesis testing. In R. F. Engle and D. McFadden (Eds.), Handbook of Econometrics, Volume 4 of Handbook of Econometrics, Chapter 36, pp Amsterdam, the Netherlands: Elsevier. 24

25 Pagan, A. and A. Ullah (1999). Nonparametric Econometrics. Cambridge, UK: Cambridge University Press. Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. International Economic Review 47(2), Sklar, A. (1959). Fonctions de répartition a n dimensions et leurs marges. Publications de l Institut de Statistique de L Université de Paris 8, Smith, M. D. (2003). Modelling sample selection using archimedean copulas. Econometrics Journal 6(1), Solon, G. (1992). Intergenerational income mobility in the united states. American Economic Review 82(3), Solon, G. (1999). Intergenerational mobility in the labor market. In O. Ashenfelter and D. Card (Eds.), Handbook of Labor Economics, Volume 3A, Chapter 29, pp Amsterdam, The Netherlands: Elsevier. Solon, G. (2002). Cross-country differences in intergenerational earnings mobility. Journal of Economic Perspectives 16(3), Trivedi, P. K. and D. M. Zimmer (2007). Copula Modeling. Foundations and Trends in Econometrics. Now Publishers Inc. Zimmer, D. M. and P. K. Trivedi (2006). Using trivariate copulas to model sample selection and treatment effects: Application to family health care demand. Journal of Business & Economic Statistics 24,

26 2 (x2 Family [ C(u, v; γ) ] c(u, v; γ) Range(γ) Tail Dep. Gaussian ΦBv Φ 1 (u), Φ 1 (v); γ (1 γ 2 ) 1 2 exp [ 1 2 (1 γ2 ) 1 (x 2 + y 2 2γxy) ] 1 + y 2 ) ] ( 1, 1) L = 0, U = 0 exp [ Frank 1 γ log { 1 + [exp( γu) 1][exp( γv) 1] exp( γ) 1 } γ[exp( γ) 1] exp[ γ(u+v)] (, ) L= 0, U = 0 {[exp( γu) 1][exp( γv) 1]+exp(γ) 1} 2 Clayton γ + [ 1 γ ] [ ( u v γ ) 1 (1 + γ)(uv) γ 1 ( u γ + v γ 1 ) 2 γ 1 (0, ) L= 2 γ 1,U= 0 1 ] 1 (uv) 1 (ũṽ) γ 1 (ũ γ + ṽ γ ) γ 1 (ũ γ + ṽ γ ) γ 1 + γ 1 C(u, v; [1, ) L = 0, U = 2 2 Gumbel exp γ) γ 1 (ũ γ +ṽ γ ) 2 1 γ Table 1: Properties of Parametric Copula Families : x =Φ 1 (u), y =Φ 1 (v). : ũ = log(u), ṽ = log(v). 26

27 Son s log-earnings Father s log-earnings Mean S.D Q Q Q Q Q # individuals Table 2: Summary Statistics of Son s and Father s Log-earnings 27

28 Sons age Father s age # years observed Mean S.D Min Max # individuals Table 3: Summary Statistics of Control Variables 28

29 Est. S.E. Est. S.E. Est. S.E. Est. S.E. Constant Father s log income (Father s log-earnings/10) (Father s log-earnings/10) (Father s log-earnings/10) Son s age (Son s age/10) Father s age (Father s age/10) # years observed Adjusted R # individuals Table 4: OLS Results of Intergenerational Earnings Elasticity Note: Robust standard errors are reported. 29

30 Est. S.E. (B.S.) log-like AIC Dev.(%) LTD RTD Gaussian Frank Clayton Sv. Clayton Gumbel Sv. Gumbel Table 5: Estimation Results of Copula Parameters Note: For all cases, the number of individuals is N = 2006 and the number of copula parameters is unity. B.S. denotes the bootstrap standard errors with 500 repetitions. 30

31 Est. S.E. (B.S.) log-like AIC Dev. (%) LTD RTD Frank-Gauss Frank Gauss Prob(Frank) Frank-Clayton Frank Clayton Prob(Frank) Frank-Gumbel Frank Gumbel Prob(Frank) Clayton-Gumbel Clayton Gumbel Prob(Clayton) Table 6: Estimation Results of Mixed-copula Parameters Note: For all cases, the number of individuals is N = 2006 and the number of copula parameters is three. B.S. denotes the bootstrap standard errors with 500 repetitions. For the Frank-Clayton mixture, B.S. is note reported since iterations often fail to converge during bootstrapping. 31

32 X i < q 25 q 25 X i < q 50 q 50 X i < q 75 q 75 X i Empirical Pr(Y i < q 25 X i ) Pr(q 25 Y i < q 50 X i ) Pr(q 50 Y i < q 75 X i ) Pr(q 75 Y i X i ) Gaussian Pr(Y i < q 25 X i ) Pr(q 25 Y i < q 50 X i ) Pr(q 50 Y i < q 75 X i ) Pr(q 75 Y i X i ) Frank-Gumbel Pr(Y i < q 25 X i ) Pr(q 25 Y i < q 50 X i ) Pr(q 50 Y i < q 75 X i ) Pr(q 75 Y i X i ) Table 7: Empirical vs. Model-based Transition Probabilities 32

33 Son s log earnings Q90 Q50 Q10 Q10 Q50 Q Father s log earnings Figure 1: Scatter Plot of Son s and Father s Log-earnings 33

34 Conditional mean functions Elasticities (derivatives) Son s log earnings Q90 Q50 Q10 Mix. copula Local poly. Quartic OLS Q10 Q50 Q90 Elasticity Mix. copula Local poly. Quartic OLS Q10 Q50 Q Father s log earnings Father s log earnings Figure 2: Conditional Means Estimated by Copula, LPR, and Polynomial OLS 34

35 Son s log earnings Frank Gumbel mix. copula mean Q90 Q50 Q10 Q10 Q50 Q Father s log earnings Elasticity Frank Gumbel mix. copula elasticity Q10 Q50 Q Father s log earnings Son s log earnings Local polynomial mean Q90 Q50 Q10 Q10 Q50 Q Father s log earnings Elasticity Local polynomial elasticity Q10 Q50 Q Father s log earnings Son s log earnings Quartic OLS mean Q90 Q50 Q10 Q10 Q50 Q Father s log earnings Elasticity Quartic OLS elasticity Q10 Q50 Q Father s log earnings Figure 3: 90 % Confidence Intervals of Alternative Conditional Mean Estimates Note: The intervals are obtained by 500 bootstraps per evaluation point and asymptotic normal distributions. 35

36 Pr(Y > q 0.25 X) dpr(y > q 0.25 X) dx Conditional probability Q10 Q50 Q Father s log earnings Pr(Y > q 0.5 X) Marginal effect Q10 Q50 Q Father s log earnings dpr(y > q 0.5 X) dx Conditional probability Q10 Q50 Q90 Marginal effect Q10 Q50 Q Father s log earnings Father s log earnings Pr(Y > q 0.75 X) dpr(y > q 0.75 X) dx Conditional probability Q10 Q50 Q90 Marginal effect Q10 Q50 Q Father s log earnings Father s log earnings Figure 4: 90 % Confidence Intervals of Conditional Probability Estimates by Copula Note: The intervals are obtained by 500 bootstraps per evaluation point and asymptotic normal distributions. 36