Economics Letters 59 (1998) 77–82

Making mobility visible: a graphical device Mark Trede* ¨ Wirtschafts- und Sozialstatistik, Universitat ¨ zu Koln ¨ , Albertus-Magnus-Platz, 50923 Koln ¨ , Germany Seminar f ur Received 19 August 1997; accepted 4 December 1997

Abstract This paper suggests a method of depicting mobility processes of income or earnings. It makes use of nonparametric quantile regression based on bivariate kernel density estimation. The method is illustrated with a comparison of German and US income mobility.  1998 Elsevier Science S.A. Keywords: Income mobility; Nonparametric quantile regression; Kernel density estimation JEL classification: D31; D63

1. Introduction The literature on income mobility has been growing rapidly since more, better, and longer panel data sets are available. It is widely recognized that merely looking at cross-sections is not sufficient to give a telling picture of the income distribution. Dynamic aspects have to be taken into account. The question at the heart of mobility measurement is: Who moves where? A well established but not entirely satisfactory instrument to answer this question is the transition matrix giving the probability of moving to a certain income class in the next period conditional on this period’s income class. Even if the number of classes is restricted to as few as five classes the transition matrix is not easily comprehended at first sight. Consequently a number of scalar measures mapping transition matrices to the real unit interval have been developed to summarize the entire matrix into a single mobility index. Naturally, a great amount of information is (deliberately) lost in this process. There are other approaches to measuring income mobility. Earlier studies often used the coefficient of correlation of incomes in two periods, see Atkinson et al. (1992) for a comprehensive overview of the literature. Shorrocks (1978) proposed a new mobility index which measures the relative reduction of inequality when the accounting period is extended. This index was generalized by Maasoumi and Zandvakili (1986) and is frequently applied in empirical work, see e.g. Maasoumi and Zandvakili (1986), Burkhauser and Poupore (1997). This paper proposes a graphical device to help the naked eye understand the mobility process easily. The basic idea is to depict the nonparametrically estimated continuous equivalent of the transition matrix, which is found by nonparametric quantile regression methods. *Tel.: 149 221 4702283; fax: 149 221 4705074; e-mail: [email protected] 0165-1765 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 98 )00015-9

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The paper is structured in the following way. Section 2 describes the nonparametric quantile regression theory. Section 3 applies the method to income data from Germany and the US. The pictures provide a clue to the well documented but puzzling result of higher income mobility in Germany. Section 4 concludes.

2. Nonparametric quantile regression Quantile regression has been introduced in the parametric framework of the linear regression model by Koenker and Bassett (1978). Instead of estimating the mean conditional on the values of the explanatory variables they suggest estimating conditional quantiles. If the conditional median is estimated the method is equivalent to least absolute errors regression. In many situations the assumption of linearity may not be reasonable and nonparametric methods are preferable. There are various approaches to nonparametric quantile regression, including spline smoothing (Koenker et al., 1992) and kernel estimation (Abberger, 1997). The latter method is used in this paper since it is widely applied in nonparametric investigations of income distributions, see e.g. Pudney (1993); Jenkins (1995) and Schluter (1996). Let Y1 denote income in the first period and Y2 income in the second period. The joint cumulative density function F( y 1 , y 2 ) gives the proportion of the population with incomes Y1 # y 1 and Y2 # y 2 . Assume that the bivariate density f( y 1 , y 2 ) of incomes in the first and second period exists. The incomes are not required to be nonnegative. The cumulative density function of income in the second period (Y2 ) conditional on income in the first period (Y1 5 y 1 ) is F( y 2 uy 1 ) 5

E

y2

2`

f(tuy 1 ) dt 5

E

y2

f( y 1 , t) ]]] dt, 2` f( y 1 )

(1)

`

21

where f( y 1 )5e2` f( y 1 , y 2 ) dy 2 is the marginal distribution of Y1 . The inverse F ( puy 1 ) of F( y 2 uy 1 ) gives the p-quantiles of Y2 conditional on Y1 5y 1 . As the exact form of f( y 1 , y 2 ) is unknown, it has to be estimated. Since there are no satisfactory parametric specifications of two-dimensional income distributions, nonparametric methods are called for. Let there be n observations on income in both periods; y 1i and y 2i are incomes of individual i51,...,n, in the first and second period, respectively. Nonparametric quantile regression is based on nonparametric density estimation in the same way as ordinary (mean) nonparametric regression is. The unknown densities in (1) are substituted by their estimates. The starting point is the estimated bivariate density

OS

n y 1 2 y 1i y 2 2 y 2i 1 ˆ y , y ) 5 ]] f( K ]]] K ]]] , 1 2 nh 1 h 2 i51 h1 h2

DS

D

(2)

where h 1 , h 2 are the bandwidths, and K(?) is the kernel. For simplicity, the kernel is assumed to be multiplicative, but more general specifications are possible. Substituting (2) into (1) yields

M. Trede / Economics Letters 59 (1998) 77 – 82

y 2y y 2y O KS]]]D G S]]]D h h ˆ y uy ) 5 ]]]]]]]]], F( y 2y O K S]]] D h 1

1i

1

i

2

2

2i

2

1

1

79

(3)

1i

1

i

z

where G(z)5e2` K(t) dt is the cdf of the kernel function. Sometimes the conditional cdf is estimated by y 2y O K S]]]D x h ˆ y uy ) 5 ]]]]]]]], F( y 2y O K S]]] D h 1

2

1i

h y 2i #y 2 j

1

i

1

1

i

1i

1

where x is the indicator function, see e.g. Abberger (1997). I will use (3) since it is explicitly derived from the joint density. In practice the differences are likely to be small. It is straightforward to invert (3) numerically with respect to y 2 . We arrive at F 21 ( puy 1 ), the p-quantile of Y2 , which might be treated as a function of y 1 . Drawing various p-quantiles as functions of y 1 results in a picture of the mobility process. What does the picture look like and how can we read it? Consider the left diagram in Fig. 1 for example (which will be described in more detail in the next section). On the abscissa there is income in the first period, normalized to have a unit median. The five lines show the 10, 30, 50, 70, and 90 per cent quantiles conditioned on income in the first period; in addition the 45 degree line of constant income is included. It can be seen, for instance, that roughly 10 per cent of the people with income 1.5 in the first period end up with income less than 1 in the second period while roughly 10 per cent managed to increase their income to more than 1.8 in the second period. The picture indicates where people move to. If there were perfect mobility (in the sense of independence from the starting period)

Fig. 1. Quantile regression of (relative) income in 1985 and 1989 on (relative) income in 1984, Germany.

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the quantile lines would be horizontal, income in the initial period would not matter. The further apart these lines, the higher is inequality in the second period. The other extreme is total immobility, in which case all the quantile lines would coincide: income in the second period would be completely determined by income in the first period. If these lines coincided with the 45 degree line there would be no change in the marginal distribution either. Thus, both the distance from each other, and the slopes of the quantile lines provide information about income dynamics. Depending on the kind of mobility one is interested in, there are several ways to normalize incomes. If growth mobility should be neglected both incomes may be normalized to have unit mean or median. There are however good reasons to take growth mobility into account (see e.g. Maasoumi, 1997). In this case incomes may simply be left as they are, or they may be divided by the mean or median income of, say, the first period. The latter approach is preferable if the mobility processes of various countries are to be compared. Using the median for normalization is advisable if the data quality is poor since, in contrast to the mean, the median is robust. The graphs of the conditional quantiles are similar in nature to transition matrices but avoid one of their main drawbacks, namely the arbitrary construction of income classes. Transition matrices have another property that is sometimes regarded as a weakness and that conditional quantiles have as well: they do not take into account what is happening between the two periods under consideration. If at all, this might only matter if the two periods are far apart.

3. An empirical illustration: mobility in Germany and the United States Recent comparative studies on income mobility in Germany and the United States indicate a higher mobility in Germany, see e.g. Burkhauser and Poupore (1997). This finding is in contrast to common wisdom stipulating that the ‘‘credentialized’’ nature of the German labour market as well as its joint union-management wage bargaining process would bring about relatively stable income profiles (Burkhauser and Poupore, 1997). The data are from the PSID-SOEP equivalent data file (Burkhauser et al., 1995). Incomes are nominal annual postgovernment household incomes equivalized by the OECD equivalence scale (a detailed description of the data can be found in Schwarze, 1995). There are a few observations with implausibly low incomes. All observations with annual equivalized income less than 1000 DM or 450 US$ in one of the years under consideration are deleted. The remaining number of observations for the two-year period is 12 308 for Germany and 18 141 for the US. For the six-year period the numbers are 9604 and 15 239. In order to make the graphs of both countries comparable income in every year is normalized by the 1984 median of the respective country. Hence growth mobility is not excluded from the analysis. Figs. 1 and 2 depict the mobility process for both countries. Each figure consists of two graphs. The left graph gives the 10, 30, 50, 70, and 90 per cent quantiles of (normalized) income in 1985 conditional on (normalized) income in 1984; the right one gives the quantiles of income in 1989. Hence, short-run mobility is shown on the left, long-run mobility on the right. The most salient feature of the figures is the divergence of quantile lines when we move from short-run to long-run mobility. This clearly indicates that long-run mobility is higher in both countries. But there are a number of other interesting findings, some of which are not apparent at first glance.

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Fig. 2. Quantile regression of (relative) income in 1985 and 1989 on (relative) income in 1984, United States.

First, the shape of the quantile bundles is similar in both countries. A closer look reveals, however, that the German bundle is flatter than the US one. This is true both for short-run and long-run mobility, but the effect is more pronounced with long-run mobility. The angle between the bundles is about 58 for short-run mobility and about 108 for long-run mobility. The flatter quantile lines in Germany indicate higher mobility as the dependence of income on the starting position is smaller. Second, the overall median income is increasing during the observation period in both countries as can be seen e.g. by the upward shift of the 50 per cent quantile line. Notice, however, the different slope of this line: in Germany more than half of the relatively rich people actually experienced a decline between 1984 and 1985, whereas more than half of the relatively poor experienced an increase in income. In contrast, the 50 per cent line for the US virtually coincides with the 45 degree line. Third, the quantile bundles are fanning out. The higher initial income the less accurate the income forecast. Almost anything except severe poverty can happen to the rich if we consider the long term. Even in the short term the bottom 10 per cent line is rather far apart from the rest of the quantile lines. The pictures do not show the quantile regressions beyond incomes of more than 2.5. The reason is the scarcity of observations in that area. Nonparametric methods need a large number of observations to work properly. The issue of statistical significance is not dealt with in this paper. Including confidence bands of quantile lines would mess up the graphs and is therefore not advisable. It would, however, be interesting to investigate whether there are simple functional forms (maybe even linear functions) that capture the quantile lines. Parametric quantile regression is, of course, more efficient than nonparametric quantile regression if the regression function is correctly specified.

4. Conclusion This paper presents an easy graphical way to make the mobility process visible. It is similar in nature to transition matrices but avoids the problematic issue of building arbitrary income classes. The nonparametric quantile regression is based on kernel density estimation.

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Comparing the mobility pictures of Germany and the US we find that mobility is higher in Germany. This finding is in accordance with other comparative studies (e.g. Burkhauser and Poupore, 1997). Since the mobility pictures are not related to any mobility index one may conclude that higher mobility in Germany is not just an artefact of the particular way in which mobility is measured but genuinely existent.

References Abberger, K., 1997. Quantile smoothing in financial time series. Statistical Papers 38, 125–148. Atkinson, A.B., Bourguignon, F., Morrisson, C., 1992. Empirical Studies of Earnings Mobility. Harwood, Chur. Burkhauser, R.V., Butrica, B.A., Daly, M.C., 1995. The Syracuse University PSID-GSOEP equivalent data file: A product of cross-national research. Cross-National Studies in Aging, Program Project Paper No. 25, Syracuse University. Burkhauser, R.V., Poupore, J.G., 1997. A cross-national comparison of permanent inequality in the United States and Germany. Review of Economics and Statistics 79, 10–17. Jenkins, S.P., 1995. Did the middle class shrink during the 1980? UK evidence from kernel density estimates. Economics Letters 49, 407–413. Koenker, R., Bassett Jr, G., 1978. Regression quantiles. Econometrica 46, 33–50. Koenker, R., Portnoy, S., Ng, P., 1992. Nonparametric estimation of conditional quantile functions. In: Dodge, Y. (Ed.), L1 -Statistical Analysis and Related Methods. North-Holland, New York. Maasoumi, E., 1997 (forthcoming). On mobility. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York. Maasoumi, E., Zandvakili, S., 1986. A class of generalized measures of mobility with applications. Economics Letters 22, 97–102. Pudney, S., 1993. Income and wealth inequality and the life cycle: A non-parametric analysis for China. Journal of Applied Econometrics 8, 249–276. Schluter, C., 1996. Income dynamics in Germany—evidence from panel data. Working Paper. Schwarze, J., 1995. Simulating German income and social security tax payments using the GSOEP. Cross-National Studies in Aging, Program Project Paper No. 19, Syracuse University. Shorrocks, A.F., 1978. Income inequality and income mobility. Journal of Economic Theory 19, 376–393.