PRELIMINARY AND INCOMPLETE [DO NOT CITE WITHOUT PERMISSION] Estimating the Intergenerational Elasticity and Rank Association in the US: Overcoming the Current Limitations of Tax Data Bhashkar Mazumder* Federal Reserve Bank of Chicago and University of Bergen March, 2015 Abstract: The intergenerational economic mobility literature has begun to use estimators of rank mobility in addition to the intergenerational elasticity (IGE). I argue that both estimators are useful for providing insight but for different mobility concepts. The influential work by Chetty et al (2014) focuses on rank mobility estimators and their choice is primarily motivated by measurement concerns with the IGE rather than for conceptual reasons. I argue that those measurement concerns are largely misplaced and instead reflect fundamental limitations of the intergenerational samples that can currently be constructed with U.S. administrative tax data from the IRS. Despite the seeming advantages of extremely large samples of administrative data, I use small samples of survey data from the PSID to demonstrate that it is the age structure, limited panel dimension, and measurement error in administrative data rather than the use of the IGE that leads to the measurement problems encountered by Chetty et al. Using the PSID data I also demonstrate that the national estimates of the IGE in family income in the U.S. are greater than 0.7 and are vastly higher than virtually all current estimates and substantially higher than those produced by Chetty et al. The father-son income elasticity estimates are consistent with those of Mazumder (2005) and suggest that those earlier estimates were not driven by imputations but rather by the availability of longer time spans to measure permanent economic status of parents. I also highlight several other lesser-known studies that find evidence consistent with Mazumder (2005) that are imputation-free. Finally, I show that when using a more ideal sample in the PSID, that the rank-rank slope estimates are around 0.5 and significantly higher than the estimate of 0.341 found by Chetty et al using the IRS data. *I thank Andy Jordan for outstanding research assistance. The views expressed here do not reflect those of the Federal Reserve Bank of Chicago or the Federal Reserve system. I. Introduction Inequality of opportunity has become a tremendously salient issue for policy makers across many countries in recent years. The sharp rise in inequality has given rise to fears that economic disparities will persist into future generations. This has resulted in a heightened focus on the literature on intergenerational economic mobility. This body of research which is now several decades old seeks to understand the degree to which economic status is transmitted across generations. Of course, a critical first step in understanding this literature and correctly interpreting its findings is having a sound understanding of the measures that are being used and what they do and do not measure. This paper will focus on two prominent measures of intergenerational mobility: the intergenerational elasticity (IGE) and the rank-rank slope and discuss several key conceptual and measurement issues related to these estimators in the context of the U.S. The IGE has a fairly long history of use in economics dating back to papers from the 1980s (add cites) and many notable advances have been made in terms of measurement and issues concerning life-cycle bias (e.g. Solon, 1992; Mazumder, 2005a; Haider and Solon, 2006).1 In recent years a number of papers have started to use rank based measures of intergenerational mobility (Dahl and Deliere, 2009; Bhattacharya and Mazumder, 2011, Corak et al., 2014, Chetty et al, 2014, Mazumder, 2014 and Bratberg et al., 2015). However, the motivation for using rankbased measures has differed among these studies. Dahl and Deliere (2009) and Chetty et al. (2014) have emphasized measurement problems with the IGE as their reason for shifting to ranks. The other studies, in contrast, have generally emphasized other conceptual aspects of the rank based measures such as the ability to: distinguish upward versus downward movements, make subgroup comparisons, and identify nonlinearities in intergenerational mobility. 1 Reviews of this literature can be found in Solon (1999) and Black and Devereaux (2011). Given the recent shift in the literature, it would be useful to better understand the recent criticisms of the IGE and to see what they imply for their use going forward. To address this, I first discuss the conceptual differences between the estimators and separate this discussion from the measurement concerns. In short, I argue that conceptually, both measures can provide useful insights about different aspects of mobility. I argue that there are clearly certain questions that are best answered by the IGE and for that reason researchers should continue to use the IGE as at least one tool for measuring intergenerational mobility. Rank estimators are also valuable because they can be used to examine other aspects of mobility such as geographic differences within a country or racial differences in a way that the IGE cannot (Mazumder, 2014). In terms of measurement, I argue that the criticisms of the IGE are context specific. When the structure of an intergenerational dataset is limited in terms of the observed age ranges of parents or children, or if the panel dimension for the data is relatively small, then the IGE estimates are likely to less robust than rank-based estimates. We should be clear, though that this is a function of the available data and is not an intrinsic problem with the estimator in all contexts. Further, the various measurement studies over the last 25 years provide tremendous insight and various tools that can be used to appropriately adjust the estimates. This is particularly important when evaluating some of the claims made in the very influential paper by Chetty et al (2014). They argue that using the IGE leads to dramatically different results depending on how they code instances of zero earnings among the children in their sample. They suggest that this is due to nonlinearities in the log-log relationship. They show that their rank mobility estimates do not suffer from this issue and in large part it is for this greater robustness that they prefer those estimates for characterizing intergenerational mobility at the national level. I will argue that this apparent sensitivity of the IGE estimates is due to the limitations of the tax data that is currently available rather than an intrinsic feature of the estimator. For all of its tremendous advantages such as having huge sample sizes, the IRS data is fundamentally limited in a few key respects. First, children’s income is only measured over 2 years at a relatively early point in the life cycle (around age 30) at a time in which unemployment was quite high. Second, parents’ income is measured relatively late in life (around age 50) and only for up to 5 years. Third, recent research has established that administrative data can be worse than survey data, particularly at the bottom end of the income distribution and can actually introduce measurement error (e.g. Abowd and Stinson, 2013 and Hokayem et al., 2012, 2015). I use the PSID to demonstrate the implications of these data limitations, empirically. First, I use the PSID to construct an intergenerational sample where both kids and parents income is observed over many years and is centered over the prime working years in both generations. The time span also covers a much larger portion of the lifecycle than the IRS records. I estimate the IGE using this close to “ideal” sample and then show how the estimates change if I impose the same kinds of data limitations that exist in the IRS data. The results of this exercise show that the data limitations lead to very dramatic differences in estimates. For example, imposing the data restrictions in the IRS data leads to IGE estimates from the PSID with respect to family income that are about 50 to 60 percent of the size of the estimates with the complete data. Moreover, even when I impose the same data limitations found in the IRS data in the PSID I find that the estimates are significantly higher in the PSID than in the tax data –which runs exactly counter to what one would expect if measurement error was reduced with administrative data. This suggests that researchers should use caution when interpreting the results only from U.S. administrative tax data and look to compare them to results with survey data. Perhaps the most significant contribution of the paper is to show that the magnitude of the IGE estimates when using many years of income data centered over the prime working years in both generations are vastly higher than almost all previous estimates in the literature. For example, the estimates of the IGE with respect to family income are centered at 0.7 and are consistently in the 0.6 to 0.8 range. Surprisingly, only one prior study has constructed a PSID sample centered over the prime working years and covering long time spans to directly estimate the IGE.2 Although Chetty et al emphasize their national rank mobility estimates, they argue that their estimates of the IGE in family income of 0.344 are preferable to previous estimates. For example, they argue that the estimate of the IGE in father-son earnings of 0.6 in Mazumder (2005) is likely due to imputations imposed in Mazumder’s data. The analysis here suggests that the findings in Mazumder (2005) can easily be replicated using publicly available survey data that requires no imputations. Obtaining these estimates simply requires using samples with the appropriate ages and long time spans of available income data centered around the prime working ages in both generations. I also point to other studies in the literature that yield findings consistent with Mazumder (2005) that do not require imputations (e.g. Mazumder, 2005b; Nilsen et al (2012); and Mazumder and Acosta, 2014). A final exercise uses the PSID data to estimate the rank-rank slope which is one of the measures used by Chetty et al (2014). Once again the estimates are significantly larger with the PSID (around 0.5) than what one obtains when using the IRS data (0.341). This suggests that while the rank-rank slope may be more relatively robust to the data limitations of the IRS sample 2 See Mazumder and Acosta (2014). than the IGE, it is still far from ideal and suggests that the intergenerational mobility even by rank-based measures may be overstated by the tax data. One clear conclusion to be drawn from this paper is that researchers should continue to use the IGE if that is the conceptual parameter of interest and when their intergenerational samples have the appropriate panel lengths and age structure. Even when the ideal data is not available, researchers can still attempt to assess the extent of the bias based on prior papers in the literature that propose methodological fixes. Substantively, the results of this paper show that intergenerational mobility in the US is substantially lower than what one would think based on using the currently available IRS data irrespective of which measure is used. The rest of the paper proceeds as follows. Section 2 describes the conceptual differences between the estimators. Section 3 describes measurement issues with IGE estimation and describes the structure of an “ideal” dataset. It then compares this ideal dataset with the IRSbased intergenerational sample used by Chetty et al (2014) and a close to ideal sample that can be constructed with publicly available PSID survey data. Section 4 presents the results when using the PSID to show the effects of imposing the limitations with the tax data. Section 5 concludes. II. Conceptual Issues The concept of regression to the mean over generations has a long and notable tradition going back to the Victorian era social scientist Sir Francis Galton who studied among other things the rate of regression to the mean in height between parents and children. Modern social scientists have continued to find this concept insightful as a way of describing the rate of intergenerational persistence in a particular outcome and to infer the rate of mobility as the flip side of persistence. In particular economists have focused on the intergenerational elasticity (IGE). The IGE is the estimate of obtained from the following regression: (1) y1i = + y0i + i where y1i is the log income of the child generation and y0i is the log of income in the parent generation.3 The estimate of provides a measure of intergenerational persistence and 1 - can be used as a measure of mobility. One way to interpret the elasticity in practical terms is to consider what it implies about how many generations it would take for a family living in poverty to attain close to the national average household income level. If for example, the IGE is around 0.60 as claimed by Mazumder (2005a) then it would take 6 generations (150 years). On the other hand if the IGE is around 0.34 as claimed by Chetty et al (2014) then it would take just 3 generations.4 Clearly, the two estimates have profoundly different implications on the rate of intergenerational mobility and if the rate of regression to the mean is what we are interested in knowing then the IGE is what we estimate. The concept of regression to the mean is also widely used in other aspects of economics such as the macroeconomic literature on differences in percapita income across countries (e.g. Barro and Sala-i-Martin, 1992). The rank–rank slope on the other hand is about a different concept of mobility, positional mobility. It asks, for example, what the expected difference in ranks between members of two different families would be if the families differed in the parent generation by 10 percentiles. One can imagine that depending on the shape of the income distribution the descendants of a family in poverty might experience relatively rapid positional mobility but slower regression to Often the regression will include age controls but few other covariates since is not given a causal interpretation but rather reflects all factors correlated with parent income 4 This exercise is based on Mazumder (2005) and considers a family of four with two children under the age of 18 whose income is $18,000 (under the poverty threshold in 2013 of around $24,000). Mean household income in 2013 was approximately $73,000. The calculation specifically examines how many generations before the expected income of the descendants of this family would be within 5% of the national average. 3 the mean.5 How are the two measures related? Chetty et al (2014) point out that that the rankrank slope is very closely related to the intergenerational correlation (IGC) in log income. They and many others have also shown that the IGE is equal to the IGC times the ratio of the standard deviation of log income in the child generation to the standard deviation of log income in the parent generation: (2) This relationship is sometimes taken to imply that a rise in inequality would lead the IGE to rise but not affect the IGC and that therefore, the IGC may be a preferred measure that avoids a “mechanical” effect of inequality. By extension one might also prefer the rank-rank slope if one accepts this argument. Several comments are worth making here. First, in reality the parameters are all jointly determined by various economic forces. In the absence of a structural model one cannot meaningfully talk about holding “inequality” fixed. For example, a change in might cause inequality to rise, rather than the reverse, or both might be altered by some third force such as rising returns to skill. The mathematical relationship shown in (2) does not substitute for a behavioral relationship and so it does not make sense to pretend that we can separately isolate inequality from the IGE. Second, even if it was the case that the IGC or rankrank slope was a measure that was independent of inequality, that doesn’t mean that society shouldn’t continue to be interested in the rate of regression to the mean. I would argue that it is precisely because of the rise in inequality that societies are increasingly concerned about intergenerational persistence and so incorporating the effects of inequality may actually be critical to understanding the rates of mobility that policy makers want to address. 5 In the example above, one can imagine that it may take significantly longer for the family with income of $18,000 to attain the mean income level of $73,000 than the median income level of $52,000. In addition to providing useful information about positional mobility the rank-rank slope has other attractive features. Perhaps it’s most useful advantage over the IGE is that it can be used to measure mobility differences across subgroups of the population. This is because the IGE estimated within groups is only informative about persistence or mobility with respect to the group specific mean whereas the rank-rank slope can be estimated based on ranks calculated based on the national distribution. Chetty et al (2014) were able to use this to characterize mobility for the first time at an incredibly fine geographic level. Mazumder (2014) used other “directional” rank mobility measures to compare racial differences in intergenerational mobility between blacks and whites in the U.S. However, for characterizing intergenerational mobility at the national level both the IGE and the rank-rank slope are suitable depending on which concept of mobility the researcher is interested in studying. III. Measurement Issues and the Ideal Intergenerational Sample Measurement Issues The prior literature on intergenerational mobility has highlighted two key measurement concerns that I will briefly review here. The first issue is regarding measurement error or transitory fluctuations in income. In an ideal setting the measures of y1 and y0 in equation (1) would be measures of lifetime or permanent income but in most datasets we only have short snapshots of income that can contain noise and attenuate estimates of the IGE. Solon (1992) showed that using a single year of income as a proxy for lifetime income of fathers can lead to considerable bias relative to using a 5 year average of income. Using the PSID, Solon concluded that the IGE was 0.4 “or higher”. Mazumder (2005a) used the SIPP matched to social security earnings records and showed that using even a 5-year average can lead to considerable bias and estimated the IGE to be around 0.6 when using longer time averages of fathers earnings (up to 16 years). Mazumder argues that the key reason that a 5-year average is insufficient is that the transitory variance in earnings tends to be highly persistent and appeals to the findings of U.S. studies of earnings dynamics that support this point. Using simulations based on parameters from these other studies, Mazumder shows that the attenuation bias from using a 5-year average in the data is close to what one would expect to find based on the simulations. In a separate paper, that is less well known, Mazumder (2005b) showed that if one uses short term averages in the PSID and uses a Hetereoscedastic Errors in Variables (HEIV) estimator that adjusts for the amount of measurement error or transitory variance contained in each observation, then that the PSID adjusted estimate of the IGE is also around 0.6. This latter paper is a useful complement because unlike the social security earnings data used by Mazumder (2005a) the PSID data is not topcoded and doesn’t require imputations. Chetty et al (2014) has contended that the larger estimates of the IGE in Mazumder (2005a) was due to the nature of the imputation process rather than due to larger time averages of fathers’ earnings. The second critical measurement concern in the literature concerns lifecycle bias best encapsulated by Haider and Solon (2006). One aspect of this critique concerns the effects of measuring children’s income when they are too young. Children who end up having high lifetime income often have steeper income trajectories than children who have lower lifetime incomes. Therefore if income is measured at too young an age it can lead to an attenuated estimate of the IGE. Haider and Solon show that this bias can be considerable and is minimized when income is measured at around age 40. A related issue is that transitory fluctuations are not constant over the lifecycle but instead follow a u-shaped pattern over the lifecycle (Baker and Solon, 2003; Mazumder, 2005). This implies that measuring parent income when they are either too young or (especially) when they are too old can also attenuate estimates of the IGE. While there are econometric approaches one can use to correct for lifecycle bias, one simple approach would be to simply center the time averages of both children’s and parents income around the age of 40. Using this approach with the PSID, Mazumder and Acosta estimate the IGE to be around 0.6. Further, Nilson et al (2012), using Norweigian data show that both time averaging and life-cycle bias play a role in attenuating IGE coefficients. It should be noted that Nilsen et al find that these biases matters despite using administrative data like Chetty et al.6 Comparisons of Intergenerational Samples To better understand the limitations with currently available intergenerational samples in the US with respect to these measurement issues, it is useful to think about what an ideal sample would look like. In an ideal setting we would want to construct an intergenerational sample where income is measured for both generations throughout the entire working life cycle, say between the ages of 25 and 55.7 If we take 2012 as an end point that would be mean that for the children’s generation we would want cohorts of children who were born in 1957 or earlier for full lifecycle coverage. For the 1957 cohort we would measure their income between 1982 and 2012. For the 1956 cohort we would measure income between 1981 and 2011. We could then continue to work backwards to collect data on earlier cohorts. Suppose that for the parent generation, the mean age at the time the child is born is 25. Then for the 1957 cohort we would collect income data from 1957 to 1987 and from 1956 to 1986 for the parents of the earlier birth cohorts, and so on. With such a dataset in hand we would be confident that we would have measures of lifetime income that are error-free and would also be free of lifecycle bias. The top portion of Figure 1 shows a visual representation of this ideal dataset. 6 Chetty et al speculate that perhaps time averaging and life cycle bias don’t matter in their robustness checks because of their use of administrative data. I discuss later why their robustness checks are of limited value. 7 The precise end points are debatable but one might want to ensure that most sample members have finished schooling and that most sample members have not yet retired. Unfortunately, for most countries, including the US, it is difficult to construct an intergenerational dataset with income data going back to the 1950s.8 Still, we can come somewhat close to this ideal sample with publicly available survey data in the Panel Study of Income Dynamics (PSID). The PSID began in 1968 and started collecting income data beginning with 1967 for a nationally representative sample of about 5000 families. The 1957 cohort would have been 11 years old at the time the PSID began so this cohort along with those born as early as 1951 would have been under the age of 18 at the beginning of the survey. The approach I take in this paper is to construct time averages of both parent and child income centered around the age of 40. For parents, these averages include income obtained between the ages of 25 and 55 and for children these averages include income obtained between the ages of 35 and 45. The middle portion of Figure 1 shows a rough approximation of the PSID sample that I will use.9 Relative to the ideal sample, the PSID sample is close in several regards. Since it covers the 1967 to 2010 period it is able cover vast stretches of the lifecycle for both generations. For example, for the 16 cohorts born between 1951 and 1965, income can be measured for all years that cover the age range between 35 and 45. For the cohorts born between 1967 and 1975, their parent’s income can also be measured through the ages of 25 and 55. Now let us contrast this with the limitations faced by Chetty et al (2014) in their analysis of currently available IRS data. First the tax data is currently only digitized going back to 1996, which is nowhere near as far back as the ideal dataset would require (1957), or even what is available in the PSID (1967). Therefore, there is no birth cohort for whom the income of parents 8 The SIPP-SER data used by Mazumder (2005) and Dahl and Deliere (2008) meets some but not all of these requirements. 9 Currently I am not restricting all of the assumptions shown in Figure 1. For example cohorts born before 1951 are being included. can be measured for the entire 31 year time span between the ages of 25 and 55. Furthermore, the authors chose to limit the analysis to just a 5-year average between 1996 and 2000. This is likely due to the skewed age structure of parent income. The mean age of fathers in their sample in 1996 is reported to be 43.5 with a standard deviation of 6.3 years. This implies that over the 5 years from 1996 through 2000, roughly 24 percent of the father-year observations used in constructing the average would be when fathers are over the age of 50.10 This is an age at which the transitory variance in income is quite high. They also report that prior to 1999 they record the income of non-filers to be zero. Therefore for about 3 percent of observations in three of the five years used in their average they impute zeroes to the missing observations.11 For the children in the sample, the data limitations are even more severe. Chetty et al use cohorts born between 1980 and 1982 and measure their income in 2011 and 2012 when they are between the ages of 29 and 32. For this age range, simulations from Haider and Solon (2006) suggest that there would be around a 20 percent bias in the estimated IGE compared to having the full lifecycle. A further complication is that their measures are taken in 2011 and 2012 when unemployment was relatively high and labor force participation quite low. They report that they throw out about 17 percent of observations from the poorest families due to their having zero income over those 2 years. If their sample also included 29 to 32 year olds over several decades which also included many boom years then this would be less of a concern. Finally, there is a concern about whether administrative income data adequately captures true income, particularly at the low and the high ends of the income distribution. For example, at the lower end of the 10 This example assumes the data is normally distributed. In 2000, more than a third of the observations would be when fathers are over the age of 50. 11 See footnote 14 of Chetty et al. (2014). They show that this has no effect on their rank mobility estimates but they do not show how the IGE estimates change. Further, measuring income from 1999 to 2003 only worsens the attenuation bias in the IGE resulting from measuring fathers at late ages. distribution, tax data could miss forms of income that go unreported to the IRS such as black market income. At the higher end, tax avoidance behavior could lead to an under-reporting of income. Hoyakem et al (2012, 2015) find that administrative tax data does a worse job than survey data in measuring poverty. Abowd and Stinson (2013) argue that it is preferable to treat both survey data and administrative data as containing error. To their credit, Chetty et al (2014) attempt to conduct some sensitivity checks to these issues, but as discussed above, their data are not well suited to doing effective robustness checks for the IGE measure, because the panel length of the data relative to the lifecycle in each generation is poorly matched. For example, when they increase the length of the time average of parent income by using more years after 2000, they must necessarily increase the attenuation bias from using later ages in the lifecycle. In one specification, they restrict their sample to just look at parents whose income is measured before the age of 50 but then they can only look at 5 year averages. They also try to measure the effects of parent income on children whose income is measured at a later age by using a separate set of tax data derived from much smaller crosssections of data from the IRS’ SOI data which covers cohorts born as far back as 1970. Unfortunately, these cross-sections only contain single year measures of parent income which are of course heavily attenuated. One also has to make heavy parametric assumptions in order to infer that the bias from using income from this short-panel data would necessarily apply when considering lifetime income. In their appendix, Chetty et al (2014) also argue that if they conduct simulations like Mazumder (2005a) but instead use a smaller share of earnings variance due to transitory fluctuations, then they do not find estimates of attenuation bias that are consistent with Mazumder’s results. However, Chetty et al do not provide evidence explaining what parameter values they use and how they are derived. In principle, Chetty et al could have directly estimated a model of earnings dynamics with their IRS data using a large sample covering men of all ages to directly provide evidence on this point. IV. PSID Data I restrict the analysis to father-son pairs as identified by the PSID’s Family Identification Mapping System (FIMS). I construct two parallel sets of samples. The first considers the labor income of the household head and the second considers the labor income of the entire family. Labor income is not simply wage income but also incorporates other sources of income such as self-employment. Observations marked as being generated by a ‘major’ imputation are set to missing. Yearly income observations are deflated to real terms using the CPI. In the PSID the household head is recorded as having zero income if their income was actually zero or if their income is missing, so one cannot cleanly distinguish true zeroes. For the analysis using the income of household heads, I use only years of non-zero income when constructing time averages of income. When using the PSID constructed measures of family income this problem does not arise as we found no instances of zero family income for the samples we constructed. The approach to estimation in this study is slightly different than in most previous PSID studies of intergenerational mobility. Rather than relying on any one fixed length time average for each generation and relying on parametric assumptions to deal with lifecycle bias (e.g. Lee and Solon, 2009), instead I measure an entire matrix of IGE’s for many combinations of lengths of time averages that are all centered around age 40. I will present the full matrix of estimates along with weighted averages across entire rows and columns representing the effects of a particular length of the time average for a given generation. For example, rather than simply comparing the IGE from using a ten-year average of fathers income to using a five year average of fathers’ income for one particular time average of sons income, I can show how it is affected for every combination of time average of sons’ income and for the weighted average across these combinations. V. Results IGE Estimates Table 1 shows the estimates of the IGE between fathers and sons when using the income of household heads. The first entry of the table at the upper left shows the estimate if we use just one year of income of fathers and one year of income for the sons when they are closest to age 40 and also are within the age-range constraints described earlier. This estimate of the IGE is 0.376 with a standard error of (0.043) and utilizes a sample of 2189 father-son pairs. One point immediately worth noting is that this estimate using the smallest possible time average centered around age 40 is higher than the 0.344 found by Chetty et al (2014). Moving across the row, the estimates gradually include more years of income between the ages of 35 and 45 for the sons. For the most part the estimates don’t change much and are all in the range of 0.34 and 0.39 except when using a 10 year average. Moving across the rows, the sample size gradually diminishes as enlarging the time average for the sons comes at the expense of sample size as an increasingly fewer number of sons have will income available for a higher length of required years. At the end of the row I display the weighted average across the columns, where the estimates are weighted by the sample size. For the first row the weighted average is 0.357. Moving down the rows for a given column, the estimates gradually increase the time average used to measure fathers’ income and as a consequence also reduces the sample size. For example, if we move down the first column and continue to just use the sons’ income in one year measured closet to age 40 and now increase the time average of fathers’ income to 2 years, the estimate rises to 0.454 as the sample falls to 2143. Using a five year average raises the estimate to 0.619 (N=1926). Increasing the time average to 10 years increases the estimate 0.690 (N=1502). Using a 15 year average raises the estimate further to 0.706 (N=1041). This pattern of rising estimates with a rising time average of fathers’ income, however, is not monotonic and is not true for every column. For most columns, in fact the time average tends to plateau quite quickly with time averages of around 5 to 10 years. The weighted average for each column is displayed in the bottom row. For example, the weighted average for first column is 0.599. A few points are worth making. Since expanding the time average in either dimension reduces the sample size it risks making the sample less representative. The implications on the estimates, however, are quite different for whether we increase the time average for the sons’ generation or for the fathers. For the fathers, increasing the time average tends to raise estimates. This is a consistent with a story in which larger time averages reduce attenuation bias stemming from mis-measurement of parent income (Solon, 1992; Mazumder, 2005). This also accords with standard econometric theory concerning mis-measurement of the right hand side variable. On the other hand, econometric theory posits that mis-measurement in the dependent variable typically should not cause attenuation bias. Indeed, increasing the time average of sons’ income has little effect. But crucially, this is because we have centered the time average of income in each generation so that the lifecycle bias which induces “non-classical” measurement error in the dependent variable (Haider and Solon, 2006) may already be accounted for. By this reasoning one might consider the estimates in the first column to be the most useful since they allow one to see how a reduction in measurement error in the father’s income affects the estimates while simultaneously minimizing life cycle bias and keeping the sample as large as possible. A more conservative view would be to use the weighted average in the final column that takes into account the possible effects of incorporating more years of data on sons’ income while also giving greater weight to estimates with larger samples. Using the first approach would suggest that the father-son IGE in own income may be as high as 0.7, while the latter approach would suggest that it is closer to 0.6. In either case the estimates are similar to those found in Mazumder (2005) without requiring imputation. It is important to note that in estimating the IGE, estimates may differ based on the concept of income being used. For example, in the landmark study by Solon (1992) using the PSID, Solon estimates the IGE in income to be 0.413 but the IGE in family income to be 0.483. More generally, estimates of the IGE in family income have tended to be larger than similar estimates that only use the labor income of one individual in each generation.12 Table 2 presents an analogous set of results as Table 1 using family income. [Add discussion of Table 2] Rank-Rank Slope Estimates 12 Examples include: References Chetty, Raj, Nathaniel Hendren, Patrick Kline, and Emmanuel Saez. 2014. “Where is the land of Opportunity? The Geography of Intergenerational Mobility in the United States”. Quarterly Journal of Economics, 129(4): 1553-1623. Dahl, Molly and Thomas DeLeire. 2008. “The Association between Children’s Earnings and Fathers’ Lifetime Earnings: Estimates Using Administrative Data.” Institute for Research on Poverty, University of Wisconsin-Madison. Mazumder, Bhashkar. 2014. “Black-white differences in intergenerational economic mobility in the United States.” Economic Perspectives, 38(1). Nybom, Martin amd Jan Stuhler. 2015. “Biases in Standard Measures of Intergenerational Dependence.” Table 1: Estimates of the father-son IGE in household head income Time Average of Sons' Income (years) Time Avg. Fath. Inc. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 wgt avg. 1 0.376 2 0.343 3 0.350 4 0.341 5 0.355 6 0.357 7 0.332 8 0.344 9 0.387 10 0.468 (0.043) (0.037) (0.040) (0.042) (0.044) (0.052) (0.052) (0.046) (0.049) (0.050) 2189 0.454 1867 0.405 1611 0.399 1381 0.392 1137 0.399 877 0.391 620 0.374 488 0.400 374 0.429 255 0.518 (0.056) (0.049) (0.050) (0.052) (0.058) (0.066) (0.066) (0.050) (0.063) (0.054) 2143 0.536 1826 0.482 1581 0.482 1358 0.480 1118 0.501 862 0.507 607 0.502 478 0.508 369 0.552 251 0.610 (0.051) (0.038) (0.039) (0.036) (0.036) (0.038) (0.046) (0.047) (0.052) (0.063) 2078 0.607 1768 0.529 1534 0.530 1315 0.505 1081 0.528 829 0.521 579 0.515 458 0.516 352 0.563 238 0.601 (0.044) (0.039) (0.040) (0.039) (0.040) (0.042) (0.049) (0.053) (0.058) (0.067) 2003 0.619 1710 0.547 1480 0.556 1265 0.525 1035 0.542 790 0.530 550 0.538 436 0.541 334 0.591 225 0.609 (0.046) (0.040) (0.042) (0.041) (0.043) (0.045) (0.050) (0.054) (0.062) (0.073) 1926 0.632 1644 0.562 1419 0.570 1209 0.536 984 0.548 746 0.526 514 0.529 402 0.535 304 0.579 201 0.617 (0.046) (0.041) (0.043) (0.042) (0.044) (0.045) (0.049) (0.051) (0.058) (0.076) 1866 0.659 1592 0.589 1372 0.592 1164 0.552 942 0.566 710 0.543 485 0.549 380 0.549 287 0.578 189 0.622 (0.050) (0.044) (0.046) (0.045) (0.047) (0.049) (0.052) (0.053) (0.061) (0.084) 1783 0.646 1521 0.565 1313 0.575 1111 0.536 894 0.549 669 0.518 451 0.501 352 0.509 264 0.557 169 0.600 (0.052) (0.046) (0.047) (0.048) (0.050) (0.052) (0.056) (0.058) (0.069) (0.098) 1685 0.660 1431 0.572 1234 0.572 1035 0.527 828 0.531 606 0.503 403 0.512 313 0.528 232 0.589 148 0.664 (0.056) (0.046) (0.047) (0.047) (0.047) (0.051) (0.061) (0.064) (0.074) (0.103) 1611 0.690 1369 0.596 1178 0.604 985 0.554 783 0.543 569 0.514 373 0.535 286 0.569 213 0.627 136 0.713 (0.062) (0.049) (0.050) (0.050) (0.051) (0.055) (0.066) (0.069) (0.083) (0.116) 1502 0.654 1269 0.566 1081 0.575 900 0.524 712 0.528 507 0.501 325 0.516 243 0.523 180 0.565 110 0.676 (0.063) (0.048) (0.049) (0.050) (0.052) (0.057) (0.068) (0.073) (0.087) (0.121) 1411 0.635 1193 0.561 1012 0.573 839 0.510 665 0.515 469 0.474 296 0.452 220 0.448 158 0.491 98 0.626 (0.059) (0.053) (0.053) (0.055) (0.058) (0.062) (0.076) (0.082) (0.099) (0.125) 1335 0.652 1123 0.564 949 0.571 779 0.502 611 0.506 423 0.465 258 0.459 186 0.427 131 0.477 81 0.570 (0.061) (0.053) (0.053) (0.054) (0.056) (0.062) (0.074) (0.080) (0.097) (0.104) 1248 0.674 1050 0.583 884 0.602 720 0.535 558 0.537 379 0.476 222 0.474 156 0.438 107 0.493 64 0.635 (0.063) (0.056) (0.055) (0.057) (0.060) (0.066) (0.081) (0.084) (0.103) (0.103) 1171 0.706 987 0.589 828 0.606 667 0.545 510 0.536 341 0.453 192 0.423 134 0.376 93 0.422 55 0.518 (0.069) (0.059) (0.058) (0.059) (0.062) (0.070) (0.092) (0.097) (0.126) (0.114) 1041 879 730 587 444 287 151 98 62 31 0.599 0.526 0.531 0.496 0.505 0.483 0.478 0.484 0.528 0.593 Wgt. Avg. 0.357 0.412 0.505 0.543 0.560 0.569 0.590 0.570 0.573 0.600 0.571 0.553 0.554 0.580 0.586 Table 2: Estimates of the father-son IGE in family income Time Average of Sons' Income (years) Time Avg. Fath. Inc. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 wgt avg. 1 0.579 2 0.659 3 0.650 4 0.644 5 0.640 6 0.641 7 0.630 8 0.639 9 0.619 10 0.566 (0.042) (0.043) (0.038) (0.039) (0.038) (0.040) (0.040) (0.047) (0.048) (0.056) 2290 0.613 2031 0.688 1806 0.675 1577 0.672 1359 0.675 1108 0.681 864 0.671 667 0.675 533 0.663 392 0.607 (0.047) (0.048) (0.043) (0.045) (0.044) (0.045) (0.045) (0.049) (0.048) (0.063) 2234 0.662 1982 0.727 1762 0.716 1538 0.713 1323 0.698 1076 0.702 835 0.693 640 0.710 515 0.685 379 0.629 (0.048) (0.050) (0.045) (0.047) (0.046) (0.047) (0.049) (0.053) (0.052) (0.069) 2194 0.686 1944 0.720 1731 0.701 1508 0.707 1296 0.683 1049 0.709 810 0.719 617 0.724 494 0.692 364 0.671 (0.046) (0.054) (0.048) (0.048) (0.046) (0.052) (0.053) (0.058) (0.062) (0.077) 2099 0.708 1856 0.745 1645 0.722 1433 0.730 1229 0.706 992 0.727 760 0.730 573 0.729 457 0.679 333 0.668 (0.047) (0.058) (0.050) (0.051) (0.048) (0.054) (0.054) (0.060) (0.061) (0.077) 2043 0.719 1802 0.725 1594 0.732 1383 0.742 1182 0.721 950 0.752 720 0.751 540 0.756 433 0.701 314 0.688 (0.050) (0.053) (0.055) (0.054) (0.052) (0.059) (0.057) (0.064) (0.067) (0.084) 1964 0.709 1726 0.704 1523 0.710 1315 0.727 1122 0.719 896 0.743 675 0.747 503 0.760 399 0.687 285 0.688 (0.052) (0.055) (0.057) (0.056) (0.054) (0.061) (0.059) (0.068) (0.071) (0.092) 1861 0.709 1631 0.697 1436 0.703 1236 0.723 1048 0.714 829 0.743 620 0.746 457 0.765 360 0.691 255 0.710 (0.055) (0.058) (0.060) (0.059) (0.056) (0.064) (0.061) (0.071) (0.076) (0.099) 1792 0.723 1573 0.697 1383 0.710 1185 0.727 1000 0.721 786 0.752 584 0.750 426 0.757 332 0.696 234 0.719 (0.057) (0.062) (0.066) (0.064) (0.060) (0.071) (0.066) (0.074) (0.082) (0.106) 1683 0.743 1468 0.718 1283 0.733 1097 0.737 918 0.737 715 0.776 522 0.772 382 0.800 295 0.772 207 0.815 (0.055) (0.059) (0.064) (0.064) (0.061) (0.073) (0.067) (0.074) (0.085) (0.105) 1580 0.748 1376 0.730 1196 0.738 1015 0.754 847 0.751 655 0.802 473 0.773 339 0.813 257 0.761 179 0.784 (0.059) (0.064) (0.068) (0.070) (0.064) (0.076) (0.075) (0.081) (0.093) (0.108) 1501 0.774 1301 0.753 1123 0.754 945 0.769 782 0.764 599 0.828 421 0.785 297 0.831 221 0.817 156 0.865 (0.061) (0.066) (0.070) (0.072) (0.066) (0.080) (0.077) (0.083) (0.093) (0.110) 1418 0.766 1226 0.742 1056 0.759 882 0.777 725 0.770 547 0.838 374 0.780 258 0.802 193 0.792 133 0.853 (0.064) (0.069) (0.074) (0.075) (0.069) (0.083) (0.079) (0.087) (0.100) (0.118) 1341 0.796 1159 0.766 993 0.777 821 0.798 668 0.773 497 0.853 330 0.778 221 0.786 164 0.770 110 0.822 (0.070) (0.075) (0.080) (0.083) (0.075) (0.093) (0.087) (0.100) (0.117) (0.138) 1207 0.825 1040 0.789 884 0.803 718 0.812 579 0.767 419 0.871 267 0.782 171 0.755 125 0.739 81 0.754 (0.073) (0.079) (0.086) (0.089) (0.083) (0.104) (0.100) (0.108) (0.122) (0.145) 1084 930 784 632 509 359 219 137 95 60 0.705 0.719 0.718 0.725 0.713 0.742 0.726 0.737 0.698 0.686 Wgt. Avg. 0.630 0.663 0.699 0.702 0.720 0.730 0.718 0.716 0.722 0.745 0.754 0.775 0.772 0.788 0.804 Table 3: Estimates of the father-son IGE in family income imposing data limitations Time Average of Sons' Income (years) Time Avg. Fath. Inc. 1 2 3 4 5 6 7 8 9 10 wgt avg. 1 0.351 2 0.383 3 0.404 4 0.413 5 0.427 6 0.423 7 0.433 8 0.453 9 0.428 10 0.477 (0.033) (0.027) (0.027) (0.029) (0.031) (0.033) (0.037) (0.042) (0.042) (0.039) 3481 0.438 3058 0.442 2705 0.462 2386 0.468 2122 0.488 1854 0.488 1622 0.499 1403 0.521 1215 0.495 1030 0.515 (0.028) (0.029) (0.030) (0.032) (0.032) (0.033) (0.036) (0.040) (0.040) (0.042) 3391 0.466 2986 0.466 2645 0.485 2333 0.489 2075 0.504 1813 0.511 1589 0.520 1373 0.547 1189 0.516 1011 0.534 (0.029) (0.029) (0.031) (0.033) (0.033) (0.035) (0.037) (0.041) (0.041) (0.044) 3285 0.462 2896 0.460 2574 0.484 2270 0.487 2019 0.501 1764 0.503 1549 0.518 1339 0.546 1158 0.518 984 0.536 (0.032) (0.031) (0.032) (0.034) (0.034) (0.035) (0.037) (0.041) (0.041) (0.043) 3109 0.456 2754 0.470 2451 0.483 2171 0.485 1934 0.485 1690 0.492 1487 0.497 1286 0.550 1111 0.515 941 0.517 (0.042) (0.036) (0.037) (0.038) (0.041) (0.045) (0.051) (0.046) (0.047) (0.051) 2923 0.498 2613 0.508 2335 0.521 2081 0.522 1857 0.523 1628 0.531 1431 0.538 1235 0.579 1064 0.551 902 0.561 (0.037) (0.035) (0.034) (0.035) (0.036) (0.037) (0.039) (0.044) (0.044) (0.047) 2725 0.513 2448 0.516 2208 0.524 1975 0.529 1771 0.529 1556 0.537 1369 0.537 1177 0.581 1013 0.552 855 0.557 (0.042) (0.038) (0.038) (0.038) (0.039) (0.040) (0.042) (0.047) (0.048) (0.050) 2476 0.540 2234 0.535 2032 0.538 1832 0.534 1644 0.534 1452 0.546 1282 0.544 1094 0.583 935 0.553 786 0.559 (0.045) (0.041) (0.040) (0.040) (0.041) (0.042) (0.043) (0.049) (0.050) (0.053) 2278 0.552 2070 0.549 1895 0.542 1719 0.546 1556 0.546 1384 0.556 1223 0.550 1039 0.590 889 0.562 750 0.557 (0.042) (0.042) (0.039) (0.042) (0.043) (0.043) (0.044) (0.049) (0.051) (0.054) 2051 0.558 1876 0.550 1724 0.540 1563 0.545 1422 0.544 1263 0.553 1113 0.544 948 0.588 808 0.565 676 0.552 (0.045) (0.045) (0.042) (0.045) (0.047) (0.047) (0.047) (0.051) (0.051) (0.055) 1829 1679 1554 1407 1286 1145 1010 861 743 620 0.472 0.479 0.492 0.496 0.504 0.509 0.514 0.549 0.520 0.534 Wgt. Avg. 0.408 0.472 0.495 0.492 0.487 0.526 0.532 0.543 0.553 0.552 Table 4: Estimates of the father-son rank-rank slope in family income Time Average of Sons' Income (years) Time Avg. Fath. Inc. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 wgt avg. 1 0.453 2 0.489 3 0.502 4 0.512 5 0.533 6 0.550 7 0.576 8 0.572 9 0.546 10 0.526 (0.022) (0.023) (0.024) (0.025) (0.026) (0.028) (0.030) (0.033) (0.037) (0.044) 2290 0.454 2031 0.486 1806 0.502 1577 0.513 1359 0.535 1108 0.555 864 0.576 667 0.567 533 0.550 392 0.526 (0.022) (0.024) (0.025) (0.026) (0.027) (0.028) (0.031) (0.034) (0.037) (0.044) 2234 0.457 1982 0.484 1762 0.502 1538 0.513 1323 0.532 1076 0.554 835 0.574 640 0.564 515 0.538 379 0.517 (0.022) (0.024) (0.025) (0.026) (0.027) (0.028) (0.031) (0.035) (0.039) (0.047) 2194 0.450 1944 0.465 1731 0.478 1508 0.491 1296 0.507 1049 0.535 810 0.556 617 0.546 494 0.521 364 0.508 (0.023) (0.024) (0.025) (0.026) (0.027) (0.029) (0.032) (0.036) (0.041) (0.050) 2099 0.452 1856 0.469 1645 0.482 1433 0.495 1229 0.510 992 0.536 760 0.552 573 0.546 457 0.519 333 0.502 (0.023) (0.024) (0.026) (0.027) (0.027) (0.030) (0.033) (0.037) (0.041) (0.051) 2043 0.458 1802 0.471 1594 0.486 1383 0.499 1182 0.516 950 0.541 720 0.554 540 0.550 433 0.524 314 0.513 (0.024) (0.025) (0.027) (0.028) (0.028) (0.031) (0.034) (0.038) (0.042) (0.052) 1964 0.451 1726 0.461 1523 0.474 1315 0.490 1122 0.511 896 0.534 675 0.549 503 0.548 399 0.514 285 0.504 (0.024) (0.026) (0.028) (0.029) (0.029) (0.032) (0.036) (0.039) (0.045) (0.056) 1861 0.451 1631 0.456 1436 0.468 1236 0.485 1048 0.506 829 0.530 620 0.544 457 0.544 360 0.508 255 0.514 (0.025) (0.027) (0.029) (0.030) (0.031) (0.033) (0.037) (0.041) (0.047) (0.058) 1792 0.451 1573 0.449 1383 0.464 1185 0.477 1000 0.497 786 0.520 584 0.540 426 0.547 332 0.516 234 0.532 (0.025) (0.028) (0.030) (0.031) (0.032) (0.035) (0.039) (0.042) (0.048) (0.058) 1683 0.443 1468 0.439 1283 0.451 1097 0.460 918 0.484 715 0.506 522 0.527 382 0.538 295 0.513 207 0.533 (0.026) (0.029) (0.031) (0.033) (0.034) (0.037) (0.042) (0.046) (0.053) (0.063) 1580 0.436 1376 0.432 1196 0.442 1015 0.453 847 0.474 655 0.497 473 0.510 339 0.525 257 0.508 179 0.519 (0.027) (0.030) (0.033) (0.034) (0.036) (0.039) (0.045) (0.049) (0.057) (0.068) 1501 0.448 1301 0.445 1123 0.454 945 0.464 782 0.485 599 0.509 421 0.536 297 0.567 221 0.579 156 0.611 (0.028) (0.031) (0.034) (0.035) (0.036) (0.040) (0.046) (0.049) (0.054) (0.065) 1418 0.446 1226 0.443 1056 0.455 882 0.467 725 0.487 547 0.518 374 0.546 258 0.560 193 0.594 133 0.641 (0.029) (0.032) (0.035) (0.036) (0.038) (0.041) (0.047) (0.053) (0.058) (0.072) 1341 0.457 1159 0.451 993 0.456 821 0.466 668 0.480 497 0.516 330 0.549 221 0.537 164 0.580 110 0.639 (0.030) (0.034) (0.037) (0.039) (0.040) (0.044) (0.050) (0.059) (0.065) (0.084) 1207 0.460 1040 0.448 884 0.450 718 0.452 579 0.455 419 0.492 267 0.502 171 0.454 125 0.495 81 0.552 (0.032) (0.036) (0.039) (0.042) (0.045) (0.050) (0.058) (0.070) (0.077) (0.100) 1084 930 784 632 509 359 219 137 95 60 0.451 0.462 0.475 0.487 0.507 0.532 0.552 0.551 0.532 0.528 Wgt. Avg. 0.511 0.511 0.510 0.492 0.494 0.497 0.489 0.485 0.481 0.469 0.459 0.475 0.476 0.476 0.461 Table 5: Estimates of the father-son rank-rank slope in family income imposing data limitations Time Average of Sons' Income (years) Time Avg. Fath. Inc. 1 2 3 4 5 6 7 8 9 10 wgt avg. 1 0.371 2 0.394 3 0.430 4 0.442 5 0.452 6 0.453 7 0.470 8 0.483 9 0.478 10 0.493 (0.018) (0.020) (0.021) (0.022) (0.023) (0.024) (0.025) (0.026) (0.028) (0.029) 3481 0.383 3058 0.406 2705 0.441 2386 0.455 2122 0.465 1854 0.466 1622 0.482 1403 0.493 1215 0.490 1030 0.503 (0.018) (0.020) (0.021) (0.022) (0.023) (0.024) (0.025) (0.027) (0.028) (0.030) 3391 0.387 2986 0.409 2645 0.444 2333 0.458 2075 0.467 1813 0.470 1589 0.488 1373 0.500 1189 0.494 1011 0.509 (0.018) (0.020) (0.021) (0.022) (0.023) (0.024) (0.025) (0.026) (0.028) (0.030) 3285 0.380 2896 0.400 2574 0.436 2270 0.449 2019 0.457 1764 0.459 1549 0.479 1339 0.488 1158 0.485 984 0.493 (0.019) (0.020) (0.021) (0.023) (0.023) (0.024) (0.025) (0.027) (0.028) (0.031) 3109 0.388 2754 0.407 2451 0.442 2171 0.456 1934 0.463 1690 0.468 1487 0.491 1286 0.501 1111 0.496 941 0.504 (0.019) (0.021) (0.022) (0.023) (0.024) (0.025) (0.026) (0.027) (0.029) (0.031) 2923 0.400 2613 0.419 2335 0.453 2081 0.464 1857 0.472 1628 0.474 1431 0.496 1235 0.505 1064 0.504 902 0.513 (0.020) (0.022) (0.023) (0.024) (0.024) (0.025) (0.026) (0.028) (0.029) (0.032) 2725 0.394 2448 0.411 2208 0.442 1975 0.457 1771 0.463 1556 0.466 1369 0.484 1177 0.495 1013 0.495 855 0.504 (0.021) (0.023) (0.024) (0.025) (0.026) (0.026) (0.027) (0.029) (0.031) (0.033) 2476 0.396 2234 0.412 2032 0.439 1832 0.448 1644 0.453 1452 0.458 1282 0.476 1094 0.484 935 0.479 786 0.491 (0.022) (0.024) (0.025) (0.026) (0.026) (0.027) (0.028) (0.031) (0.032) (0.035) 2278 0.407 2070 0.418 1895 0.440 1719 0.452 1556 0.457 1384 0.464 1223 0.482 1039 0.488 889 0.483 750 0.490 (0.023) (0.024) (0.025) (0.027) (0.027) (0.028) (0.029) (0.031) (0.033) (0.036) 2051 0.408 1876 0.420 1724 0.436 1563 0.443 1422 0.447 1263 0.452 1113 0.468 948 0.474 808 0.472 676 0.474 (0.024) (0.026) (0.027) (0.029) (0.029) (0.029) (0.031) (0.033) (0.034) (0.037) 1829 1679 1554 1407 1286 1145 1010 861 743 620 0.389 0.408 0.440 0.453 0.460 0.463 0.482 0.492 0.488 0.498 Wgt. Avg. 0.433 0.445 0.449 0.440 0.448 0.458 0.450 0.444 0.449 0.443 Figure 1: Comparison of Intergenerational Samples Parent Income “Ideal” PSID 1987, age 55 1986, age 54 . . . 1958, age 26 1957, age 25 2005, age 55 2004, ages 54-55 . . . 1968, ages 25-42 1967, ages 25-41 Child Cohort 1957 Child Income 2012, age 55 2011, age 54 . . . 1983, age 26 1982, age 25 1951-1975 2010, ages 35-45 2009, ages 35-45 . . 1988, ages 35-37 1987, ages 35-36 1986, age 35 1980-82 2012, ages 30-32 2011, ages 29-31 IRS Figure 2: Effects of time averaging on father-son IGE in household head income 0.8 0.7 0.6 IGE 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Length of Time Average of Father's Income Weighted Average One year of Sons' Income Figure 3: Effects of time averaging on father-son IGE in Family income 0.900 0.800 0.700 IGE 0.600 0.500 0.400 0.300 0.200 0.100 0.000 1 2 3 4 5 6 7 8 9 10 11 Length of Time Average of Father's Income Weighted Average One year of Sons' Income 12 13 14 15
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