Biases in Standard Measures of Intergenerational Income Dependence

A. The Elasticity

A useful way to summarize the intergenerational log-income relationship is the conditional expectation function (CEF). The coefficient from a linear regression of y^* on x^*, provides the best (in a minimized mean squared error sense) linear approximation for its slope. Much of the literature concerns the estimation of this intergenerational elasticity.⁷ However, usage of short-run instead of true lifetime incomes yields a consistent estimator of

This coefficient may differ from the true elasticity for two reasons. First, it may be attenuated by classical measurement error. If u and v are assumed to be uncorrelated to true values and each other, Equation 3 reduces to the textbook errors-in-variables formula , where rr_x = Var(x^*)/[Var(x^*) + Var(u)]. Elasticity estimates based on annual incomes can thus be corrected for classical error if plausible estimates of rr_x are available. Importantly, only classical error in parental income generates such attenuation bias. In addition, heterogeneity in income profiles may cause the covariance between error and true values to vary systematically over age, introducing life-cycle bias (Jenkins 1987; Haider and Solon 2006).

As we considered left-hand side measurement error, the bias shown in Figure 1 is exclusively due to life-cycle effects. Life-cycle bias is prone to rise with the age of offspring—going from negative to positive, crossing zero at some intermediate age—if those with higher lifetime incomes tend to have higher income growth rates, such that the dispersion of log income within each age group increases with age (Creedy 1988). But the evolution of cross-sectional inequality over age may differ across populations, making comparative analyses difficult.⁸ The attenuating effect of right-side measurement error is illustrated in Figure 2, which compares the benchmark elasticity against estimates based on annual incomes of sons at the specified age and fathers at age 45, with or without correction for attenuation bias using estimates of rr_x. As expected, the estimates corrected for classical errors in Figure 2 are very similar to the estimates by age based directly on fathers’ lifetime measures in Figure 1. Figure A1 in the Appendix plots additional estimates based on multiyear average incomes to illustrate that this attenuation reduces only slowly if longer averages instead of single-year incomes are used (consistent with Mazumder 2005).

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Figure 2 Mobility Measures, Benchmark vs. Annual (BHS ME)Notes: Each figure plots the benchmark estimate based on lifetime incomes for both generations against estimates based on annual income of sons (varying by age) and fathers’ income at age 45 (both-hand side measurement error). The sample and thus the benchmark estimates vary over age as we drop persons with missing annual observations also for estimation of the benchmark. Panels A–C additionally plot estimates by age that are adjusted using corrections for measurement error described in Section V. Estimates in Panels A and B are corrected for classical measurement error using reliability ratios computed directly in the data, while Panel C uses the correction factor provided by the generalized model in Equation 9.

B. The Log-Linear Correlation

The Pearson correlation coefficient, , abstracts from changes in the variance of (log) income over generations. Under classical errors, the usage of proxy income yields , with the reliability ratios rr_x defined as above and . Correlation estimates are thus attenuated by classical errors in both parent and offspring income, while only the former affects the elasticity. This distinction is of practical importance; correlation estimates are much lower than elasticity estimates under left-hand side measurement error (Figure 1) but more similarly attenuated under measurement error on both sides (Figure 2). While using multiyear averages for parents is now common practice when estimating the elasticity, it is for the correlation equally important also to address transitory noise in offspring income.

To our knowledge, no direct evidence exists on life-cycle effects in intergenerational correlations, but arguments made with respect to other measures can be adapted. Grawe (2006) and Haider and Solon (2006) document how the tendency of early-career (latecareer) income gaps to understate (overstate) lifetime differences generate life-cycle bias in elasticity estimates. Following Haider and Solon, capture this insight in the linear projection y = λy^* + w and assume that w is uncorrelated to x^*, such that

We can illustrate some new insights by rewriting Equation 4

Correlation estimates should thus be sensitive to life-cycle bias at very early ages, where differences in current and lifetime income can be negatively correlated, such that λ < 0. We expect less sensitivity at mid and late age: As correlations are invariant to positive monotone linear transformations of the variables, λ > 0 only enters the reliability ratio. The ratio goes to zero as λ → 0 or one as λ→∞. While correlation estimates are thus expected to increase with λ over age, they should, in contrast to the elasticity, remain attenuated also at later ages.

Our empirical findings support these implications. Correlation estimates suffer from strong life-cycle bias at young age but remain comparatively stable beyond age 30, under both left (Figure 1) and both-side (Figure 2) measurement error. Interestingly, correcting for classical errors using estimates of rr_x and rr_y, while indeed addressing attenuation, also escalates life-cycle effects. Equation 5 provides an explanation: the reliability ratio tends to rise with λ and thus with age, such that estimates based on a classical errors-in-variables model understate this ratio at late ages. Figure A1 in the Appendix illustrates that estimates of the Pearson correlation remain strongly attenuated even when five- and nine-year averages instead of annual incomes are used.

C. The Rank Correlation

Denote the ranks (normalized to the unit interval) of x^* and y^* in their respective distribution by and . Denote observed ranks by

where are the errors in ranks. The Spearman rank correlation,

measures the extent to which one variable tends to increase with the other, without requiring that relationship to be linear.

Classical measurement error attenuates log-linear measures through its effect on the variance of observed incomes, but the variances of observed and true ranks are by definition equal. The distribution of a rank-ordered variable is uniform and, as of the normalization to the unit interval, has a variance of 1/12. However, even errors that are random in the underlying values generate nonclassical errors in ranks: as top (bottom) ranks cannot be overstated (understated), the correlation between errors in ranks and true ranks is negative.⁹ Formally, we have , and thus . Appendix Figure A2, Panel A provides an illustration by plotting the mean of the errorev at age 30 (that is, the deviation of annual from lifetime rank) separately for each five-percentile bin in the distribution of true ranks . As at other ages, the relationship is systematic and strongly negative.

A textbook errors-in-variables model is therefore not suitable for ranks. To capture the relationship between observed and true ranks we instead adapt the model proposed by Haider and Solon (2006) to formulate a generalized errors-in-variables model in ranks. Let

where λ_y is the slope coefficient in a linear projection of . By construction, is uncorrelated to , and we maintain the simplifying assumption that it is likewise uncorrelated to . From the definition of ranks it follows that λ_y £ 1. This slope coefficient provides a linear representation of the expected measurement error in as a function of . A λ_y close to one implies very little measurement error, whereas a λ_y close to zero implies that carries virtually no information about . Let λ_x denote the analogous slope coefficient in a projection of .

As noted above, Haider and Solon proposed this generalization of the textbook errors–invariables model to capture a particular form of nonclassical measurement error in log incomes. But the model can provide a useful summary also of the distribution of errors in income ranks. First, it captures in a parsimonious way that true values and errors must be negatively correlated in ranks; the slope coefficient λ_y will be less than one under any form of measurement error.¹⁰ In particular, in ranks the generalized is more suitable than the textbook model even when the underlying errors in log values are classical. Second, the linear functional form describes the distribution of ranks quite well. Appendix Figure A2 illustrates these points by plotting the mean deviation of income at age 30 from lifetime income against the latter, for both ranks (Panel A) and log income (Panel B). As at other ages, the linear functional form of the generalized errorsin- variables model provides a better fit in ranks than in log income.

Under errors on the left side, we then have

where we substituted Equation 9 and simplified. As λ_y < 1 (see Footnote 10), the rank correlation is understated. The case with right-side errors is analogous, and with errors on both sides we have

As the Pearson correlation, the Spearman correlation is thus subject to attenuation bias from errors in both parental and offspring income. But is it less or more attenuated? Whereas rank correlations are less sensitive to outliers in the tails of the distribution, and to the treatment of zero incomes in particular, it does not necessarily follow that they are generally less sensitive to measurement error.¹¹ However, the probability density function of the errors in log income is negatively skewed in our data (and in other populations; see Guvenen et al. 2015), with frequent extreme (low-income) observations in the far left tail. Their influence is limited in ranks but large in logs.¹² Rank correlations are thus expected to be less attenuated than log-linear measures.

Moreover, while the Pearson correlation is invariant to a linear spread in log-income profiles over age, the rank correlation is invariant to any rank-preserving spread. We may thus expect that life-cycle bias is even less problematic for the Spearman than for the Pearson correlation.

The results in Figures 1 and 2 support both these arguments. The gap between benchmark and annual estimates is substantially smaller for the rank correlation than the log-linear measures of dependence, and between the early 30s and the late 40s amounts to about 10 percent or less (left-side measurement error, Figure 1) or 20 percent or less (errors on both sides, Figure 2) of the benchmark. And while the rank correlation is strongly underestimated at young ages, as those with high lifetime income tend to have relatively low income (ranks) at the beginning of their career, it is less affected by lifecycle effects at later ages. The slope of annual estimates over age is nearly flat around midage, in stark contrast in particular to the intergenerational elasticity. However, these findings do not imply that measurement error is not a concern. Rank correlations remain attenuated even when multiyear averages instead of single-year incomes are used (see Figure A1 in the Appendix), although the bias turns negligible when long averages around midage are used. In addition, even small biases can be a concern if those correlations tend to vary less across groups than log-linear measures. For example, in their comparison of income mobility in the United States, Canada, and Sweden, Corak, Lindquist, and Mazumder (2014) report estimates that vary between 0.25 and 0.40 for the elasticity but only between 0.24 and 0.30 for the Spearman rank correlation. Even small biases thus still can be consequential for comparative purposes.

Equations 10 and 11 also point toward a procedure to reduce attenuation bias in rank correlations that may be useful in applied research. As errors in ranks are negatively correlated to true rank, standard methods do not apply.¹³ However, we argued that the generalized errors-in-variables model in ranks captures this negative relationship well, and therefore can be used to correct for the attenuation bias if estimates of λ_x and λ_y at the right age are available. Figure 2 provides such correction based on direct estimates of Equation 9 and the corresponding equation for parents. While not eliminating (lifecycle) bias at young age, the correction yields estimates that are nearly equal to the benchmark from around age 30. The problem is then how to estimate these coefficients in real-world applications, when lifetime incomes are not observed in the intergenerational data.

One option is to import such estimates from external evidence, ideally from data covering a similar population as the one under study. In the absence of such data one can in principle import estimates from other populations. However, Haider and Solon (2006) advise caution in doing so for the generalized model in log incomes because data sources may not be comparable and life-cycle trajectories may differ between cohorts and countries. Indeed, Nybom and Stuhler (2016) find that this coefficient follows a different and steep age profile in a Swedish cohort (growing from 0.8 to 1.5 between age 30 and 50), which suggests that the usage of external estimates can lead to large errors. However, the situation is more favorable for the generalized model in ranks. We noted that ranks are less sensitive to life-cycle effects, and this is reflected also in the age pattern of λ_x and λ_y in our data (see Figure A3 in the Appendix). Estimates vary only between 0.7 and 0.9 between the early 30s and age 60, are close to 0.8 over most of this age range, and do not vary substantially between the two generations. A rule-of-thumb correction based on the generalized errors-in-variables model is therefore less sensitive to the precise age at measurement in the case of ranks than log incomes. However, differences between populations and income concepts still may matter. For example, the share of zeros is much higher in U.S. tax data than in the data that we use here, and we may plausibly expect λ_x and λ_y to be smaller in this case.

Alternatively, the coefficients λ_x and λ_y can be estimated if two income observations are available per person. The slope coefficient from a regression of an income rank observation on another observation equals

such that the square of this coefficient will eliminate bias if the errors and are uncorrelated.¹⁴ Right-side or both-side measurement error can be treated accordingly. Of course, in practice the errors are possibly serially correlated (see Mazumder 2005), and the intergenerational process also may violate the assumption that they are uncorrelated to . The correction will therefore not be exact in practice. However, as shown in Figure 3, we find that its application based on consecutive income observations still reduces the bias on average by more than 50 percent between age 30 and 50.¹⁵ The bias is not fully eliminated because the errors and are positively (serially) correlated, such that an estimate of λ based on Equation 12 is biased upward. Usage of more distant observations will reduce this bias, but in our data the correlation between the errors turns slightly negative when ten-year distances are used, leading to an overcorrection of the bias. To remain conservative, it is thus advisable to use observations that are not more than a few years apart.

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Figure 3 Correction for Measurement Error in the Rank Correlation (LHS ME)Notes: The figure plots the benchmark estimate based on lifetime incomes for both generations against estimates based on annual income of sons (varying by age). Using two consecutive income observations, we compare estimates based on the rank of the two-year average with estimates that are based on the formal correction procedure that we describe in Section V. The sample and thus the benchmark estimates vary slightly over age as we drop persons with missing annual observations also for estimation of the benchmark.

Figure 2 also shows that the formal correction based on two adjacent observations produces better results than using simple averages of these observations. However, it does not address the bias at younger ages, where nonclassical errors from life-cycle effects are the main concern. The procedure is therefore most useful if two prime-age income observations per individual are available, as is quite often the case in modern survey data sets.

D. Transition Matrices and Copula

To further characterize the joint distribution of parent and child income, researchers often estimate transition matrices that capture mobility across quantile-based income classes (Zimmerman 1992; Jäntti et al. 2006). Interest often centers on specific elements, for example, on the persistence in the very top or bottom (such as “poverty traps”) or on the probability to rise from the lowest to the highest class (“rags to riches”). With sufficient data we can distinguish more classes and thus approximate the copula, the joint distribution of parent and child percentile ranks r = 1‚ ::: ‚ 100. As the copula is a key determinant of any mobility measure (see Chetty et al. 2014a; 2014b), it is valuable in a wide range of applications. However, little is known about the estimability of intergenerational copulas and transition matrices in short income data. Dahl and DeLeire (2008) note that transition matrices vary less across studies and samples than estimates of the elasticity, suggesting that they may be less sensitive to sampling choices. Still, simulations by O’Neill, Sweetman, and Van de gaer (2007) suggest that the bias can become more severe in the tails of the distribution.

Assume that only a short-run income measure is observed for each offspring. Using the law of total probability, we can rewrite the probability that its rank is equal to i conditional on the true parent rank being j as

Assuming that the error in log offspring income is classical, we have . Bayes theorem implies that , and thus

where the first right-hand term is the true probability and the second term is the bias. For illustration, consider a transition probability on the main diagonal (i = j), such that

The terms in square brackets are always negative if —that is, if the probability for offspring to have a certain rank is largest when their parent has the same rank. This assumption appears reasonable and does indeed hold in our data, as shown below. We thus have : the diagonal elements of a transition matrix are understated under classical errors. It follows that off-diagonal elements are on average overstated, as was the case for the bottom-to-top transition probability considered in Figures 1 and 2. Moreover, we may expect considerable differences within each category. The bias will tend to be larger in the corners of the transition matrix if is particularly large for i = 1 (where ranks are bounded from below) or i = 100 (ranks bounded from above), and decreases in the difference between r and i. And given the frequent occurrence of low-income observations in annual incomes we expect that long-distance transitions such as top to bottom are overestimated to a larger degree.

These theoretical predictions suggest that the observed robustness of rank correlations may not necessarily extend to other rank-based measures. We can test each of them by estimating the copula of parent and offspring incomes. Analysis of the full distribution is data intensive. Our sample size can be increased tenfold if we measure parental incomes from 1968 instead of 1960, such that father’s incomes are observed only from age 41 onward (see Section III). For the nonlinear analysis this tradeoff is worth while, in particular since income ranks are robust to monotone spreads of income profiles over the life cycle.¹⁶

Consider first if usage of annual income observations misrepresents the conditional expectation of offspring rank. Figure 4 plots the benchmark estimate of the conditional expectation (black line) against estimates based on annual income for sons at ages 25, 30, and 40 (Panel A) and annual incomes for both fathers and sons (Panel B). In contrast to the pattern reported in Chetty et al. (2014a), the CEF is not linear in our data. Their argument that rank-rank are advantageous to log-log relationships because only the former are linear may therefore not extend to other populations or income definitions. In particular, our measure contains income from business and capital realizations, which may contribute to the shape of the rank-rank relationship at the top. We find that parental rank is particularly consequential at the very top and bottom of the distribution. (See also Dahl and DeLeire 2008, who find a similar pattern for earnings in the United States.) For example, average offspring rank rises by four percentiles when moving from the secondhighest to the top percentile of parents; the increase is similarly steep when moving from the lowest to the second-lowest percentile. The CEF has a much lower slope in between the tails, pointing to particular mechanisms of income transmission among the very poorest and richest.¹⁷

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Figure 4 Son’s Expected Rank Conditional on Father’s Rank, Benchmark vs. Annual IncomeNotes: Figures plot son’s expected rank in the income distribution conditional on father’s rank (in percentiles). The series are based on sons’ and fathers’ lifetime income (black solid line), sons’ annual incomes and fathers’ lifetime income (left-side measurement error, Panel A), or fathers’ income at age 45 (both-side measurement error, Panel B). We distinguish between estimates based on son’s income at age 25, at age 30, and at age 40. For ease of exposition we plot the rolling average over three percentiles of fathers’ rank.

Figure 4a shows that the CEF is poorly approximated by annual incomes at age 25, while over a wide range of percentiles it is well approximated already at age 30. However, not in the tails: average offspring rank is understated (overstated) by up to 10 percent in the top (bottom) five percentiles of the parent distribution. The underestimation in the top becomes negligible when incomes around age 40 are used. In contrast, the prospects of the very poorest are consistently overestimated across all ages in our data. These patterns are amplified when only short snapshots are observed for both generations. Figure 4b plots estimates based on annual income of sons and fathers at age 45. Average offspring rank is now overstated by about 30 percent in the bottom percentiles of the parent distribution. Because even parents with high lifetime income can occasionally have very low annual incomes, it is particularly hard to identify those fathers with the lowest lifetime incomes from short income spans; the CEF is particularly biased in this range.¹⁸

Figure 5 plots the copula of parent and offspring long-run incomes. The density is highest along the diagonal: As assumed in our derivation above, the likeliest position in the income distribution of children is the position of their parents. Because ranks are bounded, this probability is particularly high for the offspring with the very poorest and (especially) richest parents. The probability to reach the top percentile in the income distribution if the father was in the top percentile is 12.3 percent in our sample; their probability to reach the top five percentiles is above 37 percent. To yield a sensible scaling for the rest of the copula we thus censored the top percentile in Figure 5.

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Figure 5 Joint Density of Son’s and Father’s Rank (Benchmark)Notes: The figure plots the copula, the joint density distribution of son’s and father’s income ranks (in percentiles), using lifetime incomes for both generations based on 20 × 20 data points, interpolated. Under statistical independence each cell has expected density 0.25 and color light grey. Dark grey and black indicates excess densities, while light grey and white indicates densities that are lower than what we would have under independence. Densities along the diagonal capture immobility, off-diagonal densities mobility.

Figure 6 illustrates the bias that results from using annual income for sons at age 40 (Panel A) or annual incomes for both sons (at age 40) and fathers at age 45 (Panel B). To illustrate its size, the difference between the annual-based and the benchmark copula is expressed as a fraction of the cell probability that is expected under statistical independence. A number of patterns emerge. First, usage of annual incomes leads to underestimation along and overestimation away from the diagonal—in line with our theoretical arguments. Second, the biases are largest in magnitude in the tails of the distribution. While the inheritances of poverty and top incomes are often of particular interest, their estimates are also the most inaccurate in short data. Our findings are thus consistent with simulation results by O’Neill, Sweetman, and Van de gaer (2007), who suggest that classical measurement error leads to overestimation of mobility in particular in the tails of the distribution. Considerably mismeasured is also the extent of long-distance downward mobility. As even sons from top-income parents have occasionally low annual incomes, their probability to fall to the bottom is substantially overstated in annual data.¹⁹ We thus find that the effect of measurement error on the copula has a quite logical structure. Researchers need to exercise particular caution when studying long-distance mobility, the inheritance of poverty, or the inheritance of top incomes.

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Figure 6 Bias in Joint Density of Son’s and Father’s RankNotes: The figures plot the biases in the copula, the joint density distribution of son’s and father’s income ranks (in blocks of five percentiles each, interpolated). We use income at age 40 for sons and lifetime income (leftside measurement error, Panel A) or income at age 45 (both-side measurement error, Panel B) for fathers. The color intensity indicates the magnitude of the bias, that is, the density difference compared to when lifetime incomes are used for both generations. For comparability across the copula, we scale the density difference in each 5x5 cell by the density that we would expect under statistical independence (1/400). For example, a density difference of 0.5 implies that the density was 1/800 higher in annual than in lifetime data.