Approximating Income Distribution Dynamics Using Aggregate Data

This paper proposes a methodology to approximate individual income distribution dynamics using only time series data on aggregate moments of the income distribution. Under the assumption that individual incomes follow a lognormal autoregressive process, this paper shows that the evolution over time of the mean and standard deviation of log income across individuals provides sufficient information to place upper and lower bounds on the degree of mobility in the income distribution. The paper demonstrates that these bounds are reasonably informative, using the U.S. Panel Study of Income Dynamics where the panel structure of the data allows us to compare measures of mobility directly estimated from the micro data with approximations based only on aggregate data. Bounds on mobility are estimated for a large cross-section of countries, using data on aggregate moments of the income distribution available in the World Wealth and Income Database and the World Bank's PovcalNet database. The estimated bounds on mobility imply that conventional anonymous growth rates of the bottom 40 percent (top 10 percent) that do not account for mobility substantially understate (overstate) the expected growth performance of those initially in the bottom 40 percent (top 10 percent).

This paper proposes a methodology to approximate individual income distribution dynamics using only time series data on aggregate moments of the income distribution. Under the assumption that individual incomes follow a lognormal autoregressive process, this paper shows that the evolution over time of the mean and standard deviation of log income across individuals provides sufficient information to place upper and lower bounds on the degree of mobility in the income distribution. The paper demonstrates that these bounds are reasonably informative, using the U.S. Panel Study of Income Dynamics where the panel structure of the data allows us to compare measures of mobility directly estimated from the micro data with approximations based only on aggregate data. Bounds on mobility are estimated for a large cross-section of countries, using data on aggregate moments of the income distribution available in the World Wealth and Income Database and the World Bank's PovcalNet database. The estimated bounds on mobility imply that conventional anonymous growth rates of the bottom 40 percent (top 10 percent) that do not account for mobility substantially understate (overstate) the expected growth performance of those initially in the bottom 40 percent (top 10 percent).

Introduction
Understanding the extent of mobility -defined here as changes in individuals' relative incomesis crucial to interpreting movements over time in average incomes in different parts of the income distribution. For example, the policy implications, and even the political acceptability, of a given change in average income in the top 1 percent of the income distribution depends crucially on whether the identity of those in the top income group is stable over time, or instead whether some of the initially rich fall out of the top group and are replaced with those who were initially poorer. This distinction also matters at the lower end of the income distribution. Consider for example the World Bank's declared goal of promoting "shared prosperity", defined as growth in the bottom 40 percent of a country's income distribution. 1 This emphasis on the bottom 40 percent sits well with the World Bank's focus on the conditions of the poorest in a country. However, to the extent that the role of such a goal is to evaluate the efficacy of interventions targeted towards the initially-poor members of the bottom 40 percent, it is necessary to know whether membership in this group is stable over time, or whether the initially-poor beneficiaries of these interventions become richer and move out of the bottom 40 percent and are replaced by those who started out richer and became relatively poorer.
The ability to document and analyze income distribution dynamics at the individual or household level is severely constrained by the scarcity of panel data sets that track individual or household incomes over time. 2 In advanced economies such as the United States, long and high-quality panel data on household incomes is available in the Panel Study on Income Dynamics (PSID). However, such high-quality long panel data sets in developing countries are rare. Instead, it is much more common for survey data to be available in the form of repeated cross-sections, with different samples of individuals appearing in each round of the survey. In cases where researchers have access to record-level survey data, it is possible to deploy a variety of pseudo-panel estimation techniques to obtain approximations to income distribution dynamics at the individual level. 3 In many cases, however, access to record-level data from 1 See for example Basu (2013). 2 The different datasets we work with in this paper have different units of observation at the micro level, including individuals, households, and tax units. They also differ in whether they measure consumption or income. For terminological convenience we will refer to income distribution dynamics at the individual level wherever it is possible to do so without confusion.
3 the survey is restricted, making it difficult to deploy pseudo-panel techniques that rely on micro data from repeated cross sections. In yet other cases, particular for older datasets such as historical income tax records, the micro data may simply no longer exist, and therefore cannot be analyzed to measure mobility directly. In these cases, researchers may only have access to summary statistics on the distribution of income such as the mean and the Gini coefficient, or alternatively grouped data on the proportion of individuals in different income brackets, but not the record-level data from the entire micro dataset itself. This paper proposes and implements a method for approximating individual-level income distribution dynamics in these situations where only aggregate summary statistics on the distribution of income are available. Specifically, we consider a situation in which the researcher only has access to time series data on the mean of the income distribution, as well as some measure of dispersion of income across individuals, such as the Gini coefficient or the income share of a particular subgroup. We assume that the underlying surveys are sufficiently comparable that the observed aggregate moments characterize the same population across the successive surveys.
Our key identifying assumption is that individual incomes follow an autoregressive lognormal process with individual fixed effects. Under this data generating process, the joint distribution of individual incomes in any two periods is fully characterized by the mean and standard deviation of log incomes in the two periods (which can be inferred from the observed aggregate moments of the income distribution), and a mobility parameter characterizing the covariance of individual incomes between the two periods (that we cannot observe directly in the aggregate data). This mobility parameter depends on two underlying parameters of the data generating process: the autoregressive coefficient on log income, and the cross-sectional variance of the individual effects. We show that the limited information embodied in the evolution over time of the observed aggregate moments is sufficient to allow us to estimate the autoregressive coefficient on log income as well as to place bounds on the variance of the individual effects. This in turn permits us to place bounds on the extent of mobility in the income distribution, even though we do not observe income dynamics at the individual level.
In order to assess the usefulness of these bounds, we use record-level data from the PSID that permits us to directly measure mobility using micro-level panel data. We then discard all of the information in the record-level data, and use only the two time-series on the mean and standard deviation of log income to retrieve bounds on mobility based on aggregate data. We show that these bounds contain the point estimates obtained using the record-level data, and moreover are reasonably tight.
Encouraged by these findings, we apply our methodology to two large cross-country datasets. The first 4 of these is the World Wealth and Income Database (WID), constructed by Thomas Piketty and his collaborators 4 , which reports summary statistics on average income and top income shares compiled from tabulated data on tax records. From this database, we retrieve time-series data on mean income and the bottom 90% income share for 19 countries with reasonably complete annual time series data during the post-World War II period. For the median country in this sample, we have 60 years of annual data on aggregate moments on which to base our estimates of mobility. The second data source is the compendium of household survey data for a large set of mostly developing countries in the World Bank's PovcalNet database. This dataset provides summary statistics on the distribution of income (or consumption, depending on the survey) for over 1400 household surveys in a large cross-section of developing countries, in some cases extending back to the 1980s. We retrieve data on mean income (or consumption) and the Gini coefficient for these surveys, and restrict attention to the set of 28 countries for which we have at least 10 surveys. The median country in this sample has 16 household surveys spanning a period of 17 years.
We apply our methodology to find bounds on mobility for the countries in these two datasets.
Our estimates confirm that among the high income countries in the WID database, the US ranks as the country with the lowest income persistence, while countries with high income persistence include all of the Scandinavian countries. Interestingly, when we correlate our estimates with GDP per capita, we find that lower income persistence is associated with higher income. This observation applies both to the WID and the PovcalNet estimates. As we discuss in more detail below, this observation is not an obvious consequence of our assumptions, and suggests that countries with greater persistence must have other characteristics conducive to more rapid growth.
We use these estimated bounds on mobility to generate bounds on the difference between "anonymous" growth rates of group average incomes (that do not require membership in the group to be stable over time) with corresponding "non-anonymous" growth rates (that track the performance of the same initial group of individuals over time). When we compare estimates of anonymous income growth rates among the bottom 40% (the World Bank's measure of Shared Prosperity) to their non-anonymous counterparts, the difference is substantial: on average, non-anonymous growth rates exceed anonymous ones by four percentage points in the WID and three percentage points in PovcalNet in annual terms. This means that by tracking shared prosperity anonymously, policy makers could inadvertently overlook the 5 successes with which originally poor households have been able to increase their incomes. This ordering is reversed when we examine growth of average incomes at the top of the income distributionanonymous top incomes grow faster than non-anonymous top incomes do. Also here the magnitude of the differences we observe is considerable, at over 10 percentage points in the WID and 8 percentage points in PovcalNet. This suggest that anonymous top income growth rates, while a good indicator of changes in inequality in society, are not well-suited for estimating the expected income success of those who occupy the top of the income distribution at any given point in time.
Our paper provides an alternative approach to the large literature on estimating income distribution dynamics using pseudo-panel data techniques that track the evolution of cohorts over time. 5 This literature circumvents the need for true panel data that tracks individuals over time, but it does assume that the researcher has access to microdata from repeated series of cross-sectional household surveys. In contrast, our approach does not require access to any record level data, but instead shows how the parameters governing individual income distribution dynamics can be retrieved from time series data on aggregate moments only. 6 The obvious advantage of this is that by reducing the data requirements to a minimum, it significantly expands the number of countries where this approach can be applied. It also makes it possible to obtain estimates for historic periods for which aggregate moments have been preserved, but record level data will be hard if not impossible to come by. The price we pay for working with time series rather than pseudo-panel data is that we have to work with a much smaller number of observations. We make the most of the data that we have by relying on finite sample estimation methods (Andrews, 1993), and optimizing the trade-off between bias and precision. Finally, while we are by no means the first to notice that anonymous and non-anonymous growth rates can in principle diverge widely in the presence of mobility, we are -to our knowledge -the first to be able to provide estimates of the gap between the two for a large cross-section of countries. 7 The rest of this paper proceeds as follows. In Section 2 we state our main assumptions regarding the lognormal data generating process for individual incomes, and show how mobility is fully 6 characterized by the observed aggregate moments of the income distribution, together with a single key unobserved mobility parameter. In Section 3, we show how to obtain bounds on this mobility parameter using only information embedded in the evolution over time of the aggregate moments of the income distribution. Section 4 validates our methodology in the PSID data, and Section 5 provides estimates of mobility for a large cross-section of countries based only on the aggregate summary statistics on the distribution of income available in the WID and PovcalNet databases. In Section 6 we apply the lognormality assumption and the estimates of mobility from the previous sections to document the differences between anonymous and non-anonymous growth rates of group average incomes. Section 7 concludes.

Lognormal Income Distribution Dynamics
Throughout the paper, we rely on the following assumption regarding the process generating individual incomes: Assumption A1: The logarithm of income of individual at time , , is generated by the following autoregressive process: where is an autoregressive parameter satisfying 0 ≤ < 1; is a common factor; the innovations and are independent and normally distributed with zero mean and variances and ; and initial income is = + + .
Although very simple, this data generating process gives rise to non-trivial income distribution dynamics through the interplay of two forces. On the one hand, realizations of the idiosyncratic shock generate changes in individuals' relative incomes over time. On the other hand, there are two sources of persistence over time in relative incomes: the autoregressive term , and the individual effect .
Overall income distribution dynamics will reflect the balance of these sources of changes and persistence in relative incomes. Note also that we allow the common factor, , and the variance of the idiosyncratic errors, , to vary over time, but we require the autoregressive parameter and the variance of the individual effect to be stable over time.
Since the innovations in log income, as well as initial log income, are assumed to be normally distributed, log incomes are normally distributed in every period, as summarized in the following proposition: Proposition 1: Assumption A1 implies that log individual incomes and are jointly normally distributed: (2) ~ , , where and denote the cross-sectional mean and variance of log income at time , and 0 ≤ ≡ + ≤ 1.

Proof: See Appendix A
Proposition 1 states that log incomes in any two consecutive periods are normally distributed. 8 The proof in the appendix generalizes this to the case of any two non-consecutive periods, which will be useful when applying our methodology to irregularly-spaced survey data. In addition to establishing lognormality of income at all points in time, Proposition 1 introduces a key composite parameter which summarizes the comovement of individual incomes over time. This composite parameter can be interpreted as the OLS estimator of the slope of a regression of individual income on lagged individual income, i.e. = ( , ) . 8 individual incomes are tracked over time is 1 − . For this reason, we will refer to the composite parameter as a "mobility parameter", as it is the workhorse empirical measure of relative mobility.
Let Φ(. ) denote the cumulative normal distribution function, and define the quantile function ( ) = + Φ ( ), which returns the log income level associated with each percentile of the income distribution. Define the random variable | ( ) as log income at time of an individual who was at the percentile of the income distribution at time − 1. The probability distribution of | ( ) fully characterizes the mobility prospects of an individual who starts out at the percentile of the income distribution, and is summarized in Proposition 2.
Proposition 2: Given Assumption A1, | ( ) is distributed normally with the following mean and variance: (3) Proposition 2, which follows immediately from applying the properties of conditional mean and variance of the bivariate normal distribution to Equation (2), shows a key feature of the lognormal data generating process in Assumption A1 --the distribution of | ( ) depends only on aggregate moments (the mean and variance of log income), and a single mobility parameter . In the following section, we show that it is possible to retrieve empirical bounds on using only information in the evolution over time of the aggregate moments of the income distribution. This in turn implies that we can empirically characterize income mobility using only data on aggregate moments of the income distribution, i.e. using only the time series data on and .
Proposition 2 also helps to clarify the relative mobility interpretation of . To see this, subtract ( ) = + Φ ( ) from both sides of Equation (3) to obtain: This expression decomposes the expected change in log income of an individual starting at the percentile of the income distribution at time − 1 into two components. The first term − corresponds to the change in the mean of log income, which by definition contributes equally to everyone's income growth, and thus leaves relative incomes unchanged. This term can be thought of as capturing absolute mobility. The second term ( − 1) Φ ( ) corresponds to the expected change in relative income, and thus captures relative mobility. In the benchmark case of = 1, the expected change in relative income is zero, i.e. there is no relative mobility in expectation. The lower is , the greater is relative mobility since the expected change in relative incomes becomes larger in absolute value. In particular, when < 1, individuals starting out in the bottom half of the income distribution at time − 1 will expect to see faster-than-average income growth (since when < 0.5 we have Φ ( ) < 0 and ( − 1) Φ ( ) > 0). That is, the initially poor (in expectation) get richer in relative terms. Conversely, an individual starting in the top half of the income distribution with > 0.5 can expect to have income growth below the average. That is, the initially rich (in expectation) get poorer in relative terms.
The mobility parameter also governs uncertainty about changes in relative incomes. From Equation (4)  , and therefore less uncertainty about relative incomes.

Estimating Upper and Lower Bounds on Mobility Using Aggregate Data
A key feature of the data generating process in Assumption A1 is that it implies a simple autoregressive processes for the evolution over time of the aggregate moments of the income distribution, as summarized in the following proposition: Proposition 3: Assumption A1 implies that the mean and variance of log income follow: (6) = +  (7) as + = + + .
Inserting these approximations into Equations (6) and (7) results in the following system of two equations: (8) = + + + (9) = + + + Given time series data on the mean and variance of log income, we can obtain an estimate of by simply regressing on its lag and a time trend (i.e. from Equation (8)), or we can obtain an estimate of by regressing on its lag and a time trend (i.e. from Equation (9)). In some of our empirical applications, notably the PSID and the PovcalNet data, the available time series are quite short, rarely longer than 20 years, and frequently shorter. This has two implications for our estimation strategy. First, the short time series raises concerns about small-sample bias in the estimation of . Specifically, Andrews (1993) shows that the OLS estimator of the autoregressive coefficient in a linear AR(1) process around a deterministic trend is biased downwards in finite samples, and that this bias can be substantial. Andrews (1993) proposes a bias-corrected estimator that addresses this problem, though at the cost that the biascorrected estimator has higher variance than the OLS estimator. We balance this tradeoff between bias and precision by taking a linear combination of the OLS estimator and the bias-corrected estimator that minimizes mean squared error. We apply this procedure to obtain an estimator of from Equation (8), and an estimator of from Equation (9). Second, the scarcity of time series data on aggregate moments suggests that it is important to combine information from the dynamics of and into a single estimate of . We do this by taking a mean squared error-minimizing linear combination of the estimators based on Equation (8) and Equation (9), to arrive at a single estimate of reflecting the dynamics of both the mean and the variance of log income. Further details on the estimation strategy are in Appendix B.
Given this estimate of , which we denote , we next obtain bounds on the mobility parameter = + . Note that is an increasing function of the variance of the individual effect .
Thus, a lower bound on can be obtained by setting = 0, i.e. = . This lower bound corresponds to the benchmark of the highest degree of mobility consistent with our estimate of based on aggregate data, since we have turned off any additional persistence coming from the individual effect.
We obtain an upper bound , i.e. a lower bound on mobility, by finding a corresponding upper bound on the variance of the individual effect. To do this, note that the variance of the idiosyncratic component of the error term in Equation (7) must be weakly positive, i.e. ≥ 0. Using Equation (7) and the estimate of , this implies that ≤ ( − ). Given our assumption that and are stable over the estimation sample, this upper bound must hold for every period ∈ , where represents the time periods that comprise the estimation sample. This means that the tightest possible upper bound for the variance of the individual effect is ≤ min Inserting this into the expression for , and recalling that our data generating process implies that ≤ 1, we have = min 1, + .
Note that the bounds and depend only on the aggregate moments of the income distribution, and , as well as the estimate , which as discussed above can also be obtained using only aggregate moments. This means that we can obtain bounds on the mobility parameter and in turn approximate individual-level income distribution dynamics using only aggregate data. Whether these bounds are useful, in the sense of delineating a reasonably narrow range of values for , is an empirical question to which we turn in the remainder of the paper.

Comparing Actual and Approximate Mobility in the PSID
In this section, we verify that our methodology for placing bounds on the mobility parameter provides reasonably informative bounds on true mobility, using data from the US Panel Study of Income Dynamics (PSID). We first estimate bounds on by applying the approach described in the previous sections to the time-series of the cross-sectional mean and standard deviation of log income in the PSID.
We then take advantage of the panel structure of the PSID micro data to estimate the mobility parameter directly, and compare these micro estimates with the bounds obtained using only aggregate data.
We work with annual rounds of the PSID between 1977 and 1997. 9 The unit of observation in the PSID is the household, and we take nominal family income per capita deflated by the national consumer price index as our measure of real income per capita. Since measures of inequality are sensitive to extreme observations, we discard a small number of household-year observations corresponding to implausibly low and implausibly high per capita incomes. 10 We then compute the mean and variance of log per capita income, using the sampling weights provided in the PSID, and apply our methodology for estimating and obtaining bounds on to the resulting two time-series of aggregate moments. Table 1 summarizes our results. Panel A reports the estimates and standard errors of the autoregressive parameter . We first report estimates based on the dynamics of the mean of log income (Equation (8)) and the variance of log income (Equation (9)). The three columns correspond to the OLS estimates, the small sample bias-corrected OLS estimates, and the MSE-minimizing linear combination of the two. Comparing the first two columns suggests that the bias correction is important -the bias corrected estimates of are substantially larger than the OLS estimates, increasing from 0.65 to 1 (based in the dynamics of ), and increasing from 0.76 to 0.96 (based on the dynamics of ). However, the 9 After 1997, the PSID switches to biannual frequency. We also considered working with a biannual version of the PSID from 1977 to the present. However, over this longer time series, there is clear evidence of a structural break in the time series for the standard deviation of log income. Accommodating this trend break in a series of only 17 biannual observations led to estimates of based on Equations (8) and (9) that were highly imprecise. For this reason, we work with the shorter 1977-1997 time period with annual data. In the following subsection, we allow for structural breaks when estimating Equation (8) and (9) using longer time series on aggregate moments available in the World Wealth and Income Database. 10 Specifically, we drop households log per capita income below $400 or above $440,000, corresponding to the bottom 0.8 percent and top 0.04 percent of the household-year observations in the raw data.
comparison also reveals that the bias-corrected estimates are much less precise than the OLS estimates.
Recognizing this bias versus precision tradeoff, we calculate a MSE-minimizing combination of the OLS and bias corrected-OLS estimates, reported in the third column. The last column reports the MSEminimizing weight on the OLS estimator, which is 1.07 for the first equation, and 0.80 for the second. 11 This results in MSE-minimizing estimates closer to the OLS estimates, at 0.63 and 0.80 in the two equations. These are our preferred estimates of based on the observed dynamics of the mean and variance of log income.
Despite the short time series, these estimates are reasonably precise, with standard errors of 0.07 and 0.17 respectively. 12 However, since these combined estimates minimize MSE, they remain downward-biased. Comparing the MSE-minimizing estimates in the third column with the bias-corrected OLS estimates in the second column suggests this downward bias is non-trivial, at 0.37 for the estimate based on the dynamics of the mean of log income, and at 0.20 for the estimate based on the dynamics of the variance of log income. To further improve the precision of our estimates, we combine the two singleequation MSE-minimizing estimates into a single estimate by taking an MSE-minimizing linear combination of the two, which is reported in the last row of Table A. This results in our final estimate, which is = 0.81 with a standard error of 0.18.
In the bottom panel of Table 1 we report our upper-bound estimate of the variance of the individual effect given our preferred estimate of = 0.81, which is = 0.021. To put this estimate in perspective, it is useful to consider a benchmark version of Equation (7) in which the variance of the idiosyncratic shock to income is constant, i.e. = . In this case, the cross-sectional variance of log income converges to a steady-state value * = ( ) + ( ) , and ( ) / * can be interpreted as the share of steady-state inequality due to the variation in the individual effect. In our sample, the average over time of the variance of log income is 0.73, and using this as an estimate of * together with our preferred estimate of results in an upper bound of 83 percent of steady state inequality is due to the variance of the individual effect.
In the next two columns of the bottom panel of Table 1, we report the lower bound and the average over time of the upper bound on the mobility parameter, , which are 0.81 and 0.97, respectively. In Figure 1, we display the time-evolution of our estimated bounds on mobility based on the aggregate moments. Recall that the lower bound is , which is constant over time, while the upper bound is equal to = min 1, + and varies over time with the observed data on .
Purely for visual reference, the dashed line indicates the midpoint of the range between the lower and upper bound estimates.
The key question of interest is how these bounds on the mobility parameter compare with actual mobility as measured at the household level. To answer this question, recall that is simply the slope coefficient from an OLS regression of household-level log per capita income on its lagged value.
Given 15 The second issue concerns the interpretation of year-to-year fluctuations in household per capita income. These fluctuations in part reflect changes in household size, as well as changes in the number of income earners in the household, which may not be well-captured by our simple lognormal data generating process. While it is possible to directly observe household size, it is not possible to cleanly identify fluctuations in household income per capita due to changes in the number of income earners in the household. 13 Beyond these concerns, there are also perennial thorny questions concerning the effects of measurement error in family income on the estimates of , which could lead to different biases depending on its correlation with income and over time. 14 One crude way of partially addressing both these concerns simultaneously is to filter our estimation sample for influential observations. We do this using a Cook's distance criterion, and drop all household-year observations in our pooled baseline regression sample corresponding to the top one percent of observations on the Cook's distance statistic.
The bottom panel of Figure 1 superimposes these two alternative micro estimates of on the same grey shaded region corresponding to the bounds on mobility based on aggregate data. Both variants result in slightly higher micro estimates of , with both averaging to 0.86, as opposed to 0.83 in the baseline. These alternative micro estimates of now fall closer to the center of the range based on the macro data, and particularly so in the middle decade of our 20-year sample. Overall, the picture that emerges from Figure 1 is that the bounds on mobility we can estimate using only data on the evolution of aggregate moments of the income distribution are reasonably narrow, and for the most part include estimates of mobility estimated from the micro panel data.

Cross-Country Estimates of Mobility Using Only Aggregate Data
Encouraged by the results of the previous section, which show that our approach delivers reasonable bounds on mobility in the PSID where mobility can be directly observed in the record-level panel data, in this section we obtain bounds on the mobility parameter in two multi-country datasets where we have data only on aggregate moments of the income distribution. We work with two such datasets. The first is the World Wealth and Income Database (WID), which assembles estimates of top 13 This is because the PSID collects information on family income and income of the household head, but does not collect data separately on incomes of other household members. 14 The limited available information based on linking survey incomes with administrative data in the US suggests that the overall effect of measurement error on mobility measures is small (see Jantti and Jenkins (2015), Section 10.4.1), although these authors caution that this evidence is best interpreted as a reflection of how little is known about this issue.
income shares based on tabulations of income tax records for a large number of countries, based on the work of Anthony Atkinson, Thomas Piketty, and their collaborators. 15 We retrieve time series data on mean income and the income share of the bottom 90 or 99 percent of income earners (depending on data availability), for a set of 19 countries with long annual time series data on these aggregate moments in the post World War II period. 16 We convert these two summary statistics into the mean and variance of log income using our maintained assumption of lognormality. 17 The WID predominantly contains data from advanced economies, and our set of 19 countries from this source includes only three developing countries: China, India, and Mauritius.
Our task of estimating based on the evolution over time of these aggregate moments is complicated by the fact that the time period covered for many of these countries is long, and likely spans some structural breaks. Most of the countries in our sample have data beginning shortly after World War II, and visual inspection of trends in the mean and variance of log income suggest a trend break in these series around the 1970s for many countries (see Figure 2). Three countries in our sample (United States, France, and Germany) also have fairly long time series data prior to World War II, where again it seems plausible that income dynamics may have been different relative to the post-war period. We therefore divide the data into sub-periods, and allow for different time trends by sub-period. For the three countries with pre-World War II data, we consider the available data up to 1939 as one distinct period. For the post-World War II period, we allow the data to select a single structural break in the time trends. Figure 2 illustrates this process of identifying trend breaks for different time periods for the three countries in our sample with pre-World War II data. The left and right panels of the figure report the time series of the mean and standard deviation of log income, with the trend lines for different sub-periods superimposed.
We impose the restriction that is the same over the entire time-series available for each country, but we allow the intercept and time trend in Equations (8) and (9) to differ across the sub-periods for each country. 18 We compute OLS and bias-corrected OLS estimates of based on the evolution over time of the mean of log income, and based on the evolution over time of the variance of log income, for each country. We also compute the MSE-minimizing weighted average of the two for each equation separately, and the MSE-minimizing weighted average of these across the two equations to arrive at our preferred estimate of for each country.
Our second application of this approach draws on the PovcalNet database maintained by the relying on the assumption of lognormality, we convert these series into series for the mean and variance of log income, and then we implement the procedure described in Section 3 to obtain estimates of and upper and lower bounds on mobility. Because the available time series in PovcalNet is much shorter than in the WID, we do not allow for trend breaks when estimating Equations (8) and (9). A visual inspection of the time-series of the mean and variance of log income for these countries shows no obvious signs of structural breaks in the data. As with the WID dataset, we generate OLS and bias-corrected OLS estimates of , as well as an MSE-minimizing combination of the two, for each of the 30 surveys.
We report the results of these two empirical applications of our methodology in Tables 2 and 3.
In the first column of both tables we present our preferred estimate of for each country. The underlying OLS and bias-corrected OLS estimates from the equations for the evolution of the mean and variance of log income, and the corresponding MSE-minimizing weighted averages, are reported in Appendix Tables 18 To minimize the influence of a small number of observations corresponding to large swings in the mean and variance of log income, we eliminate from our estimation sample observations for which Cook's Distance exceeds 0.2 and/or the Studentized residuals exceed 3.5. 19 Most of the countries we selected from the PovcalNet database have annual household surveys, but a few have regularly surveys once every two or three years. We annualize our estimates of and for these countries to make them comparable to those based on annual data. Expressions for the irregularly-spaced versions of our main results are detailed in Appendix A. B1 and B2, together with details on the estimation sample and the MSE-minimizing weights. The next two columns of Tables 2 and 3 report our upper-bound estimate of the variance of the individual effect, and its contribution to steady-state inequality, and the final two columns report the lower bound and the average over time of the upper bound on the mobility parameter, and .  Since these shocks are independent over time, they have only transitory effects on income, and as a result mobility is higher.

Application: Anonymous and Non-Anonymous Growth in a Cross-Section of Countries
Popular discussions of trends in inequality frequently refer to income growth rates of "the rich" (defined as the top X% of the income distribution) and income growth rates of "the poor" (defined as the bottom Y% of the income distribution). One prominent example is the World Bank's declared goal of promoting "shared prosperity", defined as growth in the bottom 40 percent of the income distribution.
Similarly, a body of work by Thomas Piketty and his collaborators has drawn widespread attention to trends in "top incomes", defined as average incomes in the upper percentiles of the income distribution.
Measuring and interpreting the income growth rates of such population subgroups is complicated by the basic data problem that motivates this paper: true panel data tracking individuals over time is rarely available. Absent such panel data, group-average growth rates typically are calculated based on repeated cross sections. For example, the growth rate of "the poor" would be calculated by comparing average incomes in the bottom Y% at two points in time. Since growth rates calculated in this way do not track individuals over time, they are referred to as "anonymous" growth rates. In contrast, "nonanonymous" growth rates track the same individuals in an initial reference group over time, and can be very different from their "anonymous" counterparts when there is mobility in the income distribution. 20 20 We are by no means the first to notice this distinction --see Jenkins and Van Kerm (2006), Grimm (2007), Van Kerm (2009), and Bourguignon (2011) for discussions of the difference between anonymous growth incidence curves and their non-anonymous counterparts. The novelty in this section of our paper is that we are able to compute estimates of the difference between anonymous and non-anonymous growth rates for a large sample of countries, using our estimates of mobility based only on aggregate moments of the income distribution.
However, given the scarcity of true panel data, these non-anonymous growth rates are rarely observed directly.
This distinction can have significant policy implications as well, precisely because it reflects the underlying degree of mobility in the income distribution. For example, rapid growth in top incomes might be more politically acceptable if it were accompanied by significant mobility of individuals into and out of the top income bracket. Similarly, if the policy objective is to track the effectiveness of an intervention aimed at raising incomes in the poorest Y% of the population, conclusions could be quite different using the anonymous and non-anonymous growth rates, since the latter track the performance of the initial beneficiaries of the intervention while the former does not.
In this section we obtain analytical expressions for the difference between anonymous and nonanonymous growth rates implied by the lognormal data generating process described in Assumption A1.
These expressions depend only on aggregate moments and the mobility parameter . We then use our bounds on from the previous section to obtain bounds on the difference between anonymous and non-anonymous growth rates in a cross-section of countries, even though the true panel data required to track individuals over time is not available for most of these countries.
We begin by defining the anonymous and non-anonymous versions of the growth incidence curve.
The anonymous growth incidence curve, as defined in Ravallion and Chen (2003) In contrast, the non-anonymous growth incidence curve tracks the expected growth rate of individuals at each percentile of the initial income distribution, i.e.

(11)
This expression is the same as Equation (5), as is the intuition. The second term measures the extent of mobility in the income distribution. In the benchmark case of no mobility (in expectation), i.e. = 1, the non-anonymous growth incidence curve is flat, and expected income growth is the same at every point in the initial income distribution. When the mobility parameter < 1, individuals starting out below the mean of log income expect higher-than-average growth, while those starting out above the mean expect lower-than-average growth.
The difference between the two growth incidence curves is: It is straightforward to verify that our data generating process implies − ≥ 0, and so the anonymous growth incidence curve will fall below the non-anonymous growth incidence curve for the bottom half of the income distribution (since Φ ( ) < 0 when < 0.5). 21 That is, the anonymous growth incidence curve understates the actual growth rate of any individual starting out in the bottom half of the income distribution. Symmetrically, the anonymous growth incidence curve overstates the actual growth rate of any individual starting out in the top half of the income distribution. Note that the difference between the two growth incidence curves is monotonic in at every percentile .
Therefore, the upper and lower bounds for obtained in the previous section will map into upper and lower bounds on the difference between the two growth incidence curves. Figure 5 illustrates the difference between anonymous and non-anonymous growth incidence curves using the PSID microdata and our estimates of mobility based on the aggregate moments of this dataset from Section 4. The left and right panels of the graph show the 10-year average annual growth incidence curves over the periods 1977-1987 and 1987-1997. In each panel, the dashed line shows the anonymous growth incidence curve. In both periods, the anonymous growth incidence curve is weakly upward-sloping, reflecting the increase in overall inequality over this time. The two solid lines show the non-anonymous growth incidence curves based on the actual micro data, calculated in two different ways.
The thin line is a kernel-weighted local polynomial smoothed average of the actual growth rates of all 21 This is because the positive semi-definiteness of the covariance matrix in Equation (2) requires ≥ .

22
individuals at each percentile of the initial income distribution. The bold line is the lognormal version of this curve, i.e. Equation (11), but using the estimate of based on the PSID micro data. Interestingly, this lognormal version of the growth incidence curve tracks the non-parametric smoothed estimate of the growth incidence curve fairly closely, suggesting that our assumption of lognormality provides a reasonable approximation to the true income distribution dynamics in this particular setting. As expected, the non-anonymous growth incidence curves indicate much higher (lower) growth for those initially in the bottom (top) half of the income distribution. Finally, we plot the lower and upper bounds on the growth incidence curve based on our bounds on calculated using only aggregate data. These bounds include the two versions of the actual non-anonymous growth incidence curve based on micro data almost everywhere. It is also useful to note that they exclude the anonymous growth incidence curve everywhere except for very close to the middle of the income distribution where the anonymous and non-anonymous growth incidence curves are very close to each other.
We next consider growth rates of group average incomes, such as the bottom 40 or the top 10 percent of the income distribution. The anonymous and non-anonymous growth rates of group average incomes are simply weighted averages of the corresponding anonymous and non-anonymous growth incidence curves, as summarized in the following proposition:

Proof: See Appendix A.
It follows immediately from Proposition 4 and Equations (10) and (11) that the difference between the anonymous and non-anonymous growth incidence curves for group average incomes is: That is, the difference between the two growth rates of group average incomes is also a weighted average of the difference between the anonymous and non-anonymous growth incidence curve. As before, the difference between the anonymous and non-anonymous growth incidence curves depends on − ≥ 0. Whether the anonymous growth rate exceeds or is smaller than the non-anonymous rate now depends on the sign of the integral Φ ( ) ( ) . While a closed-form solution for this integral exists and is given in the proof of Proposition 4 in Appendix A, this term cannot be signed in general. However, for the particular case where 0 ≤ < ≤ 0.5, it is clear that Φ ( ) ( ) ≤ 0 since Φ ( ) ≤ 0 for 0 ≤ ≤ 0.5. This means that for any subgroup in the bottom half of the income distribution, the anonymous group average growth rate is lower than the corresponding non-anonymous growth rate. Naturally, this argument is symmetric: when 0.5 ≤ < ≤ 1, the anonymous growth rate of group average incomes for any subgroup in the top half of the income distribution exceeds the corresponding non-anonymous growth rate.
We can quantify the difference between anonymous and non-anonymous growth rates using our estimates of the mobility parameter based on aggregate data from the WID and PovcalNet databases. We begin by calculating the average annual anonymous growth rate of the bottom 40 and top 10 percent over the most-recent 10-year period available for each country. These anonymous growth rates would be the ones typically reported when true panel data are not available. We then compute the lower and upper bounds on the corresponding annual average non-anonymous growth rates, using the bounds on reported in Tables 2 and 3. The results of this calculation are summarized in Figure 6.
The top and bottom panels report results from the WID and PovcalNet samples, respectively. In each panel, we plot the anonymous growth rate on the horizontal axis, the non-anonymous growth rate on the vertical axis, and the diagonal line corresponds to the 45-degree line. The points above the 45-degree line correspond to growth rates of the bottom 40 percent, reflecting our result that non-anonymous growth rates exceed their anonymous counterparts for group averages within the bottom half of the income distribution. Conversely, the points below the 45-degree line correspond to growth rates of the top 10 percent, for which the non-anonymous growth rate is smaller than the anonymous growth rate. The triangles (for the bottom 40 percent) and squares (for the top 10 percent) indicate the midpoint of our range for the non-anonymous growth rate, and the vertical lines display the range between the lowerand upper-bound estimates of the non-anonymous growth rates. The anonymous growth rates and the bounds on the anonymous growth rates are reported by country in Tables 4 and 5. 24 The most striking feature of Figure 6 and Table 4 is the magnitude of the gap between the nonanonymous and anonymous growth rates. For example, in the WID database, the mean difference between the midpoint of our estimated range for the non-anonymous growth rate, and

Conclusions
This paper demonstrates the feasibility of approximating individual-level income distribution dynamics when the researcher only has access to time-series data on aggregate moments of the income distribution such as the mean and the Gini coefficient or top income shares. Our key identifying assumption is that individual incomes follow an autoregressive lognormal process with individual fixed effects. We show that the limited information embodied in the time-series of these aggregate moments is sufficient to place bounds on the extent of mobility in the income distribution, even though we do not observe income dynamics at the individual level. An empirical application using data from the PSID confirms that these bounds generally contain the point estimates that are obtained using the record-level panel data, and moreover are reasonably tight. Encouraged by these findings, we apply our methodology to two large cross-country datasets, namely the WID (including mostly high income countries) and the World Bank's PovcalNet (including mostly developing countries). Some of the cross-country patterns we observe in estimates of mobility seem quite plausible given our priors. For example, among the high income countries, the Scandinavian countries and much of Europe show relatively high levels of income persistence, while the United States, Singapore and Taiwan rank among the countries with low levels of income persistence. When comparing estimates between the WID and PovcalNet, our estimates suggest 25 that income persistence is lower in the developing world. However, this comparison is difficult to interpret given the differences between the two databases. Specifically, a possible interpretation is that our OLS estimates of , as well as the MSE-minimizing combination of the OLS and bias-corrected OLS estimates, have greater small-sample downward bias in the PovcalNet sample where the available time series is much shorter than in the WID sample.
When we correlate our estimates of income persistence with per capita GDP across countries, we find a negative correlation within the WID and PovcalNet samples -higher persistence is associated with lower incomes. The reason for this is not obvious, and it is unlikely to be mechanical. Note that if countries were identical in all parameters but the income persistence parameter, then lower persistence would be associated with lower average income (since lower persistence implies both a lower mean and lower cross-sectional variance of log income in the steady state, and therefore lower mean income given our lognormality assumption). This seems to suggest that countries with lower income persistence differ from countries with higher persistence in other ways that benefit their income growth.
Our approach to approximating individual-level distribution dynamics with aggregate data makes it possible for the first time to compare anonymous and non-anonymous growth rates in a wide cross section of countries. We show that the anonymous growth rate of the bottom 40 percent (corresponding to the World Bank's definition of "shared prosperity") understates the growth rate of those initially in the bottom 40 percent by 3 to 4 percentage points per year. This implies that the anonymous growth rate is a poor approximation to the success of those initially in the bottom of the income distribution, and is not a good tool for tracking the effects of policy interventions that were targeted to the initially-poor. The exact opposite holds true when tracking growth at the top end of the income distribution. Growth of top incomes will be larger when measured anonymously. Our estimates suggest that for the top 10% the difference is as large as 10 percentage points in annual terms. Thus, tracking anonymous top income shares substantially overstates the expected income growth rates realized by the average individual in the top end of the distribution.
Approximating income distribution dynamics, under the assumption that individual incomes follow a lognormal autoregressive process, can also be applied to an inter-generational context. The advantage of working with analytical approximations in this case is that it provides a means of rationalizing some of the empirical regularities that have been reported in recent studies on inter-generational mobility. One prominent example of such a regularity is that higher levels of inter-generational mobility are more likely to be observed in countries with lower levels of income inequality (see e.g. Corak, 2013; jointly determine mobility. This in turn allows us to verify how much growth it would take to offset a rise in inequality, and use this framework to interpret the results of a recent study by Chetty et al. (2017) which reports a dramatic decline in absolute mobility in the United States over the last 60 years. In a separate study we also use our lognormal framework and empirical bounds on mobility to investigate the welfare consequences of cross-country differences in mobility, using the welfare metrics suggested by Atkinson and Bourguignon (1982) and Gottschalk and Spolaore (2002).

Panel B: Estimates of Mobility Parameter
Notes: The first two rows of panel A reports estimates of based on the dynamics of the mean of log income (Equation (8)) and the variance of log income (Equation (9)). OLS refers to ordinary least squares and BC-OLS refers to small-sample biascorrected estimates using the procedure in Andrews (   Notes: The first two columns report our preferred estimate of and its standard error. This estimate is an MSE-minimizing weighted average of the estimates based on the evolution over time of the mean and variance of log income. The underlying estimates and MSE-minimizing weights are reported in Appendix Table B1. The third and fourth columns report the upper bound on the variance of the individual effect ( ), and its share in the the steady-state variance of log income. The remaining columns report the lower bound on (i.e. the preferred estimate of from Column 1), and the average over time of the upper bound on .  Notes: The first two columns report our preferred estimate of and its standard error. This estimate is an MSE-minimizing weighted average of the estimates based on the evolution over time of the mean and variance of log income. The underlying estimates and MSE-minimizing weights are reported in Appendix Table B2. The third and fourth columns report the upper bound on the variance of the individual effect ( ), and its share in the the steady-state variance of log income. The remaining columns report the lower bound on (i.e. the preferred estimate of from Column 1), and the average over time of the upper bound on .

4: Anonymous and Non-Anonymous Growth Rates of the Bottom 40 and Top 10 Percent from the World Income and Wealth Database
Notes: This table reports anonymous growth rates and bounds on non-anonymous growth rates for the bottom 40 percent (left panel) and top 10 percent (right panel). Growth rates are calculated as annual averages over the most recently-available ten years of data for each country. For details on the time period covered by the available data see Appendix Table B1. Within each panel, the first column reports the anonymous growth rate, and the second and third columns report the lower and upper bounds on the non-anonymous growth rate. The fourth column in each panel reports the difference between the midpoint of the bounds on the non-anonymous growth rate and the anonymous growth rate.  . Growth rates are calculated as annual averages over the most recently-available ten years of data for each country. For details on the time period covered by the available data see Appendix Table B2. Within each panel, the first column reports the anonymous growth rate, and the second and third columns report the lower and upper bounds on the non-anonymous growth rate. The fourth column in each panel reports the difference between the midpoint of the bounds on the non-anonymous growth rate and the anonymous growth rate.

Figure 1: Estimates of Mobility from the PSID Based on Micro and Aggregate Data
Notes: This figure reports estimates of the mobility parameter based on aggregate moments and true panel micro data, using annual rounds of the Panel Study of Income Dynamics (PSID) over the period 1977-1997. In both panels, the grey-shaded area is the range between the lower and upper-bound estimates of based on the evolution over time of the crosssectional mean and variance of log income, and the dashed line marks the mid-point of the grey-shaded range. The solid lines report three alternative estimates of mobility based on micro panel data. All three are based on a series of cross-sectional OLS regressions of log income on lagged log income at the household level. The "Baseline" estimates use data for all households in each year. The "Balanced Panel" estimates use only the subset of households available in all 21 years. The "Cook's-D" estimates drop households in the top 1 percent of the distribution of Cook's Distance statistic (a measure of influence) in the baseline OLS estimates.

Mean of Log Income Variance of Log Income
Notes: This figure plots the evolution over time of the mean (left column) and standard deviation (right column) of log income for selected countries in our WID sample. The vertical red lines delineate the subperiods corresponding to trend breaks, and the fitted lines show the estimated linear trends by subperiod. The first period for each country corresponds to all available pre-World War II data. The year of the post-war trend break is selected country-by-country and separately for the mean and variance of log income, based on the supremum of the Wald statistics for the null hypothesis that the pre-and post-break trends are the same.

Figure 3: Cross-Country Differences in Estimated Mobility
Autoregressive Parameter

Mid-Point of Range for
Notes: This figure plots the point estimate of (top panel) and the average over time of the mid-point between the upper and lower bounds on (bottom panel) on the vertical axis, against the logarithm of real GDP per capita in 2000 on the horizontal axis. In each panel, the red circles correspond to developing countries, and consist of the countries in our PovcalNet sample, as well as China, Mauritius, and India from the WID sample. The blue squres correspond to advanced economies in the WID sample.

Figure 4: Estimated Mobility Trends in the United States
Notes: This figure reports the evolution over time of the estimated bounds on the mobility parameter for the United States. The grey-shaded region contains the range between the lower and upper bounds on , and the dashed line indicates the midpoint of the range. The two shorter superimposed time series are the estimates of actual mobility based on the PSID microdata, as described in Section 4.

Figure 5: Anonymous and Non-Anonymous Growth Incidence Curves, PSID
Notes: This figure plots growth incidence curves for the US, based on the PSID microdata and using the bounds on mobility based on aggregate data. The horizontal axis corresponds to percentiles of the income distribution and the vertical axis plots the average annual growth rate over the period 1987-1997 at each percentile. The dashed line reports the anonymous growth incidence curve. The two solid lines report two versions of the actual non-anonymous growth incidence curve based on the PSID micro data. The thin line is a local polynomial smoothed average of the growth rates of households at each percentile of the initial income distribution. The bold line is a lognormal approximation to the non-anonymous growth incidence curve, i.e. Equation (11), using the estimate of obtained using the PSID microdata. The shaded region corresponds to the region between the lower and upper bounds on the non-anonymous growth incidence curve based only on aggregate data.

Estimates Based on PovcalNet Data
Notes: This figure plots average annual anonymous growth rates (on the horizontal axis) against average annual nonanonymous growth rates (on the vertical axis). The solid diagonal line represents the 45-degree line. The top panel reports results from the WID sample, while the bottom panel reports results from the PovcalNet sample. The blue triangles above (below the 45-degree line represent the mid-point estimate of the non-anonymous growth rate of the bottom 40 percent (top 10 percent), over the most recent 10-year period available for each country. The vertical lines represent the range from the lower to the upper-bound estimates of the non-anonymous growth rate.

Appendix A: Proofs
In the main text we assumed for notational convenience that survey data are available in consecutive periods and − 1. In reality however, survey data often are available at irregular frequencies that differ over time and across countries, and in the empirical part of the paper we work with such irregularly-spaced data. In this appendix we provide proofs for the case of irregularly-spaced data, i.e. for two surveys available in periods and − . The propositions as stated in the main text obtain for the special case of = 1.

Preliminaries
Iterating the data generating process in Equation (1) backwards for periods results in: where ≡ ∑ and ̃ ≡ ∑ . For the case = − and = − , and using the assumption on initial income in Assumption A1, Equation (15) becomes:

Proof of Proposition 1:
Given Equation (16) which states that is a linear combination of , … , and , and Assumption A1 that these shocks are jointly normally distributed, it follows that is normally distributed for all − ≥ 0. Equation (16)

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Setting = 1 and adopting the more compact notational convention that , = gives = + as in the main text.

Proof of Proposition 2
From the irregularly-spaced version of Proposition 1 we have:

Proof of Proposition 3
Taking unconditional expectations of both sides of Equation (15) gives the following irregularlyspaced analog of Equation (6).
Taking unconditional variances of both sides of Equation (15) gives the following irregularly-spaced analog of Equation (7): For the particular case of average incomes in the bottom percent this expression simplifies to while for the particular case of average incomes in the top percent this expression simplifies to Finally, the fact that ( , ) can also be represented as: ( , ) = ( ( , , )), where ( , , ) satisfies: ≤ ( , , ) ≤ , follows directly from the fact that ( ) is a monotonic function of z.

Appendix B: Details of OLS, Bias-Corrected, and MSE-Minimizing Estimates of
This appendix described our approach to estimating the autoregressive coefficient in log individual incomes using aggregate data. Let , denote the OLS estimator of based on Equation (8), and let , denote the OLS estimator of based on Equation (9). Given that the available time series is short, these estimators will exhibit downwards finite-sample bias. We therefore also generate corresponding bias-corrected estimators , and , using the procedure suggested in Andrews (1993). At the core of this procedure is the fact that the distribution of the OLS estimator is exclusively a function of the true autoregressive parameter and the sample size, and thus is independent of the parameters that describe the distribution of the error term and the time-trend. We refer the interested reader to Andrews (1993) for a proof. We take advantage of this result, as suggested by Andrews (1993), by computing the median bias of the OLS estimator as a function of for each sample size separately (using numerical simulations), and then inverting this function to obtain a median-unbiased estimator for .
Comparing the OLS and bias-corrected estimators highlights a tradeoff between bias and variance: while the OLS estimator is substantially downward biased, the bias-corrected estimator is much less precisely estimated. We address this tradeoff by defining the following two weighted averages of the OLS and bias-corrected estimators:

Appendix Table B1: Estimation Details for WID Sample
Notes: This table reports details of the estimates of in the WID sample. The first four columns indicate the first and last years of the sample for each country in the post-World War II period, together with the estimated trend break dates for the mean and variance of log income. Note that three countries shown in Figure 2 also have data for a pre-World War II subperiod. The next two sets of four columns report the OLS, bias-corrected OLS, the MSE-minimizing weighted average of the two, and the MSE-minimizing weight on the OLS equation, for dynamics of the mean and variance of log income, respectively. The last two columns report the MSE-minimizing weighted average of the estimates of from the two equations, together with the MSE-minimizing weight on the estimate from the equation for the dynamics of the mean of log income. Standard errors are reported below the coefficient estimates for the OLS, bias-corrected OLS, and MSE-minimizing weighted average estimators. Notes: Notes: This table reports details of the estimates of in the WID sample. The first three columns indicate the first and last years of the sample for each country, and the frequency of the surveys (every 1, 2 or 3 years). The next two sets of four columns report the OLS, bias-corrected OLS, the MSE-minimizing weighted average of the two, and the MSE-minimizing weight on the OLS equation, for dynamics of the mean and variance of log income, respectively. The last two columns report the MSE-minimizing weighted average of the estimates of from the two equations, together with the MSE-minimizing weight on the estimate from the equation for the dynamics of the mean of log income. Standard errors are reported below the coefficient estimates for the OLS, bias-corrected OLS, and MSE-minimizing weighted average estimators. Missing OLS estimates for the equation for the variance of log income correspond to cases where the OLS estimates are negative. In these cases the estimates are based on the bias-corrected estimator only.