Life-cycle variation in the association between current and lifetime earnings

TL;DR: This article found that the relationship between current and lifetime earnings departs substantially from the textbook errors-in-variables model in ways that vary systematically over the life cycle, which can enable more appropriate analysis of, and correction for, errors in variance bias in any research that uses current earnings to proxy for lifetime earnings.

Abstract: Researchers in a variety of important economic literatures have assumed that current income variables as proxies for lifetime income variables follow the textbook errors-in-variables model. In our analysis of Social Security records containing nearly career-long earnings histories for the Health and Retirement Study sample, we find that the relationship between current and lifetime earnings departs substantially from the textbook model in ways that vary systematically over the life cycle. Our results can enable more appropriate analysis of, and correction for, errors-in-variables bias in any research that uses current earnings to proxy for lifetime earnings. (JEL D31, D91)

Summary

I. Introduction

• Many influential economic studies have recognized that the use of current income as a proxy for long-run income can generate important errors-in-variables biases.
• These data provide nearly career-long earnings histories, which are based on relatively accurate administrative data and pertain to a broadly representative national sample.
• In section II, the authors develop simple models to illustrate some important aspects of the association between current and lifetime earnings and to demonstrate the implications for errors-in-variables biases in applied econometric research.

II. Models

• Throughout this section, the authors will suppress intercepts by expressing all variables as deviations from their population means.
• One familiar implication of that restriction is that, if it y proxies for i y as the dependent variable in a linear regression equation, ordinary least squares (OLS) estimation of that regression equation consistently estimates the equation's slope coefficients.
• These oft-used results no longer apply if the textbook errors-in-variables model incorrectly characterizes the relationship between current and lifetime income.
• In part A of this section, the authors explain their reasons for suspecting that the slope coefficients in regressions of current income variables on lifetime variables vary systematically over the life cycle and do not generally equal 1.
• In part B, the authors show how such departures from the textbook model alter the standard results on errors-in-variables bias.

A. Life-cycle variation

• Several fragments of evidence suggest that the association between current and lifetime income variables varies over the life cycle.
• In his words, "the correlations are quite low -and in some cases even negative -up to around 25 years of age and are rather high after 35 years of age.
• If son's log annual earnings as a proxy for the dependent variable obeyed the textbook errors-in-variables model, the estimated intergenerational elasticity would have the same probability limit regardless of the age at which the son's earnings were observed.
• The main thing to note about this result is that, contrary to the textbook errors-invariables model, t λ generally does not equal 1.
• If the worker with higher lifetime earnings has a steeper earnings trajectory, then the current earnings gap between the two workers early in their careers tends to understate their gap in lifetime earnings (and could even have the opposite sign).

B. Implications for errors-in-variables biases

• In accordance with the discussion in the preceding subsection, the authors do not assume the textbook errors-invariables model in equation ( 1).
• See section 4 of Angrist and Krueger (1999) for an excellent overview of errors in variables, including non-classical measurement error.
• First, with plausible departures from the textbook errors-in-variables assumptions, the familiar textbook results about OLS and IV estimation are overturned.
• Second, some of the estimation inconsistencies from using log annual earnings as a proxy for log lifetime earnings can be summarized with just two simple parameters: the slope coefficient in the "forward regression" of log annual earnings on log lifetime earnings and the slope coefficient in the "reverse regression" of log lifetime earnings on log annual earnings.
• In section IV, the authors will estimate those two parameters and examine how they vary over the life cycle.

III. Data and Methods

• A. Data Most U.S. studies of the relationship between current and lifetime income variables have been based on longitudinal survey data from only a limited portion of the respondents' careers.
• For the 821 men in their sample, table 1 displays the median observed earnings, the percentage in the sample with zero earnings, the taxable limit, and the percentage with earnings at the taxable limit for each year from 1951 to 1991.
• As their earnings grow over their careers, however, the taxable limit becomes more constraining, especially in the years when the taxable limit is low relative to the general earnings distribution.
• Their observation of earnings usually has been limited to relatively short segments of the life cycle.
• Because of the right-censorship, however, the authors are forced instead to estimate the joint distribution in a way that imputes the censored right tails of the annual earnings distributions.

B. Econometric methods

• As explained above in section II.B, their ultimate goal is to summarize the association between annual and lifetime earnings in terms of two types of parameters.
• Because of the censorship of the Social Security earnings data at the taxable limit, however, the authors cannot observe the exact value of annual earnings in the cases where earnings are right-censored and furthermore, in those cases, they also cannot compute the present value of lifetime earnings.
• Second, drawing from that estimated joint distribution, the authors generate a simulated sample of uncensored earnings histories, for which they can calculate the present discounted value of lifetime earnings.
• To implement the simulation, the authors need the estimated autocovariance matrix to be positive semi-definite (as the true one must be).
• Finally, for this sample of 4,000 individuals, the authors apply OLS to the regression of each year's log annual earnings on the log of the present value of lifetime earnings, and thereby produce a t λ ˆ for each year from 1951 to 1991.

IV. Empirical Results

• In the first step of their estimation procedure, the Tobit analysis described above results in a 41 41× estimated autocovariance matrix for log annual earnings from 1951 to 1991.
• B showed that theoretically the errors-in-variables bias could be either an attenuation bias or an amplification bias.
• The authors empirical results, however, confirm the conventional presumption that using current earnings to proxy for lifetime earnings as a regressor induces an attenuation bias.
• Because most previous analyses of earnings dynamics, however, have excluded observations of zero earnings, the authors supplement their main analysis with another that excludes the zeros, codes positive earnings less than $50 as $25, and estimates one-limit Tobits with only right-censorship.

V. Summary and Discussion

• All of their analyses tell the same story: contrary to the textbook errors-in-variables model usually assumed in applied research, the slope coefficient in the regression of log current earnings on log annual earnings varies systematically over the life cycle and is not generally equal to 1.
• The literature, however, has given much less attention to the left-side measurement error from using short-run proxies for son's lifetime earnings.
• An important implication is that many estimates of the intergenerational earnings elasticity have been subject to substantial attenuation inconsistency from left-side measurement error in addition to the well-known inconsistency from right-side measurement error.
• The only departure from the estimation procedure used in their main analysis is that, in the bootstrap replications, the authors use a different method for imposing positive semi-definiteness of the autocovariance matrix.
• This change greatly reduces the computational time, and the authors have verified that the resulting positive semi-definite matrix is very similar to what would be obtained using the previous method.

D. Estimates for five-year averages of log earnings

• Many intergenerational earnings mobility studies have attempted to reduce errorsin-variables bias by averaging father's log earnings over multiple years.
• To explore the extent to which such averaging reduces bias, in figure A1 the authors repeat the analysis in figure 2 except that the new estimates of t θ are for five-year averages of log annual earnings, rather than for single years.
• The observation plotted for age 30 is based on a five-year average for ages 28-32.
• Nevertheless, although the estimates of t θ usually exceed 0.7 over a wide age range from 26 to 46, they never exceed 0.8 by much.
• This finding strongly supports the conclusion of Mazumder (2001 Mazumder ( , 2005) ) that even estimates based on five-year averages of the earnings variable for fathers are subject to a substantial errors-in-variables bias.

Figures

Table 1. Descriptive Statistics for Nominal Annual Earnings Covered by Social Security

Figure 2. Main Estimates of tλ and tθ Estimates of tλ

Figure 1. Illustrative Example of Log Annual Earnings and Log Annuitized Lifetime Earnings

Table 2. Estimated Autocorrelations in Log Annual Earnings, 1975-1984

Figure 3. Alternative Estimates of tλ and tθ Estimates of tλ

Full text

NBER WORKING PAPER SERIES
LIFE-CYCLE VARIATION IN THE ASSOCIATION
BETWEEN CURRENT AND LIFETIME EARNINGS
Steven Haider
Gary Solon
Working Paper 11943
http://www.nber.org/papers/w11943
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
January 2006
The authors gratefully acknowledge grant support from the National Institute on Aging (2-P01 AG 10179).
They also are grateful for advice from the American Economic Review editor and referees, John Bound,
Charlie Brown, Art Goldberger, Nathan Grawe, Jacob Klerman, Luigi Pistaferri, Matthew Shapiro, Mel
Stephens, Bob Willis, Jeff Wooldridge, and seminar participants at the University of Michigan, American
University, the University of California at Berkeley, the University of California at Davis, Harvard’s
Kennedy School, the National Bureau of Economic Research, Ohio State University, Penn’s Wharton School,
the Society of Labor Economists, the University of Toronto, and Western Michigan University. The views
expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau
of Economic Research.
©2006 by Steven Haider and Gary Solon. All rights reserved. Short sections of text, not to exceed two
paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given
to the source.

Life-Cycle Variation in the Association between Current and Lifetime Earnings
Steven Haider and Gary Solon
NBER Working Paper No. 11943
January 2006
JEL No. D3, J3
ABSTRACT
Researchers in a variety of important economic literatures have assumed that current income
variables as proxies for lifetime income variables follow the textbook errors-in-variables model. In
an analysis of Social Security records containing nearly career-long earnings histories for the Health
and Retirement Study sample, we find that the relationship between current and lifetime earnings
departs substantially from the textbook model in ways that vary systematically over the life cycle.
Our results can enable more appropriate analysis of and correction for errors-in-variables bias in a
wide range of research that uses current earnings to proxy for lifetime earnings.
Steven Haider
Department of Economics
Michigan State University
East Lansing, MI 48824
haider@msu.edu
Gary Solon
Department of Economics
University of Michigan
Ann Arbor, MI 48109
and NBER
gsolon@umich.edu

Life-Cycle Variation in the Association between Current and Lifetime Earnings
I. Introduction
In the year 2003 alone, the American Economic Review’s refereed issues
contained 14 articles reporting regression analyses involving individual or family income
variables, and the May Proceedings issue contained almost that many again. In some
cases, the income variables were dependent variables; in others, they were regressors
used to explain dependent variables ranging from child health in the United States to
borrowing and lending behavior in Ghana. Without exception, the measured income
variables were short-term values even though, in most cases, it appeared that the relevant
economic construct was a longer-term value.
Many influential economic studies have recognized that the use of current income
as a proxy for long-run income can generate important errors-in-variables biases.
Perhaps the most famous examples are the seminal studies by Modigliani and Brumberg
(1954) and Friedman (1957), which analyzed the properties of consumption functions
estimated with current rather than permanent income variables as the regressors. Another
instance is the literature (e.g., Lillard, 1977) suggesting that inequality as measured in
cross-sections of annual earnings overstates the inequality in lifetime earnings. A recent
offshoot of that literature – exemplified by Gottschalk and Moffitt (1994), Haider (2001),
and Baker and Solon (2003) – has attempted to partition the upward trend in earnings
inequality into persistent and transitory components. Still another recent example is the
burgeoning literature on intergenerational income mobility (surveyed in Solon, 1999),
which has found that the association between parents’ and children’s long-run income is

2
susceptible to dramatic underestimation when current income variables are used as
proxies for long-run income.
Nevertheless, applied researchers often ignore the distinction between current and
long-run income. Most researchers who do attend to the issue assume the textbook
errors-in-variables model and impute the noise-to-signal ratio by estimating restrictive
models of income dynamics on the basis of short panels of income data spanning only a
segment of the life cycle.
1
In this paper, we reconsider the appropriateness of the
textbook errors-in-variables model, and we find that it does not accurately characterize
current earnings as a proxy for lifetime earnings. Thanks to a remarkable new data set,
we are able to generate detailed evidence on the association between current and lifetime
earnings, including its evolution over the life cycle, without having to resort to an
arbitrary specification of the earnings dynamics process.
Our empirical analysis uses the 1951-1991 Social Security earnings histories of
the members of the Health and Retirement Study sample. Despite some limitations
discussed in section III, these data provide nearly career-long earnings histories, which
are based on relatively accurate administrative data and pertain to a broadly
representative national sample. In section II, we develop simple models to illustrate
some important aspects of the association between current and lifetime earnings and to
demonstrate the implications for errors-in-variables biases in applied econometric
research. In section III, we describe the data set and our econometric methods. In
section IV, we present our evidence on the connections between annual and lifetime
earnings. Section V summarizes our findings and illustrates their usefulness with a brief
application to intergenerational earnings mobility.
1
See Mazumder (2001) for a relatively sophisticated recent example.

3
II. Models
Following Friedman (1957), most analyses of current income variables as proxies
for unobserved lifetime income variables have adopted the textbook errors-in-variables
model
(1)
it i it
y y v
= +
where
it
y is a current income variable, such as log annual earnings, observed for
individual
i
in period
t
;
i
y is a long-run income variable, such as the log of the present
discounted value of lifetime earnings; and
it
v
, the measurement error in
it
y as a proxy for
i
y
, is assumed to be uncorrelated with
i
y (and each of its determinants). Often, the
current income variable
it
y has been adjusted for stage of life cycle with a regression on
a polynomial in age or experience or by subtracting out the cohort mean. Throughout this
section, we will suppress intercepts by expressing all variables as deviations from their
population means.
The textbook errors-in-variables model in equation (1) is effectively a regression
model that assumes the slope coefficient in the regression of
it
y on
i
y equals 1. One
familiar implication of that restriction is that, if
it
y proxies for
i
y as the dependent
variable in a linear regression equation, ordinary least squares (OLS) estimation of that
regression equation consistently estimates the equation’s slope coefficients. Another
well-known implication is that, if
it
y proxies for
i
y as the sole explanatory variable in a
simple regression equation, the probability limit of the OLS estimator of the equation’s
slope coefficient equals the true coefficient times an attenuation factor equal to
( )/[ ( ) ( )]
i i it
Var y Var y Var v
+ .