Intergenerational Income Elasticities,Instrumental Variable Estimation,and Bracketing Strategies

Abstract

The fact that the intergenerational income elasticity (IGE)—the workhorse measure of economic mobility—is defined in terms of the geometric mean of children’s income generates serious methodological problems. This has led to a call to replace it with the IGE of the expectation, which requires developing the methodological knowledge necessary to estimate the latter with short-run measures of income. This article contributes to this aim. The author advances a “bracketing strategy” for the set estimation of the IGE of the expectation that is equivalent to that used to set estimate (rather than point estimate) the conventional IGE with estimates obtained with the ordinary least squares and instrumental variable (IV) estimators. The proposed bracketing strategy couples estimates generated with the Poisson pseudo–maximum likelihood estimator and a generalized method of moments IV estimator of the Poisson or exponential regression model. The author develops a generalized error-in-variables model for the IV estimation of the IGE of the expectation and compares it with the corresponding model underlying the IV estimation of the conventional IGE. By considering both bracketing strategies from the perspective of the partial-identification approach to inference, the author specifies how to construct confidence intervals for the IGEs, in particular when the upper bound is estimated more than once with different sets of instruments. Finally, using data from the Panel Study of Income Dynamics, the author shows that the bracketing strategies work as expected and assesses the information they generate and how this information varies across instruments and short-run measures of parental income. Three computer programs made available as companions to the article make the set estimation of IGEs, and statistical inference, very simple endeavors.

Keywords

intergenerational economic mobility intergenerational elasticity of expected income generalized error-in-variables model measurement error instrumental variables partial identification set estimation intersection bounds

1. Introduction

The research on economic mobility and the transmission of economic advantages across generations of the past four decades has relied heavily on the intergenerational income elasticity (IGE). The IGE has been extensively estimated, both to assess levels of income and earnings mobility within countries (for reviews, see Corak 2006; Mitnik, Bryant, and Grusky 2018:9–18; Solon 1999:1778–84) and to conduct comparative analyses of income and earnings mobility across geographic areas, demographic groups, and time periods (e.g., Aaronson and Mazumder 2008; Björklund and Jäntti 2000; Bloome and Western 2011; Chadwick and Solon 2002; Hertz 2005, 2007; Mayer and Lopoo 2008). The IGE has also been used, among many other purposes, to theoretically model and empirically study the impact of educational policies and labor-market and political institutions on the intergenerational persistence of economic status (e.g., Bratsberg et al. 2007; Ichino, Karabarbounis, and Moretti 2011; Landersø and Heckman 2017; Solon 2004). The well-known Great Gatsby curve, which shows a negative relationship between income inequality and economic mobility across countries, is also based on the IGE (e.g., Corak 2013). As Mitnik and Grusky (this volume, pp. 47–95) point out, “there is simply no other measure of economic mobility that comes close to the IGE in popularity” (p. 48).

Sociologists working in the field of social stratification and mobility have long been interested in the transmission of income advantages from parents to children.¹ Thus, it is unsurprising that some of the early sociological research on social stratification based on the Wisconsin Longitudinal Study (Hauser 1979; Hauser et al. 1975; Tsai 1983) produced or underpinned some of the very first estimates of the earnings IGE in the United States (see Becker and Tomes 1986, Table 1). Certainly, this line of sociological work was later overshadowed by research focusing on other indices of advantage—educational achievement, occupational status, and social class—but things have changed markedly in more recent times, with sociologists again paying extensive attention to income advantages.

Indeed, in the past 15 years or so, and focusing exclusively on research in which the earnings or income IGE plays a central role, sociologists have (1) paid close attention to empirical studies based on the IGE when assessing levels of intergenerational mobility and equality of opportunity in the United States and other countries (Beller and Hout 2006; Fox, Torche, and Waldfogel 2016; Torche 2015a, 2015b; Winship 2017), (2) explicitly recognized that those studies belong squarely in the field of social stratification (Grusky 2008, 2014), and (3) actively contributed to the empirical literature based on the IGE and to the methodological literature involving the IGE.² For instance, among the empirical contributions, Mayer and Lopoo (2004, 2005) and Bloome and Western (2011) used the IGE to study trends in income mobility in the United States by gender and race; Bloome (2015) relied on the IGE to examine the relationship between cross-sectional economic inequality and mobility across generations; Torche (2011) and Zhou (2019) used the IGE to study the equalizing power of a college degree in the intergenerational context; Mitnik, Bryant, and Weber (2019) used the IGE and tax data to show that, in the United States, at least half of economic inequality among parents is passed on to their children; and Torche (2016), Gregg et al. (2017), Mayer and Lopoo (2008), and Esping-Andersen (2015) used the IGE to study the role of education, government spending, and welfare state policies in the transmission of economic advantages across generations.³ Similarly, recent methodological contributions by sociologists involving the IGE include those of Jencks and Tach (2006); Breen, Mood, and Jonsson (2015); Mitnik and Grusky (this volume); Torche and Corvalan (2018); Sakamoto and Wang (2019); and Winship (2017).

Mitnik and Grusky’s (this volume) methodological arguments provided the motivation for this article. Although the IGE has been the workhorse measure of economic (i.e., earnings, income) mobility for a long time, Mitnik and Grusky showed that this elasticity has been misinterpreted. The IGE has been construed as pertaining to the expectation of children’s income conditional on their parents’ income—as apparent, for instance, in its oft-invoked interpretation as a measure of regression to the (arithmetic) mean. However, it pertains to the conditional geometric mean of the children’s income.⁴ This not only makes all conventional interpretations of the IGE invalid, but it also generates serious methodological problems.

At their root, these problems are the result of a very simple fact: The geometric mean is undefined for variables that include zero in their support. As Mitnik and Grusky (this volume) showed, this (1) makes it impossible to determine the extent to which parental income advantages are transmitted through the labor market among women (as many women have zero earnings) and (2) greatly hinders research on the role marriage plays in generating the observed levels of intergenerational persistence in family income (as many people remain single or have nonworking spouses and therefore cannot be included in analyses examining the relationship between people’s parental income and the income contributed by their spouses). As a result, the study of gender and marriage dynamics in intergenerational processes has been badly hampered.

Equally important, Mitnik and Grusky (this volume) showed that, as a consequence of mobility scholars’ expedient of dropping children with zero earnings from samples (to address what is perceived as the problem of the logarithm of zero being undefined), estimation of earnings IGEs with short-run proxy earnings measures is almost certainly affected by substantial selection biases. This makes the widespread use of the IGE of men’s individual earnings as an index of economic persistence and mobility in a country a rather problematic practice.

Mitnik and Grusky (this volume) argued that these conceptual and methodological problems can be solved in a straightforward manner by simply replacing the IGE of the geometric mean—the de facto estimated IGE—with the IGE of the expectation—the IGE that mobility scholars thought they were estimating—as the workhorse intergenerational elasticity. Mitnik and Grusky called for effectuating such replacement. This requires, however, that the methodological knowledge necessary to estimate the IGE of the expectation be made available.

In effect, research on intergenerational mobility based on IGEs needs to address methodological difficulties that are (partly) different from those confronted when advantage is indexed by educational achievement, occupational status, or social class. The origin of those difficulties is that IGEs are defined in terms of the long-run incomes or earnings of parents and children, but they are almost always estimated with short-run proxy measures of income or earnings, that is, with variables affected by substantial measurement error. Mobility scholars have been centrally concerned with the biases that may arise as a result and the methodological strategies that may be used to avoid them. With respect to the conventional IGE, two central achievements in this regard are (1) the generalized error-in-variables model for the estimation of that elasticity by ordinary least squares (OLS), or GEiV model (Haider and Solon 2006; see also Nybom and Stuhler 2016) and (2) the analysis of its instrumental variable (IV) estimation with invalid instruments (Solon 1992, Appendix; see also Nybom and Stuhler 2011:12–14). As I explain in detail later, these methodological analyses entail that if various empirical assumptions hold, then the probability limit of the OLS estimator is affected by attenuation bias and the probability limit of the IV estimator is affected by amplification bias and, as Solon (1992:400) first argued, “the probability limits of the two estimators bracket the true value” of the conventional IGE. This means that OLS and IV estimates can be combined to “bracket” or, to use the terminology typically used in more recent literatures (e.g., Manski 2003, 2008), to set estimate, rather than point estimate, the conventional IGE. In other words, instead of producing a single IGE value as an estimate, a “bracketing strategy” produces a set or range of values deemed to contain the IGE.

Mitnik (2019) contributed part of the methodological knowledge necessary to estimate the IGE of the expectation with short-run income variables by advancing a generalized error-in-variables model for the estimation of this IGE with the Poisson pseudo–maximum likelihood (PPML) estimator (Santos Silva and Tenreyro 2006). Here, I take on the complementary task of developing a generalized error-in-variables model for the IV estimation of the IGE of the expectation. I focus on the additive-error version of the generalized method of moments (GMM) IV estimator of the Poisson or exponential regression model (Mullahy 1997; Windmeijer and Santos Silva 1997). I show that, under empirical assumptions fully equivalent to those made for the IV estimation of the conventional IGE, we can expect the GMM IV estimator to produce upward-biased estimates of the IGE of the expectation. By combining this result with Mitnik’s (2019) result that the PPML estimation of the IGE of the expectation with short-run income measures is affected by attenuation bias, I further show that a bracketing strategy equivalent to that used with the conventional IGE can be used to set estimate the IGE of the expectation. This strategy, and the generalized error-in-variables model for IV estimation on which it relies (which is of clear independent interest), are the first two contributions of this article.

Although bracketing strategies generate set rather than point estimates of IGEs, it is nevertheless possible to construct confidence intervals for those IGEs. This can be achieved by considering those strategies from the perspective of the partial-identification approach to inference, a very active field of statistical and econometric research in recent times (for reviews, see Canay and Shaikh 2017; Tamer 2010). I explain how to construct confidence intervals for the partially identified IGEs (which are the confidence intervals of interest), and how they differ from confidence intervals for the “identified sets” (the ranges of values that lie between the probability limits of the bound or “bracketing” estimators). I address, in particular, the issue of how to construct confidence intervals for the IGEs when the upper bound is estimated repeatedly with different sets of instruments. The third contribution of the article is therefore to show how one central aspect of statistical inference may still be carried out when using the bracketing strategies to estimate IGEs, and to connect the literature on IGEs to the literature on partial identification and set estimation (e.g., Ho and Rosen 2017; Manski 2008).

Although this does not seem to have been done before, when data sets with the necessary information are available it is possible to empirically evaluate whether a bracketing strategy works as expected. Here, I rely on a U.S. sample from the Panel Study of Income Dynamics (PSID) to examine the performance of both bracketing strategies. Using various instruments, I show that the set estimates produced with the bracketing estimators do bound the corresponding long-run IGE estimates when they are expected to do so. I also assess how the information supplied by the set estimates varies across instruments and short-run measures of parental income, and how informative the set estimates based on the minimum estimates of upper bounds across instruments are. These empirical analyses are the fourth contribution of the article.

The set estimation of IGEs—both the conventional IGE and the IGE of the expectation—can be carried out very easily with the statistical packages social scientists typically use. However, the computation of confidence intervals for the partially identified IGEs involves some difficulties, especially when it needs to account for repeated estimation of an upper bound with different sets of instruments. For this reason, I have made available with this article three Stata programs that make the set estimation of both IGEs, and the computation of confidence intervals, a very simple endeavor. This is the fifth contribution of the article.

The structure of the rest of the article is as follows. I first explain why the conventional IGE pertains to the conditional geometric mean of children’s income rather than to its conditional expectation, present the generalized error-in-variables models relevant for the set estimation of the conventional IGE, and fully specify the empirical conditions under which the bracketing strategy previously discussed in the literature works. Next, I introduce the IGE of the expectation, present the generalized error-in-variables model for its estimation with the PPML estimator advanced by Mitnik (2019), develop the new generalized error-in-variables model for its IV estimation contributed by this article, and fully specify the empirical conditions under which the new bracketing strategy I propose here works. After that, I discuss statistical inference under the partial-identification approach and present the results of the empirical analyses. The last section draws the article’s main conclusions.

2. The Conventional IGE and Its Interpretation

The standard population regression function (PRF) posited in the mobility literature, which assumes the IGE is constant across levels of parental income, is

E (\ln Y | x) = β_{0} + β_{1} \ln x,

(1)

where $Y$ is the child’s long-run income or earnings, X is long-run parental income or father’s earnings, and $β_{1}$ is the IGE as specified in the literature.⁵ As already indicated, Mitnik and Grusky (this volume) showed that this conventionally estimated IGE has been misinterpreted. Mobility scholars have interpreted it as the elasticity of the expectation of children’s income or earnings conditional on parental income, but it pertains in fact to the conditional geometric mean.

Indeed, the parameter $β_{1}$ is not, in the general case, the elasticity of the conditional expectation of the child’s income. This would hold as a general result only if $E (\ln Y | x) = \ln E (Y | x)$ . But because of Jensen’s inequality, the latter is not the case. Instead, as $E (\ln Y | x) = \ln \exp E (\ln Y | x),$ and $G M (Y | x) = \exp E (\ln Y | x)$ , equation (1) is equivalent to

\ln G M (Y | x) = β_{0} + β_{1} \ln x,

(2)

where GM denotes the geometric mean operator. Therefore, $β_{1}$ is the elasticity of the conditional geometric mean, that is, the percentage differential in the geometric mean of children’s long-run income with respect to a marginal percentage differential in parental long-run income.⁶

3. Set Estimation of the IGE of the Geometric Mean

3.1. OLS Estimation and the GEiV Model

Estimation of equation (1) by OLS, after substituting short-run income variables for the long-run variables, opens the door to the two types of biases widely discussed in the literature. First, because income- and earnings-age profiles differ across economic origins, using proxy measures taken when parents or children are too young or too old to represent lifetime differences well results in life-cycle biases (e.g., Black and Devereux 2011). Second, in the case of the parental variables, and even in the absence of any life-cycle bias, the combination of transitory fluctuations and measurement error in the measure of short-run income with respect to true short-run income produces substantial attenuation bias (see, e.g., Mazumder 2005; Solon 1999). The GEiV model provides a joint analysis of these biases (Haider and Solon 2006; see also Nybom and Stuhler 2016).

To introduce the empirical assumptions of the GEiV model, it is necessary to first introduce the following population linear projections:

\ln Z_{t} = λ_{0 t} + λ_{1 t} \ln Y + V_{t}

(3)

\ln S_{k} = η_{0 k} + η_{1 k} \ln X + Q_{k},

(4)

where $Z_{t} > 0$ is children’s income at age t; $Y > 0$ ; $λ_{0 t} + V_{t}$ is the measurement error in the logarithm of the short-run children’s variable as a measure of the logarithm of the corresponding long-run variable when $λ_{1 t} = 1$ ; $λ_{1 t}$ captures left-hand life-cycle bias and thus may be different from 1 and varies with t; $S_{k} > 0$ is parents’ income at age k; $X > 0$ ; $η_{0 k} + Q_{k}$ is the measurement error in the logarithm of the short-run parental variable as a measure of the logarithm of the corresponding long-run variable when $η_{1 k} = 1$ ; and $η_{1 k}$ captures right-hand life-cycle bias and thus may be different from 1 and varies with parents’ age.⁷

The empirical assumptions of the GEiV model are the following. For any t and k,

Cov (\ln X, V_{t}) = 0

(5)

Cov (\ln Y, Q_{k}) = 0

(6)

Cov (V_{t}, Q_{k}) = 0 .

(7)

These assumptions are expected to hold imperfectly but still as good approximations, at least when $λ_{1 t} \approx η_{1 k} \approx 1 .$ (To simplify the notation, in what follows I drop the subscripts t and k.)

In the general case, the probability limit of the “short-run OLS estimator” of the conventional IGE (i.e., the OLS estimator with the long-run income variables replaced by short-run variables), denoted by ${\tilde{β}}_{1}$ , can be obtained by substituting equations (3) and (4) in ${\tilde{β}}_{1} \equiv Cov (\ln Z, \ln S) / V a r (\ln S)$ . Because equation (4) is a linear projection and therefore $Cov (\ln X, Q) = 0,$ that probability limit is

{\tilde{β}}_{1} = β_{1} \frac{λ_{1} η_{1}}{{[η_{1}]}^{2} + V R} + \frac{λ_{1} Cov (\ln Y, Q) + η_{1} Cov (\ln X, V) + Cov (V, Q)}{V a r (\ln X) [{(η_{1})}^{2} + V R]},

(8)

where $V R = V a r (Q) / V a r (\ln X);$ I will refer to this as the “variance ratio.”

If the three empirical assumptions of the GEiV model hold, then equation (8) reduces to

{\tilde{β}}_{1} = β_{1} \frac{λ_{1} η_{1}}{{[η_{1}]}^{2} + V R} .

If, in addition, both children’s and parents’ incomes are measured at the “right points” of their life cycles, that is, if $λ_{1} = η_{1} = 1,$ then ${\tilde{β}}_{1} = β_{1} [1 / (1 + V R)]$ and the estimates obtained with the short-run OLS estimator are affected by attenuation bias. The bias, however, falls with $V a r (Q)$ , which in turn depends on the exact short-run measure of parental income that is used.

The GEiV model supplies a methodological justification for the estimation of the conventional IGE by OLS with proxy variables that satisfy some conditions (Nybom and Stuhler 2016). Indeed, the GEiV model suggests that using measures of economic status (i.e., income, earnings) obtained at specific ages should eliminate the bulk of life-cycle bias—and the available evidence indicates that estimating IGEs with parents’ and children’s information close to age 40 is the best approach (Böhlmark and Lindquist 2006; Haider and Solon 2006; Mazumder 2001; Nybom and Stuhler 2016). In the case of attenuation bias, the GEiV model (and many analyses predating it, e.g., Solon 1992) suggests pushing $V a r (Q)$ down by using a multiyear average of parents’ income, rather than a single-year measure, as the proxy measure S. There is strong evidence that the bias can be substantially reduced this way, although there is disagreement on how many years are necessary to eliminate the bulk of it (see Chetty et al. 2014:1582 and Online Appendix E; Mazumder 2005, 2016; Mitnik et al. 2018:9–14).

3.2. IV Estimation and the GEiV-IV Model

Mobility scholars usually have fewer years of information available than what most believe are needed to nearly eliminate attenuation bias. Therefore, a natural alternative is to forgo estimation by OLS and, instead, address right-side measurement error by using an IV estimator of $β_{1}$ . Here, variables such as parental education or occupational status are used as instruments for the error-ridden proxy measure of long-run parental income (e.g., Mulligan 1997; Ng 2007; Zimmerman 1992). The main concern with this strategy, however, is that the instruments typically available are most likely endogenous. In this context, IV estimates may still be useful if the sign of their asymptotic bias can be established. An analysis by Solon (1992, Appendix)—carried out under the assumption that there is no life-cycle bias—achieved that. I now present a more general version of this analysis that allows for life-cycle effects, which I refer to as the generalized error-in-variables model for the IV estimation of the conventional IGE, or GEiV-IV model. I also nest the key equation of this model within a more general expression for the probability limit of the IV estimator of that IGE.⁸

Let L (e.g., parental years of education) be the instrument used for the IV estimation of $β_{1}$ , and

\ln Y = ϱ_{0} + ϱ_{1} \ln X + ϱ_{2} L + κ

(9)

a population linear projection of $\ln Y$ on $\ln X$ and $L$ . The GEiV-IV model makes the following empirical assumptions. For any t and k,

Cov (V_{t}, L) = 0

(10)

Cov (Q_{k}, L) = 0,

(11)

at least when $λ_{1} \approx η_{1} \approx 1 .$

I show in Online Appendix A that, in the general case, the probability limit of the IV estimator of the conventional IGE, ${\overset{\cdot\cdot}{β}}_{1}$ , may be written as

\begin{matrix} {\overset{\cdot\cdot}{β}}_{1} = \frac{λ_{1}}{η_{1}} {β_{1} [1 - \frac{Cov (Q, L)}{Cov (\ln S, L)}] + ϱ_{2} \frac{SD (L)}{SD (\ln X)} [\frac{1 - {[Corr (\ln X, L)]}^{2}}{Corr (\ln X, L) + \frac{Cov (Q, L)}{η_{1} SD (\ln X) SD (L)}}]} + \frac{Cov (V, L)}{Cov (\ln S, L)}, \end{matrix}

(12)

where SD denotes the standard deviation operator (and I again dropped the subscripts t and k). Therefore, taking for granted that $0 < | Corr (\ln X, L) | < 1,$ equation (12) shows that IV estimation is consistent when the following conditions are met: (1) L is uncorrelated with the measurement errors, that is, $Cov (V, L) = Cov (Q, L) = 0;$ (2) both children’s and parents’ incomes are measured at the right points of their life cycles, that is, $λ_{1} = η_{1} = 1;$ and (3) L is a valid instrument, that is, $ϱ_{2} = 0$ . If, however, the instrument is invalid but equations (10) and (11) do hold, the probability limit of the IV estimator is

{\overset{\cdot\cdot}{β}}_{1} = \frac{λ_{1}}{η_{1}} {β_{1} + ϱ_{2} \frac{SD (L)}{SD (\ln X)} [\frac{1 - {[Corr (\ln X, L)]}^{2}}{Corr (\ln X, L)}]} .

(13)

If, in addition, $λ_{1} = η_{1} = 1$ , then equation (12) reduces to Solon’s (1992) widely cited result:

{\overset{\cdot\cdot}{β}}_{1} = β_{1} + ϱ_{2} \frac{SD (L)}{SD (\ln X)} [\frac{1 - {[Corr (\ln X, L)]}^{2}}{Corr (\ln X, L)}] .

(14)

Parental education and other similar instruments are assumed to be positively correlated with log parental income; that is, it is assumed that $Corr (\ln X, L) > 0$ . Then, equation (14) indicates that we can expect ${\overset{\cdot\cdot}{β}}_{1}$ to be larger or smaller than $β_{1}$ , depending on the sign of $ϱ_{2}$ . Under the additional substantive assumption that $ϱ_{2} > 0$ , IV estimates with years of parental education and similar instruments have been interpreted as upper-bound estimates of the conventional IGE.

3.3. Empirical Assumptions of the Bracketing Strategy

On the basis of the last result, Solon (1992) suggested the bracketing strategy: combining OLS and IV estimates to bracket (i.e., set estimate) the true value of the conventional IGE. The analyses conducted here indicate that this strategy relies on nine empirical assumptions: (1) the assumptions of the GEiV model (i.e., equations 5, 6, and 7); (2) the assumptions of the GEiV-IV model (i.e., equations 10 and 11); (3) the life-cycle assumptions $λ_{1} = η_{1} = 1$ ; (4) the invalid-instrument assumption $ϱ_{2} > 0;$ and (5) an auxiliary empirical assumption, that is, $0 < Corr (\ln X, L) < 1$ .

4. The IGE of the Expectation

Because of the conceptual and methodological problems affecting the conventional IGE, Mitnik and Grusky (this volume) have called for redefining the workhorse intergenerational elasticity. This entails replacing the PRF of equation (1) with a PRF whose estimation delivers estimates of the IGE of the expectation in the general case. Under the assumption of constant elasticity, that PRF can be written as

\ln E (Y | x) = α_{0} + α_{1} \ln x,

(15)

where $Y \geq 0$ , $X > 0,$ and $α_{1} = d \ln E (Y | x) / d \ln x$ is the percentage differential in the expectation of children’s long-run income with respect to a marginal percentage differential in parental long-run income. Crucially, (1) all interpretations incorrectly applied to the conventional IGE are correct or approximately correct under this formulation (see Mitnik and Grusky, this volume, Section 5.1), (2) the IGE of the expectation is fully immune to the “zero problem” and the concomitant selection bias affecting the estimation of the IGE of the geometric mean, and (3) the IGE of the expectation is well suited for studying the role of marriage in the intergenerational transmission of economic advantages (see Mitnik and Grusky, this volume, Section 5.2).

The IGE of the expectation is not the only mobility measure immune to the zero problem. Most notably, the increasingly popular rank-rank slope (RRS), a measure of relative-position mobility and of the transmission of relative-position advantages across generations (see Chetty et al. 2014; Dahl and DeLeire 2008), is well defined when income variables include zero in their support. This may suggest that mobility scholars can rely on the RRS to circumvent the problems affecting the conventional IGE and therefore that analyses based on the IGE of the expectation are unnecessary.

Nothing of the sort is the case. First, although the RRS is a clearly useful measure, it does not measure the same concept as does the IGE of the expectation—in other words, both of these measures are “mobility measures” in a very broad sense, not two alternative approaches for measuring the same thing. Second, although it is somewhat easier to deal with left- and right-side life-cycle biases, as well as right-side attenuation bias, in the estimation of the RRS than in the estimation of intergenerational elasticities (Mazumder 2016; Nybom and Stuhler 2017), estimation of the RRS is affected by its own unique problem, left-side attenuation bias (Nybom and Stuhler 2017). This makes estimation of the RRS when only one annual measure of children’s income is available to compute their ranks (a very common situation) an unadvisable course of action. Third, because ranks, unlike money flows, are not additive across income sources—and, more generally, because family-income rank cannot be derived in any way from the ranks of family members in the individual-earnings distribution—it is not possible to use the RRS to study the “channels” (e.g., labor market, marriage market) through which the intergenerational transmission of family-income advantages occurs; this can be easily done, however, with the IGE of the expectation (see Mitnik and Grusky, this volume, Section 5.2). Finally, intergenerational elasticities can be easily embedded, and have been frequently embedded, within policy-relevant theoretical models (for the conventional IGE, see, e.g., Bénabou 2000; Durlauf and Seshadri 2018; Solon 2004; for the IGE of the expectation, see Mitnik 2018), but there are not yet any comparable models embedding the RRS and, more crucially, developing them seems a quite daunting task. It follows that mobility scholars need both the RRS and the IGE of the expectation in their toolbox.

5. Set Estimation of the IGE of the Expectation

5.1. PPML Estimation and the GEiVE Model

After substituting short-run for long-run income measures in equation (15), the IGE of the expectation can be estimated using several approaches. Here, I assume that estimation is based on the PPML estimator.⁹ To introduce the empirical assumptions of Mitnik’s (2019) generalized error-in-variables model for the estimation of the IGE of the expectation with that estimator, or GEiVE model, I first introduce the following population linear projections:

Z_{t} = θ_{0 t} + θ_{1 t} Y + W_{t}

(16)

\ln S_{k} = π_{0 k} + π_{1 k} \ln X + P_{k},

(17)

where $Z_{t} \geq 0$ is children’s income at age t; Y is as defined earlier; $θ_{0 t} +$ $W_{t}$ is the measurement error in the children’s short-run income variable as a measure of the corresponding long-run variable when $θ_{1 t} = 1$ ; $θ_{1 t}$ captures left-hand life-cycle bias and thus may be different from 1 and varies with t; and equation (17) is the same as equation (4) but I have used a different notation for the parameters and the error term, both to avoid confusion and because the populations covered by the two equations will be different as long as there are children whose short-run income is zero.¹⁰ Indeed, unlike in the case of the GEiV model, in which $Z_{t} > 0,$ in the GEiVE model $Z_{t} \geq 0,$ that is, because the left-side measurement error concerns the income variable rather than its logarithm, the model does not preclude that the observed income variable is zero, and therefore children without any short-run income (e.g., those who are unemployed the year in which their incomes are measured) are included in all analyses.

The empirical assumptions of the GEiVE model are the following. For any t and k,

Cov (W_{t}, \ln X) = 0

(18)

Cov (P_{k}, Y) = 0

(19)

Cov (W_{t}, P_{k}) = 0 .

(20)

As in the case of the GEiV model, these assumptions are expected to hold imperfectly but still provide good approximations, at least when $θ_{1 t} \approx π_{1 k} \approx 1 .$ (As before, I omit the subscripts t and k in the following to simplify the notation.)

Mitnik (2019) showed that, at the level of approximation provided by second-order Taylor-series expansions, the probability limit of the “long-run PPML estimator” of the IGE of the expectation (i.e., the PPML estimator with long-run income variables), is

α_{1} \approx C_{α_{1}} - {[{(C_{α_{1}})}^{2} - V_{α_{1}}]}^{\frac{1}{2}},

(21)

where $C_{α_{1}} = {[Cov (Y, \ln X)]}^{- 1}$ and $V_{α_{1}} = 2 {[V a r (\ln X)]}^{- 1}$ .¹¹ His analyses also entail that, in the general case, the probability limit of the “short-run PPML estimator” of the IGE of the expectation (i.e., the PPML estimator with short-run income variables substituted for the long-run variables), denoted by ${\tilde{α}}_{1}$ , is

{\tilde{α}}_{1} \approx C_{{\tilde{α}}_{1}} - {[{(C_{{\tilde{α}}_{1}})}^{2} - V_{{\tilde{α}}_{1}}]}^{\frac{1}{2}},

(22)

where

\begin{matrix} C_{{\tilde{α}}_{1}} = {[θ_{1} π_{1} Cov (Y, \ln X) + θ_{1} Cov (Y, P) + π_{1} Cov (W, \ln X) + Cov (W, P)]}^{- 1} \\ V_{{\tilde{α}}_{1}} = 2 {V a r (\ln X) [{(π_{1})}^{2} + V R]}^{- 1}, \end{matrix}

and $V R = V a r (P) / V a r (\ln X)$ is again the variance ratio.¹²

If the three empirical assumptions of the GEiVE model (i.e., equations 18, 19, and 20) hold, $C_{{\tilde{α}}_{1}}$ reduces to

C_{{\tilde{α}}_{1}} = {[θ_{1} π_{1} Cov (Y, \ln X)]}^{- 1} .

If, in addition, $θ_{1} = π_{1} = 1$ , that is, if the children’s and parents’ incomes are measured at the right points of their life cycles, $C_{{\tilde{α}}_{1}} = C_{α_{1}}$ and $V_{{\tilde{α}}_{1}} = 2 {V a r (\ln X) [1 + V R]}^{- 1}$ . Then, as $\partial {\tilde{α}}_{1} / \partial V R < 0$ , the estimates obtained with the short-run PPML estimator are affected by attenuation bias.

The GEiVE model provides a methodological justification for estimating the IGE of the expectation with the PPML estimator and proxy variables that satisfy some conditions (exactly as the GEiV model does for the estimation of the conventional IGE by OLS). To minimize life-cycle biases, the GEiVE model suggests researchers use measures of children’s and parents’ economic status obtained at ages in which $θ_{1} \approx 1$ and $π_{1} \approx 1$ , respectively. Mitnik’s (2019) empirical results suggest that, as in the case of the conventional IGE, this happens when both parents’ and children’s incomes are measured close to age 40. The GEiVE model also indicates that researchers should use measures of average parental income over several years, rather than annual measures, so as to reduce $V a r (P)$ as much as possible. Mitnik’s (2019) empirical results suggest that, with survey data, it is necessary to use at least 13 years of information to eliminate the bulk of that bias.

5.2. IV Estimation and the GEiVE-IV Model

As I pointed out earlier, in most cases mobility scholars have relatively few years of parental information available, so we can expect their estimates of the IGE of the expectation with the PPML estimator to be affected by (potentially substantial) attenuation biases. As in the case of the conventional IGE, an obvious alternative is to estimate the IGE of the expectation with an IV estimator. There are several such estimators that could be used, but the additive-error version of the GMM IV estimator of the Poisson or exponential regression model (Mullahy 1997; Windmeijer and Santos Silva 1997), which I call the GMM-IVP estimator, appears preferable.¹³ I show next that this estimator is upward biased, asymptotically, when used with the instruments typically available to mobility scholars.

To this end, I advance a generalized error-in-variables model for the IV estimation of the IGE of the expectation, the GEiVE-IV model, in which I make empirical assumptions fully comparable to those of the GEiV-IV model.¹⁴ The empirical assumptions are the following. For any t and k,

Cov (W_{t}, L) = 0

(23)

Cov (P_{k}, L) = 0 .

(24)

As usual, the assumptions are expected to hold imperfectly but still as good approximations, at least when $θ_{1 t} \approx π_{1 k} \approx 1$ . (In what follows, I drop the subscripts t and k.)

As with the PPML estimator, a closed-form expression for the probability limit of the GMM-IVP estimator is not available. In the case of the former estimator, Mitnik (2019) addressed this problem by deriving the approximate closed-form expression I introduced above from the population moment problem solved by the probability limit of the estimator. This approach proves less convenient in the case of the GMM-IVP estimator, so here I work directly with the relevant population moment conditions. That is, I compare the population moment problems solved by (1) the probability limit of the PPML estimator with long-run income variables and (2) the probability limit of the GMM-IVP estimator with short-run income variables (the “short-run GMM-IVP estimator”) and the instruments typically available.¹⁵

The probability limit of the PPML estimator with long-run variables, $α_{1}$ , solves the population-moment condition

\frac{E (X^{α_{1}} \ln X)}{E (X^{α_{1}})} = E (Y \ln X)

(25)

(Mitnik 2019:13), and ${\overset{\cdot\cdot}{α}}_{1},$ the probability limit of the short-run GMM-IVP estimator, solves

\frac{E (S^{{\overset{\cdot\cdot}{α}}_{1}} L)}{E (S^{{\overset{\cdot\cdot}{α}}_{1}})} = E (Z L) .

(26)

¹⁶

Using equations (16) and (17) to substitute S and Z out in equation (26) yields

\begin{matrix} \frac{E (X^{π_{1} {\overset{\cdot\cdot}{α}}_{1}} \ln X)}{E (X^{π_{1} {\overset{\cdot\cdot}{α}}_{1}})} \\ \approx E (Y \ln X) {\frac{1 + F ({\overset{\cdot\cdot}{α}}_{1}) [θ_{1} Cov (Y, \ln X) + V a r (\ln X) \frac{θ_{1} Cov (L, Ψ) + Cov (L, W)}{Cov (L, \ln X)}]}{1 + Cov (Y, \ln X)}}, \end{matrix}

(27)

where

F ({\overset{\cdot\cdot}{α}}_{1}) \approx \frac{1 + 0.5 {{[π_{1} {\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (\ln X) + {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (P)}}{1 + 0.5 {[π_{1} {\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (\ln X)} \frac{π_{1} Cov (\ln X, L)}{π_{1} Cov (\ln X, L) + Cov (L, P)}

(28)

and $Ψ = Y - \exp (α_{0}) X^{α_{1}}$ is an additive error based on equation (15) (see Online Appendix A for the derivation of equations 26, 27, and 28).

Equations (27) and (28) show the (approximate) population moment problem solved by the probability limit of the short-run GMM-IVP estimator of the IGE of the expectation in the general case. Together, equations (25), (27), and (28) provide a counterpart to equation (12). As the latter does for the linear IV estimator of the IGE of the geometric mean, the former (1) identify the various factors determining the probability limit of the GMM-IVP estimator of the IGE of the expectation with short-run variables and (2) indicate that, in the general case, this estimator may be upward or downward inconsistent.

The term $F ({\overset{\cdot\cdot}{α}}_{1})$ may be larger, equal, or smaller than 1. Under the assumptions of the GEiVE-IV model, however, equation (28) reduces to $F ({\overset{\cdot\cdot}{α}}_{1}) \approx 1 + \frac{0.5 {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (P)}{1 + 0.5 {[π_{1} {\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (\ln X)} > 1$ , and equation (27) becomes

\begin{matrix} \frac{E (X^{π_{1} {\overset{\cdot\cdot}{α}}_{1}} \ln X)}{E (X^{π_{1} {\overset{\cdot\cdot}{α}}_{1}})} \\ \approx E (Y \ln X) {\frac{1 + [θ_{1} Cov (Y, \ln X) + V a r (\ln X) \frac{θ_{1} Cov (L, Ψ)}{Cov (L, \ln X)}] [1 + \frac{0.5 {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (P)}{1 + 0.5 {[π_{1} {\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (\ln X)}]}{1 + Cov (Y, \ln X)}} . \end{matrix}

(29)

If in addition $θ_{1} = π_{1} = 1,$ the last equation reduces to

\begin{matrix} \frac{E (X^{{\overset{\cdot\cdot}{α}}_{1}} \ln X)}{E (X^{{\overset{\cdot\cdot}{α}}_{1}})} \approx E (Y \ln X) + V a r (\ln X) \frac{Cov (L, Ψ)}{Cov (L, \ln X)} [1 + \frac{0.5 {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (P)}{1 + 0.5 {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (\ln X)}] \\ + Cov (Y, \ln X) \frac{0.5 {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (P)}{1 + 0.5 {[{\overset{\cdot\cdot}{α}}_{1}]}^{2} V a r (\ln X)}, \end{matrix}

(30)

where I have used $E (Y \ln X) = 1 + Cov (Y, \ln X) .$

Equations (25) and (30) provide a counterpart to equation (14). Indeed, as $\partial E (X^{{\overset{\cdot\cdot}{α}}_{1}} \ln X) {[E (X^{{\overset{\cdot\cdot}{α}}_{1}})]}^{- 1} / \partial {\overset{\cdot\cdot}{α}}_{1} > 0$ (see Online Appendix A), and safely assuming that $Cov (Y, \ln X) > 0,$ these equations show that the probability limit of the short-run GMM-IVP estimator is an upper bound for the IGE of the expectation when the invalid instrument is positively correlated with the error term. The equations also show that the upper bound is tighter when (1) the covariance between the instrument and the error term, $Cov (L, Ψ),$ is smaller; (2) the slope of the linear projection of $L$ on $\ln X$ , $Cov (L, \ln X) / V a r (\ln X)$ , is larger; and (3) the short-run measure of parental income is less noisy, that is, $V a r (P)$ is smaller relative to $V a r (\ln X)$ .

5.3. Empirical Assumptions of the Bracketing Strategy

The GEiVE-IV model just advanced provides a foundation for combining PPML and GMM-IVP estimates with the goal of set estimating the IGE of the expectation. The bracketing strategy relies in this case on the following empirical assumptions: (1) the assumptions of the GEiVE model (i.e., equations 18, 19, and 20); (2) the assumptions of the GEiVE-IV model (i.e., equations 23 and 24); (3) the life-cycle assumptions $θ_{1} = π_{1} = 1$ ; (4) the invalid-instrument assumption $Cov (L, Ψ) > 0$ ; and (5) the auxiliary empirical assumption $Cov (\ln X, L) > 0$ . These assumptions are strictly isomorphic to those I identified in the case of the bracketing strategy for the IGE of the geometric mean.

6. Constructing Confidence Intervals for Partially Identified IGEs

Under the empirical assumptions of the bracketing strategies discussed in the previous two sections, the probability limits of the OLS and IV estimators, and the probability limits of the PPML and GMM-IVP estimators, provide lower and upper bounds for the long-run IGEs of the geometric mean and expectation, respectively. This means these IGEs are only “partially identified” (see, e.g., Manski 2003) by data on short-run incomes and the instruments: Even if we had an unlimited number of observations, we would not be able to learn the true values of the long-run IGEs. For each IGE, we can only aim at learning the range of values consistent with those data, that is, the identified set defined by the relevant probability limits. The second difficulty is, of course, that we do not have an unlimited number of observations; rather, we need to estimate the bounds from a finite sample. This means we have to consider the uncertainty regarding the estimated bounds as well as the uncertainty due to partial identification.

Previous research using the bracketing strategy reported separate confidence intervals for the bounds estimated by the OLS and IV estimators, that is, for the bounds of the identified set. However, when we use a bracketing strategy, we would like to provide just one confidence interval that (1) pertains to the partially identified long-run IGE and (2) reflects uncertainty due to partial identification and sampling variability. A seemingly plausible approach would be to use as lower bound the lower bound of the confidence interval associated with the OLS or PPML estimate, and as upper bound the upper bound of the confidence interval associated with the relevant IV estimate. This, however, would generate a confidence interval for the identified set (see Stoye 2009:1300), not for the IGE itself. The probability that this interval covers the IGE is at least as large as the probability that it covers the identified set (Imbens and Manski 2004, Lemma 1). In other words, the suggested confidence interval is too conservative.

The intuition for why this is the case is that, asymptotically, the width of the identified set is large compared with the sampling error. Therefore, if the true IGE is not close to the lower bound, then the risk that the lower-bound estimate will be larger than the true value can be ignored. Likewise, if the true parameter is not close to the upper bound, then the risk that the upper-bound estimate will be lower than the true value can be ignored. As the true value cannot be close to both bounds, the noncoverage risk is effectively one-sided. Denoting the probability of type I error by $α$ , this entails that, asymptotically, a $100 (1 - α) %$ confidence interval (e.g., a 90 percent confidence interval) for the identified set is a $100 (1 - \frac{α}{2}) %$ confidence interval (e.g., a 95 percent confidence interval) for the IGE.

This suggests using $100 (1 - 2 α) %$ confidence intervals for the identified sets as $100 (1 - α) %$ confidence intervals for the IGEs.¹⁷ This approach for constructing confidence intervals, however, may produce shorter confidence intervals than under point identification (see Online Appendix C). To address this problem, and following Imbens and Manski (2004), $100 (1 - α$ )% confidence intervals for the long-run IGEs can be constructed as follows:

C I_{IGE} (α) \equiv [{\hat{IGE}}^{l} - c_{IGE} (α) \hat{SE} ({\hat{IGE}}^{l}), {\hat{IGE}}^{u} + c_{IGE} (α) \hat{SE} ({\hat{IGE}}^{u})]

where $c_{IGE} (α)$ solves

Φ (c_{IGE} (α) + R W_{IGE}) - Φ (- c_{IGE} (α)) = 1 - α;

$IGE = β_{1}, α_{1}$ ; the superscripts l and u identify the lower-bound (OLS or PPML) and upper-bound (IV or GMM-IV) estimators, respectively; $R W_{IGE} = \frac{{\hat{IGE}}^{u} - {\hat{IGE}}^{l}}{\max (\hat{SE} ({\hat{IGE}}^{l}), \hat{SE} ({\hat{IGE}}^{u}))}$ is (an estimate of) the relative width of the identified set; SE is the standard error operator; and $Φ (.)$ denotes the cumulative distribution function of the standard normal distribution. With 95 percent confidence intervals, $c_{IGE} (α)$ is close to $Φ^{- 1} (0.90) \approx 1.64$ when the width of the identified set is large compared with sampling error, and it is equal to $Φ^{- 1} (0.95) \approx 1.96$ with point identification.

So far, I have assumed that the upper bound is estimated only once. However, the upper bound may be estimated repeatedly, with different sets of instruments. As the probability limit of the IV estimator is different in each case, the identified set is the intersection of all the sets that can be formed by combining one of these probability limits with the probability limit of the lower-bound estimator. In this “intersection-bounds context,” the two-step (2S) estimator that selects the minimum of the IV estimates across instruments is a consistent estimator of the upper bound (Nevo and Rosen 2012). But as this estimator is based on multiple estimators, each with its own distribution, the construction of confidence intervals needs to take into account the sampling uncertainty about all upper-bound estimates, not just the estimate that happens to be the minimum in the current sample. Closely related approaches for constructing valid confidence intervals in the intersection-bounds context have been developed by Nevo and Rosen (2012, Section IV) and Chernozhukov, Lee, and Rosen (2013). In Online Appendix C, I provide a step-by-step explanation of how to apply Nevo and Rosen’s approach to the problem at hand.

Although the 2S estimator of the upper bound is consistent, it is not asymptotically unbiased; in fact, no estimator involving minimization or maximization can be unbiased (Hirano and Porter 2012). Chernozhukov et al. (2013) proposed an alternative three-step (3S) estimator that is consistent and “half-median unbiased.” In our context here, this means the estimator has the property that at least half of its values across samples are above the true upper bound. The 3S estimator also selects the minimum of the estimates of the upper bound, but only after correcting them (this is the added step); I explain how to do this correction in Online Appendix C. The risk with the 2S estimator is that in finite samples it may produce significantly downward biased estimates, yet the alternative 3S estimator may tend to be overly conservative. In the empirical analyses of the next section, I report estimates based on both estimators.

As companions to this article, I have made available three Stata programs that compute the confidence intervals just discussed. Two of these programs are meant to be used when the upper bound is estimated with only one instrument. One of them set estimates the IGEs and computes the confidence intervals (which minimizes the amount of coding that is necessary), and the other computes the confidence intervals after the models have been estimated by the researcher (which provides more flexibility). The third Stata program set estimates the IGEs with multiple sets of instruments and computes the appropriate confidence intervals. With these programs, both set estimation and inference are very simple tasks.¹⁸

7. Empirical Analyses

The main goal of this section is to empirically assess whether the bracketing strategies work as the generalized error-in-variables models lead us to expect, examine the information supplied by the bounds the strategies generate, and explore how the bounds vary when different instruments and short-run measures of parental income are used. The analyses are preceded by a brief description of the data and the estimation approaches used.

7.1. Data and Estimation

The empirical analyses are based on a PSID sample that makes it possible to construct an approximate measure of long-run parental income but not children’s long-run income.¹⁹ However, as I will explain, this sample still sheds light on questions of central interest for this article. The sample includes information on children born between 1966 and 1974, for which 25 years of parental data centered on age 40 (obtained when the children were between 1 and 25 years old) are available. Children observed in the PSID when they were between 35 and 38 years old are included in the sample. I use information on children’s average family income when they were 35 to 38 years old; parents’ family income, age, and years of education when the children were 1 to 25 years old; and fathers’ occupation when the children were growing up (as reported by the latter). I do not use information on earnings because I only estimate family-income IGEs (in part, to minimize the selection bias that results when children with zero income or earnings are dropped from samples when estimating the IGE of the geometric mean). Table 1 presents descriptive statistics; Online Appendix D provides additional information on the sample and variables and explains in more detail why I focus exclusively on family-income IGEs.

Table 1.

Descriptive Statistics (Unweighted Values)

Child’s gender (% female)	51.7
Child’s age
Mean	36.6
Standard deviation	0.6
Child’s family income
Mean	$91,625
Standard deviation	$88,398
Average parental age
Mean	40.1
Standard deviation	6.7
Average parental income
Mean	$78,858
Standard deviation	$64,185
Parents’ years of education at child age 15
Mean	21.8
Standard deviation	7.0
Household head’s years of education at child age 15
Mean	12.4
Standard deviation	3.0
Father’s occupation (%)
Management occupations	10.2
Business operations specialists	0.6
Financial specialists	1.8
Computer and mathematical occupations	1.1
Architecture and engineering occupations	3.3
Life, physical, and social science occupations	0.9
Community and social services occupations	0.9
Legal occupations	0.4
Education, training, and library occupations	2.7
Arts, design, entertainment, sports, and media occupations	1.6
Health care practitioners and technical occupations	1.6
Protective service occupations	2.8
Food preparation and serving occupations	0.5
Building and grounds cleaning and maintenance occupations	1.5
Personal care and service occupations	0.4
Sales occupations	8.6
Office and administrative support occupations	3.0
Farming, fishing, and forestry occupations	1.9
Construction trades	10.2
Extraction workers	0.4
Installation, maintenance, and repair workers	7.5
Production occupations	12.8
Transportation and material moving occupations	10.2
Military-specific occupations	2.5
Doesn’t know, refused	11.6
Not applicable	1.5
N	827

Note: Monetary values in 2012 dollars (adjusted by inflation using the Consumer Price Index for Urban Consumers–Research Series). The average parental age and income pertain to when the children were 1 to 25 years old. The occupation is coded as “not applicable” when there was no father or surrogate, the father was deceased, or the father never worked.

I use the OLS and two-stage least squares (TSLS) estimators to estimate the IGE of the geometric mean of children’s family income, I use the PPML and GMM-IVP estimators to estimate the IGE of the expectation of children’s family income, and I use sampling weights and compute robust standard errors in all cases. I construct the confidence intervals for the partially identified IGEs as explained in the previous section.²⁰

I use the following instruments in my analyses: (1) parents’ total years of education when the child was 15 years old, and in the time period covered by each short-run parental-income measure, (2) the household head’s years of education when the child was 15 years old, and in the time period covered by each short-run parental-income measure, and (3) the father’s occupation. I use both time-varying and at-age-15 parental-education variables as instruments because, in the data sets used by mobility scholars, parental education is sometimes available all years in which parental income is measured and sometimes only for when children were some specific age, usually in the 12-to-16 range. It is thus important to examine IV estimates generated with both types of parental-education variables. Online Appendix D explains why I use parents’ education and household head’s education as instruments but not father’s education, which is the typical approach in the literature.

The relationship between long-run and short-run measures of income varies with the age at measurement; for this reason, it is customary to include polynomials on children’s and parents’ ages as controls when estimation is based on short-run measures. However, because all IGE estimates I report are based on a sample in which the variation in children’s ages is very small, controlling for children’s age is unnecessary (see the next paragraph for a second reason for proceeding this way). Mitnik et al. (2015:34) argued that the age at which parents have their children is not exogenous to their income, that parental age is causally relevant for children’s life chances, and that insofar as we want persistence measures to reflect the gross association between parental and children’s income we should not control for parental age. Here I present estimates from models without controls for parental age, but estimates from models with such controls are very similar.

A measure of children’s long-run family income is not available, so in all analyses, regardless of whether they pertain to short- or long-run IGEs, I use children’s family income when they were 36 to 38 years old as their income measure. This is equivalent to making $λ_{1} = θ_{1} = 1$ , $V = W = 0$ (for all children), and $Cov (L, V) = Cov (L, W) = 0$ by construction. As a result, in the empirical analyses I am not able to assess any aspect of the generalized error-in-variables models pertaining to left-side measurement error. Nevertheless, I can still draw clear conclusions regarding right-side measurement error and the bracketing strategies.

7.2. Results

Figures 1 and 2 pertain to the IGE of the geometric mean. They present information that allows us to assess the qualitative implications of the relevant generalized error-in-variables models and how the bracketing strategy works when parental income is measured at different parental ages. Figures 3 and 4 do the same for the IGE of the expectation. Tables 2, 3, and 4 summarize the results obtained with the bracketing strategies and present the estimates of the long-run IGEs. Table 2 reports the results obtained with what I call “ideal bracketing strategies”: It displays IGE estimates pertaining to the average parental ages at which the measurement-error slopes $η_{1}$ and $π_{1}$ are equal to 1. These estimates are computed by interpolation, and therefore confidence intervals are not available. Tables 3 and 4 show estimates based on measures of parental income centered on the years when children were age 13 (and parents’ average age was close to 40), for the IGE of the geometric mean and the IGE of the expectation, respectively. Assessing the performance of the bracketing strategies in this context is of eminent practical interest, as mobility scholars normally do not know the ages at which the measurement-error slopes are equal to 1 with their data. So, when possible with those data, they simply use estimates based on income measures taken when parents and children are close to age 40 as their best guess (a guess informed by results obtained with other, potentially quite different, data). I will refer to bracketing strategies implemented this way as “feasible bracketing strategies.”²¹

Table 2.

Long-Run Estimates and Set Estimates When Measurement-Error Slopes Are Equal to 1

	Parental Information
	Long-Run	One Year	Three Years	Five Years	Seven Years
IGE of geometric mean of children’s income
Long-run estimate	0.70 (0.58–0.82)
Set estimates, with short-run income instrumented by
Parents’ education when income was measured		0.46–0.79	0.55–0.81	0.58–0.84	0.60–0.86
Household head’s education when income was measured		0.46–1.11	0.55–1.11	0.58–1.09	0.60–1.09
Parents’ education when child was 15 years old		0.46–0.90	0.55–0.91	0.58–0.91	0.60–0.91
Household head’s education when child was 15 years old		0.46–1.05	0.55–1.02	0.58–1.01	0.60–1.01
Father’s occupation when child was growing up		0.46–0.76	0.55–0.74	0.58–0.78	0.60–0.79
Average parental age	40.5	37.0	37.6	37.8	38.1
IGE of children’s expected income
Long-run estimate	0.60 (0.45–0.74)
Set estimates, with short-run income instrumented by
Parents’ education when income was measured		0.46–0.74	0.52–0.75	0.55–0.77	0.56–0.78
Household head’s education when income was measured		0.46–0.95	0.52–0.93	0.55–0.93	0.56–0.92
Parents’ education when child was 15 years old		0.46–0.80	0.52–0.80	0.55–0.81	0.56–0.81
Household head’s education when child was 15 years old		0.46–0.92	0.52–0.89	0.55–0.90	0.56–0.90
Father’s occupation when child was growing up		0.46–0.69	0.52–0.70	0.55–0.73	0.56–0.74
Average parental age	40.5	37.0	37.6	37.8	38.1

Note: The set estimates were computed by interpolation. Point and set estimates are in boldface type, and 95 percent confidence intervals (for the long-run estimates only) are in parentheses. IGE = intergenerational income elasticity.

Table 3.

Long-Run Estimate and Set Estimates of the IGE of the Geometric Mean When Average Parental Age Is Close to 40

	Parental Information
	Long-Run	One Year	Three Years	Five Years	Seven Years
Long-run estimate	0.7 (0.58–0.82)
Set estimates, with short-run income instrumented by
Parents’ education when income was measured		0.40–0.76(0.29–0.90)	0.52–0.79(0.44–0.94)	0.55–0.78(0.47–0.90)	0.57–0.78(0.48–0.90)
Household head’s education when income was measured		0.40–0.93(0.29–1.12)	0.52–0.93(0.44–1.11)	0.55–0.98(0.47–1.17)	0.57–0.98(0.48–1.17)
Parents’ education when child was 15 years old		0.40–0.82(0.29–0.98)	0.52–0.82(0.44–0.97)	0.55–0.82(0.47–0.97)	0.57–0.82(0.48–0.97)
Household head’s education when child was 15 years old		0.40–0.93(0.29–1.11	0.52–0.93(0.44–1.11)	0.55–0.92(0.47–1.10)	0.57–0.93(0.48–1.10)
Father’s occupation when child was growing up		0.40–0.80(0.29–0.97)	0.52–0.82(0.44–0.99)	0.55–0.83(0.47–1.00)	0.57–0.84(0.48–1.01)
η₁ estimates		1.10(1.02–1.18)	1.08(1.01–1.15)	1.07(1.02–1.13)	1.08(1.03–1.12)
Set estimates based on all upper-bound estimates
Two-step estimator		0.40–0.76(0.29–0.94)	0.52–0.79(0.44–0.98)	0.55–0.78(0.47–0.94)	0.57–0.78(0.48–0.94)
Three-step estimator		0.40–0.82(0.29–0.94)	0.52–0.85(0.44–0.98)	0.55–0.83(0.47–0.94)	0.57–0.83(0.48–0.94)

Note: The set estimates are based on measures of parental income centered on the years the children were 13 years old. Point and set estimates are in boldface type, and 95 percent confidence intervals are in parentheses. IGE = intergenerational income elasticity.

Table 4.

Long-Run Estimate and Set Estimates of the IGE of the Expectation When Average Parental Age Is Close to 40

	Parental Information
	Long-Run	One Year	Three Years	Five Years	Seven Years
Long-run estimate	0.6 (0.45–0.74)
Set estimates, with short-run income instrumented by
Parents’ education when income was measured		0.40–0.72(0.31–0.86)	0.48–0.74(0.39–0.87)	0.51–0.73(0.41–0.87)	0.52–0.70(0.41–0.84)
Household head’s education when income was measured		0.40–0.85(0.31–1.04)	0.48–0.84(0.39–1.00)	0.51–0.85(0.41–1.01)	0.52–0.84(0.41–0.99)
Parents’ education when child was 15 years old		0.40–0.77(0.31–0.94)	0.48–0.75(0.39–0.91)	0.51–0.74(0.41–0.89)	0.52–0.72(0.41–0.87)
Household head’s education when child was 15 years old		0.40–0.85(0.31–1.04)	0.48–0.84(0.39–1.01)	0.51–0.84(0.41–1.00)	0.52–0.82(0.41–0.98)
Father’s occupation when child was growing up		0.40–0.68(0.31–0.82)	0.48–0.68(0.39–0.83)	0.51–0.69(0.41–0.82)	0.52–0.70(0.41–0.85)
π₁ estimates		1.10(1.02–1.18)	1.08(1.01–1.15)	1.07(1.02–1.13)	1.08(1.03–1.12)
Set estimates based on all upper-bound estimates
Two-step estimator		0.40–0.68(0.31–0.87)	0.48–0.68(0.38–0.88)	0.51–0.69(0.41–0.87)	0.52–0.70(0.41–0.88)
Three-step estimator		0.40–0.74(0.31–0.87)	0.48–0.74(0.39–0.88)	0.51–0.75(0.41–0.87)	0.52–0.76(0.41–0.88)

Figure 1.

Estimates of the intergenerational income elasticity (IGE) of the geometric mean of children’s income with the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators.

Figure 2.

Estimates of the intergenerational income elasticity (IGE) of the geometric mean of children’s income with the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators.

Figure 3.

Estimates of the intergenerational income elasticity (IGE) of children’s expected income with the Poisson pseudo–maximum likelihood estimator (PPML) and generalized method of moments instrumental variables estimator of the Poisson or exponential regression model (GMM-IVP).

Figure 4.

Figures 1 to 4 each include four panels. The results presented in each panel are based on a different set of short-run measures of parental income. In the top left panels, parental income was measured when the children were 1 or 2 or 3 . . . up to 25 years old. In the other three panels, the short-run measures of parental income are multiyear averages. In the top right panels, they are three-year averages, centered when the children were 2 or 3 or 4 . . . up to 24 years old. In the bottom panels the measures of parental income are five- and seven-year averages, centered similarly. In all cases, the age in the horizontal axis is the average age of parents in the sample. The bottom curve in each panel shows the relationship between OLS- or PPML-based IGE estimates and average parental ages; the top curve shows the relationship between estimates of the relevant parental measurement-error slope and those ages. In Figures 1 and 3, the two middle curves in each panel show the relationship between IV IGE estimates obtained when parental income is instrumented by the at-age-15 parental education variables and average parental ages. In Figures 2 and 4, the middle curve in each panel is similar but pertains to IV IGE estimates generated with father’s occupation as the instrument. As the short-run income measures rely on more years of information, from 1 to 7, estimates in all figures become less affected by transitory income fluctuations and the shapes of the curves become progressively clearer.

Income-age profiles vary in a well-known manner across people with different levels of human capital (which is strongly associated with parental income). We therefore expect that $η_{1}$ and $π_{1}$ will increase with parental age and will be smaller than 1 when parents are younger and larger than 1 when parents are older (see Haider and Solon 2006). Figures 1 to 4 fully confirm these expectations. In addition, consistent with what the two IV generalized error-in-variables models predict (see equations 13 and 29), the figures also show a clear inverse relationship between the measurement-error slopes and the IV IGE estimates: The former increase, whereas the latter fall with parents’ age.²² Similarly, consistent with what the GEiV and GEiVE models predict (see Mitnik 2019:10, 16), the OLS- and PPML-based IGE estimates first increase (at very low values of the measurement-error slopes) but then decrease with parents’ age. Those slopes are equal to 1 when the parents are, on average, somewhat younger than 40 (i.e., at average ages between 37.0 and 38.1; see Table 2), and the slopes are in the 1.07-to-1.10 range at age 40 (see Tables 3 and 4).

In all figures, the long-run IGE, represented by the darker gray horizontal lines, is the IGE of children’s family income, when they were 36 to 38 years old, with respect to the (approximate) long-run family income of their parents. As explained earlier, for the purposes of the analyses here, the former income is assumed to be children’s true long-run income. Moreover, under the assumptions of the GEiV and the GEiVE models, and given what we know from previous research about the children’s ages at which $λ_{1} = 1$ and $θ_{1} = 1$ (e.g., Haider and Solon 2006; Mitnik 2019), the estimates reported as long-run estimates in the figures and tables—0.7 for the IGE of the geometric mean, 0.6 for the IGE of the expectation—are likely quite close to the estimates that would be obtained if children’s long-run income were available.

Comparing the long-run-IGE line with the IGE curves based on short-run income measures in the four figures makes apparent that the ideal bracketing strategies work as expected: In all 16 panels, without exception, the point estimate of the long-run IGE is covered by the corresponding set estimate (i.e., it is bracketed by the OLS and TSLS estimates or by the PPML and GMM-IVP estimates, as applicable), when the relevant measurement-error slope is equal to 1. This is confirmed by Table 2, which also includes results obtained with the time-varying instruments. This table also makes clear that, within IGE concepts, the location of the set estimates generated by the bracketing estimators (represented by the sets’ midpoints), as well as their width, vary substantially across instruments and parental-income measures. This variation is driven, first, by the fact that, consistent with the implications of the GEiV and GEiVE models, the OLS and PPML estimates tend to increase with the number of years of information used to compute parental income.²³ Second, there is substantial variation across instruments—some instruments generate much tighter upper bounds than others—and this variation is almost perfectly correlated across measures of parental income and IGE concepts. Father’s occupation and parents’ education when income was measured provide the tightest bounds, both household-head education variables provide the loosest bounds, and the upper bounds obtained with the at-age-15 parental-education variable are in between.

A similar analysis applies when we focus on the results of estimating the IGEs with the feasible bracketing strategies. Although all short-run estimates tend to be somewhat lower close to age 40 than at the ages at which the measurement-error slopes are equal to 1—the midpoints of the “feasible set estimates” are, on average, about 7.5 percent lower than the midpoints of the “ideal set estimates”—the former still cover, without any exception, the long-run estimates. Moreover, although the feasible set estimates are shifted downward, their widths are very similar to, and are highly correlated with, the widths of the ideal set estimates.²⁴ More generally, the short-run estimates bracket the long-run estimates at essentially all parental ages (between ages 29 and 53).

The foregoing shows that the upper bounds different invalid instruments provide may be markedly different. It also suggests that, when implementing a feasible bracketing strategy, in addition to using a short-run measure of parental income centered around age 40 or so, and based on as many years of parental information as possible, mobility scholars should generate IV IGE estimates with multiple sets of instruments rather than rely on just one instrument or set of instruments.

The set estimates that result from the approach just suggested, using the 2S and 3S estimators, are shown at the bottom of Tables 3 and 4. They put the IGE of the geometric mean of children’s family income in the 0.57 to 0.78 (2S estimator) and 0.57 to 0.83 (3S estimator) ranges, compared with the long-run IGE estimate of 0.7. Similarly, they put the IGE of children’s expected family income in the 0.52 to 0.70 (2S estimator) and 0.52 to 0.76 (3S estimator) ranges, compared with the long-run IGE estimate of 0.6. Therefore, it is clear that the bracketing strategies provide highly informative set estimates for both long-run IGEs, even with the potentially conservative 3S estimator. At the same time, it is important to keep in mind that the confidence intervals for the long-run IGEs generated by the bracketing strategies are noticeably larger than those generated by the long-run estimators: 0.48 to 0.94 compared with 0.58 to 0.82, for the IGE of the geometric mean, and 0.41 to 0.88 compared with 0.45 to 0.74, for the IGE of the expectation. This is a consequence of the fact that the confidence interval of a partially identified parameter reflects not only sampling variability but also that the location of the parameter within the identified set cannot be determined by the data, regardless of the size of the sample.

In the previous paragraph I focused on the set estimates based on short-run income measures that average seven years of parental information. Often, fewer years of information are available; in some countries, the data sets used to estimate IGEs include only one year of parental information. Tables 3 and 4 show that although the set estimates obtained with the 2S and 3S estimators are wider, they are still highly informative when they are based on five- or even three-year parental-income measures: In the latter case, the set estimates are 0.52 to 0.79 (2S estimator) and 0.52 to 0.85 (3S estimator), for the IGE of the geometric mean, and 0.48 to 0.68 (2S estimator) and 0.48 to 0.74 (3S estimator), for the IGE of the expectation. The set estimates are much wider, however, when annual income measures are used: In this case, they put the first IGE in the 0.40-to-0.76 or 0.40-to-0.82 ranges, and the second IGE in the 0.40-to-0.68 or 0.40-to-0.74 ranges, depending on the estimator.

7.3. Discussion

The results of the empirical analyses make clear that the generalized error-in-variables models underlying the bracketing strategies provide a very good account of the relationships between (1) long-run IGE estimates; (2) the short-run IGE estimates generated by the OLS, PPML, and IV estimators; (3) parents’ ages at which their income is measured; and (4) years of parental information used. Crucially, the results show that both the ideal and the feasible versions of the bracketing strategies work exactly as those models lead us to expect. The empirical analyses also confirm that, as predicted by the GEiV and GEiVE models, the lower bounds generated by the bracketing strategies become tighter as additional years of information are used to compute the short-run parental income measures.

Different invalid instruments can be expected to be differentially correlated to the logarithm of long-run parental income, and to the error terms of the long-run-IGE population regression functions (i.e., the error terms associated with equations 1 and 15). Therefore, we should also expect them to lead to different IV estimates and to provide different upper bounds for the set estimates of the IGEs. Nevertheless, the magnitude of the differences revealed by the empirical analyses is quite striking. This suggests that mobility scholars should put a good amount of effort into searching for “best invalid instruments,” as this may have a large payoff in terms of the tightness of the upper bounds supplied by the IV estimators. This may involve looking for additional instruments, beyond those typically used by mobility researchers (i.e., parental education and occupation), and exploring the effects of alternative functional forms (e.g., entering an instrument in levels or in logarithms), as this has been very consequential in some contexts (Reiss 2016).

In the empirical analyses, I simply focused on the instruments commonly used in the previous literature, but the set estimates generated by the bracketing strategies proved to be highly informative for both IGEs as long as they were based on short-run parental income measures relying on at least three years of information. At the same time, the fact that the confidence intervals for the long-run IGEs generated by those strategies are rather wide underscores the fact that obtaining satisfactory levels of precision with these strategies requires samples substantially larger than those required to obtain satisfactory levels of precision with the individual short-run estimators that the strategies combine.

8. Conclusions

The IGE conventionally estimated in the mobility literature pertains to the conditional geometric mean of children’s income, which is at odds with the interpretations imposed on its estimates. In addition, the conventional IGE makes studying gender and marriage dynamics in intergenerational processes a very difficult enterprise and leads to IGE estimates affected by selection biases. For these reasons, Mitnik and Grusky (this volume) have called for replacing it with the IGE of the expectation. This requires that the methodological knowledge necessary to estimate this IGE with short-run income variables is made available.

This article contributes to this goal by advancing a generalized error-in-variables model for the IV estimation of the IGE of the expectation that makes empirical assumptions entirely comparable to those made for the IV estimation of the conventional IGE. Analogously to what its counterpart for the IV estimation of the IGE of the geometric mean does, this model (1) provides an account of the relationship between the ages at which children’s and parents’ income are measured and the GMM-IVP estimates of the IGE of the expectation and (2) entails that when the measurement-error slopes are equal to 1, estimation of the IGE of the expectation with the GMM-IVP estimator is upward inconsistent.

By combining the latter result with Mitnik’s (2019) result that the PPML estimation of the IGE of the expectation with short-income measures is downward inconsistent in the same context, I have proposed a bracketing strategy fully equivalent to that used to set estimate the conventional IGE. The proposed bracketing strategy couples short-run estimates generated with the PPML and GMM-IVP estimators to generate a set estimate of the long-run IGE of children’s expected income. As in the case of the IGE of the geometric mean, the feasible version of this strategy relies on estimates obtained with short-run income measures pertaining to when children and parents are close to 40 years old.

Previous research that estimated bounds for the conventional IGE with the OLS and IV estimators reported separate confidence intervals for those bounds. In contrast, by considering the bracketing strategies from the perspective of the partial-identification approach to inference, I have specified how to construct confidence intervals for the partially identified long-run IGEs, in particular when the upper bound is estimated multiple times with different sets of instruments.

The results of the empirical analyses with PSID data are fully consistent with the qualitative implications of the generalized error-in-variables models underlying the bracketing strategies (both the new strategy proposed here and the strategy previously used in the literature). Crucially, those analyses evaluated the performance of the feasible bracketing strategies by comparing their set estimates with point estimates of long-run IGEs. This indicated that those strategies work exactly as expected, and that the set estimates they generate may be highly informative.

Supplemental Material

Mitnik_iv_estimation_supplementary_materials – Supplemental material for Intergenerational Income Elasticities, Instrumental Variable Estimation, and Bracketing Strategies

Supplemental material, Mitnik_iv_estimation_supplementary_materials for Intergenerational Income Elasticities, Instrumental Variable Estimation, and Bracketing Strategies by Pablo A. Mitnik in Sociological Methodology

Footnotes

ORCID iD

Pablo A. Mitnik

Notes

Author Biography

Pablo A. Mitnik is a social science research scholar at the Center on Poverty and Inequality at Stanford University. He conducts research on intergenerational mobility, labor markets, economic inequality, and statistical methods.

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