## Abstract

It has recently been argued that the intergenerational income elasticity (IGE) ubiquitously estimated in the economic mobility literature should be replaced by the IGE of expected income. This article advances a generalized error-in-variables model for the estimation of the latter IGE with short-run proxy measures of income and the Poisson pseudo-maximum likelihood estimator of constant elasticities. Empirical analyses with data from the panel study of income dynamics offer clear support for the account of lifecycle and attenuation biases the model provides. Together, the model and the associated empirical evidence supply a methodological justification for the estimation of the IGE of the expectation with proxy income variables that satisfy some conditions. This eliminates the main obstacle for making this IGE the workhorse elasticity of the economic mobility field, which would in turn dissolve the selection bias problem generated by the current expedient of dropping children with zero income from samples in order to make estimation of the conventional IGE possible.

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## Notes

These selection biases are with respect to the true IGE of the conditional geometric mean of children’s income.

Zeros are much less of a problem for short-run parental measures, as (a) they typically are multiyear averages (more on this later), (b) parents with zero income for extended periods of time are unlikely to be able to raise their kids themselves, and (c) there are good reasons to use as parental income the income of the family in which the child was raised, which is unlikely to be zero (see Hertz 2007:35). In addition, dropping parents with zero income is in principle unproblematic as it does not involve selecting on the dependent variable.

Note that children with zero income or earnings can be unproblematically kept in the sample when estimating the IGE of expected income. This is particularly apparent when Eq. (3) is rewritten as \(Y=\mathrm{exp}\left({\alpha }_{0}+{\alpha }_{1}\mathrm{ln}X\right)+{\Psi },\) where \(E\left({\Psi }|\mathrm{ln}x\right)=0\).

In the general case, the measurement errors are equal to \({\lambda }_{0t}+({\lambda }_{1t}-1)\mathrm{ln}Y+{V}_{t}\) (children) and \({\eta }_{0k}+({\eta }_{1k}-1)\mathrm{ln}X+{Q}_{k}\) (parents).

In some cases, a multiyear average of the logarithm of parental income has been used, which implies that the geometric mean of parental income over those years is the proxy measure

*S*.Nybom and Stuhler (2016) also found that lifecycle bias is very close to zero around age 40.

PML estimators are consistent regardless of the actual distribution of the error term, provided that the mean function is correctly specified (Gourieroux et al. 1984).

Santos Silva and Tenreyro (2006, 2011, Forthcoming) have argued that the PPML estimator should be preferred over other consistent estimators of constant-elasticity models of expected outcomes, as both theoretical arguments and simulations indicate that the NLLS estimator is inefficient (often to the point of being useless in empirical applications) and highly sensitive to outliers, whereas the gamma PLM estimator is very sensitive to measurement error. By contrast, the PPML estimator “is reliable in a wide variety of situations” and therefore “has the essential characteristics needed to make it the new workhorse for the estimation of constant-elasticity models” (Santos Silva and Tenreyro 2006:649); in addition, it behaves well even when the share of zeros is very large (Santos Silva and Tenreyro 2011).

The same is true of any other estimator that could be used to estimate the IGE of expected income.

In the general case, the measurement error for children is \({\theta }_{0t}+({\theta }_{1t}-1) Y+{W}_{t}\). For parents, it is \({\pi }_{0k}+ {P}_{k}\) when \({\pi }_{1k}=1\) and \({\pi }_{0k}+\left({\pi }_{1k}-1\right)\mathrm{ln}X+{P}_{k}\) in the general case.

This entails no loss of generality because it can always be achieved by changing the monetary units used to measure income, i.e., by dividing the children’s income variable by its mean and the parental income variable by the exponential of the mean of its logarithmic values minus 1.

The PSID collects income information referring to the calendar year prior to the survey. Unless I indicate otherwise, I always refer to “income years” rather than “survey years.” The PSID switched to biannual data collection in survey year 1997. For cohorts that became 56 years old in years in which data are not available, I require that the children be present in the PSID at age 55.

In order to be able to compute a reasonable approximation to long-run income and earnings, the analyses exclude children with fewer than 12 years of income or earnings (as relevant).

For the 1953 and 1952 cohorts, parental information refers to when the children were 14-20 and 15–20 years old, respectively, rather than 13–20 years old.

Ideally, I would use just one sample to study left-side and right-side biases jointly rather than separately. Unfortunately, no feasible PSID sample allows one to construct approximate long-run income measures for both parents and children. More generally, apart from the (restricted access) Swedish registry data used by Nybom and Stuhler (2016), which includes measures of family income but not of earnings, there is no other panel data that could be used with that purpose.

With the statistical packages typically used by social scientists, it is as easy to estimate the IGE of the expectation by PPML using microdata as it is to estimate the conventional IGE by OLS using such data. See Online Appendix, G, for how to estimate the former IGE with the statistical package Stata.

In the literature, controls other that ages are typically not included. The reason is that an IGE does not measure a causal effect. Rather, it is descriptive measure (e.g., Aronson and Mazumder 2008:146; Bjorklund and Jantii 2011, esp. secs. 4 and 6; Hertz 2007:26; Stuhler 2012:2), comparable in nature to, for instance, the Gini coefficient.

The estimates underlying the curves (and standard errors) as well as additional estimates from models with controls for parental age can be found in the Online Appendix, F.

A similar point was made by Nybom and Stuhler (2016: 264) for the conventional IGE. They suggested averaging income information across years (within children) when estimating the conventional IGE. This can be expected to have the same effect as pooling years of information into one sample (as I do here).

I substituted the correlation for the covariance for scaling purposes.

The estimates underlying the curves (and standard errors) as well as additional estimates generated with measures of parental income based on 1–25 years of information (at various parental ages) can be found in the Online Appendix, F.

Reliance on an asymptotic value is implicit in arguments along the lines that the fact that the differences between IGE estimates based on

*n, n*+ 1 and*n*+ 2 years of information are small and decreasing means that*n*+ 2 years are enough to eliminate the bulk of attenuation bias.

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## Funding

This study was partially funded by The Pew Charitable Trusts and The Russell Sage Foundation. “Measuring Economic Mobility with Tax-Return Data: Toward an IRS Platform” (Principal Investigators: Pablo Mitnik and David B. Grusky), 2011–2013, $176,736.

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Mitnik, P.A. Estimating the intergenerational elasticity of expected income with short-run income measures: a generalized error-in-variables model.
*Empir Econ* **65**, 2779–2803 (2023). https://doi.org/10.1007/s00181-023-02442-6

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DOI: https://doi.org/10.1007/s00181-023-02442-6

### Keywords

- Intergenerational economic mobility
- Intergenerational elasticity of expected income
- IGE
- Poisson pseudo-maximum likelihood estimator
- Measurement error in nonlinear models
- Panel study of income dynamics (PSID)