Partial Identification of Economic Mobility: With an Application to the United States

Abstract

The economic mobility of individuals and households is of fundamental interest. While many measures of economic mobility exist, reliance on transition matrices remains pervasive due to simplicity and ease of interpretation. However, estimation of transition matrices is complicated by the well-acknowledged problem of measurement error in self-reported and even administrative data. Existing methods of addressing measurement error are complex, rely on numerous strong assumptions, and often require data from more than two periods. In this article, we investigate what can be learned about economic mobility as measured via transition matrices while formally accounting for measurement error in a reasonably transparent manner. To do so, we develop a nonparametric partial identification approach to bound transition probabilities under various assumptions on the measurement error and mobility processes. This approach is applied to panel data from the United States to explore short-run mobility before and after the Great Recession.

Keywords:

• Measurement error
• Mobility
• Partial identification
• Poverty
• Transition matrices

ACKNOWLEDGMENTS

The authors are grateful for helpful comments from the editor, Rajeev Dehejia, two anonymous referees, Hao Dong, Elira Kuka, Essie Maasoumi, Xun Tang, and conference participants at Texas Econometrics Camp XXII and the LACEA-LAMES 2018 Annual Meeting.

Notes

1 See Washington Post (October 6, 2016) at https://www.washingtonpost.com/news/wonk/wp/2016/10/06/striking-new-research-on-inequality-whatever-you-thought-its-worse/?utm\_term=.83d37c53195b.

2 In closely related work, Vikström, Ridder, and Weidner (2018) study the partial identification of treatment effects where the outcomes are conditional transition probabilities. In their setup, measurement error is not considered. Rather, point identification fails even under randomized treatment assignment as treatment assignment is not guaranteed to be independent of potential outcomes in future periods conditional on intermediate outcomes. Our approach is also similar to Molinari (2008); she studies the partial identification of the distribution of a discrete variable that is observed with error.

3 Available at http://faculty.smu.edu/millimet/code.html.

4 The level of income inequality in the U.S. has followed a U-shaped pattern over the past century (Picketty and Saez 2003; Kopczuk, Saez, and Song 2010; Atkinson and Bourguignon 2015).

5 Within the partial identification literature, our analysis is most closely related to Molinari (2008), who posits a direct misclassification approach to bound the distribution of a discrete variable in the presence of misclassification errors, and studies of partial identification of treatment effects under nonrandom selection and misclassification of treatment assignment (e.g., Kreider and Pepper 2007, 2008; Gundersen and Kreider 2008, 2009; Kreider et al. 2012).

6 For example, if K = 5, then the cutoff points might correspond to quintiles within the two marginal distributions of $y_0^{\ast }$ and $y_1^{\ast }$.

7 In contrast, “perfect” mobility may be characterized by origin-destination independence, implying $p_{\mathit{kl}}^{\ast }=1/K$ for all k, l, or by complete rank reversal, implying $p_{\mathit{kl}}^{\ast }=1$ if $k+l=K+1$ and zero otherwise. See Jäntti and Jenkins (2015) for discussion.

8 Note, while the probabilities are conditional on X, the cutoff points $\zeta$ are not. Thus, we are capturing movements within the overall distribution among those with X = x.

9 ${\theta }_{(k,l)}^{(0,0)}$ may be strictly positive even though income is misreported in either or both periods (i.e., $y_{\mathit{it}}\ne y_{\mathit{it}}^{\ast }$ for at least some i and t) as long as the misreporting is not so severe as to invalidate the observed partitions (i.e., $k^{\prime }=k$ and $l^{\prime }=l$ regardless). Throughout the article, we use the term measurement error to refer to errors in observed income ($y_{\mathit{it}}\ne y_{\mathit{it}}^{\ast }$) and misclassification to refer to errors in the observed partitions ($k^{\prime }\ne k$ and/or $l^{\prime }\ne l$).

10 The expression in (5) is identical to that in Gundersen and Kreider (2008, p. 368) when K = 2.

11 In the interest of brevity, we focus attention from here primarily on the unconditional transition matrix. We return to the conditional transition matrix in Section 3.3.

12 For example, if $P_{0,1}^{\ast }$ is a 2 × 2 poverty transition matrix and all individuals over-report their income by a constant amount, then rank preservation will hold. However, some individuals may now be incorrectly classified as above the poverty line. Instead, Assumption 1 allows measurement error to be unrestricted as long as true poverty status is observed for all observations.

13 As suggested by an anonymous reviewer, two additional restrictions might also be considered in conjuction with Assumption 2. First, one might impose independence between the misclassification probabilities in the initial and terminal periods. This implies that the misclassification probabilities $${\theta }_{(k,l)}^{(k^{\prime }-k,l^{\prime }-l)}=\mathrm{Pr}(y_0\in k^{\prime },y_1\in l^{\prime },y_0^{\ast }\in k,y_1^{\ast }\in l)$$ simplify to $${\theta }_{(k,l)}^{(k^{\prime }-k,l^{\prime }-l)}={\alpha }_k^{k^{\prime }-k}•{\beta }_l^{l^{\prime }-l},$$ where ${\alpha }_k^{k^{\prime }-k}$ (${\beta }_l^{l^{\prime }-l}$) is the probability of being observed in partition $k^{\prime }$ ($l^{\prime }$) in the initial (terminal) period when the true partition is k (l). This restriction reduces the number of misclassification parameters from $K^2(K^2-1)$ to $2K(K-1)$. Second, one might wish to assume the misclassification probabilities are time invariant, implying ${\alpha }_k^{k^{\prime }-k}={\beta }_k^{k^{\prime }-k}$ $\forall k$. This restriction further reduces the number of misclassification parameters to $K(K-1)$. Both assumptions are quite strong. The former restriction requires that individuals’ misclassification probabilities are independent of their income history. However, one might suspect different misreporting propensities, say, for an individual who finds him/herself in poverty for the first time versus someone who has been in poverty throughout his/her lifetime. The latter restriction assumes that data accuracy and other sources of measurement error such as stigma are constant over the analysis period. In the interest of brevity, we leave the consideration of such restrictions to future work.

14 See Chetverikov, Santos, and Shaikh (2018) for a recent review of the use of shape restrictions in economics.

15 Note, there is no assurance that the bounds under Assumption 5, but without Assumption 6, will be narrower than the corresponding bounds without Assumption 5.

16 A fourth measure derived from the transition matrix is the immobility ratio, attributable to Shorrocks (1978). The measure is given by $$\text{IR}=\frac{K-\text{tr}\left(P_{0,1}^{\ast }\right)}{K-1},$$ where $\text{tr}(·)$ denotes the trace of a matrix. Since the trace is a function of multiple elements of the matrix—one from each row and column—bounds on IR using the upper and lower bounds on the diagonal elements of the trace under Assumption 2 are not sharp. They are sharp under Assumption 2. Future work may wish to consider sharp bounds on IR under arbitrary errors.

17 We employ sub-sampling (without replacement) rather than an m-bootstrap (with replacement), where m < N, as sub sampling is valid under weaker assumptions (Horowitz 2001). Noneless, our Stata code allows for both options. Moreover, we set $m=N/2$ as it is unlikely that an optimal, data-driven choice of m is available (or computationally feasible in the present context). Politis, Romano, and Wolf (1999, p. 61) stated that “subsampling has some asymptotic validity across a broad range of choices for the subsample size” as long as $m/N\to 0$ and $m\to \infty$ as $N\to \infty$. Martínez-Muñoz and Suáreza (2010, p. 143) note that setting $m=N/2$ is “typical.”

18 Since a K × K transition matrix entails the estimation of $K(K-1)$ free parameters, one might be concerned with issues related to multiple hypothesis testing depending on the nature of the hypotheses being considered. While not considered here, our code does allow for a Bonferonni correction if one so chooses.

19 There is no need to adjust income for household size when estimating the poverty transition matrix since the poverty threshold already accounts for differences in household composition.

20 OECD equivalized household income for an individual household is defined as $Y/N$, where Y is total household income, $N=1+0.7(A-1)+0.5C$, and A (C) is the total number of adults (children) in the household.

21 OECD-modified equivalized household income for an individual household is defined as Y/N, where Y is total household income, $N=1+0.5(A-1)+0.3C$, and A (C) is the total number of adults (children) in the household.

22 The 2004 panel contains 10,503 households observed in the initial and terminal periods. Two observations are dropped due to negative household income. The remainder are dropped because the household head is outside the 25–65 year old age range. The 2008 panel contains 21,616 households observed in the initial and terminal periods. Eighty-eight observations are dropped due to negative or missing household income. The remainder are dropped because the household head is outside the 25–65 year old age range.

23 In all cases, we use 25 replicate samples for the subsampling bias correction and 100 replicate samples to construct 90% Imbens–Manski (2004) confidence intervals via subsampling using $m=N/2$ without replacement. For brevity, we do not report bounds based on all possible combinations of restrictions. Unreported results are available upon request.

24 Throughout the analysis, poverty status is measured only at the initial and terminal period. Thus, for example, “remaining in poverty” does not mean a household is necessarily in poverty continuously over the four-year period. For expositional purposes, however, we describe the results in terms of remaining in or out of poverty.

25 Throughout the discussion of the results, unless otherwise noted, we focus on the point estimates for simplicity. The confidence intervals are generally not much wider than the point estimates of the bounds.

26 For brevity, not all combinations are presented. Full results are available upon request.

27 For brevity, Table 8 displays only the 90% confidence intervals and not the point estimates of the bounds. In addition, only the results for the individual panels are provided. All results are available upon request.