The Model

To explain how selection on observables and selection on unobservables generate sample-selection bias and how I adjust for it, I introduce a simple model of height and military enlistment. As is common in settings where researchers are concerned about selection on unobservables, I use a Tobit type-II model (Amemiya Reference Amemiya1985), which is an empirical application of the Roy model used by Bodenhorn, Guinnane, and Mroz (Reference Bodenhorn, Guinnane and Mroz2014, Reference Bodenhorn, Guinnane and Mroz2017, p. 185) to illustrate their concerns regarding selection on unobservables.

Suppose that the height of individual i from birth cohort t, h_it, is determined by

(1) $${h_{it}} = {\gamma _t} + {{{\bf{x'}}}_{it}}\theta + {\varepsilon _{it}},$$

where γ_t are cohort-specific intercepts, x_it is a vector of covariates affecting both height and military enlistment, and ɛ_it are unobserved components in the determination of height. Let y_it be an indicator variable equal to one if individual i from birth cohort t enlists and zero otherwise. Suppose that individuals enlist if and only if their latent utility of enlistment, $y_{it}^*$, is greater than zero. Let $y_{it}^*$ be determined by

(2) $$y_{it}^* = {\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k} + {\mu _{it}},$$

where α_t are cohort-specific intercepts, x_it is (as in Equation (2)) a vector of covariates affecting both military enlistment and height, z_it is a vector of covariates affecting military enlistment but not height, and u_it are unobserved components in the determination of military enlistment.Footnote ⁴ Note that the coefficients β_k and δ_k in Equation (2) are indexed by k to indicate that they are permitted to vary by cohort group (i.e., k could represent either the 1832–1846 birth cohorts, who were old enough to serve in the Civil War, or the 1847–1860 birth cohorts, who were not). I impose this structure because the nature of the enlistment decision likely differed between the cohorts that were eligible for Civil War service and those that were not. The variables y_it, x_it, and z_it are observed regardless of enlistment status,Footnote ⁵ while ɛ_it, u_it, and $y_{it}^*$ are never observed. The main challenge is that h_it is observed only if y_it = 1—that is, height is observed only if individual i enlists.

Under standard assumptions,Footnote ⁶ it is possible to write the probability that individual i enlists, given his observable characteristics (his conditional probability of military enlistment), as

(3) $$P({y_{it}} = 1|{{\bf{x}}_{it}},{{\bf{z}}_{it}};t) = G({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k}).$$

These assumptions also permit height for enlisters to be written as

(4) $${h_{it}} = {\gamma _t} + {{{\bf{x'}}}_{it}}\theta + \Omega ({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k}) + {\xi _{it}},$$

where $\Omega ({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k}) = E({\varepsilon _{it}}|{{\bf{x}}_{it}},{{\bf{z}}_{it}},{y_{it}} = 1;t)$ and ξ_it is an error term that is orthogonal to u_it. Thus, enlister i’s height is the sum of three components—the average height in the population for all individuals with the same observables $({\gamma _t} + {{{\bf{x'}}}_{it}}\theta )$, the difference in average height between enlisters and the whole population with the same observables ($\Omega ({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k})$), and a “well behaved” error term (ξ_it). Note that if ɛ_it and u_it are uncorrelated (i.e., there is no selection on unobservables) then Ω(·) = E(ɛ_it|x_it,z_it;t) = 0 and the average height of enlisters is equal to the average height in the population for individuals with the same observables.

One notable difference between my model and the diagnostic test proposed by Bodenhorn, Guinnane, and Mroz (Reference Bodenhorn, Guinnane and Mroz2017) is that I model the choice of whether or not to enlist in the military as a once-per-lifetime decision whereas they focus on the dynamic decision of military enlistment within a cohort. This is simply a matter of two different approaches to the same problem. Fundamentally, Bodenhorn, Guinnane, and Mroz (Reference Bodenhorn, Guinnane and Mroz2017, p. 173) are concerned, as am I, with a situation “in which an individual enters the sample, in part, due to the unmeasured characteristics that are related to the outcome of interest.” Their focus on the relationship of military enlistment with within-cohort changes in economic conditions is intended as a way to diagnose the presence of sample-selection bias using only the selected sample, which is not the fundamental concern of their article. Thus, although I do not directly address Bodenhorn, Guinnane, and Mroz’s (Reference Bodenhorn, Guinnane and Mroz2017) specific concern regarding changing stature over time within a birth cohort, I do address the broader concern that unobservables determined both height and enlistment, generating time-varying sample-selection bias.

The Empirical Challenge

The goal of this article is to learn the unconditional average height of birth cohort t, E(h_it|t), for all t ϵ {1832,…,1860}. If the heights of a random sample of the population were observed, it would be possible to estimate E(h_it|t) simply by computing averages for each cohort or by regressing heights on a series of birth cohort indicators (with no controls). However, because height data are available only for military enlisters, and because non-random selection into military service generates sample-selection bias, it is impossible to accurately estimate E(h_it|t) by this simple approach. Such selection comes in two forms—selection on observables and selection on unobservables.Footnote ⁷

Selection on observables stems from the impact of x_it on both military enlistment and height. If observable characteristics impact the probability of military enlistment, then their distribution in the military will be different from their distribution in the population. If these characteristics also affect height, then the distribution of military heights will also differ from that of the population. For example, if residents of urban areas are both shorter than the population average and more likely to join the military, then they will be over-represented in the military, which will, in turn, be shorter than the whole population. If all bias is due to selection on observables (i.e., ɛ_it and u_it are uncorrelated, so Ω(·) = 0), then Equation (4) shows that selection is random, conditional on observables. Thus, the average height of enlisters with a given set of observable characteristics is equal to the average height of the population with the same observables. Such selection can then be addressed by re-weighting the military data so that its distribution of observables matches that of the population. This is a standard approach in the historical heights literature (e.g., Fogel et al. Reference Fogel, Engerman and Floud1983, p. 454) and can be achieved by computing weights from aggregate data or (as I do later) from estimates of conditional enlistment probabilities from Equation (3).Footnote ⁸

The formal basis for selection on unobservables is also illustrated in Equation (4). Enlisters’ average heights, conditional on all observables, differ from those of the whole population by $\Omega ({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k})$, which is non-zero when ɛ_it and u_it are correlated with one another. Such a correlation might arise if, for instance, an unobserved adverse health shock in childhood harmed an individual’s labor market prospects, making him more likely to enter the military, and also reduced his terminal height relative to others with his same observable characteristics. Re-weighting cannot address this bias because re-weighting requires selection into the military to be random, conditional on observables; in this case, the military will over-represent shorter individuals for a given set of covariates. Addressing selection on unobservables requires the estimation of $\Omega ({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k})$ through the estimation of Equation (4) for the selected sample. Once estimated, this term can be removed, leaving the adjusted average heights of enlisters equal to the average height of the population with the same observables. Any selection on observables can then be addressed by re-weighting. I discuss the intuition of this approach later.

While sample-selection bias may cause the average height of military enlisters to differ from the average height of the population, it does not bias a naively estimated trend in average height unless its magnitude changes over time. For instance, if selection on unobservables caused military enlisters to always be one inch shorter than the population, then the trend in heights of military enlisters would be the same as the trend in heights of the population. Bodenhorn, Guinnane, and Mroz (Reference Bodenhorn, Guinnane and Mroz2014) argue that selection on unobservables might have changed over time due to economic growth in the antebellum period: as economic growth improved the attractiveness of the civilian sector relative to the military sector, only those with increasingly poor civilian labor market opportunities would choose to enlist; if success on the civilian labor market was positively correlated with height (for instance, if both were affected by childhood health shocks, as in the example noted earlier), then enlisters would be more negatively selected over time.

Intuition of the Correction for Selection on Unobservables

The historical heights literature typically corrects for selection on observables, but not for selection on unobservables (see detailed discussion by Bodenhorn, Guinnane, and Mroz Reference Bodenhorn, Guinnane and Mroz2017, pp. 173–4, 187–9). In so doing, the literature typically assumes (implicitly or explicitly) that ɛ_it and u_it are uncorrelated. In this article, I impose no such assumption, allowing ɛ_it and u_it to be potentially correlated. I then adjust observed heights for sample-selection bias induced by selection on both observables and unobservables using the methods discussed earlier.

How is it possible to recognize and correct for sample-selection bias induced by selection on unobservables? The key insight is that the magnitude of the bias will vary with the probability of military enlistment conditional on observables, as defined in Equation (3).Footnote ⁹ To see this, suppose that there is negative selection on unobservables (i.e., a negative correlation between u_it and ɛ_it) that arises, for example, from unobserved adverse health shocks in childhood making the military relatively more attractive and reducing terminal height. Suppose that there is one group of potential enlisters whose probability of enlistment conditional on observables is close to zero (i.e., some combination of observable characteristics in the population that is almost never observed to enlist). For concreteness, this group might be thought of as the children of wealthy craftsmen.Footnote ¹⁰ These individuals enlist only if they experienced a particularly strong adverse health shock, and, thus, the members of this group observed enlisting are shorter than the whole population with the same observable characteristics. On the other hand, suppose that there is a group of men whose conditional enlistment probability is close to one (i.e., some combination of observable characteristics in the population that is almost always observed to enlist). For concreteness, this group might be thought of as the children of poor unskilled laborers, for whom enlistment is more attractive than their civilian labor market options. They enlist regardless of their childhood health, and enlisters from this group are approximately the same height as the whole population with the same observable characteristics.

Thus, in the presence of negative selection on unobservables into the military height sample, there would be a positive correlation between the probability of enlistment implied by observables, and height, after conditioning on observables. The presence of such a correlation is how selection on unobservables is recognized, and precisely what is tested for by including $\Omega ({\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k})$ in Equation (4). The precise nature of the relationship of height with the enlistment probability identifies (in the statistical sense) Ω(·), enabling the determination of the magnitude of sample-selection bias.

Why are the exclusion restrictions required? That is, why is it necessary to include z_it in the military enlistment decision, but not the height determination equation, or to allow β and δ to differ by cohort group? The example earlier showed that selection on unobservables creates a correlation between the probability of enlistment and height, after conditioning on observables. But if enlistment probability is a function only of observables affecting height, then there is no variation in this probability after conditioning on observables, and, thus, no correlation can be found. The exclusion restrictions provide variation in enlistment probability while conditioning on all observables affecting height. In other words, they create different enlistment probabilities for individuals whose height-determining covariates are the same, implying that their average heights should be the same if there is no selection on unobservables.

To be concrete, let z_it represent (as it will later) political ideology. If ideology does not affect height, then all individuals with the same observables (other than ideology) should be of the same average height regardless of ideology. But negative selection on unobservables would cause individuals whose ideology renders them relatively likely to enlist to be observed to be taller than otherwise identical individuals whose ideology renders them relatively unlikely to enlist. As in the earlier example, individuals whose ideology renders them relatively unlikely to enlist do so only if they experienced a strong adverse health shock, whereas those whose ideology renders them relatively likely to enlist do so even without such shocks.

Permitting the effect of covariates on enlistment probability to differ by cohort group, but ruling out such a difference in the effect on height gives a similar advantage. Whenever the correlation of height with a covariate differs between cohort groups, this is evidence of selection on unobservables. For instance, if the conditional urban height penalty differs between cohort groups, this indicates the presence of sample-selection bias because Equation (1) permits no such variation; the only way that it could arise is if urban residence had a different impact on enlistment probability across the two cohort groups, and if this different enlistment probability led to different magnitudes of selection on unobservables.

In light of this intuition, it is instructive to consider how my model admits changing bias from selection on unobservables over birth cohorts. Because Ω(·) is a function of the same single index ${\alpha _t} + {{{\bf{x'}}}_{it}}{\beta _k} + {{{\bf{z'}}}_{it}}{\delta _k}$ as is the conditional enlistment probability, selection changes whenever conditional enlistment probability changes. There are, therefore, three mechanisms by which selection can change over cohorts: changes in the distribution of x_it and z_it over cohorts, differences between cohorts in the cohort-specific intercept α_t, and variation in β_k and δ_k across cohort groups. As discussed earlier, only changes in enlistment probability generated by the exclusion restrictions are useful in identifying Ω(·) (in a statistical sense);Footnote ¹¹ but any differences in enlistment probability over birth cohorts (generated by any of these three sources of variation) are translated into differences in sample-selection bias induced by selection on unobservables.Footnote ¹²

Military Height Data

I collected military height data for the 1832–1860 birth cohorts from two sources that have previously been used to study the Antebellum Puzzle (e.g., Fogel Reference Fogel, Engerman and Gallman1986, Table 9.6). The first is the Union Army Project (Fogel et al. Reference Fogel, Costa and Haines2000), which provides information collected at the time of entry into the Union Army (during the Civil War). I collected data from this source for individuals born between 1832 and 1846, for whom height, birth year, and age at height measurement are known.Footnote ¹⁴ It is not possible to use the Union Army data to extend the height series past the birth cohort of 1846 because younger birth cohorts would have been too young to serve during the Civil War and the Union Army was disbanded after the war.Footnote ¹⁵ To extend the height series, I collected records of enlistments in the Regular U.S. Army for the birth cohorts of 1847–1860 from the Register of Enlistments. This source contains the records of enlistments occurring between 1798 and 1914, though the birth cohorts of 1847–1860 largely enlisted after the Civil War in the 1860s, 1870s, and 1880s. To make my results comparable to those of most other studies of military stature in the United States, I restrict attention to native-born whites and exclude individuals born in the West region of the United States (e.g., Fogel Reference Fogel, Engerman and Gallman1986; Zehetmayer Reference Zehetmayer2011).Footnote ¹⁶ I also exclude enlistments before age 18.Footnote ¹⁷

Understanding the distinction between the Union and Regular Armies is crucial. The Union Army was a special temporary force raised during the Civil War (1861–1865). It was a citizen’s army, comprising at some point of nearly half of the eligible population, and reaching a peak strength of about one million men (Weigley Reference Weigley1967, p. 267). Enlisters in the Union Army were drawn from all walks of life and were largely inspired to enlist by patriotism and wartime fervor (Weigley Reference Weigley1967). The Regular Army is the same professional U.S. Army that exists today. During the Civil War, enlistment in either the Regular or the Union Army was possible, but the vast majority of individuals serving during the war joined the Union Army.Footnote ¹⁸ Indeed, despite identical pecuniary incentives for enlistment, there was difficulty in maintaining the strength of the Regular Army during the Civil War. To my knowledge, no prior work exists on the composition of enlisters in the Regular Army during the Civil War (historians understandably focus on the Union Army). Between 1865 and 1898, all Army enlistment was in the Regular Army, which never exceeded a strength of 60,000 men (Department of Defense 1997, Table 2-11) and only included about 2 percent of the eligible population in the 1847–1860 cohorts. Enlisters in the postbellum Regular Army were generally drawn from lower social strata and enlisted largely due to a lack of civilian alternatives (Coffman Reference Coffman1986; Foner Reference Foner1970; Weigley Reference Weigley1967).

The revealed preference for service in the Union Army when enlistment was possible in both it and the Regular Army is evidence of substantial differences between the two forces. C. Joseph Bernardo and Eugene H. Bacon (1955, pp. 201–6) hypothesize that the Union Army was preferred because, as compared to the Regular Army, it had a shorter term of service, less rigorous discipline, and the freedom to elect officers. Conditions of service in the postbellum Regular Army were also poor in comparison to those experienced in the Union Army (Coffman Reference Coffman1986, pp. 328–9). Moreover, while volunteers in the Union Army were held in high esteem by the public, the Regular Army (outside of the war years) was more poorly perceived.

These differences underlie my decision to allow the effect of covariates on the military enlistment decision to differ between the 1832–1846 and the 1847–1860 cohort groups: the former group faced wartime patriotic motivations for enlistment and had the option to enlist in the Union Army; the younger cohorts only had the option to enlist in peacetime in the Regular Army. These differences also make concrete precisely the selection problems that concern Bodenhorn, Guinnane, and Mroz (Reference Bodenhorn, Guinnane and Mroz2017) and that this article addresses. This clear change in the incentive for military enlistment at the end of the Civil War makes it natural to suspect that the birth cohorts old enough to serve in the Civil War (the 1832–1846 cohorts) might have been differently selected than were those enlisting in the postbellum period (the 1847–1860 cohorts), even after conditioning on their observable characteristics. In particular, the worse conditions of service and contemporary reports (e.g., Coffman Reference Coffman1986, p. 329) of the poor labor market attributes of enlisters suggest that selection into military service may have become more negative after (as compared to during) the war.

Selection into military service may also have differed between birth cohorts within each of these cohort groups. With the enlistments in the Union Army taking place over a short period and attracting individuals from many different birth cohorts (and, thus, ages), it is possible that enlistment may have appealed differently to individuals of different ages. Similarly, postbellum enlistment in the Regular Army may have appealed differently to individuals in different birth cohorts depending on the state of the civilian labor market in their prime years (as Bodenhorn, Guinnane, and Mroz Reference Bodenhorn, Guinnane and Mroz2014 argue). Drawing a consistent trend in stature over time despite these changes is difficult. This is the heart of Bodenhorn, Guinnane, and Mroz’s (Reference Bodenhorn, Guinnane and Mroz2017) concern, and addressing it by permitting selectivity to differ is the contribution of the present article.

Census Data

Data on the covariates x_it for military enlisters were collected through linkage of their military data with U.S. census records from their childhood and adolescence. Enlisters in the Union Army sample have already been linked to the U.S. Censuses of 1850 and 1860 by Fogel et al. (2000), but no previous linkage exists for the Regular Army enlisters whose information I extracted from the Register of Enlistments. I, therefore, linked the latter group to the U.S. Censuses of 1860–1880 (using the procedure in Online Appendix E.1) and transcribed census information for the linked individuals and their households. For both the Union Army and the Regular Army I retain only individuals for whom census information could be located. Except for height and age of height measurement, all data pertain to the individual or his household as observed in the census in which he was aged 9–18, or to his county of residence in that census.Footnote ¹⁹ Census linkage provided information on the property ownership of the enlister’s household, his place of residence, the size and composition of his household, the occupations of the members of his household, and his school attendance.

The use of census-linked data raises two concerns. One is that non-random failure to link enlisters to the census may introduce bias—that is, the linked data might not be representative of the military data as a whole. I study this possibility in Online Appendix E.2. Based on the fact that the trends in stature in the linked data and in the complete collection of military data are nearly identical, I conclude that my results are unlikely to be affected by selection into the linked sample.Footnote ²⁰ Another concern stems from the fact that, unlike the Union Army data, which are hand-linked by genealogists, my linkage of the Regular Army data is automated. Martha Bailey et al. (Reference Bailey, Cole and Henderson2017) have recently shown that the rates of false links arising from automated linkage may be high.Footnote ²¹ To address this concern, I repeat my main analysis limiting the data to either hand matches (i.e., all of the Union Army data, where false positives are not a concern) or to exact automated matches, which are less likely to generate false positives. This exercise is discussed in Online Appendix G, where I show that limiting the sample in this way does not meaningfully affect the results.

The census-linked military data form the two choice-restricted samples—one for the 1832–1846 birth cohorts, generated by linking Union Army enlisters to the census, and one for the 1847–1860 birth cohorts, generated by linking the Regular Army enlisters to the census. For each of these two samples, a supplementary sample of the covariates of the population at risk for enlistment is required. To create such samples, I collected information on the covariates x_it for a random sample of the population at risk for enlistment for the 1832–1860 birth cohorts from the public use samples of the 1850, 1860, and 1870 censuses (Ruggles et al. Reference Ruggles, Genadek and Goeken2015). I again restrict attention to native-born (outside of the West region) white males observed between ages 9 and 18. Dividing the random census sample along the same birth cohorts creates two supplementary samples.

For both the choice-restricted and supplementary samples I limit attention to individuals residing (at ages 9–18) in either non-seceding states or Virginia (i.e., I omit residents of Confederate states other than Virginia). I impose this restriction because residents of the excluded states are only rarely observed in Union Army, making it difficult to estimate their average heights. I impose the same restriction on the Regular Army data, which would have included individuals from these states, for comparability.

I also collected county-level data on agricultural production and population from Steven Manson et al. (Reference Manson, Schroeder and Van Riper2017).

Voting Data

The Civil War was fought over the issues of slavery and preservation of the Union. Similarly, after the Civil War, one of the Regular Army’s main duties was Reconstruction—the military occupation of the South. It is not hard to imagine that military service during these periods might have been attractive to those who had been opposed to slavery or supportive of preservation of the Union. Although it is not possible to observe political ideology, it is possible to observe a proxy—county-level voting patterns in the U.S. Presidential Election of 1860, which also centered on the issues of slavery and preservation of the Union. Thus, it is likely that these voting patterns are informative regarding the military enlistment decision.

Election data have previously been used to predict military enlistment and desertion in the mid-nineteenth-century United States. Dora L. Costa and Matthew E. Kahn (Reference Costa and Kahn2003) relate voting data from the elections of 1856 and 1860 to the probability of desertion from the Union Army, finding that enlisters from counties with greater support for Republican candidates were less likely to desert. Similarly, Costa and Kahn (Reference Costa and Kahn2007) use voting data from the 1864 election to measure a community’s support for the Civil War, finding that deserters from communities with greater support for the war were more likely to migrate after the war, and were more likely to settle in more anti-war communities, as measured by voting. The relevance to desertion suggests relevance to enlistment. Shari Eli, Laura Salisbury, and Allison Shertzer (Reference Eli, Salisbury and Shertzer2018) show that voting patterns in 1860 are predictive of the enlistment decisions of Kentuckians in the Civil War and of the subsequent migration of Civil War veterans.

Based on the likely impact of ideology, as measured by voting patterns, on military enlistment, I use county-level voting data for the Presidential Election of 1860 (ICPSR 1999) for identification of the sample-selection model (that is, to act as the excluded variable z_it). In particular, I focus on the fraction of the individual’s county of residence (at the time that he is observed in the census between ages 9 and 18) voting for Abraham Lincoln in 1860. To be a valid exclusion restriction, this variable must satisfy two conditions. First, it must be related to military enlistment—that is, it must actually generate variation in the probability of enlistment. Given the earlier discussion, and the fact that Lincoln represented one extreme on the issue of slavery, it is plausible that voting patterns in this election should be related to military enlistment. The second requirement is that ideology (as proxied by voting) should be excludable from the determination of height; that is, conditional on all the observed covariates of height, voting patterns should be unrelated to height. Intuitively, the fact that I control for the available socioeconomic variables makes it likely that any factors that would cause voting and health to be correlated will be captured. Both relevance and excludability will be formally explored in the empirical analysis.