Selection on Observables

Consider the example of trying to determine the average height of the (northern) US population and the unconditional difference in average stature between Midwesterners and Northeasterners from a sample of military data.Footnote ⁹ Without information on how the sample was formed, it is impossible to determine whether the average stature of the military reflects that of the population, or whether any observed difference in average stature between regions reflects a true difference in the heights of the populations of each region, sample-selection bias induced by differences in selection into the sample across regions, or some combination of these two forces. That is, in the absence of information about how individuals came to enter the military, it is impossible to draw conclusions regarding population average stature from the average stature of the sample. In some countries’ data, this challenge is overcome by conscription: if everyone (or a randomly selected group) were required to serve in the military, then observed heights could be taken as representative of those of the population.

However, if military enlistment was the product of individual choice, as in Britain and the United States in the nineteenth century, then the translation of the average stature observed in the sample to the average stature of the population, and thus the determination of the Northeast–Midwest difference in average stature, is less straightforward. To see this, consider the following example: (1) each region is divided into an urban and a rural sector; (2) ruralists are, on average, taller than urbanites; (3) there is no difference in the average heights of individuals of the same sector across regions; (4) the only other determinant of height is genetic variation that is the same in each region-sector and averages away in random samples; and (5) the fraction of the population that is rural is greater in the Midwest than in the Northeast, implying greater average stature in the Midwest than in the Northeast. Panel A of table 1 presents an example of average heights satisfying these conditions. These are the true average heights—what the researcher wishes to learn but does not observe.

Table 1. Example height distributions

Notes: Panel A describes the population of interest. Columns 1 and 2 describe the average heights of each region-sector, and columns 3 and 4 describe the distribution of each region’s population across these sectors (so that each row sums to one). Column 5 of panel A shows the true average height of each region, and thus the true difference in average heights between regions. Columns 3 and 4 of panel A are observed, but the other columns are not. Panel B describes the observed population—the military enlisters. The contents of all five columns are observed, but because the greater tendency of urbanites to enlist causes columns 3 and 4 to differ from panel A, the observed average height of each region and the difference between them does not match the true difference in panel A.

Consider first the extreme case in which only urbanites enlist in the military. Both regions’ average heights would thus be understated, leading the researcher to underestimate the average stature of the population. A less extreme case allows both urbanites and ruralists to enlist, but retains the greater tendency for urbanites to enlist relative to ruralists. Such an example is illustrated in panel B of table 1. As the observed data would overrepresent urbanites relative to the population, the average stature of each region as observed in the enlistments would again understate the true stature, leading to an underestimate of the average stature of the population. If the urban status of enlisters is observed, then this is an example of selection on observables because the variable driving the nonrepresentativeness of the data (urban or rural status) by impacting both height and the probability of enlistment is observed.

The first pattern that sample-selection bias creates in a data source is evident from this example.

Pattern 1.Selection on observables occurs whenever an observable characteristic that affects the outcome of interest is over- or underrepresented in the sample relative to the population of interest—that is, whenever an observable characteristic that affects the outcome also affects entrance into the sample.

Such selection on observables would also affect the estimated Northeast–Midwest height difference. In the extreme example of enlistment only by urbanites, the observed heights of Northeasterners and Midwesterners would be the same despite the true Midwestern height advantage. In the less extreme example of panel B of table 1, the regional differences in the sample would also not reflect regional differences in the population. In this example, selection on observables causes the observed difference in the heights of Midwesterners and Northeasterners (0.80 inches in panel B) to differ from the actual difference (1.00 inch in panel A).Footnote ¹⁰

The researcher must make two determinations to ascertain whether selection on observables is likely present. The first is whether any given characteristic affects entry into the sample. Comparing the potentially selected sample to the random sample of the population (one of the two additional pieces of information discussed in the preceding text) enables the researcher to determine the factors affecting entry into the sample. If such a population sample is not available, it is possible to compare sample fractions to population fractions,Footnote ¹¹ or to use theoretical or other knowledge of the environment in question if no other data are available.

The second is whether a given factor affects the outcome of interest. This is often known on theoretical grounds. It can also be determined from the observed data. If there is no selection on unobservables, then, as in the example, the sample is random conditional on the observables, and a simple regression analysis can reveal the relationship between observables and the outcome.Footnote ¹² The lack of selection on unobservables in this example implies that the sample within each region-sector is random and observed heights represent actual heights in that region-sector. The only problem is that the fractions of each sector in the sample differ from those in the population. If the population fractions are known (as in this example), then it is possible to compute true average stature by combining observed stature for each region-sector with its population fraction. That is, in table 1, the researcher can compute population average heights using panel B’s height data in columns 1 and 2, and panel A’s fractions of the population in columns 3 and 4.Footnote ¹³

As a result of the relatively small data requirements to do so, sample-selection bias induced by selection on observables is relatively simple to recognize and address. Indeed, this is commonly done in anthropometric history (e.g., Fogel Reference Fogel, Engerman and Gallman1986; Fogel et al. Reference Fogel, Engerman, Floud, Friedman, Margo, Sokoloff, Steckel, Trussell, Villaflor and Wachter1983).

Selection on Unobservables

If enlisters’ sector were not observed by the researcher, then the example in table 1 would be a case of selection on unobservables because an unobserved factor (in this case, sector) affects both height and entrance into the sample. The researcher would observe only the height difference in column 5 of panel B of table 1, and would not know how much of this difference is a true difference and how much is the product of selection on unobservables. More fundamentally, the researcher would have no information on whether the average height of the sample reflects that of the population. Such selection can arise even if there is no selection on observables, or even if a sample overrepresents portions of the population that the researcher is interested in studying.Footnote ¹⁴

This bias can be better illustrated with another example. For this example, remove the urban–rural distinction so that all individuals are in the same sector and the distribution of heights is the same in each region. Instead, assume the following: (1) individuals differ in their wages, and only those with wages below a particular threshold enlist in the military; (2) lower wages imply lower stature; (3) average wages are higher in the Northeast; and (4) the relationship between height and wages is the same in each region once accounting for regional differences in wages. Figure 1 illustrates this example.Footnote ¹⁵ Higher wages in the Northeast are evident from the rightward shift of its wage–height relationship relative to that of the Midwest. The same distribution of heights in each region is illustrated by the same range of each line (and an implicit assumption of a uniform distribution along the line).

Figure 1. Hypothetical relationship of wages, heights, and military enlistment.

The most important assumption made in this example is that only individuals below a certain wage threshold are observed.Footnote ¹⁶ The intuition would be analogous if it were instead assumed that enlisters were all above a particular threshold (this would generate positive selection rather than negative selection as in this example). However, it is crucial to assume that enlistment comes from one extreme or the other of the wage distribution. All the analysis in the following text (as well as the method of Heckman Reference Heckman1979) would fail if enlistment, for instance, came only from both extremes of the wage distribution (but not its center), or excluded its extremes.

Figure 1 shows that the relationship between wages and stature implies that only the shorter members of each region tend to join the military, leading the researcher to understate the average stature of the population from the observed data. Moreover, figure 2 shows that Midwesterners are more likely to enlist, as evidenced by the greater share of the Midwest’s line that is below the cutoff for enlistment $\overline w $. Higher wages in the Northeast imply that there is a range of heights such that the wages are low enough to enlist in the Midwest but not in the Northeast (the range $B$ to $C$). This would make the Midwest appear taller in the enlistments data—Midwesterners would have an observed average height of ${1 \over 2}\left( {C + A} \right)$ while Northeasterners would have an observed average height of ${1 \over 2}\left( {B + A} \right)$—even though it was assumed above that the distribution of height is the same in each region. If wages are not observed, then this is a case of selection on unobservables.Footnote ¹⁷

Figure 2. Hypothetical relationship of wages, heights, and military enlistment with hawks and doves.

This example provides the second pattern generated by sample-selection bias in the data.

Pattern 2.Selection on unobservables that differs across groups creates differences across groups in the probability of entering the sample.

It is important to note that this pattern is merely suggestive. Different probabilities of entering the sample (i.e., different enlistment probabilities) need not imply selection on unobservables. If wages were unrelated to height, then there would be a different enlistment probability in each region but no selection on unobservables. But under the assumption of selection from only one extreme, there cannot be selection on unobservables without such a difference in the probability of entering the sample.

The value of this pattern is that it is typically easy to check for it in most data sources. It is essentially the same as determining whether selection on observables has occurred, but the observable is the indicator of group. It is not possible, however, to determine what the likely direction of selection bias is, though it may be possible to guess based on outside knowledge of the institutional environment.

But in the absence of additional information, there is ultimately no way for the researcher to know whether the average stature of the sample over or understates that of the population. There is also no way for the researcher to know whether the difference in stature observed between regions in this example is because of different incentives to enlist (the true reason) or because of differences in health between regions, as the literature on historical heights has usually interpreted such results. All that the researcher knows is that Northeasterners in the data are shorter on average than Midwesterners in the data.

A more definitive check for the presence of sample-selection bias, as well as the ability to determine the direction of the bias induced by selection on unobservables, is possible with additional information if the information satisfies certain conditions. Continuing the example depicted in figure 1, make the following additional assumptions: (1) the population is divided between hawks and doves; (2) hawk–dove status in the population and in the military is observable; (3) the division between hawks and doves is independent of height and wage so that the distribution of heights and wages in each region is the same between hawks and doves; and (4) the threshold wage for hawks’ enlistment is higher than that of doves.Footnote ¹⁸ This is a simplification of the idea that ideology played a role in driving military enlistment in the Civil War (Zimran Reference Zimran2019). The crucial assumption here is that hawks and doves differ only in their likelihood of entering the sample and not in their heights. Hawk–dove status is known as an excluded variable.Footnote ¹⁹ This excluded variable is the essential piece of information that enables the researcher to uncover selection on unobservables.

The value of the hawk–dove division in addressing the sample-selection problem stems from the following insight, illustrated in figure 2. While doves enlist only if their wages are below ${\overline w^D}$, hawks with wages in the range $[{\overline w^D},\,{\overline w^H}]$ also enlist (as well as hawks with wages below ${\overline w^D}$). This implies that hawks have a higher probability of enlistment. It also implies that observed hawks (in the military) would be taller than observed doves (again, in the military) in each region despite there being no relationship of hawk–dove status with height in the population: hawks in the Northeast who are observed in the military include individuals of heights $A$ to $D$, while observed doves in that region include only those of heights $A$ to $B$; similarly, observed hawks in the Midwest include individuals of heights $A$ to $E$, while observed doves in that region include only those of heights $A$ to $C$. That is, hawk–dove status has no relationship to height in the population; but because it affects the military enlistment decision, bringing individuals with wages between ${\overline w^D}$ and ${\overline w^H}$ into the sample, the observed average height of hawks is greater than that of doves.

This observed difference in height between hawks and doves is the third pattern created by sample-selection bias.

Pattern 3.Selection on unobservables causes a variable that affects selection into the sample but is unrelated to outcome of interest in the population to be related to the outcome in the sample.

Pattern 3 is essentially the logic underlying the “diagnostic test” proposed by Bodenhorn et al. (Reference Bodenhorn, Guinnane and Mroz2017). The implicit assumption made in that case is that, within a birth cohort, year of enlistment should be unrelated to population height, but is related to the probability of entering the sample. But pattern 3 is more general than that of Bodenhorn et al. (Reference Bodenhorn, Guinnane and Mroz2017), which would capture selection on unobservables only from the mechanism on which they focus—changing incentives for enlistment over birth cohorts.Footnote ²⁰

As with Bodenhorn et al.’s (Reference Bodenhorn, Guinnane and Mroz2017) test, pattern 3 can be identified in the selected sample alone under the assumption that the excluded variable affects entry into the sample and not the outcome. That is, only one of the two added pieces of information—the excluded variable—is necessary. But data on the population of interest can be used to test the assumption that the excluded variable affects entry into the sample. Specifically, comparison of the excluded variable in the sample and the population can more definitively determine that the variable in question is related to the probability of entering the sample. It is not possible to determine whether this variable has no relationship to the outcome in the population, so it must be assumed. This assumption is important because if the two were related in the population, then the hawks’ height premium in the sample could also reflect an actual height premium for hawks in the population as much as a role of ideology in driving enlistment and thus selection on unobservables.

This excluded variable can also reveal the direction of the bias induced by selection on unobservables. As illustrated in figure 2, military enlisters are negatively selected on their unobservables—that is, only the shortest enlist. But this is not directly observed. Nonetheless, the fact that hawks are more likely to enlist than are doves, and that observed hawks are taller than observed doves within each region, indicates that the selection on unobservables uncovered by pattern 3 must be negative. Under the crucial assumption of selection from a single extreme (of the wage distribution), hawks draw in a greater fraction of their respective populations to enlistment. The fact that doing so brings in taller individuals implies that enlistment must be primarily from the bottom of the height distribution. Had hawks instead been observed to be shorter than doves, that would indicate that selection into military service was positive.Footnote ²¹ This is the fourth pattern created by sample-selection bias in the data.

Pattern 4.If individuals whose value of the excluded variable makes them more likely to enter the sample are observed to be taller in the selected sample, then selection on unobservables is negative, and vice versa.

This pattern is again something that can be determined from only the selected sample if the effect of the excluded variable on the probability of entering the sample is known or assumed. But as with pattern 3, data on the population at risk to enter the sample enable the researcher to compare the selected sample to the population and more definitively to determine whether and how the excluded variable affects entrance into the sample.

Finally, suppose that there is a third group of individuals (“zealots”) who enlist regardless of their wage, and continue to assume that the membership in the hawks, doves, or zealots is observed and unrelated to height. In this case, it is possible to learn the true heights of each region simply from the zealots. More generally, the bias in the observed height of each region is decreasing as the probability of entering the military increases from doves to hawks to zealots (where there is no selection on unobservables) and a greater fraction of the group is observed. The fifth pattern induced by sample-selection bias is generated by this example.

Pattern 5.The more predisposed individuals are to be observed on the basis of their observable characteristics, the less is the sample-selection bias induced by selection on unobservables among these individuals. If there are individuals whose observable characteristics so strongly predispose them to enlist that their unobservables are unimportant, then there is no sample-selection bias among these individuals.

This pattern shows that it is sometimes possible to solve the sample-selection problem by using only a limited portion of the data, though inference from this smaller sample will be less precise due to the smaller sample size. In general, a sample of the population at risk for observation is necessary to determine if any portion of the sample has sufficiently high population of entering the sample to perform such an analysis.

Patterns 2–4 can also shed light on how bias from selection on unobservables affects the Northeast–Midwest height difference in the sample. Pattern 2 showed that selection on unobservables that differed between regions created differences in the probability of entering the sample across regions. But the researcher could not be certain that such differences indicated selection on unobservables because there was no way of knowing whether the lines in figure 1 were upward sloping (leading to negative selection on unobservables) or flat (implying no selection on unobservables)—that is, whether higher wages are associated with greater height. However, with patterns 3 and 4 revealing negative selection on unobservables into observation, the researcher can conclude that the lines are upward sloping—wages and height are positively correlated. The greater probability for Midwesterners to be observed than Northeasterners thus draws in people of higher wage in the Midwest, and implies more negative selection into observation in the Northeast than in the Midwest. The Midwest’s height premium is thus overstated. Indeed, despite there being no such premium by assumption, the different patterns of enlistment cause observed Midwesterners to appear taller in the observed data.

Multivariate Settings

In the preceding discussion, I have made the simplifying assumption that there are no observable characteristics that affect both the outcome and the probability of entering the sample. This assumption is helpful in clarifying the intuition and deriving the patterns, but it is unrealistic in practice. Relaxing this assumption requires some clarification of patterns 2, 3, and 4 to fit a multivariate context.

Pattern 2 used a difference between regions in the probability of entering the sample to suggest the presence of selection on unobservables that differs between regions. In a multivariate setting, it is possible for selection on unobservables to differ between regions without a difference in the probability of entering the sample by region. Instead, the difference that would arise would be in the conditional selection probability—the probability that an individual is observed given his observable characteristics. Selection on unobservables that differs between regions would result in a different distribution of these probabilities by region, which would generally, but not necessarily, result in a difference in the fraction of each region that is observed in the sample. Thus, in a multivariate setting, pattern 2 should be considered suggestive, and more information can be gleaned from examining the conditional selection probabilities, as will be done in the empirical exercises in the text that follows.

Pattern 3 allows the researcher to detect selection on unobservables by looking for a correlation in the sample between the outcome and the excluded variable. In a multivariate setting, in which variables other than the excluded variable affect entrance into the sample, it is the relationship between this variable and the outcome, conditional on all observables affecting the outcome, that is important.Footnote ²² Failure to control for observables might spuriously create a relationship. This is a relationship that can be tested using only the selected sample, though again data on the population can establish the relevance of the excluded variable to entrance into the sample.

Pattern 4 allows the researcher to determine the direction of the sample-selection bias induced by selection on unobservables from the sign of the correlation of the outcome and the excluded variable. With other observables, the relationship, as mentioned in the preceding text, is conditional on other observables.Footnote ²³

Sources

The data for this analysis are taken from four main sources. The first is the potentially selected sample including the stature data (the outcome of interest) and covariates for military enlisters. It is based on Records of the Adjutant General’s Office (1861–65). Data from this source are the products of two collections, each of which provides a random sample of enlisters in the Union Army, including data on stature, age at enlistment, date of enlistment, place of birth, place of enlistment, and occupation at the time of enlistment. The first is Fogel et al.’s (Reference Fogel, Costa, Haines, Lee, Nguyen, Pope, Rosenberg, Scrimshaw, Trussell, Wilson, Wimmer, Kim, Bassett, Burton and Yetter2000) Union Army Project, which provides information on a random sample of 16,285 enlisters. The second is Cuff’s (Reference Cuff2005) data set, which adds information on an additional 10,304 enlisters from the state of Pennsylvania.Footnote ²⁴ The total number of observations is thus 26,589.Footnote ²⁵ The oversampling of Pennsylvanians is a form of selection on observables. Because this bias is not that which typically concerns scholars in historical heights (because it is not generated by individuals’ choices regarding enlistment), I simply weight all analyses so that the distribution of states of enlistment in the data matches that of the Union Army (Gould Reference Gould1869).Footnote ²⁶ I limit the data to native-born white males in the birth cohorts of 1820 to 1846, who were born and lived in the Northeast and Midwest, and who were at least 18 years old at the time of enlistment. Because the place of enlistment will be treated as the place of residence in the following analysis, I also exclude individuals who enlisted in a state other than the state of their regiment.Footnote ²⁷

The second source provides the description of the observable characteristics of the population at risk for military enlistment but not their height. Specifically, it is the 1 percent sample of the 1860 US Census (Ruggles et al. Reference Ruggles, Genadek, Goeken, Grover and Sobek2015). When applying the same filtering criteria as applied to the military data, this data set includes 28,205 individuals. It provides information on age, place of residence, and occupation.Footnote ²⁸

The third source is a collection of county-level data from the Census of 1860, provided by Manson et al. (Reference Manson, Schroeder, Van Riper and Ruggles2017), which gives information on county-level agricultural and manufacturing production and capital stocks, wealth, and population density. This information is assigned to individuals in the census sample based on their county of residence and to individuals in the military data based on their county of enlistment.

The fourth and final main source (ICPSR 1999) provides data on the excluded variable—voting patterns in the presidential election of 1860. The main variable of interest in this case is the share of each county’s vote cast for Abraham Lincoln, the Republican candidate. These data are assigned to individuals in each sample in the same way as the county data from Manson et al. (Reference Manson, Schroeder, Van Riper and Ruggles2017). As the variable affecting entrance into the sample but assumed to have no direct effect on the outcome, these voting data are crucial to the exercises that follow and to implementing the insights of patterns 3 and 4 mentioned previously. The impact of the political ideology represented by the vote for Lincoln on military enlistment will be demonstrated empirically in the following paragraphs (and has also been shown by Costa and Kahn Reference Costa and Kahn2003, Reference Costa and Kahn2007; Eli et al. Reference Eli, Salisbury and Shertzer2018; Zimran Reference Zimran2019), and is unsurprising given that the Civil War was fought over the same issues that defined the 1860 election. The lack of a direct effect of ideology on height is untestable and must be assumed.Footnote ²⁹ It is justified (as in Zimran Reference Zimran2019) by the claim that any association between the two would likely be the product of socioeconomic characteristics, which can be included as controls in any regression.

Summary Statistics

Figure 3 presents the distributions of observed heights of Midwesterners and Northeasterners, combining the Fogel et al. (Reference Fogel, Costa, Haines, Lee, Nguyen, Pope, Rosenberg, Scrimshaw, Trussell, Wilson, Wimmer, Kim, Bassett, Burton and Yetter2000) and the Cuff (Reference Cuff2005) data with the use of the Gould (Reference Gould1869) weights. A height premium for the Midwest is evident.Footnote ³⁰

Note: This figure presents the estimated height distributions for each region, combining both sources of military data and weighting observations to match Gould’s (Reference Gould1869) distribution of states.

Figure 3. Height distributions by region.

Table 2 presents summary statistics for variables observed for both the military and the population at risk for enlistment, as well as height, which is observed only for the military sample. Columns 1 and 2 present summary statistics for enlisters in the Midwest and the Northeast; column 3 presents difference-in-means tests comparing columns 1 and 2; column 4 presents summary statistics for all individuals in the military data; columns 5 and 6 present summary statistics for the population of the Midwest and the Northeast from the 1860 census sample; column 7 presents difference-in-means tests comparing columns 5 and 6; column 8 presents summary statistics for all census data; and column 9 presents difference-in-means tests comparing columns 4 and 8. In all cases, the enlistment data are weighted so that the distribution of states of enlistment matches the distribution presented by Gould (Reference Gould1869).

Table 2. Summary statistics

Significance levels: ^a p < 0.01. ^b p < 0.05. ^c p < 0.10.

Notes: All figures in the enlistments are weighted to match Gould’s (Reference Gould1869) distribution of states of enlistment. Standard deviations in parentheses. Standard errors, clustered by county, in square brackets.

The first row of table 2 confirms the insight given by figure 3—Midwesterners were taller than Northeasterners in the observed sample by 0.73 inches. The second row of the table compares the vote shares for Lincoln. Northeasterners’ counties of residence, both in the military and in the population, had a greater vote share for Lincoln than those of Midwesterners. This difference is about 7 percentage points in the military sample and about 4 percentage points in the population. Comparing the enlisters to the population (column 9) reveals virtually no difference between them on the basis of the voting variables, though this is only the unconditional difference.

Except for differences in the regional representation (the enlistment sample statistically significantly overrepresents the Midwest by about 7 percentage points) and in terms of birth year (enlisters are, on average, about 2.6 years younger), none of the other county-specific variables exhibits a large or statistically significant difference between the enlisting population and the census. For the only individual-level variables that are observed, the occupational indicators, there are differences between enlisters and the complete population. In particular, the enlisted sample overrepresents farmers and the unskilled, and underrepresents those with white-collar occupations. These patterns are typical of military enlistment in the nineteenth century (e.g., Margo and Steckel Reference Margo and Steckel1983; Zehetmayer Reference Zehetmayer2011; Zimran Reference Zimran2019). But the comparison between the occupations of the military and census samples is complicated by the fact that they are observed up to five years apart and thus may not be directly comparable.Footnote ³¹

Selection on Observables

Table 2 provides some suggestive evidence pertaining to pattern 1—that selection on observables arises when factors affecting height are over- or underrepresented in the sample. Column 9 of table 2 shows that, for instance, higher skill occupations are underrepresented, indicating selection on observables if occupational skill is correlated with height in the population.

A more formal test for this pattern is given in table 3, which presents two sets of regressions describing the relationship between the observable characteristics described in table 2, on the one hand, and military enlistment and observed height, on the other. Columns 1–4 present the results of probit regressions for the probability of military enlistment, with columns 1 and 2 including a Midwest indicator, and columns 3 and 4 including state-specific fixed effects.

Table 3. Relationship of covariates to enlistment probability and observed heights

Significance levels: ^a p < 0.01. ^b p < 0.05. ^c p < 0.10.

Notes: Dependent variable is an indicator for military enlistment in columns with the header “Enl” and height in inches in columns with the header “Height.” The sample in columns 1–4 includes all individuals with height data or in the census sample, excluding residents of Missouri, Minnesota, and Rhode Island. Columns 5–9 include only individuals among these who are from the military sample. All specifications include birth year fixed effects and all specifications with height as the outcome also include age-of-measurement fixed effects to standardize age of measurement to age 21. In all specifications, the enlistment data are weighted to match the distribution of states of enlistment. Standard errors in parentheses, clustered at the county level.

Due to the unusual structure of the sample, columns 1–4 are not estimated by an ordinary probit regression, though the interpretation of the coefficients is the same. In the standard setting, the researcher observes a random sample of the population with the military enlistment status of all individuals. In this setting, I observe a sample of military enlisters and their covariates and a sample of the complete population with their covariates but without information on the military enlistment decision.Footnote ³² Following Zimran (Reference Zimran2019), I use Cosslett’s (Reference Cosslett, Manski and McFadden1981) method to estimate the model; I also use Zimran’s (Reference Zimran2019) weights, reflecting the general probability of entering the sample, which are necessary for this estimation.Footnote ³³

Columns 1–4 of table 3 show that, all else equal, individuals from counties with greater manufacturing production per capita were more likely to enlist. Individuals with skilled or unskilled occupations, or who were farmers, were also more likely to enlist than were individuals with white collar occupations (the excluded group).

Columns 5–8 present OLS regressions for the correlates of height in the military data without any correction for potential bias from selection on unobservables. Columns 5 and 6 include a Midwest indicator and columns 7 and 8 include state-specific fixed effects. All four of these specifications indicate that individuals from counties with greater population density and greater agricultural output per capita tended to be shorter. Moreover, individuals reporting an occupation of farmer at enlistment were, on average, about 0.30 to 0.35 inches taller than individuals reporting other occupations.

These results relate to pattern 1. For instance, the fact that farmers were taller and more likely to enlist than the white-collar workers implies that there was a variable affecting both entrance into the sample and height, and thus there was likely to be selection on observables. It must be noted that this evidence is only suggestive, because the regressions of columns 5–8 do not correct for potential selection on unobservables. The impact of manufacturing value on enlistment also raises suspicions of selection on observables—though columns 5–8 find no relationship of this variable with height in the sample, a population-level relationship is not out of the question. Thus, it is likely that the average observed stature in the sample does not correspond to the actual stature of the population, and that the unconditional Northeast–Midwest height difference in the data is also likely affected.

This analysis requires the use of the sample describing the observable characteristics of the population at risk for military enlistment. If only the selected sample were observed, then the researcher would be able to produce only columns 5–8 and not columns 1–4 of table 3. The height advantage of farmers and of individuals from areas of lower population density and agricultural value would suggest the presence of selection on observables if the researcher had reason to believe that these variables also affected the likelihood of enlisting in the military. The other advantage that comes from the availability of data on the population at risk for military enlistment is that, as shown by Zimran (Reference Zimran2019), the estimated conditional enlistment probabilities (another term for the conditional selection probabilities in this context) from columns 1–4 can be used to correct for sample-selection bias from selection on observables through the creation of inverse probability weights.

Selection on Unobservables

Table 2 provides suggestive evidence based on pattern 2—that different probabilities of entering the sample across groups suggest different selection on unobservables across these groups. The evidence in column 9 that Midwesterners were more likely to enlist suggests that indeed there may have been differences between regions in selection on unobservables that would affect the use of the military sample in determining population average stature and in testing for a regional difference in stature. Figure 4, which plots the distributions of estimated conditional enlistment probabilities from the estimates of column 3 of table 3, confirms this result, showing a greater mean conditional enlistment probability among Midwesterners. Contemporary reports of negative selection into military enlistment (Coffman Reference Coffman1986; Foner Reference Foner1970; Weigley Reference Weigley1967), combined with these insights based on pattern 2, suggest that Northeasterners were more negatively selected than Midwesterners on unobservables, and therefore that the Midwest’s height premium may have been exaggerated in the data. This will be explored in more detail in the following paragraphs.

Note: Enlistment probability is estimated from the results of the probit regression of column 3 of table 3. The top panel describes the distribution in the sample of census data. The bottom panel describes the distribution in the sample of military enlistment data.

Figure 4. Distribution of estimated enlistment probabilities by region.

Table 3 also provides evidence informed by pattern 3. Specifically, columns 1–4 show that the vote share for Lincoln enters with a positive and statistically significant coefficient, indicating that individuals from counties that were more supportive of Lincoln were more likely to enlist. The coefficients are not directly interpretable because these are probit coefficients, but it can be shown that the coefficient 2.21 in column 3 indicates that a 10-percentage point increase in Lincoln’s vote share was associated with an increase in the probability of enlistment by 7.2 percentage points, relative to a base probability of 44.6 percent. Crucially, column 9 of table 3 shows that there is a positive and strongly statistically significant relationship between the vote share and height in the military sample. Under the assumption that voting patterns are unrelated to height in the population, these results suggest the presence of sample-selection bias induced by selection on unobservables based on the logic of pattern 3. The sample describing the population at risk for military enlistment is crucial to determining that there was in fact a positive relationship between the vote for Lincoln and the probability of entering the military.

The positive and statistically significant coefficient on the Lincoln vote share in column 9 of table 3 also speaks to pattern 4. That the vote for Lincoln is positively associated with enlistment probability (column 3) and observed height implies that the selection on unobservables suggested by pattern 3 is likely negative. That is, individuals from the bottom of the height distribution were likely overrepresented in enlistment. This suggestive finding of negative selection is consistent with the suggestion of Bodenhorn et al. (Reference Bodenhorn, Guinnane and Mroz2017), with the results of Zimran (Reference Zimran2019), and with contemporary reports of the characteristics of military enlisters (Coffman Reference Coffman1986; Foner Reference Foner1970; Weigley Reference Weigley1967).

The combination of the insights in table 3 from patterns 3 and 4 suggests that there likely was sample-selection bias caused by selection on unobservables and that this bias would cause the researcher to understate the average stature of the population from these military data.

A more common concern than whether there is bias induced by selection on unobservables is whether such a bias would affect conclusions regarding trends or differences in the outcome over time or space. For example, researchers might not seek to use height data to describe the average stature of the population as a whole—though this is done (e.g., Fogel Reference Fogel, Engerman and Gallman1986; Floud et al. Reference Floud, Fogel, Harris and Hong2011)—but instead to describe trends in average stature over time or differences over space. In this case, it is not the presence of sample-selection bias that is important, but whether it varies over time or space. Fortunately, a more detailed analysis based on patterns 3–5 can shed light on whether the Northeast–Midwest height difference in the sample can be taken as informative of a true Northeast–Midwest difference in average stature.

The top panel of figure 5 plots a nonparametric regression of height on the estimated conditional enlistment probabilities separately by region. The key feature in this graph, building on pattern 5, is that the height premium for the Midwest is present among those with conditional enlistment probabilities close to one. Pattern 5 concluded that individuals so predisposed to enlist on the basis of their observable characteristics that they do so almost regardless of their unobservables have no selection on unobservables. Patterns among these individuals can thus be taken as unaffected by sample-selection bias. The presence of a Midwestern height premium at the right extreme of the top panel of figure 5 indicates that even though there is sample-selection bias (as shown previously) it is unlikely to have produced a spurious Midwestern height premium. Indeed, the presence of a Midwestern height premium at all levels of the conditional enlistment probability, at which the level of selection on unobservables is constant,Footnote ³⁴ provides validation to the existence of a true Midwestern height premium notwithstanding the presence of selection on unobservables.

Note: The x-axis is the enlistment probability estimated in column 3 of table 3. The y-axis in the top panel is height. The y-axis in the bottom panel is residuals of height from a regression of heights on all variables in column 5 of table 3 except for state fixed effects. Each graph presents results of a separate nonparametric regression for each region.

Figure 5. Height and residuals by region and estimated enlistment probability.

The bottom panel of figure 5 changes the y-axis of the figure to be the residuals of height after a regression on all observable characteristics except region.Footnote ³⁵ This adjustment reverses the direction of the slope of the relationship of height and the conditional enlistment probability, indicating that the negative relationship in the top panel is the product of observable characteristics that drive enlistment also being associated with lower stature (i.e., of negative selection on observables). When controlling for these observables, however, the upward slope, following patterns 3 and 4, indicates negative selection on unobservables into the military in both regions. That is, those with a greater probability of enlistment (analogous to hawks in the example) were taller than those with a lower probability of enlistment (analogous to doves in the example), just as in patterns 3 and 4.

Figure 6 uses this graph to investigate in more detail how the presence of sample-selection bias induced by selection on unobservables would affect the comparison of average heights of the Northeast and the Midwest. It repeats the bottom panel of figure 5, but indicates approximately the points at which the bulk of Northeasterners and Midwesterners are located in the distribution of enlistment probability, as shown in the bottom panel of figure 4—points $A$ and $C$, respectively. The effect of selection on unobservables on estimation of the height difference between the regions can be illustrated by comparing these points. Point $A$ is (loosely) the average observed height of Northeasterners, while point $C$ is (again loosely) the average observed height of Midwesterners. A comparison of these two points yields the Midwest’s observed height advantage. But this comparison conflates two differences—the true Northeast–Midwest difference and the difference in sample-selection bias between the regions, which is greater for the Northeast at point $A$ than for the Midwest at point $C$ because of the greater enlistment probability of the Midwest. A better comparison would be of points $A$ and $B$, which compares individuals with the same enlistment probability, and thus the same degree of sample-selection bias. More generally, rather than computing the difference in heights between the Midwest and the Northeast using the distribution of enlistment probabilities in the data (figure 4), a correct comparison would be a weighted average of differences between individuals across regions with the same enlistment probability.

Note: This figure repeats the bottom panel of figure 5 but is annotated to show the danger of comparing individuals with different enlistment probabilities instead of making comparisons only for individuals with the same enlistment probabilities. Points A and B correspond to the modal enlistment probability of Northeasterners, while point C corresponds to the modal enlistment probability of Midwesterners, each from the bottom panel of figure 4.

Figure 6. Residuals of height and enlistment probability, annotated.

On the whole, then, the patterns of selection on unobservables revealed by this analysis suggest that the Midwest–Northeast height premium is likely overstated, but that there truly was a premium. This is consistent with the conclusions of Zimran (Reference Zimran2019).