Uncertainty About the Size of the Unauthorized Foreign-Born Population in the United States

Abstract

One of the most common methods for estimating the U.S. unauthorized foreign-born population is the residual method. Over the last decade, residual estimates have typically fallen within a narrow range of 10.5 to 12 million. Yet it remains unclear how sensitive residual estimates are to their underlying assumptions. Here, we examine the extent to which estimates may plausibly vary due to uncertainties in their underlying assumptions about coverage error, emigration, and mortality. Findings show that most of the range in residual estimates derives from uncertainty about emigration rates among legal permanent residents, naturalized citizens and humanitarian entrants (LNH); estimates are less sensitive to assumptions about mortality among the LNH foreign born and coverage error for the unauthorized and LNH populations in Census Bureau surveys. Nevertheless, uncertainty in all three assumptions contributes to a range of estimates, whereby there is a 50% chance that the unauthorized foreign-born population falls between 9.1 to 12.2 million and a 95% chance that it falls between 7.0 to 15.7 million.

Estimates of the size, growth, and composition of the unauthorized foreign-born population shape public debates about immigration and are important for the evaluation and administration of U.S. policies. A widely accepted report compiled by Passel and Cohn (2019) estimates that this population declined from about 12 million in 2007 to 10.5 million by 2017, largely due to a major decline in the number born in Mexico. Accurate estimates of this population can shed light on the scope and cost of proposed legislation to grant legal status to certain groups of unauthorized immigrants and help evaluate immigration enforcement efforts (Meissner and Mittelstadt, 2020).

One of the most common methods for estimating the unauthorized foreign-born population is the residual method (e.g., Warren & Passel 1987; Bean et al 2001; Baker 2021). In its most basic form, this method subtracts an estimate of the legally resident foreign-born population composed of legal permanent residents, naturalized citizens and refugees, asylees, and other humanitarian entrants (a group that we refer to as the “LNH” foreign born) observed in administrative data, from the total foreign-born population recorded in the American Community Survey (ACS) or another major national survey. After accounting for the degree to which foreign-born individuals are underrepresented in the ACS and making other adjustments, the difference yields an estimate of the unauthorized foreign born.

Residual estimates generated for the most recent ACS data years have typically fallen within a relatively narrow range no matter which research group or organization produced the estimate. For example, the Department of Homeland Security (DHS) (Baker 2021) estimated there were 11.4 million unauthorized immigrants as of January 2018 and the Pew Research Center (Pew) (Passel and Cohn 2019) estimated a population of 10.5 million as of mid-2017.

The narrow range of these estimates has contributed to media and public confidence and driven a consensus about the changing size and composition of the unauthorized immigrant population. However, the uniformity of the estimates may convey a false degree of certainty. Residual estimates rely on assumptions about emigration, mortality, and coverage error among the foreign-born population, and the precise levels of these inputs are not known with complete certainty. Despite this uncertainty, none of the research organizations that produce residual estimates have provided plausibility ranges. A plausibility range would help the demographic community evaluate whether there are meaningful differences among the various residual-methods estimates produced by organizations with longstanding track records such as DHS and Pew. It would also be important to know if the plausible range around residual estimates is so wide as to render these estimates useless for public policy debates; a high level of uncertainty would motivate future research to narrow the range.

Here, we develop an estimate of the plausible range of residual estimates of the unauthorized foreign-born population. Our overarching strategy is to examine how uncertainty in key inputs translates into uncertainty in residual estimates. In what follows, we first review our approach to calculating residual estimates. We then use a simple simulation to estimate the sensitivity of residual estimates to changes in the method’s three key assumptions: (1) the coverage error in the ACS and other nationwide surveys of the unauthorized and LNH foreign-born populations, (2) emigration rates of the LNH foreign born, and (3) death rates of the LNH foreign born.

Finally, we produce our own residual estimates using coverage error, emigration, and death rates that reflect the best available evidence, and we assess their sensitivity to a plausible range of assumptions. We find that most of the uncertainty in residual estimates derives from uncertainty in emigration rates. Residual estimates are less sensitive to assumptions about coverage error and even less sensitive to mortality assumptions. After accounting for uncertainty in all three assumptions, we estimate that there is a 50% chance that the unauthorized foreign-born population falls between 9.1 to 12.2 million and a 95% chance that it falls between 7.0 to 15.7 million.

Section 1. The Residual Method

As part of our effort to assess the uncertainty in residual estimates, we developed our own residual estimates of the unauthorized foreign-born population by age, sex, year of arrival, and country or region of birth using the best available data and methods available to us. We followed the same approach and obtained similar results as other researchers (i.e., Pew and DHS). Nevertheless, different researchers tend to rely on slightly different data sources and assumptions. We provide an overview of our method here along with a full list of data sources in Appendix A and comparisons with the different assumptions used by DHS and Pew in Appendix B.

For the purposes of developing a residual estimate, we distinguish among three foreign-born groups, shown in Box 1. The first, the unauthorized foreign-born population, U, includes those who entered the country without inspection and those who arrived legally with temporary visas (e.g., student, tourist, temporary worker) but overstayed or otherwise violated the terms of their visas. We also include foreign-born people who have received an official, temporary reprieve from deportation but otherwise resemble unauthorized immigrants demographically, for instance: Temporary Protected Status (TPS) recipients, Deferred Action for Childhood Arrival (DACA) participants, and asylum applicants with work authorization. For estimation purposes, we limit this group to those who arrived in the country in 1982 or later, with the rationale that most immigrants who arrived before 1982 would have legalized because they were eligible for amnesty under IRCA. This group of post-1982 entrants who are unauthorized cannot be identified directly in administrative records or census data.

Box 1. Components of the Foreign-born Population.

Unauthorized Foreign-born Population, Arrived 1982-present (U)

• Entered without inspection (predominantly across the U.S.-Mexico border)

• Overstayed a valid visa (e.g., tourist, student, or temporary work visa)

• Temporary Protected Status (TPS) recipients

• Deferred Action for Childhood Arrival (DACA) participants

• Asylum applicants whose claims have not yet been approved

LPR/Naturalized/Humanitarian, Arrived 1982-present (LNH)

• Naturalized citizens

  • Most LPRs are eligible to naturalize after 5 years in that status
  • Those married to U.S. citizens are eligible to naturalize after 3 years in status
• Legal permanent residents (LPRs)

  • Admitted into the United States as LPRs
  • Adjusted from unauthorized, nonimmigrant or humanitarian status

• Refugees, asylees, and other humanitarian immigrants with lawful status but who have not yet adjusted to LPR status

Other Foreign-born (O)

• Nonimmigrants visa holders (e.g., international students; H-1B, H-2A, H-2B, L and O temporary workers)

• Pre-1982 Arrivals

The second group, the LPR/naturalized/humanitarian population (LNH) includes all naturalized citizens; legal permanent residents (LPRs, or “green card” holders); and immigrants with humanitarian statuses such as refugee or asylee who have yet to adjust to LPR status. Like the unauthorized foreign-born, we limit this group to those who arrived in the country in 1982 or later. This group can be estimated using administrative data.

Finally, the third group, all other foreign-born, includes nonimmigrants admitted lawfully with temporary visas (such as international students, H-1B high-skilled workers, and H-2A agricultural workers) and all foreign-born persons who arrived in the country before 1982. Pre-1982 arrivals can be identified directly in the ACS, while nonimmigrants can be identified indirectly based on their characteristics in the ACS.

As just noted, the unauthorized foreign-born population cannot be estimated directly. However, the combined unauthorized and LNH populations (C = U + LNH) can be estimated using the ACS by excluding the “other foreign born” (nonimmigrants and pre-1982 arrivals) from tabulations of the total foreign-born population. Additionally, the number of LNH foreign born can be estimated using administrative data (LNH). Therefore, after certain adjustments are made, the unauthorized foreign-born population can be estimated by subtraction (U = C – LNH).

Estimation occurs in four steps. First, we used the ACS (Ruggles et al. 2020) to estimate the combined unauthorized and LNH populations (C), disaggregated by sex (s), region or country of birth (r), birth cohort (c), year of entry (y), and year (t). To obtain these estimates, we tabulated the foreign-born population for each demographic subgroup using the ACS. Our sample included all persons of foreign parentage born outside of the United States or outlying areas except those in the third “other foreign-born” group in Box 1. This group (i.e., nonimmigrants and those who arrived in the United States before 1982) was dropped from the sample. Nonimmigrants can be identified in the ACS and other survey data with some precision. They include noncitizens whose occupations, immigration histories, and family/household characteristics are congruent with the eligibility criteria for specific nonimmigrant visa categories. For example, international students (F-1 visa holders) can be identified based on age of arrival to the U.S., full-time school enrollment, and lack of full-time employment. H-2A workers can be identified based on years in the country, country of birth (nearly all are Mexican-born), and agricultural employment, while H-1B high-skilled workers can be identified based on their educational attainment, years in the country, and employment in certain occupations such as information technology workers, engineers, researchers, and doctors and surgeons. Totals of nonimmigrants identified in the ACS are comparable with administrative data from DHS.

The second step involves estimating the LNH foreign-born population who arrived in the country in 1982 or later. To do so, we start by compiling administrative data on the number of legal admissions or entrants for each year since 1982. We specifically use LPR admissions data from DHS, disaggregated by sex (s), region or country of birth (r), birth cohort (c), year of entry (y), and year of admission (a). We add to these data an estimate of the number of lawfully present refugees, asylees, and other humanitarian entrants who have not yet adjusted to LPR status, similarly disaggregated. We next project each admission cohort forward from year of admission (a) to the current year (t), to yield a stock estimate of the LNH population:

$$LN{H}_{\mathit{srcyat}}=A_{\mathit{srcya}}-{\sum }_{i=a}^{i=t}\left(D_{\mathit{srcyi}}+E_{\mathit{srcyi}}\right)$$ (1)

where A is the number of LNH admissions or entrants in year a, D is the annual number of deaths, and E is the annual number of emigrants. D and E are derived from a set of assumed mortality (m) and emigration (g) rates among the LNH foreign born multiplied by the size in year i of the cohort that was admitted in year a.¹ Even though the LNH foreign-born become eligible to naturalize after 5 years in LPR status (3 years if married to a U.S. citizen), we do not construct separate estimates of those who naturalized and those who remained noncitizens in our methodology; doing so would unnecessarily introduce error into the estimates due to known biases in self-reports of citizenship status (Van Hook and Bachmeier 2013; Brown et al. 2019).

Equation 1 demonstrates that the estimate of the LNH population is subject to uncertainty in assumptions about the mortality and emigration rates, and that systemic errors in assumptions about mortality and emigration rates, which are contained within the summation sign, accumulate over time. If annual emigration rate were too large, for example, this would contribute to overestimates of the annual number of LNH foreign born leaving the country, and the error in the cumulative number of emigrants would grow as time elapses since admission.

In the third step, the LNH population is subtracted from the combined unauthorized and LNH populations to yield an estimate of the unauthorized immigrant population. These estimates are disaggregated by sex (s), region or country of birth (r), birth cohort (c), year of entry (y), and year (t). An important part of this step is to adjust the estimates for the extent that the ACS underrepresents both the LNH ($e_{LNH}$) and unauthorized foreign-born populations ($e_u$). We refer to the underrepresentation of groups in the ACS as “coverage error”, whereby coverage error = (Population - ACS Estimate)/(Population). We obtain estimates of coverage error from prior research and apply them to the components of the residual estimate as shown in equation 2:

$$U_{srcyt}=\frac{C_{srcyt}-{LNH}_{srcyt}\left(1-e_{LNH}\right)}{1-e_u}$$ (2)

The LNH population (${LNH}_{srcyt})$ is derived from administrative records and therefore unaffected by coverage error. However, the combined unauthorized and LNH population ($C_{srcyt})$ is derived from ACS data and could be too low due to coverage error. Before subtracting the LNH population from the combined population, we adjust the LNH population downward (by multiplying by $1-e_{LNH})$ to reflect the number represented in the ACS. As a result, the numerator of equation 2 is the unauthorized population represented in the ACS, which we finally adjust upward by dividing by $1-e_u$ to yield the total unauthorized population. Note that higher coverage error of either the unauthorized or LNH populations would inflate the unauthorized population.

The fourth and final step involves smoothing the unauthorized population estimates to account for heaping in reported year of entry every ten years (i.e., around 1990, 2000, and 2010), to ensure continuity in entry and birth cohorts over time, and to reduce the incidence of negative population estimates.

Section 2. A Simple Simulation

Although many observers see coverage errors in the ACS and other surveys as the major challenge for residual estimates, the residual method also relies heavily on assumptions about the emigration and mortality of the LNH foreign born population—key factors in determining the size of the LNH population that is subtracted from the combined unauthorized and LNH foreign born populations to derive the estimate of unauthorized immigrants. Higher estimates of coverage error for unauthorized immigrants and higher estimates of coverage error, emigration, and mortality among the LNH foreign born all result in higher residual estimates. But which factors matter the most?

The effect of these assumptions can be assessed using a simple simulation. Imagine that one thousand LNH foreign born people are admitted each year. Similar to actual rates in the United States (Baker 2021), 1 percent of LNH foreign-born persons emigrate each year, 0.1 percent die each year, and coverage error is 1 percent among the LNH foreign born and 10 percent among unauthorized immigrants. After 35 years, 35,000 foreign-born are enumerated in a census survey and this number increases by 1.5% annually. Under these assumptions, we would estimate 6,934 unauthorized immigrants. This is referred to as the “Original Estimate” in Table 1 (Panel A, row 1).

Table 1.

Illustration of Effects of Assumptions About Coverage Error, Emigration, and Mortality

	Panel A. Unauthorized Foreign Born, 35 years After Baseline		Panel B. Percentage Difference 40, 45, and 50 Years After Baseline
	Unauthorized Foreign-born	Percentage Diff. from Original Est.	40 years	45 years	50 years
(1) Original Estimatea	6,934
If Assumptions Increased by 50%:
(2) Coverage error among unauthorized foreign born (10% to 15%)	7,341	6	6	6	6
(3) Coverage error among LNH foreign born (1.0% to 1.5%)	7,095	2	3	3	3
(4) Emigration rate among LNH foreign born (1.0% to 1.5%)	9,409	36	49	61	70
(5) Mortality rate among LNH foreign born (.0010 to .0015)	7,194	4	5	7	7

^aBased on scenario in which there are 1,000 LNH admissions each year; 10% coverage error among unauthorized immigrants; 1% coverage error, 1% emigration rate and 0.1% mortality rate among LNH foreign born; census enumerates 35,000 foreign-born 35 years after baseline; and the enumerated foreign-born population increases by 1.5% each year. LNH = LPR/naturalized/humanitarian.

Now imagine that each of the assumptions were fifty percent higher. The estimate of the unauthorized population would increase by 36% if the emigration rate for the LNH population increased by 50% (row 4), but only by 6%, 2%, and 4% if the other assumptions increased by 50% (rows 2, 3, and 5). This simulation shows that the residual estimate is especially sensitive to changes in emigration rates.

Emigration rates have the largest impact because they are applied to the LNH population each year over the 35-year projection period, so their impact accumulates over time. While mortality rates are also applied annually over 35 years, they have less impact on the result because they are much lower (0.1 percent versus 1 percent annually). This point is illustrated in Panel B, which projects the scenarios in Panel A forward in time: 40, 45, and 50 years after the initial starting point of the simulation. When the assumed emigration rate increases by fifty percent, the percentage difference from the original estimate grows over time: 49 percent after 40 years, 61 percent after 45 years, and 70 percent after 50 years. In contrast, when coverage error or mortality rates increase by fifty percent, the percentage difference from the original estimate remains low and nearly constant over time.

This simple illustration reveals an important point. Residual estimates are particularly sensitive to small changes in the emigration rate of the LNH population. Mortality assumptions could also become influential as U.S. residence increases and mortality rates rise, particularly for older immigrant cohorts.² Coverage error assumptions do not influence the estimates as much because changes in coverage assumptions are only applied once in the model; they do not accumulate over time. Of course, this simple illustration may not hold under more realistic conditions. Of note, this illustration does not account for the fact that emigration rates tend to decline with increased duration of residence, which may offset the tendency for errors in emigration rates to accumulate over time since admission. Additionally, the importance of coverage error may decline over time as the unauthorized population grows older, accrues more years of U.S. residence, and is more likely to be represented in household surveys.

Section 3. A Plausible Range of Residual Estimates Under Realistic Conditions

We next approximate the plausible range of residual estimates for the unauthorized foreign-born population under more realistic conditions. To do this, we first review, and update as necessary, prior research on coverage error, emigration, and mortality. We pay attention not just to the levels of these assumptions, but also to the degree of variation among plausible estimates, which we interpret at as an indication of uncertainty. We specifically use the standard deviation across plausible values of each assumption in prior research to produce probability distributions for each assumption. We draw random values from these distributions to use as inputs for residual estimates.

Coverage Error.

The residual method relies on coverage-error estimates for both the unauthorized and LNH foreign-born populations, in that higher levels of coverage error of either population would lead to a higher estimate of the unauthorized population. In the ACS and similar nationwide surveys, coverage error occurs when people are missed because they fail to respond to survey takers; respond but provide insufficient or inaccurate information about their demographic characteristics (in this case, their place of birth and citizenship); or because they live in nonresidential or unconventional locations. Coverage error could be particularly high among unauthorized immigrants because they may be more difficult to locate (e.g., they live in agricultural worker barracks or crowded multi-family housing units), or they may attempt to avoid detection due to fear of government authorities.

Most prior research on the coverage error for unauthorized immigrants has focused on Mexicans, the largest single national-origin group among them. Therefore, we first review the evidence about Mexicans before explaining how we extrapolate results for Mexicans to other groups. In general, this research compares the population counted in the U.S. Census or ACS with an independent estimate of the same population derived or inferred from non-census data sources, such as birth or death registrations, independent surveys, ethnographic studies of neighborhoods with large shares of unauthorized immigrants, and estimates of Mexicans living in the U.S. derived from Mexican census data. The idea is that the unauthorized foreign-born population leaves “footprints” in statistical and administrative record systems even if they do not willingly participate in official U.S. Census and survey collection efforts (Gelatt et al. 2018).

Evaluations of the rate of coverage error for the Mexican unauthorized immigrant population fell in the range of 15 to 35 percent in 1990 (Corona Vasquez 1991; de la Puenta 1992; U S General Accounting Office 1993; Van Hook and Bean 1998), and remained in this range until the middle of the 2000-2009 decade (Hill and Wong 2005; Genoni et al. 2012). Warren’s recent analysis (2020) supports these findings. He examined the decline in cohort sizes (after accounting for mortality) between the 1990 and 2000 Mexican censuses and found that about 5.5 million people left Mexico during the 1990s. The 2000 U.S. Census counted 4.5 million such individuals, implying a coverage error rate of 18 percent for 2000.

However, based on their analyses of U.S. death records and Mexican census data, Van Hook and colleagues (2014) found evidence that coverage error declined substantially during the latter half of the 2000-decade. Declining coverage error was apparently associated with substantial reductions in shorter-term unauthorized-immigrant laborers during the Great Recession (particularly in the hard-hit sectors of construction and services)—a group that is likely to be harder to count than longer term, more settled unauthorized immigrants. By 2010, coverage error rates for the unauthorized Mexican-born population were estimated to be below eight percent. These estimates are somewhat lower than the coverage error assumptions made by DHS (10%) and Pew (13%)³ in the past, although Pew now assumes similarly low levels of coverage error.

To conduct the work presented here, we updated Van Hook and colleagues’ (2014) estimates of coverage error with the latest available Mexican census data and U.S. death records. We found evidence of further declines in coverage error among women but small increases among men between 2010 and 2017. We produced these estimates by analyzing two different data sources: (1) death registrations of Mexican-born individuals in the United States and (2) net migration from Mexico based on Mexican census data. In Table 2, ranges for these estimates are displayed to reflect uncertainty in the mortality rate of Mexican immigrants and coverage error in the Mexican census. The methodology underlying these estimates is described in Appendix C.

Table 2.

Summary of Estimates of Coverage Error for Mexican Unauthorized Foreign-born adults

	2000	2005	2010	2015	2018
Average Estimates
All	16	20	8	4	5
Women 15-64	20	23	5	1	1
Men 15-64	20	19	7	8	10
Standard Deviation of Estimates
All	8	7	9	4	4
Women 15-64	9	2	10	5	5
Men 15-64	12	6	5	3	3

Note: estimates are averages and SD of high and low estimates of death registration and net migration methods.

We next extrapolated the estimates for Mexicans to non-Mexicans and adjusted levels of coverage error to account for likely variation by year and duration of residence in three ways. First, we linearly interpolated values for the years not shown in Table 2 (see Table C2). Second, we assumed that coverage error for Latinos was the same as for Mexicans, but that coverage error for non-Latinos (chiefly those from Africa, Europe, and Asia) was 25% lower than the values shown in Table 2, largely because almost all non-Latin-American unauthorized immigrants overstayed their visas rather than entering the country illegally and tend to be more highly educated and therefore living in better housing—both factors making them more likely to be represented in ACS and Census data. This is also consistent with estimates produced by the U.S. Census Bureau for 2010 (Jensen et al. 2015) showing that coverage error among the Hispanic foreign-born population is much higher than the non-Hispanic foreign-born population. Third, we assumed that recent arrivals (those with fewer than 5 years of U.S. residence) have coverage error rates that are three times as high as longer-term residents (10+ years of residence), consistent with evidence of high coverage error among recent arrivals (Van Hook et al. 2014).

To estimate uncertainty in coverage error among unauthorized immigrants, we used the standard deviations in Table 2 to produce probability distributions of coverage error from 2000 to 2018. We used a gamma function to constrain the distribution to positive values. Figure 1 provides an example of the probability distribution of coverage error among Mexican men in 2018 (average = 10, SD = 3). When averaged across all demographic groups, the mean coverage error of unauthorized immigrants was 18.9 percent (SD = 10.0 percent) in 2005; 5.8 percent (SD = 3.8 percent) in 2010; and 5.1 percent (SD = 3.2 percent) in 2018.

Figure 1.

Probability Distribution of Coverage Error Unauthorized Mexican Men, 2018

Finally, although we know less about coverage error among the LNH population, we assume that it is low given that net coverage error was virtually zero for the entire U.S. population and only 1.54 percent for all Hispanics in 2010 (Mule 2012). DHS and Pew both estimate coverage error among the LNH population to be 1.5%, so we assume the same but with a standard deviation of .5% to account for uncertainty.

Emigration.

Besides coverage error, the residual method relies on estimates of emigration among the LNH population. Emigration rates are needed to estimate how many in this population left the country following their admission. Higher levels of emigration lead to lower estimates of the LNH population and correspondingly higher estimates of the unauthorized population. Unfortunately, official government statistics on emigration from the United States have not been published since 1956, mainly due to concerns about the incompleteness and poor quality of emigration administrative records (Kraly 1998). Therefore, out of necessity, foreign-born emigration has been estimated with a variety of indirect demographic methods.

The U.S. Census Bureau estimates net emigration using a residual method (not to be confused with the residual method for estimating the unauthorized foreign born). This method compares the size of foreign-born cohorts between two decennial censuses or surveys after adjusting for mortality, yielding estimates of emigration among the entire foreign-born population. Residual-based estimates of the annual foreign-born emigration rate tend to fall between 1% and 1.2% (Warren and Peck 1980: 1.2%; Ahmed and Robinson 1994: 1.2%; and Mulder 2003: 0.9%). A limitation of this method is its inability to estimate emigration for recent entrants (i.e., those arriving during the period between the two decennial censuses). Borjas and Bratsberg (1996) overcame this problem by using immigrant-admission records collected over multiple years in place of the first census. Their estimates imply annual emigration rates of 3.8% in the first five years and 0.8% in the second five years of U.S. residence. Leach and Jensen (2014) also overcame this problem by tracking the size of immigrant entry cohorts across adjacent years of the ACS. They too found higher annual rates of emigration for recently arrived immigrants: 0.6% for all immigrants and 1.3% among those in the country less than 10 years, which implies an annual rate of about 0.4% for longer-term residents. Leach (2017) later revised these estimates upward, implying rates of 0.8%, 1.8%, and 0.5%, respectively.

Other researchers have used linked administrative records to estimate emigration levels and rates among LPRs and naturalized citizens. Jasso and Rosenzweig (1982) linked immigrant admissions data from 1971 (which contain a record for all immigrants who were granted LPR status in that year) to data from the now defunct Alien Address Report Program (AARP), finding an annual emigration rate of 2.1%. Duleep (1994) used Social Security Administration (SSA) records matched across years to estimate the emigration of all immigrants with work authorization, whereby a discontinuation in earnings across multiple years (without retirement) was interpreted as emigration. She found that about 30% of the immigrants in the SSA earnings file eventually emigrated, implying an annual emigration rate of 2.8% in the first decade of U.S. residence but less than 1% in subsequent decades. More recently, Schwabish (2009) used a similar approach to estimate emigration among immigrants in the SSA earnings file, finding somewhat lower levels of emigration: 1.3% overall and 2.3% in the first decade of U.S. residence.⁴

Our residual estimates require estimates of emigration for the LNH foreign born population. No published emigration rates perfectly align with this specific population, but we selected the emigration rates pertaining to immigrants in the SSA earnings file. Although the SSA earnings file includes some unauthorized immigrants who have fraudulent Social Security Numbers, and some classes of nonimmigrants who do not eventually adjust to LPR status, it excludes a greater share of both of these types of immigrants than does the ACS—the basis for other estimates of emigration such as those produced by the Census Bureau.⁵ This suggests that the SSA earnings file may be a more accurate source of information about emigration of the LNH foreign-born population than the ACS.

Among the SSA-based emigration rates, we chose those by Schwabish because they are the most recent and because Schwabish provided us with a prediction model of the annual probability of emigration, which we used to produce annual emigration rates broken down by age, sex, duration of residence, and country or region origin.⁶ We adjusted Schwabish’s estimates to account for annual trends in emigration. We specifically used the ACS to produce annual residual estimates by country or region of birth from 2005-2018 following Leach’s (2017) methodology. Emigration among the foreign born tended to be low in the years before the Great Recession but increased between 2007 and 2009, fell between 2010 and 2014, and then increased again after 2015. We adjusted the Schwabish estimates to account for annual fluctuations while maintaining the average probability of emigration by age, sex, and duration of residence as designated by Schwabish’s prediction model (estimates shown by region of birth, year, and duration of residence in Table 3).

Table 3.

Estimated emigration rates for LNH foreign born, by Country/Region of Birth, Year, and Duration of U.S. Residence

Region of Birth and Year	Duration of U.S. Residence
	0-4	5-9	10-14	15-19	20+
Mexico
2000-04	0.026	0.015	0.009	0.010	0.012
2005-09	0.022	0.011	0.006	0.006	0.007
2010-14	0.029	0.015	0.006	0.007	0.009
2015-18	0.035	0.019	0.007	0.008	0.009
Central America
2000-04	0.022	0.015	0.011	0.013	0.013
2005-09	0.016	0.009	0.005	0.005	0.006
2010-14	0.016	0.009	0.004	0.004	0.004
2015-18	0.016	0.009	0.004	0.004	0.004
Caribbean
2000-04	0.026	0.019	0.018	0.023	0.023
2005-09	0.018	0.012	0.007	0.009	0.011
2010-14	0.019	0.011	0.006	0.006	0.008
2015-18	0.020	0.013	0.005	0.006	0.006
South America
2000-04	0.023	0.022	0.021	0.024	0.023
2005-09	0.016	0.010	0.009	0.010	0.011
2010-14	0.017	0.009	0.003	0.003	0.004
2015-18	0.019	0.012	0.003	0.003	0.004
Europe/Canada/Oceania
2000-04	0.026	0.018	0.013	0.017	0.018
2005-09	0.030	0.016	0.011	0.011	0.013
2010-14	0.034	0.019	0.007	0.006	0.006
2015-18	0.035	0.023	0.007	0.007	0.007
Asia
2000-04	0.024	0.011	0.005	0.003	0.002
2005-09	0.032	0.018	0.009	0.006	0.003
2010-14	0.041	0.028	0.015	0.008	0.003
2015-18	0.055	0.041	0.027	0.021	0.007
Other
2000-04	0.022	0.018	0.015	0.019	0.019
2005-09	0.018	0.012	0.008	0.008	0.010
2010-14	0.018	0.013	0.005	0.005	0.005
2015-18	0.018	0.013	0.005	0.005	0.004

LNH = LPR/naturalized/humanitarian. Estimates are based on Schwabish prediction model and adjusted for trends in emigration.

To estimate the level of uncertainty in emigration among the LNH foreign born, we examined the variation in estimates in prior literature. If we consider all of the studies cited above, the standard deviation of the estimates is 0.75%. However, if we confine ourselves to studies of immigrants who attained LPR status or are present in the SSA earnings file (the group of greatest relevance), the standard deviation is 0.42%, and if only the Census studies are considered, the standard deviation drops farther to 0.26%. Because of our focus on emigration among the LNH population, we selected a moderate level of uncertainty. We center the probability distribution around the Schwabish, trend-adjusted emigration rate, and we set the standard deviation of the probability distribution at half the level of the emigration rate, and again, we use a gamma distribution to constrain the distribution to positive values. When averaged across all demographic groups, the mean emigration rate of the LNH population was 1.1 percent (SD = .53 percent) in 2005; 1.1 percent (SD = .56 percent) in 2010; and 1.8 percent (SD = 0.9 percent) in 2018.

Mortality.

Finally, the residual method relies on estimates of mortality among the LNH population. Higher mortality rates lead to a lower stock estimates of the LNH population and a higher estimate of the unauthorized population.

Most groups that produce residual estimates assume that the LNH population has the same age- and sex-specific mortality rates as the U.S. population. But given the well-documented mortality advantage of immigrants (Hummer et al. 2000; Riosmena, Kuhn and Jochem 2017), we adjusted the U.S. mortality rates downward. The adjustments were based on our analysis of the 1997-2008 National Health Interview Survey (Blewett et al. 2019). We first estimated proportional hazard Cox models predicting the hazard of dying as a function of region of birth (Latino, Asian, and other foreign-born versus U.S.-born), by sex. We then used the estimated hazard ratios (see Table 4) to adjust the mortality rates for the United States (Human Mortality Database), thus obtaining sex-, age-, and year-specific rates for Latino, Asian, and other immigrants. Uncertainty in these estimates derives primarily from sampling error, so we used the standard errors of the coefficients to determine the spread of the probability distribution of coefficients, using a normal distribution.

Tabic 4.

Cox Regression Models Predicting Mortality Hazard among U.S. Adults Age 18+

	Hazard Ratio	Coefficient	SE
Men
U.S.-born (Ref.)	---	---	---
Foreign-born
Hispanic	0.798	−0.226	0.020
Asian	0.735	−0.308	0.035
Other	0.661	−0.413	0.021
Women
U.S.-born (Ref.)	---	---	---
Foreign-born
Hispanic	0.770	−0.261	0.020
Asian	0.728	−0.318	0.035
Other	0.702	−0.354	0.021

Source: 1997-2009 NHIS-NDI (N = 772,323). Models control for age and age at intereview.

Baseline Residual Estimates.

Our assumptions lead to similar estimates as produced by others, both for the total unauthorized foreign-born population (Figure 2) and unauthorized Mexican-born population (Figure 3). On closer inspection, however, our estimates of the total tend to be higher than DHS and Pew estimates in 2005 and 2006 and lower than their estimates between 2010 and 2015. Our estimates of the unauthorized Mexican-born population follow a similar pattern, except that they closely conform with Pew estimates between 2010 and 2015. Our 2005-2018 estimates differ from the others by about 756 thousand (6.8% of the average) in the case of the total unauthorized foreign-born population, and by about 424 thousand (6.7% of the average) in the case the unauthorized Mexican-born population. Estimates by country/region of birth also differ somewhat. For example, our method estimates more Mexicans and Europeans/Canadians than the Pew method (Figure 4). We could not compare our estimates with DHS’s because of inconsistencies in country/region categories.

Figure 2. Estimates of Unauthorized Foreign-born Population Residing in the United States, 2005-2018.

Note: Our estimates are averages of 1,000 iterations wherein assumptions for coverage error, emigration, and mortality are randomly drawn from postulated distributions; the estimate for 2018 is 10.8 million. However, when assumptions are fixed at their mean levels, the estimate for 2018 is 11 million. The DHS estimates represent results from three different series (2005-2010, 2010-2015, and 2015-2018) that use slightly different data and methods.

Figure 3. Estimates of Unauthorized Mexican-born Population Residing in the United States, 2005-2018.

Note: Our estimates are averages of 1,000 iterations wherein assumptions for coverage error, emigration, and mortality are randomly drawn from postulated distributions. The DHS estimates represent results from three different series (2005-2010, 2010-2015, and 2015-2018) that use slightly different data and methods.

Figure 4. Unauthorized Foreign born by Region of Birth, 2017 or 2018.

Note: Our estimates are averages of 1,000 iterations wherein assumptions for coverage error, emigration, and mortality are randomly drawn from postulated distributions; the estimate for 2018 totals 10.8 million. However, when assumptions are fixed at their mean levels, the estimate for 2018 totals 11 million.

Are these differences meaningful, or do they fall within a range of equally plausible estimates? We turn to this question next.

Plausible Range of Residual Estimates.

To ascertain the uncertainty of residual estimates, we draw random values from the distributions of assumptions and use them to calculate residual estimates. We repeat the process one thousand times to obtain a distribution of residual estimates associated with uncertainty in underlying assumptions. To isolate the effects of each assumption, we conducted three different simulations, whereby we allow each assumption -- coverage, emigration, and mortality -- to vary while holding values of the remaining assumptions fixed at their average levels. Finally, to gauge the combined effects of uncertainty, we conducted a fourth simulation in which we allow all assumptions to vary simultaneously.

The resulting distributions of residual estimates are summarized in Table 5 and Figure 5. Table 5 displays the average residual estimates by year in the first column, and the standard deviations of the distributions for simulations that vary by coverage error, emigration, mortality, and all factors simultaneously in the remaining columns. The magnitudes of the standard deviations indicate distribution spread, and hence, the degree of uncertainty in the estimates. To further illustrate the uncertainty in the estimates due to uncertainty in all three assumptions (coverage error, emigration, and mortality), Figure 5 depicts the probability distribution of residual estimates over time from 2005 to 2018.

Table 5.

Uncertainty in Residual Estimate of Unauthorized Foreign-Born Population Due to Uncertainty in Underlying Assumptions (Standard Deviation Across 1,000 Iterations)

		Uncertainty in Residual Estimate Due To Uncertainty in:
Year	Residual Estimate	Coverage Error	Emigration	Mortality	All Factors
2005	11,713	1,778	1,342	11	2,200
2006	12,070	1,839	1,418	12	2,295
2007	12,220	1,887	1,495	13	2,380
2008	11,080	1,114	1,474	13	1,745
2009	10,591	782	1,516	14	1,568
2010	10,254	498	1,553	15	1,472
2011	10,229	484	1,632	16	1,548
2012	10,249	473	1,709	17	1,626
2013	10,301	458	1,785	18	1,702
2014	10,461	455	1,860	19	1,783
2015	10,577	445	1,933	20	1,861
2016	10,717	480	2,024	22	1,970
2017	10,800	515	2,140	23	2,100
2018	10,773	507	2,271	25	2,232

Note: Residual estimates are averages of 1,000 iterations wherein assumptions for coverage error, emigration, and mortality are randomly drawn from postulated distributions; the estimate for 2018 is 10.8 million. However, when assumptions are fixed at their mean levels, the estimate for 2018 is 11 million.

Figure 5. Plausible Range of Residual Estimates of the Unuathorized Foreign-born Population.

Note: This figure presents the mean and distribution of estimates across 1,000 iterations wherein assumptions for coverage error, emigration, and mortality are randomly drawn from postulated distributions; the mean estimate for 2018 is 10.8 million. However, when assumptions are fixed at their mean levels, the estimate for 2018 is 11 million.

The results in Table 5 show that the residual estimates are most sensitive to uncertainty in emigration rates, particularly during the 2010-18 period, and least sensitive to uncertainty in mortality rates. As discussed earlier, uncertainty about emigration in prior research led us to postulate a probability distribution with a standard deviation equal to half the emigration rate; when averaged across different demographic groups, the standard deviation of the emigration rates was about .65%. In 2018, this amount of uncertainty about emigration was associated with an estimated 2.3 million unauthorized immigrants.

In contrast, prior research on coverage error led us to postulate a probability distribution for the amount of coverage error with an average standard deviation of about 3.2% as of 2018. But because this coverage error rate is only factored in once, instead of annually over 36 years, it is associated with 507 thousand unauthorized immigrants in 2018—less than one-quarter of the uncertainty associated with emigration.

Finally, assumptions about mortality have far less impact on the estimates than emigration and coverage error. Mortality rates among the foreign-born are fairly well documented yet still subject to sampling error, leading us to postulate a narrow probability distribution. Moreover, the impact of mortality tends to be small given the youthful age structure of the immigrants in our analysis. Accordingly, we find that a one standard deviation increase in the assumed mortality rate was associated with only an additional 25 thousand unauthorized immigrants in 2018.

Looking at earlier years in the simulations, emigration has not always been the most important factor. In 2005, the uncertainty in residual estimates associated with emigration (SD = 1,352) was less than the uncertainty associated with coverage error (SD = 1,778). However, uncertainty associated with coverage error declined over time as the unauthorized population grew more settled (Van Hook et al. 2014). Additionally, uncertainty associated with emigration and mortality increased over time because errors in these factors compounded as they were repeatedly applied to each LPR admission cohort every year since admission, as we illustrated in our simple simulation in Table 1.

Finally, when uncertainty in all assumptions was considered simultaneously, the variation across estimates tended to run parallel to the most uncertain underlying assumptions, i.e., coverage error in the earlier years and emigration in the later years (shown in the last column of Table 5). Uncertainty initially peaked in 2007 (SD = 2,380), declined between 2007 and 2010 (SD = 1,472), and then increased again between 2010 and 2018 (SD = 2,232). As of 2018, the 95% confidence interval of plausible residual estimates ranged from 7.0 to 15.7 million, meaning that there is a 95% probability that the true value lies within this range (Figure 5). The interquartile range – within which half of the plausible estimates lie – is narrower, ranging from 9.1 to 12.2 million.

Conclusions

The residual method is one of the most common ways of estimating the size of the unauthorized foreign-born population, but it remains unclear how sensitive residual estimates are to uncertainty in their underlying assumptions. This makes it difficult to assess the plausible range of estimates of the unauthorized foreign-born population, and whether differences between estimates are meaningful. In this paper, we produced a new series of residual estimates using the highest-quality data we could identify, and we updated and improved assumptions about coverage error, emigration, and mortality. Beyond this, we examined the extent that residual estimates may plausibly vary due to uncertainties in their underlying assumptions about coverage error, emigration, and mortality.

The results of our simulations suggest that the estimates produced by Pew and DHS, which range from 10.5 to 12 million, may not be meaningfully different from one another. These research groups may use slightly different assumptions, but their estimates fall within a narrow plausible range of 9.1 million to 12.2 million, the interquartile range in our simulations. It would be difficult to conclude that one estimate is superior to another.

Our results also suggest that it is very unlikely that the unauthorized foreign-born population is larger than about 15.7 million. This is important in light of a recently published study (Fazel-Zarandi, Feinstein and Kaplan 2018) in which the authors expressed skepticism that a significant portion of unauthorized immigrants are counted in Census data. On the basis of an inflow-outflow estimation method, they claimed that the number of unauthorized immigrants living in the country in 2016 was much higher than estimated by the residual method—ranging from 16.7 to almost 30 million, with a midpoint of 22.1 million (Fazel-Zarandi, Feinstein and Kaplan 2018). The lower bound of their estimate (16.7 million) is outside the upper-bound of the 95-percent confidence interval produced by the residual method as described in this paper: 7 to 15.7 million. Across the 1,000 simulations varying emigration, mortality and coverage error rates conducted for our analysis, only 2 percent yielded estimates of 16 million or higher and none were as high as 22.1 million. Several commentators have already published critical evaluations of the Fazel-Zarandi, Feinstein and Kaplan study and have shown that its estimates are too high because it fails to account for the circular migration patterns of unauthorized immigrants during the 1990s (Capps et al. 2018; Gelatt, Fix and Van Hook 2018; Warren 2018). Our evaluation of the plausible range of residual estimates further supports these critiques.

Finally, our results demonstrate that most of the uncertainty in residual estimates derives from uncertainty in emigration rates among the LNH population. Coverage error assumptions matter much less, and mortality assumptions scarcely matter at all. The sensitivity of residual estimates to assumptions about emigration stems from a weakness in the residual method whereby errors in emigration (and mortality) accumulate over time. Emigration rates (and to a much lesser degree, mortality rates) determine the size of surviving LNH foreign-born cohorts living in the U.S., so that when emigration is overestimated, the unauthorized population is also overestimated. In our simulations, a one-standard deviation increase in the assumed emigration rate (or about half of a percentage point) was associated with nearly 2.3 million more unauthorized immigrants in 2018. Because error in emigration rates accumulates from the time of admission to the present, this type of error will increase in the future. Similarly, emigration errors of unauthorized immigrants compound over time in the inflow-outflow model employed by Fazel-Zarandi, Feinstein and Kaplan, greatly affecting their estimates.

Unfortunately, the United States does not collect high-quality data on emigration. Researchers have had to rely on indirect methods, which tends to lead to inconsistent and imprecise estimates. It would be very easy for emigration estimates to differ by half a percentage point or more on account of any number of seemingly arbitrary methodological decisions. For example, when Leach (2017) updated his earlier work (Leach and Jensen 2013), his estimate of the emigration rate among new arrivals (<10 years in the U.S.) increased from 1.3% to 1.8%. Moreover, it is possible—even likely—that emigration rates vary over time and across demographic groups. We attempted to account for this variation by using a prediction model to estimate emigration rates for different demographic groups and by building in trends into the Schwabish-based emigration estimates, yet very little of this potential variation in emigration has been formally studied.

In conclusion, we still view the residual method as more robust than other available methods, and we believe the strength of existing evidence supports the assumptions that have been used in generating these estimates. Even if these assumptions are slightly wrong, it is unlikely that the unauthorized immigrant population is far outside the range of current, widely used, residual estimates. However, to move the field forward, it will be important to continuously develop new and better methods and data sources on unauthorized immigrants. This will become especially important as time passes and error associated with uncertainty in emigration rates continues to accumulate. Government agencies with the ability to contact and track immigrants after their admission may offer new avenues for research and development in this area. For example, DHS may have the capacity to produce precise and detailed estimates of emigration rates based on its own administrative data but has not produced such estimates for researchers’ use.

Appendix A: Data Sources Used to Produce Residual Estimates of Unauthorized Foreign-born Population 2005-2018

Component	Data Source	Notes
(C) Combined unauthorized and LNH foreign-born population
Population, 2005-2018	2005-2018 ACS (Ruggles et al. 2020)	Estimates tabulated using a sample of foreign-born persons of foreign parentage, excluding nonimmigrants and those who arrived in the U.S before 1982. Year of arrival cohorts were smoothed using weighted three-year moving averages (with .25, .50, .25 used as weights for previous, current, and next year).
Nonimmigrants, 2005-2018	Logically imputed in ACS
(LNH) LPR, Naturalized and Humanitarian Entrants
LPR admissions 1982-2000, excluding IRCA adjustments	U.S. Immigration and Naturalization Service. “Immigrants Admitted to the United States” (1982-2000), ICPSR microdata files	Data on year of entry was missing for a large portion of adjustees who were admitted in 1998 to 2000. We imputed missing year of entry for these cases using a model with age, age-squared, country of birth, sex, and class of admission as predictors. Adjustees from North America with missing year of entry were further adjusted to match distributions on year of entry of IRCA adjustees.
IRCA adjustments to LPR status	Office of Immigration Statistics, Yearbook of Immigration Statistics, Table 4 (various years) Legalized Population Survey (LPS1)	The ICPSR microdata files of 1982-2000 admissions exclude IRCA adjustees, so we added them based on numbers reported in the Yearbook of Immigration Statistics. The LPS provided demographic detail on IRCA adjustees, which we used to allocate total IRCA adjustees by age, sex, and year of entry.
LPR admissions 2001-2018	Office of Immigration Statistics. Profiles on Lawful Permanent Residents (Detailed Tabulations) Office of Immigration Statistics, Special tabulations on year of entry.	The profiles provided estimates of admissions by country, class of entry, age, and sex. The special tabulations provided additional information on year of entry.
Refugees, asylees, and other humanitarian entrants but not yet adjusted as of 2018	2018 Yearbook of Immigration Statistics, Tables 14, 17, & 19; Department of Health and Human Services (SIV and Cuban entrants, page 38)	Counts of refugee, asylee, and other humanitarian entrants are based on DHS data. We included entrants for 2015-2018 while discounting those who had already been adjusted as LPRs.
(D) Deaths rates for LNH foreign born
Mortality rates by age, sex, and year	Human Mortality Database, U.S. death rates by age, sex and year
Relative hazard of dying for immigrants relative to overall population	1997-2009 National Health Interview Surveys and Linked Mortality Records, (Blewett et al. 2019).	See Table 4.
(E) Emigration rates for LNH foreign born
Emigration rates	Schwabish (2009)	Schwabish (2009) estimated emigration using Social Security earnings data of those who left the country. We used a prediction model based on his data (provided to us by Schwabish) to estimate emigration rates by age, sex, country of birth, and duration of residence.
Annual trends in emigration rates	American Community Survey 2001-2018, (Ruggles et al. 2020)	We used the ACS to produce residual estimates following Leach’s (2015) methodology. Results were used to adjust the Schwabish-based estimates for annual trends in emigration.
Variation in emigration rates	Leach & Jensen 2013, Table 1
(e) Coverage Error	Van Hook, et al. 2014	See Appendix C for more details

Appendix B: Comparison of Underlying Assumptions of Residual Estimates

	DHS (Baker 2021)	Pew (Passel and Cohn 2018)	This Article
Foreign-born in ACS or other survey	All foreign born in the 2014 ACS, projected forward to January 2015.	All foreign-born in the ACS each year (or in the CPS for years prior to 2005).	All foreign-born in the ACS, excluding nonimmigrants and those who arrived before 1982
Year of entry	1980-present	1980-present	1982-present
Adjustment for heaping on year of entry	three-year moving average.	Not mentioned.	three-year moving average.
Nonimmigrants	Administrative data on the average number of nonimmigrants present per day between July 1, 2014, and June 30, 2015. Counts only nonimmigrants in those in categories for which the length of stay typically exceeds two months, such as students, temporary workers, and exchange visitors.	Logically imputed in ACS, and then dropped from sample of foreign-born in the ACS.	Logically imputed in ACS, and then dropped from sample of foreign-born in the ACS.
LPR Admissions	USCIS records. For people who adjusted to a green card within the U.S., year of entry is assumed to be their last year of entry after 1980 and before obtaining a green card.	DHS records.	DHS records. Projected LPR population starts at year of admission (not year of entry) since that is the year the LPRs attained that status. However, year of entry is used in breakdowns of population by duration of residence. Each estimate year includes just half of admissions in that year because it represents a mid-year estimate.
Recent refugees, asylees and other humanitarian entrants not yet adjusted to LPR status	Administrative data on dates of entry and dates of adjustment to LPR status show that refugees take on average 2.2 years to adjust and asylees take on average 4.1 years to adjust to a green card. So they counted all refugee entrants for the 2.2 years prior to January 2015 as lawfully present immigrants and people granted asylum for the 4.1 years prior to January 2015 as lawfully present immigrants.	Identifies refugees and asylees in the ACS based on DHS data on country of birth and year of immigration to align with administrative data on these populations.	Counts of refugees, asylees and other humanitarian entrants based on DHS data (found in online tables) by country, year of immigration, and allocated age and sex. Add in up to 4 years’ worth, discounting those who had already been adjusted as LPRs.
Mortality rates for LNH foreign born	Age and sex specific mortality rates for 1999-2001 based on U.S. national vital statistics.	Not stated.	Baseline age- and sex-specific mortality rates for each year obtained from the Human mortality database for the U.S. These rates are adjusted using hazard ratios (from NHIS-linked mortality data) by race/ethnicity and nativity.
Emigration rates for LNH foreign born	Based on Ahmed and Robinson 1994 (average emigration rate of 1.1 percent per year for 1980-2014). Refugees, asylees, and other humanitarian entrants are assumed to have a migration rate of zero	Not stated.	Prediction model based on Social Security earnings data (Schwabish 2009) and adjusted for trends based on residual methodology of Leach (2015). Refugees, asylees and other humanitarian entrants are assumed to have a migration rate of zero
LNH foreign born	2.5%	Not mentioned	1.5%
Nonimmigrants	10%	N/A	N/A
Unauthorized immigrants	10%	10%-20% in the 1995-2000 CPS, with slightly larger (30% or above) adjustment for unauthorized Mexicans; 8%-13% coverage error adjustment in 2000-09 estimates; 5%-7% coverage error adjustment in 2010-16 estimates.	Mexican/LA/Carib: Children 21%, Men steady decline from 20% in 2000 to 8% in 2015, and then rise to 10% in 2018; Women, steady decline from 20% in 2000 to 1% in 2018. Others: 25% lower than Mexican/LA/Carib. All groups: recent arrivals (less than 5 years in the country) have coverage error rates that are three times higher than the rates of longer-term residents (10+ years in the country).
Other populations excluded from potential unauthorized pool	Cubans	Cubans	Cubans

Appendix C: Coverage Error

In previously published research, Van Hook and colleagues (2014) used three different methods—the death registration, net-migration, and birth registration methods—to estimate the coverage error of the Mexican immigrant population living in the United States between 2000 and 2010. They also extrapolated from these results coverage error estimates for the unauthorized Mexican-born population from about 1995 through 2010. Because each method provided a range of plausible estimates rather than a single estimate, this work is helpful for establishing both the magnitude and uncertainty of coverage error.

Here, we briefly review their methodology and update the estimates for working age men and women based on the death registration and non-migration methods through 2016. We do not update the birth registration estimates because of limited access to updated data on the number of births to Mexican-born mothers. We refer readers to the original research for a more complete description of these methods.

The Death-Registration and Net-Migration Methods

The death-registration method uses the number of Mexican immigrant deaths observed in U.S. vital statistics and age-specific death rates for the Mexican born to estimate how large the population must be for it to generate the observed number of deaths. The strength of the death registration method is that virtually all deaths in the United States are registered, and classification as Mexican origin among the foreign born appears to be consistent with reporting in surveys (Arias et al. 2010).

Specifically, if the number of deaths and a set of age-specific death rates from an independent data source are known, then the expected population for the Mexican born can be expressed as:

$$\text{E}[{{}_{\text{n}}\text{P}}_{\text{x}}]={{}_{\text{n}}\text{D}}_{\text{x}}/{{}_{\text{n}}\text{M}}_{\text{x}}$$

Coverage error for the Mexican-born population is then estimated as:

$${{}_{\text{n}}\text{r}}_{\text{x}}=(\text{E}[{{}_{\text{n}}\text{P}}_{\text{x}}]-{{}_{\text{n}}\text{P}}_{\text{x}})/\text{E}[{{}_{\text{n}}\text{P}}_{\text{x}}],$$

where _nP_x is the enumerated population age x to x+n.

Van Hook and colleagues (2014) used the death registration method to produce estimates of coverage error for the working age and elderly populations from 1995 to 2008. To update these estimates, we used the deaths for Mexican immigrants and mortality rates for Hispanics in 2007-2017 produced by NCHS (e.g., Arias and Xu 2019). The NCHS estimates pertain to all Hispanics rather than the group we focus on, the Mexican born. To adjust, we multiplied the NCHS death rates by mortality ratios (Hispanic foreign-born/Hispanic) reported by Eschbach et al. (2006). Low versus high range estimates reflect sampling error in the ratios reported by Eschbach et al.

The net-migration method compares two estimates of net migration from Mexico to the United States, one based on Mexican census data and another based on U.S. Census data (Hill and Wong 2005). The major assumption is that the Mexican census is better positioned to enumerate this population than the U.S. Census. Unauthorized immigrants may be reluctant to participate in U.S. data collection efforts because they do not want to attract attention and might worry that participating would lead to their deportation. However, this group would have no strong reason to avoid census enumerators in Mexico. This method is more sensitive to the coverage of recent migrants than the whole population.

The net-migration method involves two steps. Step 1 estimates the number of net migrants moving from Mexico to the United States based on an analysis of the demographic components of change between two censuses or surveys, first on the basis of Mexican data, and second, U.S. data. The Mexican-born population in any given census year t (P^t) equals the population from the previous census or major survey (e.g. P^t−5), projected forward while accounting only for natural increase (P^t,NI) plus net international migration (N). Net migration is thus equal to the enumerated minus the projected population:

$$\text{N}={\text{P}}^{\text{t}}-{\text{P}}^{\text{t},\text{NI}}.$$

Step 2 converts the two net annual migration estimates (based on Mexican versus U.S. data) into estimates of coverage error for all Mexican born. To do this, consider that the U.S.-resident Mexican-born population in one year of the ACS can be expressed as the population in an earlier year of the census or ACS that has been projected forward (accounting only for natural increase) plus net migration, where net migration is based on U.S. Census data:

$${{\text{P}}^{\text{t}}}_{\text{US}}={{\text{P}}^{\text{t},\text{NI}}}_{\text{US}}+{\text{N}}_{\text{US}}.$$

However, an expected population estimate can be obtained by substituting the U.S.-based estimate of net migration with the Mexican-based estimate:

$$\text{E}[{{\text{P}}^{\text{t}}}_{\text{US}}]={{\text{P}}^{\text{t},\text{NI}}}_{\text{US}}+{\text{N}}_{\text{Mx}}.$$

The estimated coverage error rate is then:

$$\text{r}=(\text{E}[{{\text{P}}^{\text{t}}}_{\text{US}}]-{{\text{P}}^{\text{t}}}_{\text{US}})/\text{E}[{{\text{P}}^{\text{t}}}_{\text{US}}].$$

Van Hook and colleagues used the net migration method to estimate coverage error for working-aged Mexican immigrants for the 1995-2000 and 2000-2005 periods. Here, we update these estimates for the 2005-2010 and 2010-2015 periods. We produce a range of estimates to reflect different assumptions about the level of coverage error in the Mexican census.

Extrapolating Coverage Error Estimates from the Mexican-born to Unauthorized Mexican-born

After estimating coverage error for the Mexican-born population, we extrapolated estimates of coverage for the unauthorized Mexican born. Our approach builds on logic that coverage error, r, is a function of the size and coverage error of its subgroups: r = 1 - {1/[p₁/(1-r₁)+p₂/(1-r₂)]}, where p₁ and p₂ are the proportions and r₁ and r₂ are coverage error rates for subgroups 1 and 2. If, for example, coverage error is 0% among lawfully present Mexicans, 30% among unauthorized immigrants, and 45% are unauthorized, then the minimum coverage error for all Mexican born = .16 = 1-{1/[.45/(1-.30)+.55]}.

The coverage error levels for the lawfully present segments of the Mexican-born population are relatively low and have narrow ranges of plausible variation. For example, there is little reason to think that coverage error for the lawfully present Mexican-born population is appreciably greater than that for Hispanics as a whole (the coverage error rate for whom in the 2010 Census was estimated at 1.5 percent) (Mule 2012).

When we combined these in the above equation with the error levels we estimate for the entire Mexican-born population along with a plausible range of coverage error for the lawfully present Mexican-born population (we assumed it ranged from 0 to 2.5 percent), we were able to solve for the levels of coverage error in the unauthorized Mexican-born population.

Results

Table C1 summarizes the results for unauthorized Mexican-born men and women ages 15-64. It presents high, low and average coverage error levels for various time points or periods based on the death registration and net migration methods. The death-registration estimates are averaged across multiple years to help smooth erratic fluctuations in the numbers of deaths, but the net-migration estimates pertain to singular years. The high and low estimates reflect uncertainty in each estimate’s underlying assumptions as described above and in prior publications (Van Hook et al 2014). The top panel displays estimates for unauthorized immigrants while assuming that the lawfully present Mexican-born population has no coverage error, while the lower panel provides estimates that assume higher (2.5%) coverage error for the lawfully present Mexican-born population.

We draw two major conclusions. First, coverage error tends to be relatively high for the Mexican-born population during the early 2000s. When we assume no coverage error among the lawfully present Mexican-born population (top panel), the average death-registration estimates are 20 percent in 2000-04, and the average net-migration estimates are 15 and 31 percent for 2000 and 2005, respectively. The estimates are somewhat lower but show a similar pattern when we assume higher rates of coverage error for the lawfully present Mexican-born population (lower panel).

Second, coverage error appears to have declined substantially during the 2000s. This trend appears in estimates based on both the death-registration and net-migration methods. By the 2010-decade, average coverage error rates for the unauthorized Mexican-born population were eight percent or less. Estimates for women appear to have dropped more than among men, however.

Van Hook and colleagues (2014) suggest that these trends may be due to declines in circular migratory patterns among unauthorized labor migrants in the years following the Great Recession. Circulatory labor migrants may be particularly difficult to capture in household surveys, which may have led to particularly high coverage error rates during the construction boom years of the early 2000s. This circulatory migration pattern was disrupted during the housing crisis of 2007, when many of the construction and service jobs filled by unauthorized immigrants disappeared. This interpretation is further supported when we examine the trends in coverage error estimated by the net-migration method. The net-migration method is particular adept at capturing coverage error among recently arrived immigrants, and it shows very high coverage error for 2000 and 2005 (higher than the death-registration estimates), followed by very low rates for 2010 and 2015.

We next extrapolated trends in coverage error for the Mexican unauthorized population from 2005 through 2017, based on an average and linear interpolation of the results in Table C1. We also estimated the standard deviation across estimates that rely on different underlying assumptions, which we used as a measure of uncertainty in the analyses for this article. The results are shown in Table C2. They show high coverage error from 2000 through 2006, declines from 2007 through 2015, and small increases in coverage error from 2015 through 2017, especially for men. Uncertainty in coverage was also high at the beginning of the time period but declined over the years.

Table C1.

Coverage error estimates of the unauthorized Mexican-born population, by method

	Death Registration Method				Net migration Method
	2000-04	2005-09	2010-14	2015-17	2000	2005	2010	2015
If Coverage error for legal Mexican immigrant adults is zero
Adults 15-64
Low	16	4	0	0	6	31	0	0
High	23	12	5	6	23	31	15	8
Average	20	8	3	3	15	31	8	4
Women 15-64
Low	19	4	0	0	13	30	0	0
High	29	16	3	2	25	32	4	4
Average	24	10	1	1	20	31	2	2
Men 15-64
Low	14	3	0	1	29	30	0	7
High	19	10	7	9	36	31	11	15
Average	17	6	3	5	33	31	6	11
If Coverage error for legal Mexican immigrant adults is 2.5 percent
Adults 15-64
Low	15	2	0	0	5	30	0	0
High	23	11	4	5	0	30	14	7
Average	19	7	1	2	2	30	7	3
Women 15-64
Low	18	2	0	0	12	0	0	0
High	28	14	1	0	24	31	2	2
Average	23	8	0	0	18	16	0	0
Men 15-64
Low	13	2	0	0	29	0	0	6
High	19	9	6	8	36	31	10	15
Average	16	5	2	4	32	16	5	10

Table C2.

Summary of Estimates of Coverage Error for Mexican Unauthorized Foreign-born adults by year

	2005	2006	2007	2008	2009	2010	2011	2012	2013	2014	2015	2016	2017
Average Estimates
All	20.4	17.9	15.4	12.9	10.4	7.9	7.2	6.4	5.6	4.8	4.1	4.3	4.5
Women 15-64	21.9	18.5	15.2	11.8	8.4	5.1	4.3	3.6	2.8	2.1	1.3	1.3	1.3
Men 15-64	17.4	15.3	13.1	11.0	8.8	6.7	6.9	7.2	7.4	7.6	7.9	8.7	9.6
Standard Deviation of Estimates
All	6.6	7.0	7.5	8.0	8.4	8.9	7.9	6.8	5.8	4.7	3.7	3.7	3.7
Women 15-64	1.6	3.3	5.1	6.9	8.6	10.4	9.2	8.1	6.9	5.8	4.6	4.6	4.6
Men 15-64	5.9	5.8	5.7	5.6	5.5	5.4	4.9	4.4	3.9	3.4	2.9	2.9	2.9