A New Residual Approach for Estimating Undocumented Populations

Abstract

Demographers generally agree that the total undocumented population in the United States increased from 3.5 million in 1990 to about 12 million in 2008 and declined from 2008 to 2010. The consensus breaks down after 2010, however, with three of the five organizations that derive annual estimates of the population showing increases from 2010 to 2018 and two organizations reporting substantial declines over the period. The primary reason for this divergence is that organizations use a variety of estimates of emigration of legal residents, and in some cases the data are decades out of date. Reliable information about trends in this population is important for developing legislative and administrative policies to reduce the population and for assessing the effectiveness of enforcement efforts. This research note describes an improved residual method for updating annual estimates of the undocumented population counted in the American Community Survey (ACS; Ruggles et al. 2022); the method incorporates a time-varying estimate of emigration. Data needed to update estimates are available in the year estimates are derived, and a new estimate can be compiled as soon as ACS data are released. The methodology and data needed to update an estimate of the undocumented population each year are described.

Reliable information about trends and characteristics of the undocumented population in the United States is important for developing legislative and administrative policies to reduce the population and to assess the effectiveness of enforcement efforts. Typically, estimates of the undocumented population are derived using the residual method in which an estimate of the legal foreign-born population is subtracted from census- or survey-based estimates of the total foreign-born population. This research note proposes the first major modification of the residual method since it was first used to estimate the undocumented population nearly four decades ago.

Typically, estimates of the undocumented population are derived by subtracting an estimate of the legal foreign-born population from census- or survey-based estimates of the total foreign-born population. The difference, or residual, is an estimate of the undocumented population. An estimate of the undocumented population is derived for an initial year, for example 2010, and the estimate is “carried forward” from year to year by repeating the residual method. The method described here improves the process of “carrying forward” the undocumented population primarily because it incorporates an improved method of estimating annual emigration. Note, however, that the proposed method still requires an estimate of the undocumented population in an initial year.

The residual method has had several successes over the past few decades. The first empirical estimates of the undocumented population – about 2.1 million counted in the 1980 census – used the residual method (Warren and Passel 1987). The residual methodology was used to accurately project the number of undocumented residents that would apply for legalization under the Immigration Reform and Control Act of 1986 (Warren 2015). Over the past few decades, the residual method (Baker 2021; Capps et al. 2020) has been the principal way to determine the size of the undocumented population in the United States. The method is currently used by four organizations: the Department of Homeland Security (DHS; Baker 2021); the Migration Policy Institute (MPI; Capps et al. 2020); the Congressional Budget Office (CBO; Heinzel, Heller and Tawil 2021); and the Pew Research Center (Pew; Passel and Cohn 2018). The Center for Migration Studies (CMS) uses a somewhat different methodology that begins with estimates for 2010 based on the residual method; see the Appendix and Warren (2019) for more information about how CMS’s estimates are derived after 2010.

As useful as the residual method has been, different estimates derived using the method began to diverge after 2010. Three of the five organizations that routinely estimate the size of the population showed increases from 2010 to 2018 and two organizations reported substantial declines over the same period (see Figure 1). The relatively narrow vertical scale of nine million to eleven million in Figure 1 was chosen to highlight differences in the slopes of the lines. If the scale was set in the 0 to eleven million range, the estimates would appear to be much more consistent, with all the estimates within a range of about plus or minus about 500,000 from each other (Van Hook et al. 2021). Nevertheless, the differences in the estimates at the beginning and end dates are substantial; as shown in Figure 1, the CMS estimate declined by 1.0 million while the CBO estimate increased by 675,000.

Figure 1.

Estimates of undocumented residents counted in the ACs, by organization: 2010 and 2018 (in 5000s).

Differences in the trends illustrated in Figure 1 are likely the result of differences in the data and methods used to estimate emigration. The residual method, as used currently, requires annual data on emigration of the foreign-born population from 1980 or 1982 to the estimate date. Much of the data currently being used to estimate emigration for residual estimates are decades out of date. Two organizations (DHS and CBO) rely on the same emigration estimates from the 1980 and 1990 censuses for their emigration rates (Ahmed and Robinson 1994). MPI uses emigration rates based on Social Security data collected from 1978 to 2003 (Schwabish 2009). CMS’s method does not require estimates of emigration, and as far as we know Pew has not published its methodology for estimating emigration.

Continued divergence of the trend lines in Figure 1 could reduce confidence in future estimates of the population and reduce confidence in the efficacy of the residual method. As Van Hook et al. (2021: 2333) observed:

[T]o move the field forward, it will be important to continuously develop new and better methods and data sources with which to estimate the number of unauthorized immigrants. This will become especially important as time passes and error associated with uncertainty in emigration rates continues to accumulate.

Research Design

The estimates of undocumented residents described below refer to the number counted in the American Community Survey (ACS). They have not been adjusted for under-enumeration; for this reason, the numbers will not match the (undercount-adjusted) estimates published by other organizations. Unless specified otherwise, all ACS data refer to noncitizens that arrived after 1981. The term emigration, or left the population, includes both voluntary emigration and removal by DHS. Finally, for brevity, the population composed of legal permanent residents (LPRs, or immigrants admitted for permanent residence), refugees, asylees, and nonimmigrants (for example, students, temporary workers, and their families) is referred to as the legal noncitizen population.

The noncitizen population counted in the ACS each year is made up of two parts: the undocumented population and the legal noncitizen population. The methodology presented here describes a new way to derive annual estimates of the legal noncitizen population. That estimate can be subtracted from the ACS count of noncitizens to estimate the undocumented population counted in the ACS.

This research note proposes the first major modification of the residual method since it was first used to estimate the undocumented population nearly four decades ago. The new approach has important advantages over the residual procedures currently being used. All the data needed to derive an estimate of undocumented residents for a new year, including estimates of emigration, are available at the time ACS data are released, or data are available in recent publications¹. Also, this new residual approach reduces the amount of data to be estimated because, unlike the current residual methodology, naturalized citizens are not part of the estimation procedure.

The new residual method described here uses current information to produce estimates of the legal noncitizen population each year for subtraction from the ACS count of noncitizens. The method simplifies the process of estimating annual changes in the legal noncitizen population. As described in the next section, only four data elements are needed to update an estimate of the legal noncitizen population, and thus the undocumented population, each year – annual ACS and DHS data, undercount rates in the ACS (available for the 2010 ACS), and life table survival rates (available each year).

Estimates of the Legal Noncitizen Population, 2010 to 2011

We begin by deriving estimates of the size of the undocumented population in 2011; later we present results for each year through 2018 using the same methodology.

In the traditional residual method, each annual immigrant entry cohort is tracked from its year of entry, as long ago as 1982, to the estimate date, subtracting estimates of emigration and deaths each year. In the approach presented here, the legal noncitizen population is updated using information collected or estimated in the same year as the estimate. That is, except for the population at the beginning of the year, no historical data are needed to derive new estimates. Table 1 shows the components of population change of the legal noncitizen population from 2010 to 2011 and the estimated undocumented population in 2011.

Table 1.

New Residual Method of Estimating the Undocumented Population in 2011.

Numbers in thousands.	All ACS data are for the population that arrived after 1981.
Component of population change	Estimate	Source of data or estimate
1. Noncitizens counted in 2010 ACS	19,907	2010 ACS
2. Undocumented pop. counted, July 2010	10,500	Beginning population
3. Legal noncitizen pop., July 2010	9,407	3 = 1 − 2
4. Immigrants, refugees, and asylees	1,065	2011 DHS Yearbooka
5. Naturalizations	−656	2011 DHS Yearbook
6. Change in nonimmigrant population	114	Compiled from DHS datab
7. Deaths	−47	2010 ACS data and survival ratesc
8. Emigration rate	−2.4%	Computed as shown in Table 2.
9. Emigration	−233	9 = [3 + sum(4 to 7)/2.0] × 8
10. Legal noncitizen population, July 2011	9,649	10 = Sum(3 to 7) + 9
11. Noncitizens counted in 2011 ACS	19,710	2011 ACS
12. Undocumented pop. counted, July 2011	10,061	12 = 11 − 10

^aDHS data reduced by 3.6 percent for undercount. See text for explanation of the reduction.

^bDerived from data in Table 25, 2019 DHS Yearbook. See Appendix Table B for method of computation.

^cComputed with crude death rate of 4.9 derived based on the age distribution of 2010 ACS data for noncitizens that arrived after 1981 and age-specific life table survival rates from Arias and Xu (2019).

The estimation procedure requires an initial population to begin the process. The estimate of 10.5 million in Table 1, row 2, was chosen for this example because it approximates the average of the figures for 2010 shown in Figure 1. Each organization that derives estimates of this population has an initial estimate of undocumented residents counted in the ACS. Note, also, that some organizations derive estimates “as of” July 1 while others are as of January 1. Likewise, for some, ACS data for noncitizens begins with 1980 and, for others, in 1982. These minor differences can easily be taken into account in compiling data to use in the estimation procedure illustrated in Table 1.

The components of change in items 4 through 7 in Table 1 are straightforward; however, item 4 requires clarification. In item 4, the number of immigrants, refugees, and asylees has been reduced by 3.6 percent to make the data compatible with other “census-level” data in the table. The undercount rate of 3.6 percent for legal immigrants used in this table is half the rate estimated for the foreign-born population (see Appendix).

An emigration rate of 2.4 percent is shown in Table 1, row 8. Explanation of the method used to compute that rate is deferred to the next section to simplify the presentation of the proposed methodology. Note, however, that it is an important component of population change, and the method of estimation is described in detail in the next section.

In Table 1, the legal noncitizen population in 2011 is derived by adding the components of population change (steps 4 to 9) to the beginning population (step 3). That population is subtracted from the 2011 ACS count of the noncitizen population (step 11). The result is an estimate of the undocumented population in 2011. These steps can be repeated for each subsequent year.

Estimated Emigration Rate for 2010 to 2011

Errors in the emigration rate in Table 1, row 8, would affect the estimated legal noncitizen population and estimates of undocumented residents in future years. If the emigration rate of legal noncitizens was too high, the estimates of the undocumented population would be too high. Conversely, if the emigration rate of legal noncitizens was too low, the undocumented population would be too low.

Emigration rates used to estimate change in the legal noncitizen population should: (1) cover the same period as the estimates and (2) be derived from information easily accessible and likely to be available on an annual basis in future years. The data used to construct Table 2 meet both criteria. The estimation procedure requires data from the two most recent ACS counts, undercount adjustments for the ACS, and life-table survival rates. All the other components of change and the emigration rate are derived from these three sets of data.

Table 2.

Methodology for Estimating the Emigration Rate of the Foreign-Born Population, 2010 to 2011.

Numbers in thousands.		All ACS data are for the foreign-born population that arrived after 1981.
Component of population change	July 2010-July 2011	July 2011	Source of data or estimate
1. Foreign-born pop., July 2010	29,725	30,236	ACS data for 2010 and 2011
2. Percent undercount	7.2%	7.1%	Jensen et al., 2015 (See Appendix)
3. Adjusted population	32,035	32,549	3 = 1 / (1.0 − 2)
4. Net change, 2010 to 2011	513		4 = Row 3, 2011 – row 3, 2010
5. Arrivals, 2010 to 2011	1,230		ACS data for 2010 and 2011a
6. Percent undercount of arrivals	14.4%		6 = 2 × 2.0
7. Arrivals adjusted for undercount	1,437		7 = 5 / (1.0 − 6)
8. Left the population, 2010 to 2011	−924		8 = 4 − 7
9. Crude death rate	4.7		2010 ACS data and survival ratesb
10. Deaths, 2010 to 2011	−151		10 = −[3 +(4/2.0)] × 9 /1000
11. Emigration, 2010 to 2011	−773		11 = 8 − 10
12. Emigration rate, 2010 to 2011	−2.4%		12 = 11 / 3

^aArrivals = [(2010 ACS, arrived 2010) + (2011 ACS, arrived in 2010 and 2011)]/2.0.

^bDerived using age distribution of 2010 ACS data for the foreign-born population that arrived after 1981 and age-specific life table survival rates from Arias and Xu (2019).

The procedure for estimating emigration in Table 2 is straightforward. The only data needed to estimate emigration for a year are net population change, arrivals, and deaths. Net population change from 2010 to 2011 was 513,000 (row 4). About 1,437,000 arrived during the year (row 7). That means 924,000 left the population during the year (row 8). Of those 924,000, 151,000 died (row 10); therefore, 773,000, or 2.4 percent, emigrated.

The estimate of undercount in the ACS is the only component of change in Table 2 that could have a sizable effect on the estimated emigration rate in Table 2. For this reason, a later section includes a sensitivity analysis of the effects of alternative estimates of undercount on the emigration rate derived here.

The undercount rate of 7.2 percent in Table 2, row 2, is based on estimates of undercount, by age, Hispanic origin, and gender (Jensen, Bhaskar and Scopilliti 2015) and data from the 2010 ACS. The rates shown in Tables A3 and A4 in that report were multiplied by corresponding data for the foreign-born population that arrived from 1982 to 2010 and were counted in the 2010 ACS. The same procedure was repeated for 2011 and each subsequent year to derive annual estimates of undercount. Details of the computations for 2010 are shown in Appendix Table A.

New arrivals in the ACS include new immigrant arrivals, undocumented residents, and temporary residents, such as students, temporary workers, and their families. These groups are likely to have higher undercount rates in their first year than the population that has resided here longer. ACS data for new arrivals in Table 2, row 6, were adjusted for undercount at twice the rate for the rest of the population, or 14.4 percent.

In recent years, the Census Bureau has used data for both “year of entry” and “residence one year ago” for deriving postcensal population estimates. The data for arrivals in Table 2, row 5, as adjusted in row 7, were derived using data from the year of entry question. The estimates produced using data from the question on residence one year ago yielded estimates of undocumented residents in 2018 that were well below those produced using data from the year of entry question.² We chose to use data from the question on year of entry to avoid underestimating the undocumented population.³

Estimates of emigration of the legal noncitizen population are needed for Table 2, but data are not available to estimate emigration for this population. The annual emigration rate of 2.4 percent shown in Table 2 was derived from data for the total foreign-born population. The discussion of emigration rates by type of population – undocumented, naturalized, and legal noncitizens – in the next paragraph indicates that the emigration rate of 2.4 percent derived for 2010 in Table 2 is an appropriate rate for legal noncitizens.

The foreign-born population of about 32 million⁴ in 2010 (see Table 2, row 3) is composed of three primary groups with roughly the same size populations: naturalized citizens (10.0 million; ACS), undocumented residents (10.5 million; Table 1, row 2), and legal noncitizens (9.4 million; Table 1, row 3). It is safe to assume that naturalized citizens have emigration rates considerably lower than the rate of 2.4 percent derived here for the total foreign-born population. It is also likely that undocumented residents have higher rates than legal noncitizens. Thus, the emigration rate of 2.4 computed in Table 2 is a good approximation of the emigration rate of the legal noncitizen population. The sensitivity analysis in a later section provides further information on the effects of alternative assumptions on the emigration rate.

Results

The data sources and estimation procedures used above for 2010 to 2011 are available each year through 2018. Table 3 extends the estimates of legal and undocumented resident beyond 2010 using the data sources and methods shown in Tables 1 and 2. The purpose of these estimates is to facilitate the sensitivity tests described in the next section.

Table 3.

Estimates of the Legal Noncitizen and Undocumented Population Counted in the ACS, 2010 to 2018.

Numbers in thousands.
Estimate	2010	2011	2012	2013	2014	2015	2016	2017	2018
Noncitizens in the ACS*	19,907	19,710	19,700	19,703	19,992	20,289	20,312	20,388	20,103
Legal noncitizen pop.	9,407	9,649	9,769	9,840	10,174	10,471	10,586	10,679	10,588
Undocumented pop. counted	10,500	10,061	9,932	9,864	9,818	9,818	9,726	9,709	9,515

Source: Same data sources and assumptions used in Tables 1 and 2.

^*Excludes Cuba.

Undocumented, as used here is defined as noncitizens counted in the ACS minus the following legal residents: LPRs (immigrants admitted for permanent residence), refugees, asylees, and nonimmigrants (for example, students, temporary workers, and their families).

Effects of Alternative Undercount Assumptions on the Estimates

The only components of change that could have more than a minor effect on the figures in Tables 1 and 2 and the estimates of undocumented residents shown in Table 3 are the adjustments of ACS arrival and population data for undercount (see Table 2). The population data were adjusted by 7.2 percent, and arrival data by 14.4 percent. Table 4 shows the effects of alternative undercount assumptions on the estimated undocumented population for one year, 2011, and for the longer term, 2018. It shows the effects of increasing or decreasing the undercount rates shown in Table 2 by 25%.

Table 4.

Effects of Alternative Undercount Rates on the Estimated Undocumented Population in 2011 and 2018.

Numbers in thousands.
Assumption	Undercount rates for:		Emigration rate (3)	Undocumented population (4)	Difference (5)	Percent difference (6)
	Population (1)	Arrivals (2)
Effects of alternative assumptions on the estimated undocumented population in 2011
Rates used	7.2%	14.4%	−2.4%	10,061	-	-
Rates × 1.25	9.0%	18.0%	−2.6%	10,089	28	0.3%
Rates × .75	5.4%	10.8%	−2.3%	10,034	−27	−0.3%
Effects of alternative assumptions on the estimated undocumented population in 2018
Rates used	6.7%	13.5%	−3.4%	9,515	-	-
Rates × 1.25	8.4%	16.9%	−3.5%	9,694	180	1.9%
Rates × .75	5.1%	10.1%	−3.3%	9,340	−175	−1.8%

Note. Undocumented population refers to the number counted in the ACS.

Table 4, column 1, shows alternative undercount rates for the ACS population data, and column 2 shows alternative undercount rates for arrivals in the ACS. Column 3 shows emigration rates for 2011 and 2018 under these alternative assumptions. Columns 4 and 5 show the estimated undocumented population counts in 2011 and 2018 and how much each would change if the alternative assumptions were used. Finally, column 6 shows the percent change in the estimated undocumented population.

The data shown in Table 4 indicate that annual estimates of undocumented residents derived using the new residual method are not sensitive to large changes in the assumptions about undercount in the ACS. If the rates used to adjust the population and arrival data in Table 2 for undercount were 25 percent higher or were reduced by 25 percent, the emigration rates would change by just 0.25 percent in 1 year (see Table 4). After 8 years, these large alternative rates would change the estimated undocumented population by approximately 180,000, or about 1.9 percent.

Effects of Changes in Completeness of Coverage in the ACS

Each year, the ACS sample is controlled to independent population controls, including age, sex, and Hispanic origin to maintain consistency with the previous census count. The Census Bureau does not derive these independent estimates for the foreign-born population. As a result, completeness of coverage of the foreign-born population can decline over the decade.⁵ This section discusses possible effects of declining coverage in the ACS on residual estimates of the undocumented population.

First, we should note that in the estimates derived for Table 3, the estimated completeness of coverage of the foreign-born population in the ACS increased from 92.8 percent in 2010⁶ to 93.2 percent in 2018. Improvement in coverage from 2010 to 2018 in these estimates occurred because of changes in the age structure; the same age-specific undercount rates were used each year.

Setting aside changes in coverage due to changes in the age structure, we can quantify the effects of declining annual ACS coverage rates on the estimated undocumented population in 2018. In a simulation, we gradually increased the undercount rate so that it was 2.0 percent higher in 2018 than the rate for 2018 used to derive the estimates shown in Table 3. That is, the rate increased from 7.2 percent in 2010 to 8.7 percent (instead of 6.7 percent) in 2018. The result was a decrease of 152,000, or 1.5 percent, in the undocumented population estimate in 2018.

This exercise demonstrates that a decline in coverage of the foreign-born population in the ACS⁷ can lead to an underestimate of the undocumented population later in a decade. Note, however, that a change in coverage of 2.0 percent over eight years would affect the later population estimate by only 1.5 percent. The potential error caused by a similar decline in coverage would have an even greater effect on estimates derived using the traditional residual method. That is because the traditional method uses a considerably larger⁸ portion of the ACS data than in the residual method proposed here.

Effects of Possible Misreporting of Citizenship in the ACS

Misreporting of citizenship in the ACS could affect the estimates shown in Table 3. For example, if large numbers of noncitizens reported being naturalized each year, the annual estimates of undocumented immigrants derived would be too low. We used the results of a study by Van Hook and Bachmeier (2013) to test the effects of misreporting of citizenship on the estimates shown in Table 3. Consistent with the results for Mexico shown in Table 1 of that study, we tested the effect of adding 125,000⁹ to the ACS count of noncitizens each year. This raised the estimated undocumented population shown in Table 3 by 125,000 each year. The annual estimates of the undocumented population shown in Table 3 would increase by about 1.1%.

Discussion

This article demonstrates a simplified and improved residual method of deriving annual estimates of the undocumented population. The new method has several advantages over the standard residual methodology. The residual method requires annual estimates of emigration and mortality for the legal foreign-born population (citizens as well as noncitizens) for 1982 to the estimate date, and in most cases the emigration data available for those estimates are decades out of date. The result is displayed in Figure 1 – disparate trend lines that would likely be closer to parallel if accurate and timely emigration rates had been available.

Van Hook et al. (2021) highlighted the importance of emigration as a component in the estimation of undocumented populations in a simulation of the effects of different components of change on future estimates. The simulation showed that, compared to other components of population change, “the residual estimate is especially sensitive to changes in emigration rates” (p.2320). The adoption of the techniques for estimating emigration described here should reduce the possible range of error in future estimates of undocumented populations.

The approach described here focuses on estimates of annual change in the legal noncitizen population compiled using currently available data. The legal noncitizen population estimated here is about half the size of the total legal resident population, and far less information is required to estimate annual change. The scope of the estimates is narrowed considerably – cohort entry data for previous years are not needed. Most importantly, the information needed to update an estimate of the legal noncitizen population is available from data collected annually in the ACS or compiled by DHS.

A decline in completeness of coverage in the ACS over a decade can cause an underestimate of the undocumented population by the end of the decade. The simulation described above indicated that a decline in coverage of 2.0 percent from 2010 to 2018 would yield an error of less than 2 percent in the estimated undocumented population in 2018. The potential error in estimates derived using the traditional residual method would be larger than for the proposed method because the traditional method uses a relatively larger share of the ACS foreign-born population.

The method of estimating emigration described here has several important advantages over those currently being used. First, the proposed emigration rate is more timely than other emigration rates available—as noted, most are decades out of date. Second, except for the mortality rate (a minor component) used in the estimate, all the data needed to estimate emigration are from current ACS data, and thus the emigration rate can be updated each year. Third, the demographic procedures and concepts used to estimate emigration for a year are straightforward: Emigration = net population change (year 2 – year 1) minus arrivals during the year minus deaths. Finally, the ACS population and arrival data used to estimate emigration are adjusted for undercount based on ACS data in the current year and undercount rates by age, sex, and Hispanic origin from the 2010 ACS.

The new residual method can be used to derive estimates of the total undocumented population, but it can also be used to derive estimates for subsets of the population. The necessary data can be compiled by country of birth, age, sex, gender, and Hispanic origin from the ACS and DHS data. Undercount rates in the ACS can be compiled by age, sex, gender, and Hispanic origin each year using ACS data and the undercount rates reported by Jensen, Bhaskar and Scopilliti (2015).

This residual method can be used to evaluate trends in undocumented population estimates regardless of the methodology used to derive the original estimates. Each organization that derives estimates can determine how well the method fits its needs, adopt the methods described here to derive new annual estimates, or use the methodology to determine whether their recent estimates are too high or low relative to their estimates for earlier years. Widespread use of the residual methodology described here would yield parallel lines in a 2020 to 2030 graph like Figure 1. Consistency in estimated trends by different organizations is important for accurately tracking changes in the undocumented population and maintaining confidence in the output of the residual method and similar types of estimation procedures.