Imputing the Legal Status of Foreign-Born Persons on 

Surveys: 

Two New Approaches* 

Dean H. Judson 

Decision Analytics 

15000 Fort Trail, Accokeek, MD 20607 (email: dhjudson@comcast.net) 

Legislation and policymaking often run ahead of our ability to produce the data needed for implementation 

and evaluation. In the case of the 2010 Patient Protection and Affordable Care Act (ACA), this has 

occurred with respect to state-level estimates of the legal status of the immigrant population. Several 

provisions that affect unauthorized immigrants and are directly relevant to state planning are included in the 

bill. For example, unauthorized immigrants are exempted from the 'individual mandate' to obtain coverage, 

are excluded from the Medicaid expansion, and are prohibited from purchasing health insurance coverage 

in federal or state health insurance exchanges. Because of provisions such as these, unauthorized 

immigrants are expected to comprise a substantial portion of the remaining uninsured population after the 

ACA is in full effect. Accordingly, accurate state-level estimates of the size of the unauthorized immigrant

population are needed as states estimate the size and characteristics of the populations eligible for the ACA 

and determine the appropriate safety net capacity for the remaining uninsured population after 2014. 

However, no Federal agency makes a state-by-state estimate of persons by legal status. Similarly, no 

specifically-state-representative Federal survey asks "legal status questions" of the foreign-born. This 

paper focuses on two experimental methods for imputing legal status of foreign-born persons. The first of 

these involves developing a regression-based imputation model from the Survey of Income and Program 

Participation and using that model on other target surveys; the other involves developing a latent class 

model to classify foreign-born persons. Some caveats and other considerations will be discussed. 

*This work was funded by the State Health Access Data Assistance Center (SHADAC) at 

the University of Minnesota, a program of the Robert Wood Johnson Foundation. 

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Introduction: The Problem of Estimating the Size and Composition of the Unauthorized Population 

in the U.S. 

The 2010 Patient Protection and Affordable Care Act (ACA) will introduce the most significant 

changes to the US health care system since the introduction of Medicare and Medicaid in 1965. Preparing 

for those changes has created many challenges at the federal, state, and community levels, as estimates of 

the size, characteristics, and location of the populations affected by the many different provisions of the 

ACA are needed. Key among the needed estimates are state-level estimates of the legal status of the 

immigrant population as several provisions exclude unauthorized immigrants. For example, unauthorized 

immigrants are exempted from the 'individual mandate' to obtain coverage, are excluded from the Medicaid 

expansion, and are prohibited from purchasing health insurance coverage in federal or state health 

insurance exchanges. Because of these provisions, unauthorized immigrants are expected to comprise a 

substantial portion of the remaining uninsured population after the ACA is in full effect. Accordingly, 

accurate state-level estimates of the size and composition of the unauthorized immigrant population are 

needed as states estimate the size and characteristics of the populations eligible for the ACA and determine 

the appropriate safety net capacity for the remaining uninsured population after 2014. 

National estimates show the numbers of foreign-born residents increasing over time, reaching 31.1 

million, or 11.1 percent of the US population by 2000 (Malone, Beluja, Costanzo, and Davis, 2003), and 

40.2 million by 2010 (Passel and Cohn, 2011). Estimates of the share of that population that is not 

“authorized” to reside in the United States have varied widely, often with little information on the data and 

methods used to generate the estimates (Walsh, 2007). Among the studies with documented methods, 

however, there is more consistency, with the most recent estimates showing 11.2 million unauthorized 

people, or about one-third of the foreign-born population of the US (Passel and Cohn, 2011). 

Obtaining state-specific estimates of the unauthorized population is challenging since no Federal 

agency makes a state-by-state estimate of persons by legal status and Federal surveys do not typically ask 

"legal status questions" of the foreign-born. This paper focuses on two experimental methods for imputing 

legal status of foreign-born persons. The first of these involves developing a regression-based imputation 

model from the Survey of Income and Program Participation and using that model on other target surveys; 

the other involves developing a latent class model to classify foreign-born persons. 

The primary purpose of this imputation exercise is to develop state-level estimates of the unauthorized 

population. A secondary, but potentially still important output, is to generate small-domain estimates of 

the number of unauthorized immigrants by key subgroups of interest, such as by demographic 

characteristics, employment, and health status . It is only a tertiary goal to place probability values on 

individual persons; we will examine record-level imputations as a way of building confidence in the 

models, but record-level imputation is not the focus here. . 

The Ambiguous Definition of ‘Legal Status’ 

There is substantial debate as to the exact legal categories that migrants fall into, and what language to use 

to describe these categories. More importantly, reports from different agencies (and sometimes from 

different parts of the same agency1

) use terminology that is either undefined or inconsistent. 

A major area of potential confusion in defining immigration status categories involves foreign-born 

residents who have pending applications. The reason is that (1) these pending applications have different 

objectives and potential immigration status outcomes, and (2) in some cases, an immigration status may be 

defined in terms of the reason(s) for estimating immigration status, rather than in strict legal terms. 

Regarding (1), for example, a pending application may involve petitioning for: 

• a legal immigrant status (such as lawful permanent resident, or LPR), 

1 See, e.g., Costanzo, et. al. (2001), footnote 4. 

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• a change of temporary or nonimmigrant status (such as a student applying to be a temporary 

worker), or 

• an extension of nonimmigrant status (seeking permission to remain in the United States for a 

longer period of admission than was originally granted). 

Regarding (2), there are legitimate reasons for interpreting the immigration status of the resident foreignborn population in multiple ways. For example, an agency may need to render a legal classification for an 

enforcement, budget, or program purpose, while policymakers may want to estimate how many foreignborn residents may qualify for or be affected by proposed legislation. Thus, there may be no single, 

uniquely correct way to estimate numbers of foreign-born residents with pending applications by 

immigration status. 

Classifying foreign-born residents with pending applications is also complicated because many of them: 

(1) have also been issued employment authorization documents, also called EADs, and 

(2) may or may not be legally present in the United States, and may or may not become legal 

immigrants (i.e., LPRs with a “green card”) or granted some other immigration status.2 

Having an EAD and/or being illegally present in the United States may complicate identifying a “correct” 

immigration status for some foreign-born residents who are asked to identify their immigration status in a 

survey. For example, those who have been issued EADs (1) have some basis for thinking of or identifying 

themselves as “documented,” but (2) may not know or identify themselves according to their correct 

immigrant statuses. This applies, of course, to direct survey questions rather than aggregate demographic 

estimates. 

There is no apparent consensus by immigration researchers or federal agencies on whether or not (or how) 

to define some foreign-born residents with pending applications as “unauthorized” (that is, illegally 

residing in the United States), because their immigration status may be ambiguous. The Department of 

Homeland Security (DHS), for example, uses the term “unauthorized” to describe foreign-born persons 

residing illegally in the United States, but includes some “unauthorized immigrants” who have pending 

applications in the legally resident foreign-born population.3 

2 Foreign-born residents who encompass a wide range of immigration statuses are eligible to obtain EADs, 

including those who may be illegally present in the United States. As background, certain aliens who are 

temporarily in the United States may file a Form I-765, Application for Employment Authorization, to 

request an Employment Authorization Document (EAD). There are more than 40 eligible categories under 

which aliens may apply for an EAD, including Temporary Protected Status, Spouse/Dependent of foreign 

government official, spouse of Class E (treaty trader or treaty investor) nonimmigrant, fiance(e) of a U.S. 

citizen, Public Interest Parolee, and certain legalization applicants. Some aliens who are issued EADs 

separately apply for, and can be expected to receive, legal permanent resident (LPR, or “green card”) or 

other status allowing them to live permanently in the United States. Others, such as the spouse of a Class L 

(intracompany transferee) temporary worker, may leave the United States after their lawful period of 

admission expires, or become overstays. An overstay is an illegal alien who was legally admitted to the 

United States for a specific authorized period but remained here after that period expired, without obtaining 

an extension or a change of status or meeting other specific conditions. Under certain circumstances, an 

application for extension or change of status can temporarily prevent a foreign visitor’s status from being 

categorized as illegal. 

3DHS defines “unauthorized immigrants” as “foreign-born persons who entered without inspection or who 

violated the terms of a temporary admission and who have not acquired LPR status or gained temporary 

protection against removal by applying for an immigration benefit” (2002 Yearbook of Immigration 

Statistics. Department of Homeland Security. Washington, D.C.: U.S. Government Printing Office, 2003, 

p. 213). These definitions are taken from a separate report, “Estimates of the Unauthorized Immigrant 

Population Residing in the United States: 1990 to 2000.” (Office of Policy and Planning. U.S. Immigration 

and Naturalization Service, January 2003), which includes “unauthorized immigrants with pending I-485 

forms—LPR status not yet official by January 1, 2000,” as part of the “legally resident foreign-born 

population—entered 1990-1999 (see Table 3, p. 18). This report is also available at 

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Since there is no apparent consensus, it is incumbent upon us to be as clear as possible in our use of 

terminology. Therefore, in order to avoid confusion, we propose the following terminology and definitions 

for describing each legal status. The major categories are in bold below. The universe of discussion is all 

foreign-born persons whose place of residence is in the United States on the estimates (July 1) or 

enumeration (April 1) day4

. 

http://www.uscis.gov/graphics/shared/statistics/publications/Ill_Report_1211.pdf (downloaded June 11, 

2006). 

4 Depending upon the data source, this will include or exclude the Group Quarters population. 

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Table 1: Legal Status Terminology 

Term Legal and procedural definition 

(Authorized) Legal 

Immigrant 

Naturalized Has obtained U.S. citizenship. 

Lawful Permanent Resident 

(LPR) 

Has applied for LPR status, and has been formally 

admitted. 

(Authorized) 

Legal Temporary 

(“Legal 

Nonimmigrant”) 

• Lawful temporary application has been 

accepted; AND 

• Terms of admission have not been violated; 

AND 

• Neither naturalized nor LPR status has been 

granted, even if application exists. 

Refugee/Asylee • Has applied for refugee or asylee status and 

been granted same; OR 

• Present in the U.S., citizen of Temporary 

Protected Status-recognized country; AND 

• Has not converted status to Legal Temporary, 

Lawful Permanent, or Naturalized. 

Residual, 

Unauthorized * or 

Other 

All other: 

• Application in process but not yet granted; 

OR 

• Entered without inspection; OR 

• Violated terms of residence. 

Within residual: 

Quasi-legal • Has applied for Lawful Permanent 

Resident, Legal Temporary status, 

Refugee, Asylee, or Temporary Protected 

Status; AND 

• Status not yet granted. 

Table notes: 

* We believe that this category definition is equivalent to the Office of Immigration Statistics (OIS) 

definition. 

We note for the record that table 1 is primarily a conceptual exercise. Few, if any, data sources literally 

correspond to these categories: Often questions sufficient to classify cases are not asked, or the data source 

is a transaction database instead of a person database, or the lag or slippage between the data source and the 

population of interest is sufficiently great so that the database cannot properly represent the population of 

interest. This lack of sufficient data is one of the greatest difficulties in estimating the foreign-born 

population by legal status. 

Traditional Approach to Estimating Legal Status: A (Brief) History of the Residual Method 

Prior efforts to estimate the size of the foreign-born population by legal status have largely relied on 

residual methods (Judson and Swanson 2011). We provide a brief review of those methods here, with a 

focus on the limitations of that approach for the objectives of this paper. 

The U. S. Census Bureau defines the foreign-born as people who are not U.S. citizens at birth--a definition 

that is used in this paper. This population consists of legal immigrants, temporary migrants, and 

unauthorized migrants as shown in the following equation (Costanzo et al., 2001): 

FB = [L – (M + E) + T + R], 

where, 

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FB = Foreign-born population; 

L = Legal immigrants; 

M = Mortality to legal immigrants; 

E = Emigration of legal immigrants; 

T = Temporary (legal) migrants; and 

R = Residual. 

The “residual” is then assumed to be foreign-born unauthorized and quasi-legal migrants. 

The preceding equation is a useful way to look at the foreign-born as a whole as well as by status.. By 

estimating the components and rearranging the equation, we obtain: 

R = FB – L + M + E – T, an estimate of the residual population. 

The use of the a residual estimate for estimating the foreign-born is primarily, but not exclusively, aimed at 

estimating those who lack legal documents (see, e.g., Passel, Van Hook, and Bean, 2004) There are 

several variations of the ‘residual” method for this purpose. Many of them are members of the “stock 

method” in that they attempt to estimate the number without legal documents as the difference between the 

non-citizen population enumerated in a census or a survey (i.e., the Current Population Survey or the 

American Community Survey) and the legally resident alien population, where enumerated unauthorized 

resident migrants = enumerated non-citizens - legally resident aliens. Others are “flow-based” residual 

methods. In general, the residual methods can be done by national origin, period of entry, age, sex, and 

depending on the level of detail available in estimates of the legal population, by state and metropolitan 

area. However, finer levels of detail create a potential problem: If the data sources for FB, L, M,E, and T 

are from different sources (e.g., Censuses, surveys, administrative systems), then for fine levels of detail 

there is the potential for a ‘negative’ residual increases, a problem that is demographically ‘embarrassing’. 

The Problem with Residual Reasoning 

Residual reasoning is easily described: Define the known elements by forming an identity equation that 

should include every member of some population. Assume one has direct estimates or counts of all 

subgroups except the subgroup for which no data source exists. Solve for the unknown and “viola” one has 

the amount which cannot be directly detected. However, when the data sources to estimate each of the 

components vary (some by a survey, some by administrative data, some by assumption or algorithm) and 

are subject to error, the logic of residual reasoning, which seems so algebraically neat and tidy, begins to 

break down. The reality is that, in estimates of the undocumented population based on residual methods, 

the residual group is assumed to capture the undocumented population along with other population groups 

not captured accurately in the source data. 

Residual reasoning is not by itself problematic: The search for dark matter and dark energy by residual 

reasoning has the same characteristics (Carroll, 2007). The residual method in demography is analogous. 

Seeing a population of foreign-born persons: 

• First eliminate those that are naturalized, 

• Second eliminate those that are lawful permanent residents, , 

• Third eliminate refugees, asylees, and legal temporary persons,: 

• Finally, conclude the remainder must be unauthorized persons. 

However, the limitation of residual reasoning is the same limitations in earlier inferences about dark matter: 

Given a choice, we would much rather have a direct measure of the population rather than an assumed 

residual. Thus, we address this problem by attempting to estimate the undocumented population directly, 

rather than by residual methods. 

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Imputation and the American Community Survey 

Our base for this study is the American Community Survey (ACS). The American Community Survey is a 

continuously fielded household survey administered by the U.S. Census Bureau. It is the successor to the 

Decennial Census Long-Form Survey and the Supplemental Survey and asks questions on a wide range of 

topics focusing on the demographics, economic situation, well-being, and program participation of U.S. 

households and their members. Its large sample size of about 3 million persons per year (drawn from a 

comprehensive frame of both private residences and group quarters) enables it to be used to generate 

statistics at state and sub-state levels, albeit estimates made at finer levels of geography than state can 

require multiple years of data. To allow independent research using ACS, the Census Bureau produces and 

releases Public Use Microdata Sample (PUMS) files. 

Because we do not “know” the legal status of foreign-born persons, and we have highlighted above our 

reservations regarding using residual methods, we propose to test imputation techniques in this context. 

Our goal in making imputations is to make a statistical “best guess” as to each person’s legal status, and 

then, to construct state- and domain-level estimates, add up those statistical “best guesses”. Imputation has 

a long history in censuses and surveys (Scheuren, 2005), and the use of imputation for census-taking and 

survey measurement is well established. There are a variety of techniques used for imputation; the specific 

technique used depends upon the particular type of variable being imputated (e.g., household count 

imputation is a different animal than item [individual variable] imputation. Before we launch into our 

proposed methods, we would like to highlight three “problems” that are addressed in our work. 

The Problem of Coverage 

It is widely believed by many researchers on the foreign born that the foreign born in general, and the 

unauthorized foreign born in particular, suffer from more extensive undercoverage in censuses and surveys 

than non-foreign-born persons. Judson (2009) documents some of these beliefs, and examines evidence 

that is almost uniformly consistent with, but does not definitively prove, such a belief. Censuses and 

surveys attempt to correct for undercoverage via post-stratification to population controls derived from 

population and housing estimates, but it is not known whether controls to age, race, sex and ethnicity 

correctly account for supposed undercoverage of the foreign born. 

This point raises a problem for deriving estimates based on imputations. If there is either a general 

coverage problem associated with the foreign born (again, in particular, the unauthorized foreign born), 

then population totals could be askew—perhaps only a little, if ACS coverage corrections are robust, but 

perhaps quite a bit if not so. This is not uniquely true for estimating levels of the foreign born only, but it is 

certainly true for this group. 

In addition, if there is any kind of ‘social desirability bias’ associated with foreign-born-related questions, 

that bias will also affect levels for estimates. For example, Passel and Cohn (2011) negate the reported 

naturalized citizenship status of some foreign-born respondents because their characteristics (specifically, 

reported year of entry5

) are not consistent with being a citizen according to U.S. immigration law. In effect, 

they believe that too many foreign-born persons are counted as ‘naturalized’, thus downwardly affecting 

the levels of non-naturalized (and hence unauthorized) as well. 

Both kinds of bias could be present in the data, thus leading to a need for corrections involving control 

totals. 

The Problem of ‘Control Totals’ 

If levels of foreign-born population estimates are potentially biased downward, and non-naturalized but 

authorized and unauthorized foreign born in particular are potentially downwardly biased (again, we use 

the word ‘potentially’ because evidence is somewhat sketchy on these beliefs), then it is natural enough to 

5 The “year of entry” is derived from the question in the CPS that asks “When did you come to live in the 

United States?” The implicit assumption underlying the question is that there is a single entry into the U.S., 

which ignores circular migration and temporary residency. 

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imagine using post-strata that are specifically designed to correct for foreign-born legal statuses. But where 

should such control totals come from? 

Passel and Cohn (2011), and other, earlier, estimates, use control totals that are themselves derived using 

the residual method; they then use iterative proportional fitting (or ‘raking’) to bring the algorithmicallyderived legal statuses into harmony with external controls. This method is common and is a near-cousin to 

the methods used for post-stratification in ‘typical’ surveys. Currently, their estimation models use the 

Current Population Survey as the ‘base’ from which LPR and other quantities are subtracted. 

The Office of Immigration Statistics (OIS; Hoefer, Rytina and Campbell, 2006; Hoefer, Rytina, and Baker 

2009) also uses a residual method to generate totals. Their base, however, is the American Community 

Survey, from which LPR and other quantities are subtracted. For the record, because the Passel and OIS 

systems are similar, their final estimates results typically are very comparable. 

Because OIS uses the same data base as this paper, are considered the “official” U.S. government 

estimates, and because OIS publishes national totals by age and sex, for this estimation system we have 

chosen to use OIS totals. However, we wish to be very ‘light handed’ in applying control totals, so as to let 

the imputations do most of the ‘talking’. We will use two approaches: 

1) The ‘simple rake factor’ (SRF) approach will control only to the total national estimate of 

unauthorized foreign born, leaving everything else to the imputations; and 

2) The ‘complicated rake factor’ (CRF) approach will control to the total national estimate by broad 

age/sex categories, leaving everything else to the imputations. 

The Problem of Variance Estimation 

Traditional residual methods have struggled mightily with the problem of uncertainty or variance 

estimation. If this year’s sum of unauthorized is 11.5 million persons and last year’s was 11.2, is that 

change ‘statistically significant’? Further, when data from surveys and administrative systems, combined 

with assumptions about emigration and mortality, what even can be said about uncertainty? Judson and 

Cornwell, 2010, made a first attempt, using ACS sampling variances and approximating administrative 

records “uncertainty” using a time-series approach, but other work in the literature has not addressed the 

uncertainty of the estimates. 

In this paper, there are two sources of uncertainty: The sampling uncertainty associated with the ACS 

design, and the imputation uncertainty associated with the probability prediction. In principle, if we treat 

the imputations as approximately independent of the sample design, for a sample total, we have: 

V (Tˆ) = V (I

ˆ

)V (S), 

where, 

V ˆ(T ) is the total population estimate variance, 

)

V (I ) is the imputation variance, and 

V (S) is the sampling variance. 

Again in principle the sampling variance is straightforward (although we will complicate matters somewhat 

in a moment). The Census Bureau provides balanced repeated replication (BRR) replicate weights and a 

formula for estimating variance for random variable X0 (X0 is the point estimate of interest): 

4

80 

2 V (X ) = (X − X ) 0 i 0

80 i=1 

For the imputations, we can treat the imputation as the output of a statistical model, and calculate it 

directly: 

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V (I ) = MSE ,

)

Model 

and merely combine the two multiplicatively. 

Let us return momentarily to the replicate weights. The general formula provided for the ACS would work 

just fine, except for one conundrum: By using the simple and complex rake factors, we have changed the 

original person weight by raking. The replicates are based on the original person weight, and thus would 

misrepresent the sampling variability in our raked estimates. This problem is found in replicate weights 

generally; as noted by Korn and Graubard (1999:34): 

“...theoretically one should recalculate the sample weights for each replicate. However the analyst may 

not have enough detailed information about these adjustments to perform these calculations...” 

This is, of course the situation here. We do not have information to replicate rake factors (either simple or 

complicated) R , for the ith replicate. Therefore, we are going to settle for a reasonable approximation, i

R = R, "i , and obtain: i 

80 80 2 80 4 2 4 2 4R 2

V (X ) =   (R X − RX ) @ (RX − RX ) = (X − X ) 0 i i 0 i 0 i 0 80 80 80 i=1 i=1 i=1

for our variance estimate. Based on empirical results in table 2.5-2 of Korn and Gruabard, 1999: 36), it 

appears that this approximation slightly inflates the variance estimate, making this estimate “conservative”. 

For the purposes of this paper, we will be focusing on point estimates rather than variance estimation; 

however, we recognize its importance to the overall system and intend to incorporate proper variance 

estimation in the near future. 

Alternate Approach #1: Using the Survey of Income and Program Participation to Develop an 

Imputation Model 

In 2006 the first author tested the use of the Survey of Income and Program Participation to develop an 

imputation model. This model takes advantage of key migration legal status questions that are asked in 

wave two, on the ‘migration’ topical module. The questions asked include: 

1) When ... moved to the U.S. to live, what was ...'s immigration status? 

A follow-up question included: 

2) Has ...'s status been changed to permanent resident? 

A person who indicated “other” (i.e., not permanent or refugee) to the first question, and “No” to the 

second question, could reasonably be considered to still be in the “other” category as of the survey date. 

Furthermore, as with Passel, certain foreign-born persons report themselves “naturalized” but with a year of 

entry too recent to likely be an accurate report. Thus the sum total of the first group “other” and not 

adjusted plus the probable misreport on citizenship, represents our target variable of undocumented 

immigrants.6 

Using this classification as a guide, a cross-sectional logistic regression model can be constructed; the 

resulting right-hand-side variables of this model, then, generate a predicted probability that the person is in 

the “other” category. This model, when applied to the American Community Survey, provides a 

probabilistic imputation, a form of so-called “cold deck” imputation (Lohr, 1999). Judson, 2006, reports 

the estimates derived from this assumption; it appeared at the time that a possible “social desirability bias” 

might generate estimates that were generically too low, relative to competing methods of estimation. 

6 In the SIPP, year of entry is derived from the question: “When did this person come to live in the United 

States?” Respondents who “came to live” in the U.S. more than once were asked to report their most recent 

year of entry. It may be that some of those who are assumed to be a probable misreport on naturalized 

status because of a “too recent” year of entry have had moved between residing in the U.S. and another 

country. 

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Alternate Approach #2: Using Latent Class Analysis to Develop an Imputation Model 

An intriguing idea was first proffered in Judson and Swanson (2010): Although a foreign-born person’s 

legal status is unknown, if a clustering of lawful and unauthorized classes exist, a natural method to attempt 

to discover those classes is latent class analysis (the statistically-principled successor to the older cluster 

analysis technique). It, too, would output a probability of belonging to each class. The latent class method 

has the inherent liability that the researcher chooses how to label each class (in this case “unauthorized” 

versus “not unauthorized”). This labeling, one must admit, is partly arbitrary; if, however, the resulting 

output is comparable across methods, then the validity of each will be increased. 

Results From the Models and Comparisons to Other Systems 

As stated in the introduction, the primary goal of these imputation exercises is to produce state-level 

estimates, with small domain estimates as a secondary goal, and individual-level verisimilitude a tertiary 

goal. Thus, we will focus on aggregate results (part 1), and only touch on individual microdata results (part 

2). Likewise, specific model specifications will not be presented here, as they have been presented 

elsewhere (Judson, 2011)7

. 

Aggregate Results 

Table 2 below, tabulates the four latent class models LC1-LC4 and SIPP imputation results, for the fifty 

states. For LC1, we have presented results for the simple rake factor (SRF) and complicated rake factor 

(CRF). Two SIPP models are presented, the first based on a right hand side specification described in 

Judson (2006, update in 2011); the second is based on submitting the same collection of right hand side 

variables to an automated “boosting” procedure—boosting being a data mining tool the selects variables 

and interaction effects automatically. (More information can be found in Schonlau [2005], or on 

http://www.schonlau.net/. A more general discussion is found in Ridgeway, 1999.) 

7 Details on model specifications are available from the authors. 

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Table 2: Survey Total Estimates by state (Model 1-SRF,1-CRF,2-CRF,3-CRF,4-CRF, SIPP, and Boosted SIPP), based on 2009 ACS 

(LC1-SRF) (LC1-CRF) (LC2-CRF) (LC3-CRF) (LC4-CRF) (SIPP) (Boosted 

SIPP) 

Total Total Total Total Total Total Total 

State 

Alabama 52,766 55,900 54,391 54,391 54,391 59,082 55,957 

Alaska 10,823 10,292 10,285 10,285 10,285 9,190 8,046 

Arizona 308,182 310,990 317,093 317,096 317,096 318,739 329,204 

Arkansas 39,644 41,376 42,195 42,196 42,196 43,957 44,683 

California 2,626,233 2,579,563 2,622,181 2,622,203 2,622,203 2,566,899 2,737,879 

Colorado 164,515 170,177 169,840 169,840 169,840 171,813 176,573 

Connecticut 120,885 121,192 121,816 121,816 121,816 116,817 111,107 

Delaware 22,872 23,884 22,953 22,952 22,952 26,168 24,171 

District_of_Columbia 22,371 22,261 22,229 22,228 22,228 23,648 22,130 

Florida 901,494 852,226 836,855 836,851 836,850 877,530 833,313 

Georgia 304,404 321,296 318,523 318,521 318,520 323,882 318,056 

Hawaii 45,075 39,122 39,417 39,417 39,417 33,986 31,511 

Idaho 30,287 31,199 31,671 31,671 31,671 30,528 30,881 

Illinois 470,551 471,308 476,907 476,908 476,908 450,244 468,617 

Indiana 89,362 94,461 93,177 93,176 93,176 98,784 93,502 

Iowa 36,974 39,699 38,710 38,710 38,710 38,384 38,119 

Kansas 62,707 66,371 65,648 65,648 65,648 68,848 65,005 

Kentucky 44,564 47,428 46,137 46,137 46,137 51,071 46,704 

Louisiana 43,472 44,963 44,133 44,132 44,132 48,234 42,690 

Maine 9,212 8,986 9,566 9,566 9,566 6,757 6,423 

Maryland 198,727 200,827 199,325 199,324 199,324 194,428 178,759 

Massachusetts 242,264 241,781 239,154 239,153 239,153 227,124 207,128 

Michigan 149,844 148,361 147,947 147,947 147,947 143,990 129,942 

Minnesota 99,245 105,356 104,678 104,677 104,677 100,107 96,652 

Mississippi 20,396 21,981 21,606 21,605 21,605 23,226 21,391 

Missouri 59,687 61,159 60,568 60,567 60,567 61,352 56,820 

Montana 4,526 4,501 4,595 4,595 4,595 4,433 3,847 

Nebraska 36,809 39,422 38,487 38,487 38,487 40,216 39,723 

Nevada 152,006 153,532 153,075 153,075 153,075 164,842 160,705 

New_Hampshire 18,231 17,851 18,045 18,045 18,045 16,463 15,259 

New_Jersey 441,543 444,570 436,008 436,004 436,004 444,634 427,844 

New_Mexico 66,708 67,345 68,605 68,606 68,606 69,134 68,706 

11 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

New_York 1,006,584 968,245 950,415 950,412 950,412 926,298 914,987 

North_Carolina 238,912 255,069 249,886 249,883 249,883 266,484 262,477 

North_Dakota 5,786 5,905 5,732 5,732 5,732 6,492 5,331 

Ohio 110,866 115,010 114,059 114,058 114,058 109,791 100,255 

Oklahoma 63,481 67,420 67,415 67,415 67,415 72,102 71,137 

Oregon 117,998 119,787 121,503 121,504 121,504 120,239 124,764 

Pennsylvania 163,302 163,497 163,780 163,779 163,779 150,333 140,531 

Rhode_Island 35,923 34,898 34,179 34,179 34,179 35,580 35,383 

South_Carolina 71,773 75,644 73,696 73,695 73,695 84,884 78,565 

South_Dakota 4,789 5,119 4,973 4,973 4,973 5,173 4,469 

Tennessee 88,361 94,774 94,105 94,105 94,105 98,138 93,770 

Texas 1,347,441 1,370,619 1,379,599 1,379,603 1,379,603 1,418,183 1,450,644 

Utah 68,551 72,520 74,017 74,017 74,017 72,127 73,349 

Vermont 4,713 4,305 4,481 4,481 4,481 2,861 2,891 

Virginia 220,842 225,510 224,088 224,087 224,087 226,234 206,909 

Washington 220,420 224,261 224,469 224,469 224,469 212,192 207,814 

West_Virginia 5,111 5,275 5,505 5,505 5,505 4,639 4,314 

Wisconsin 73,594 77,210 76,734 76,733 76,733 78,528 75,997 

Wyoming 5,173 5,548 5,544 5,544 5,544 5,213 5,067 

Observations 171305 171305 171305 171305 171305 171305 171305 

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As can be seen, despite the wide diversity of specific implementation, the point estimates at the state level 

are highly consistent with one another. Because they are so consistent with one another, we will cease 

presenting large sets of estimates, and focus on only three: Basic SIPP model, Latent class model 1, and 

Boosted SIPP model, each using the complicated rake factor. 

The fundamental question for these estimates is: Are they consistent with other published estimates, 

particular those of OIS? Table 4 presents a comparison with published tables for the largest states in the 

United States. Note that OIS does not present results for states other than those presented in table 3, so 

further comparisons with OIS are not available at this time. 

Table 3: Predicted State of Residence of the Unauthorized Immigrant Population 

State of Residence of the Unauthorized Immigrant Population 

OIS residual estimates 

State of 

residence 

Total 

California 

Texas 

Florida 

New York

Illinois 

Georgia 

Arizona 

North Carolina 

New Jersey 

Nevada 

Other states 

January 

2009 

10,750,000 

2,600,000 

1,680,000 

720,000 

550,000 

540,000 

480,000 

460,000 

370,000 

360,000 

260,000 

2,730,000 

Percent 

of total 

24% 

16% 

7% 

5% 

5% 

4% 

4% 

3% 

3% 

2% 

25% 

SIPP model-based 

estimates 

Total Estimate 

10,750,000 

 2,566,899 

 1,418,183 

 877,530 

 926,298 

 450,244 

 323,882 

 318,739 

 266,484 

 444,634 

 164,842 

 2,992,264 

Percent of

total 

24% 

13% 

8% 

9% 

4% 

3% 

3% 

2% 

4% 

2% 

28% 

Detail may not sum to totals because of rounding. 

Source: U.S. Department of Homeland Security. 

Latent Class (model 1)-

based estimates 

ACS 2009 

Total Estimate 

 10,750,000 

 2,579,640 

 1,370,620 

 852,209 

 968,236 

 471,309 

 321,289 

 311,002 

 255,061 

 444,557 

 153,532 

 3,022,544 

Percent of 

total 

24% 

13% 

8% 

9% 

4% 

3% 

3% 

2% 

4% 

1% 

28% 

Boosted SIPP modelbased estimates 

Total Estimate 

 10,750,000 

 2,741,810 

 1,437,921 

 829,632 

 919,174 

 466,011 

 317,099 

 326,542 

 261,586 

 430,723 

160,574 

 2,858,927 

Percent of 

total 

26% 

13% 

8% 

9% 

4% 

3% 

3% 

2% 

4% 

1% 

27% 

As can be seen, both methods generate very similar results for the largest states. (Note, again, that the rake 

factors did not take state into account, only total population [for the simple rake factor] and broad age/sex 

groups [for the complicated rake factor].) 

However, state totals are not the end of the matter; table 4 exhibits comparisons with published OIS data on 

period of entry, and table 5 exhibits comparisons with published OIS data on country of birth. 

13 
Table 4: Predicted Period of Entry of the Unauthorized Immigrant Population 

Period of Entry of the Unauthorized Immigrant Population 

OIS residual estimates 

SIPP model-based 

estimates 

Latent Class (model 1)-

based estimates 

Boosted SIPPbased estimates 

Period of entry Januar y Percent 

2009 Total 

 Allyears 10,750,000 

2005-2008 910,000 

2000-2004 3,040,000 

1995-1999 3,080,000 

1990-1994 1,670,000 

1985-1989 1,190,000 

1980-1984 860,000 

0 

Detail may not sum to totals because of rounding. 

Source: U.S. Department of Homeland Security. 

of 

 Allyears 

8% 2005-2009 

28% 2000-2004 

29% 1995-1999 

16% 1990-1994 

11% 1985-1989 

8% 1980-1984 

0% Before 1980 

ACS 2009 

Percent of 

Total estimate total 

10,750,000 

 4,566,277 42% 

 2,913,867 27% 

 1,310,408 12% 

 821,096 8% 

 572,967 5% 

 278,452 3% 

 286,933 3% 

ACS 2009 

Percent 

Total estimate total 

10,750,000 

 3,076,479 

 3,242,733 

 1,911,875 

 1,076,482 

 714,651 

 352,132 

 375,648 

of 

29% 

30% 

18% 

10% 

7% 

3% 

3% 

ACS 2009 

Total estimate 

10,750,000 

3,522,279 

2,934,170 

1,745,499 

1,111,204 

735,983 

344,991 

355,873 

Percent of 

total 

33% 

27% 

16% 

10% 

7% 

3% 

3% 

14 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here we begin to see some differences of note—basic SIPP, latent class, and boosted SIPP all seem to be pointing to more unauthorized persons with recent 

period of entry than that generated by the OIS residual method. Furthermore, the OIS system automatically makes anyone whose period of entry is 1979 or 

earlier authorized—they assume that the legal reforms of the 1980’s would result in regularized legal status. Further, year of entry may not be conceptualized 

exactly the same across surveys and administrative records, both of which are used in the OIS system. However, the three models suggest that there might be 

hundreds of thousands of persons still unauthorized among those whose period of entry is 1979 or earlier. 

Finally, we compare estimates by country of birth. 

Table 5: Predicted Country of Birth of the Unauthorized Immigrant Population 

Country of Birth of the Unauthorized Immigrant Population 

OIS residual estimates 

Country of 

birth 

January 

2009 

Percent of

total 

Total 10,750,000 

Mexico 6,650,000 62% 

El Salvador 530,000 5% 

Guatemala 480,000 4% 

Honduras 320,000 3% 

Philippines 270,000 2% 

India 200,000 2% 

Korea 200,000 2% 

Ecuador 170,000 2% 

Brazil 150,000 1% 

China 120,000 1% 

Other Countries 1,650,000 15% 

SIPP model-based 

estimates 

ACS 2009 

Total 

estimate 

Percent of 

total 

10,750,000 

4,865,822 45% 

 478,028 4% 

 413,356 4% 

243,045 2% 

228,521 2% 

465,762 4% 

192,292 2% 

155,081 1% 

133,521 1% 

290,833 3% 

 3,283,740 31% 

Latent Class (model-1) 

based estimates 

ACS 2009 

Total estimate 

Percent of 

total 

10,750,000 

 4,583,566 43% 

 438,653 4% 

 347,778 3% 

 205,557 2% 

 269,011 3% 

 493,209 5% 

 220,526 2% 

 135,270 1% 

 145,677 1% 

 321,372 3% 

 3,589,380 33% 

Boosted SIPP-based 

estimates 

ACS 2009 

Total estimate 

Percent of 

total 

10,750,000 

5,229,107 49% 

497,099 5% 

421,152 4% 

240,414 2% 

204,538 2% 

403,889 4% 

181,195 2% 

157,668 1% 

118,446 1% 

266,022 2% 

3,030,470 28% 

Detail may not sum to totals because of rounding. 

Source: U.S. Department of Homeland Security. 

Again we see some differences worthy of note—we see that the three models generate fewer Mexican unauthorized persons than the OIS residual method, and 

substantially more persons from other countries not on the list. The first finding is consistent with news reports averring that the recent Mexican census found 

more Mexicans, men in particular, than expected, and a suspicion is that this finding is a result of return migration and lower Mexican emigration in response to 

economic conditions (Cave, 2011). The second finding does not have the same “obvious” interpretation, but may reflect a trend in unauthorized immigration 

from other countries that, if true, could be of important policy interest. 

15 
 

 

 

 

 

 

 

 

Graphical Analyses 

Finally, we ask two questions: 

1) Do the imputation probabilities, when added up to make population estimates, make demographic sense? That is, are the people that we expect to make 

up the bulk of the unauthorized, from other sources, in fact show up in our data? 

2) Do the various imputation schemes, in particular SIPP with simple rake factor (SIPP-SRF), SIPP with complicated rake factor (SIPP-CRF), Latent Class 

with simple rake factor (LCA1-SRF), and latent class with complicated rake factor (LCA1-CRF), all hold together? 

Figure 1 summarizes the age distribution of the SIPP model-based imputations by Hispanic/not Hispanic ethnicity. 

Figure 1: Predicted Age Distribution of the Unauthorized Immigrant Population, by Hispanic Ethnicity: 

16 
 

 

 

 

 

 

As can be seen in Figure 1, probabilities peak at approximately the prime migration ages, decline both before and after, and are more concentrated amongst 

Hispanics than non-Hispanics. Note that the ‘spikiness’ in the data represents sampling variability. 

Figure 2 graphs the four models’ results by age distribution, by country of birth code. (Note that the Y-scale is in 100,000’s for readability.) 

Figure 2: : Predicted Age Distribution of the Unauthorized Immigrant Population, by Country of 

Birth 

Again, with slight exceptions, the four models are in general agreement, although there appear to be some areas of difference in Mexico at peak migration ages 

that is worth examining more closely. 

17 
 

 

 

 

 

 

Figure 3 performs the same exercise by state of residence (again, Y-scale in 100,000’s). 

Figure 3: : Predicted Age Distribution of the Unauthorized Immigrant Population, by State of Residence 

Again, broad consistency is obtained. 

Our final two figures illustrate the “small domain estimation” capabilities of these models. Out of any number of ACS domains we could have chosen to 

illustrate, we have selected two: Numbers of persons living in non-husband/wife households, and occupation code by broad occupational groups. We present the 

estimated age distribution for each group. 

18 
 

 

 

 

 

 

Figure 4 : Predicted Age Distribution of the Unauthorized Immigrant Population, by Ethnicity and Household Marital Status 

In Figure 4, “non-husband/wife” indicates persons living in non-husband/wife households, “husband/wife” the converse, and we can see some differences in the 

upper left panel and lower right panel. Substantively, the right upper and lower panels indicate that Hispanic persons, particularly those in prime migration ages, 

are far more likely to live in non-husband/wife households (lower right panel) than non-Hispanic persons are (upper right panel). 

Finally, to illustrate the power of this system, we graph something very specific--distributions of unauthorized persons by occupation group and by sex (again, Yscale in 100,000’s) in Figure 5. 

19 
 

 

 

 

 

Figure 5: Predicted Age Distribution of the Unauthorized Immigrant Population, by Occupation Group and Gender 

In this figure, occupations are: 

MGR/BUS: Managerial business, and finance; 

CMM/ENG/SCI: Computers, engineering and science; 

CMS/EDU/LGL/ENT: Counselors, education, legal, and entertainment; 

20 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MED/HLS/PRT: Medical, health Services, public protection; 

EAT/CLN/PRS/SAL: Eating, Cleaning, personal services, sales; 

OFF: Office; 

FFF/CON/EXT: Forestry, fishing, agriculture, construction, extraction; 

RPR/PRD: Repair and production; 

PRD: Production; and 

TRN/MIL: Training and military. 

As can be seen, our system estimates unauthorized persons almost exactly where anecdotal evidence says they should be (males in FFF/CON/EXT and in 

EAT/CLN/PRS/SAL; females in EAT/CLN/PRS/SAL but not FFF/CON/TEXT) and very few unauthorized persons in occupations unlikely to be attractive due 

to legal constraints. 

Microdata Results 

As noted above, record-by-record (i.e., individual observation in the survey) accuracy is only a tertiary goal of this project; however, to the extent that record-byrecord results remain consistent with one another, then the overall validity of the approach is enhanced. To this end, we merely wish to examine simple 

correlation coefficients of the raked-probability imputations, to determine whether the systems “hang together” at a record level. Table 6, below, presents these 

simple (unweighted) correlations. 

Table 6; Simple Correlations Across Models Predicting Unauthorized Immigration Status 

Boosted Boosted Base Base LCA LCA 

SIPP - SIPP - SIPP - SIPP - Model Model 

SRF CRF SRF CRF 1 - SRF 1 - CRF 

Boosted SIPP - SRF 1 

Boosted SIPP - CRF 0.8736 1 

Base SIPP - SRF 0.7683 0.7161 1 

Base SIPP - CRF 0.7387 0.7967 0.9501 1 

LCA Model 1 - CRF 0.4176 0.4073 0.5664 0.5315 1 

LCA Model 1 - SRF 0.5587 0.678 0.5553 0.7005 0.6077 1 

As can be seen, for most of the imputations, the correlations are positive (as expected) and generally are strong. Where they are lower, graphical analyses 

suggest that a failure of linearity is at issue, as the pearson correlation coefficient is a measure of linear relationship, and these imputations appear to depart from 

linearity. 

21 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Conclusions 

As can be seen, where external comparisons are available (published tables from the Office of Immigration Research), these (point) estimates generally fare 

well—both the residual-based OIS estimates and these estimates are very comparable, for the states for which OIS publishes data. The “simple rake factor” 

models, both SIPP-based and latent-class based, generate what appear to be more older unauthorized persons than would be expected based on residual methods. 

We are left with the question: Is this really true or is this a technical artifact of the different methods, or, further is this the result of a discrepancy between year of 

entry in the survey versus administrative data? 

In order to demographically harmonize the OIS and these estimates, we introduced the (so-called) “complicated rake factor”, which rakes by broad age and sex 

groups. These results are obviously demographically compatible with OIS results (by design), but they retain state-level values that are comparable to those 

obtained using the simple rake factor—no harm appears to be being done to the state-level estimates of the distribution of the undocumented population. 

Next steps in this project include variance estimation, small domain estimation results (e.g. industry, occupation) to continue to assess the validity of the results 

relative to other data sources, and further development of the statistical models (both SIPP, boosted SIPP and latent class) that form the base of this estimation 

system. 

These results, as they are, do generate novel outcomes by country of birth and period of entry—raising the question as to which set of estimates, OIS or modelbased, are more “accurate.” It is far too early to attempt to answer this question. But, we now have two completely different methods to estimate the size of the 

unauthorized population—two methods that are broadly consistent with each other, but differ in important ways—and so we now have new tools to use to try to 

understand the characteristics of that population. By taking advantage of the large sample size and relatively detailed characteristics available in the American 

Community Survey, we can say more about the unauthorized population than was possible before, providing much needed information for states as they prepare 

for the forthcoming changes under the Affordable Care Act. 

22 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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