Using Apprehensions to Infer Unauthorized Flows

There are complications with using apprehensions data to estimate the number of individuals crossing the border illegally. The more serious problem relates to the inherent nature of the data: while apprehensions provide data on the number of individuals captured at the border, they provide no direct information on those who elude capture, which is the population of interest for this study. Minor problems include misreporting of key variables (e.g., given USBP return policies, non-Mexicans have a strong incentive to claim Mexican nationality) and possible missing data during peak apprehension periods (owing to failure to record all arrests during these times).

Nevertheless, the data can be used to apply capture-recapture techniques to make inferences about the size of the undocumented population and the flow of individuals into this population at regular time intervals. The re-apprehension of individuals provides information that can, in theory, be used to make inferences about the size of the unauthorized population entering the United States successfully. Of apprehensions of Mexican men over the period from 1999 to 2009, approximately three-fifths were of individuals who were apprehended only once, one-fifth are of individuals who were apprehended twice over the period, and one-fifth are of individuals who were apprehended three or more times (Borger et el., in press). Re-apprehensions of individuals typically occur within a few days or weeks of the initial capture. Such re-apprehensions are likely part of a single crossing episode. Of apprehended Mexican men who are subsequently re-apprehended, three quarters occur within 90 days, with the vast majority of these occurring in the first 30 days.

With the appropriate data in hand, how would one describe apprehensions analytically? Consider a simple model of the apprehensions process, whose main elements are described in Table 5-2. Suppose that at time t there are M(t) individuals in Mexico who are considering crossing the U.S.–Mexico border illegally (this exercise ignores other nationalities). Suppose further that a fraction m(t) of these individuals choose to attempt illegal migration, where m(t) may be affected by economic conditions in the United States and Mexico and by the intensity of enforcement at the border. Let the probability of apprehension by USBP at time t be a(t). Upon being apprehended, an individual decides whether to cross again or to return home (the latter group having been successfully deterred from further crossing attempts). Let the probability of retrying, conditional on being apprehended, be r(t). It is likely that m(t), a(t), and r(t) will vary

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