## Statistics for Policymakers: Statistical Modeling to Address the Problem of Illegal Immigration

*David Salsburg*

###### In this inaugural article for the series “Statistics for Policymakers,” David Salsburg provides an illustration of how statistical thinking can address the U.S. immigration stalemate. *Amstat News* is inviting articles by ASA members to share what they think policymakers should know about statistics. View details about this series in the article Statistical Thinking in Policymaking.

The problem of immigration reform resembles a situation that those of us who have engaged in statistical consulting often find. One member of the team working on the problem insists one aspect be addressed first when that aspect is ill defined and often impossible to resolve. In the public discussion of immigration reform, there are politicians who insist we must first “secure our borders.” Over the years, different attempts by the federal government to improve on border security have failed to meet this ill-defined objective in the eyes of those who insist upon it.

Statistical thinking can help defuse such a stalemate. A simple model can help focus the arguments around the parameters of that model and can be resolved with adequate data—data that are often already on hand.

Immigrants who are illegally in the United States arrive in several ways. Some cross the Mexican border, some arrive legally but overstay their visas, and some are put ashore without having gone through customs. The U.S. Census Bureau has estimates of the proportion that belong in each category. So, any discussion of “securing the borders” has to deal with these three aspects. If, as I gather from reading newspaper accounts, the vast majority is about equally split between the first two, the problem becomes one of constructing two models.

In each case, suppose we model the probability of entering illegally as a function of the effort expended to prevent it. It is usually convenient to quantify the effort in terms of monetary cost. Once we have some decent bounds on the probability of entrance as a function of cost, we can multiply those probabilities by expected numbers of those seeking to cross and the discussion can become one of how much we are willing to spend to get to a minimal specific number of illegal entries.

Consider the question of crossing the Mexican-U.S. border. The Communist governments of Eastern Europe have provided us with one end of that probability function. They erected high walls with barbed wire in cities. Elsewhere, a no-man’s land was plowed up and sowed with land mines. There were high cyclone fences on either side and guard towers every couple of kilometers with orders to shoot to kill anyone who attempted to cross.

We would never resort to this in the United States, but the number of people who managed to escape in spite of these structures provides us with one basic bit of information. This is the probability that someone will cross even the most tightly controlled border.

The Israeli fence blocking off the West Bank provides another way to estimate minimum probability. Newspaper reports suggest that 2–3 would-be terrorists manage to cross each week in spite of the high-tech nature of the fence.

We can use statistical methods to get a handle on the probability that the border will be crossed for any given amount of effort put into “securing” that border. We start with an estimate of how many would cross if there were no controls. Then, we can model the process so we have the decrease in probability of crossing as a function of the amount of money put into the effort.

We have data from the last few years, when attempts were made to strengthen the border crossings, that would enable us to make those calculations. Like any other statistical approach, we would be able to estimate the most likely relationship between money expended and reduction in probability of crossing and we would have a measure of the uncertainty involved. For instance, we might learn that an increase in enforcement cost by 10% will decrease the probability of crossing from 80% to between 68% and 52% when enforcement is first started, but will decrease a probability of crossing of 40% only down to something between 27% and 39%. These hypothetical numbers describe a function with diminishing return, a situation usually found in circumstances like this.

Arguments will still be there, but the arguments will now be over the validity of the statistical estimates, the level of uncertainty that exists about them, and the amount of effort the nation can afford. If there are arguments over the validity of the data used to estimate the parameters, the model can be expanded by the use of Bayesian methods that force the contenders to quantify their arguments.

The problem of expired visas could be modeled the same way. It would be necessary to model each type of visa separately, and there may be a problem in finding enough reliable data, but the basic methodology would be the same.