## A Stochastic Model of the Long-Range Financial Status of the OASDI Program—September 2004

II. EQUATION SELECTION AND PARAMETER ESTIMATION

### C. IMMIGRATION

Total immigration is defined here as legal immigration minus legal emigration plus net other immigration. Each component is modeled separately.

### 1. Legal Immigration

Legal immigration is defined as persons lawfully admitted for permanent residence into the United States.1 The level of legal immigration largely depends on legislation which basically serves to define and establish limits for certain categories of immigrants. The Immigration Act of 1990, which is currently the legislation in force, establishes limits for three classes of immigrants: family-sponsored preferences, employment-based preferences, and diversity immigrants. However, no numerical limits currently exist for immediate relatives of U.S. citizens.

Historical data for legal U.S. immigration for years 1901 through 2002 are from the U.S. Citizenship and Immigration Services.2 Legal immigration averaged nearly one million per year from 1900 through 1914, then decreased substantially to about 23,000 in 1933. Since the mid-1940s, legal immigration increased steadily to over one million in 2002.

An ARMA(4,1) equation was selected and parameters were estimated using the entire range of historical data. The R-squared value was 0.92. Figure II.2 presents the actual and fitted values. The modified equation is:

IMt = IMtTR +1.08imt-1 −0.54imt-2 +0.69imt-3 −0.31imt-4 + εt +0.49εt-1.     (3)

In this equation, IMt represents the annual level of legal immigration in year tIMtTR represents the projected level of legal immigration from the TR04II in year t; imt represents the deviation of the annual level of legal immigration from the TR04II value in year t; and εt represents the random error in year t.

### 2. Legal Emigration

Legal emigration is defined as the number of persons who lawfully leave the United States, and are no longer considered to be a part of the Social Security program. Although annual emigration data are not collected in the United States, the U.S. Census Bureau estimates that the level of emigration for the past century roughly totaled one-fourth of the level of legal immigration.

Using the Census estimates as an approximate guide, the parameters of Equation (3) are multiplied by one-fourth.3 The modified equation is:

EMt = EMtTR +0.27emt-1 −0.13emt-2 +0.17emt-3 −0.08emt-4 + εt +0.12εt-1.     (4)

In this equation, EMt represents the annual level of legal emigration in year tEMtTR represents the projected annual level of legal emigration from the TR04II in year t; emt represents the deviation of the annual level of legal emigration from the TR04II value in year t; and εt represents the random error in year t.

### 3. Net Other Immigration

Net other immigration is defined as the annual flow of persons into the United States minus the annual flow of persons out of the United States who do not meet the above definition of legal immigration or legal emigration. Thus, net other immigration includes unauthorized persons and those not seeking permanent residence.

Since complete data does not exist for net other immigration, we rely on indirect measurements from the U.S. Census Bureau for our estimate. The Census Bureau accomplishes this by comparing two consecutive decennial census populations, applying known components of change, and assigning the residual to net other immigration. The annual level of net other immigration is assumed to follow a random walk. The modified equation is of the form:

ΔΟt = ΔΟtTRεt;        Ο2003 = Ο2003TR.              (5)

In this equation, ΔΟt represents the change in the annual level of net other immigration in year t;  ΔΟtTR represents the projected annual change in the level of net other immigration consistent with the TR04II in year t; and εt represents the random error in year t. The equation is initialized with Ο2003 = Ο2003TR, the level of net other immigration in 2003 from the TR04II.

### Footnotes—

1 For more detailed information, refer to the Yearbook of Immigration Statistics -