- SAIPE combines estimates from the regression with direct estimates from the Current Population Survey (CPS) in a way that varies the importance given to the direct CPS estimates from state to state depending upon their reliability.
- SAIPE multiplies estimates of poverty ratios by estimates of the population to provide estimates of the numbers of poor people.
- SAIPE controls the state estimates of the number of poor people so that the total agrees with the direct CPS national estimates.
- SAIPE uses data from the prior census (1990) in the regression in the form of residuals from auxiliary cross-sectional regressions.
- Because the Department of Education requires estimates of the number of "related children age 5 to 17 in families in poverty," and not all children 5 to 17 are "related children," there are two sets of equations for children ages 5 to 17.
- SAIPE estimates the total number of poor people as the sum of estimates derived from a set of age-specific equations.

*Bayesian Estimation Techniques.*
The models SAIPE used to estimate 1997 income and poverty at the
state level employ both direct survey-based estimates of 1997 income and poverty from the
March 1998 CPS and regression predictions of income and poverty based on administrative and
census data. We combine the regression predictions with the direct sample estimates using
Bayesian techniques. The Bayesian techniques weight the contribution of the two
components (regression predictions and direct estimates) on the basis of their relative precision.

The regression model used to develop the regression predictions is postulated for the true, unobserved poverty ratios or median income, but it is fitted to the CPS direct estimates allowing for the sampling error in the data. If the variance of the error term in this regression model (the model error variance) were known, then the Bayesian estimate for each state would be a weighted average (shrinkage estimate) of the state's regression prediction and direct CPS estimate. The two weights in this average add to 1.0, with the weight on the direct estimate computed as the model error variance divided by the total variance (model error variance plus sampling error variance). In this average, the larger the sampling variance of a direct sample estimate, the smaller its contribution to the Bayesian estimate, and the larger the contribution from the regression prediction. Since the model error variance is unknown, the Bayesian approach averages the shrinkage estimates computed over a plausible range of values of the model error variance, weighting the results for each of these values according to the posterior (conditional on the data) probability distribution of the model error variance developed from the Bayesian calculations. The result is generally very close to what one gets by estimating the model error variance by the mean of its posterior distribution, and computing the corresponding shrinkage estimate. Technical details of the Bayesian approach are discussed in the paper, "Accounting for Uncertainty About Variances In Small Area Estimation," (Bell 1999) in the Published Papers section of this web site.

*Poverty Ratios and Numbers of Poor People.*
In deriving state-level estimates of the numbers
of poor people of various ages, we use regression models with poverty ratios for those ages as
the dependent variables. We multiply Bayesian predictions of poverty ratios from these models
by estimates of
the noninstitutional population of the appropriate ages to obtain model estimates of the
numbers of poor people.

The poverty ratios used in the state-level models are not the official poverty rates because we use the noninstitutional population as the denominator rather than the poverty universe (for a discussion of poverty universe differences, see Denominators for Model-Based State and County Poverty Rates). For related children in families, we use ratios of the number of related children in families in poverty to the number of children in the noninstitutional population. We use these poverty ratios because of the difficulty of deriving estimates of the size of the poverty universe and the number of related children in families.

We derive the estimates of the noninstitutional population by age from the U.S. Census Bureau's annual intercensal population estimates for states. We use these estimates, instead of the estimates of population obtained directly from the CPS, because the CPS controls survey weights only to estimates of the population age 16 and over at the state level, and we are making estimates for more specific age groups.

While we have multiplied model-based poverty ratio estimates by population estimates at the state level, we have not addressed the county-level estimation in the same way, because the estimates of the populations of counties by age are likely to be much less stable than the state population estimates, and little is known about their error structure. Thus, for counties, we directly model (logarithms of) CPS estimates of the number of poor people.

*Controlling to the National Estimates.*
After converting the Bayesian estimates of poverty ratios to
estimates of numbers of poor, the last step in the process is controlling these model-based state
estimates of the number of poor people to the direct national estimate of number poor based on the
CPS. We do not control estimates of state median household income to
the national median because the estimation model does not produce the entire household income
distribution, which would be required to do so.

*Using Estimates from the Prior Census in the Models.*
The prior census estimates are used
to construct regression predictor variables for each of the age-specific poverty ratio
models and the median income model. For the 1997 model for the poverty ratio for
people age 65 and over, and for median income, the corresponding estimate from the 1990
census is used as a predictor variable. For the 1997 poverty ratio models for the age
groups 0-4, 5-17, and 18-64, the prior census estimates determine census residuals
that are used as regression predictor variables in the models. These census residuals
are obtained by fitting a cross-sectional model for 1989, using the 1990 census estimated
poverty ratio for the age group as the dependent variable and the 1989 values
of the regression predictors from the administrative data as the independent variables.
The residuals from these cross-sectional regressions identify states in which the selected
predictors tend to either overestimate or underestimate poverty, as measured by the census.

*Estimating the Total Number of Poor People.*
We derived the estimate of the total number of poor people in a state
by summing the separate model-based estimates of the number of poor people by age (not limited to
related children). The age groups with separate models were 1) people under 5 years of age,
2) people age 5 to 17 years, 3) people age 18 to 64 years, and 4) people age 65 years and over. Summing state-level
estimates from separate models for these groups produces superior estimates of the total relative to a single state-level model for the total number of poor.

**The Model for the Poverty Ratio of People Ages 5 to 17**

The model of 1997 state poverty ratios for related children age 5 to 17 years in families employs the following predictors:

- an intercept term;
- the 1997 "tax return poverty rate" for children -- the percentage of child exemptions entered on returns for which the adjusted gross income falls below the official poverty threshold for a family of the size implied by the number of exemptions on the return;
- the 1997 "nonfiler rate" for people under age 65 -- the difference between the estimated population under age 65 and the number of exemptions under age 65, expressed as a percentage of the population under age 65;
- the 1997 food stamp participation rate -- the adjusted, monthly average number of participants of all ages in the food stamp program over the 12 months July 1997 - June 1998 as a proportion of the total population; and
- residuals from a regression of the 1990 census poverty ratios for 1989 for all people 5 to 17 years of age on the 1989 values of the above variables.

For further information on these variables, go to Information about Data Inputs.

The dependent variable is the 1997 state estimate of the ratio of poor related children age 5 to 17 years to the noninstitutional population of that age from the March 1998 CPS.

The residuals from the 1990 census identify states in which the selected predictors tend to either overestimate or underestimate poverty, as measured by the census. Note that this is the only predictor variable that refers specifically to the group age 5 to 17 years.

To derive the 5 to 17 year old component of the total number of poor in 1997, we estimate an equation with the same predictor variables as above using the 1997 state estimates of the ratio of the number of poor children age 5 to 17 (related and unrelated) to the noninstitutional population age 5 to 17 as the dependent variable.

We average the regression predicted poverty ratio for each state with the CPS direct sample estimates using Bayesian techniques, and transform the results into estimated numbers of poor by multiplying them by the appropriate estimate of the noninstitutional population (see the discussion of poverty rates and numbers of poor above in this section). Finally, we ratio-adjust the estimated number poor for each state to the direct CPS national estimate.

**The Model for the Poverty Ratio of People under Age 5**

The model of 1997 state poverty ratios for people under age 5 employs the following predictors:

- an intercept term;
- the 1997 "tax return poverty rate" for people under age 65 years -- the number of exemptions under age 65 years which are on returns for which the adjusted gross income falls below the official poverty threshold for a family of the size implied by the number of exemptions on the return divided by the population for that age group;
- the 1997 "nonfiler rate" for people under age 65 -- the difference between the estimated population under age 65 and the number of exemptions under age 65, expressed as a percentage of the population under age 65;
- the 1997 food stamp recipiency rate -- the adjusted monthly average number of participants of all ages in the food stamp program over the 12 months July 1997 - June 1998 as a proportion of the total population; and
- residuals from a regression of the 1990 census poverty ratios for 1989 for all people under age 5 years on the 1989 values of the above variables.

For further information on these variables, go to Information about Data Inputs.

The dependent variable is the 1997 state estimate of the ratio of the number of poor children under 5 years to the noninstitutional population of that age from the March 1998 CPS. Note that only the residuals from the previous census refer precisely to the target age group. We average the regression predicted poverty ratio for each state with the CPS direct sample estimates using Bayesian techniques, and transform the results into estimated numbers of poor by multiplying them by the appropriate estimate of each state's noninstitutional population. Finally, we ratio-adjust the estimated number poor for each state to the direct CPS national estimate.

**The Model for the Poverty Ratio of People Ages 18-64**

The model of 1997 state poverty ratios for people age 18 to 64 years employs the same predictors as the model for people under age 5, except that the 1990 census residuals are specific to people 18 to 64 years of age instead of children under age 5. The dependent variable is the 1997 state estimate of the ratio of the number of poor people age 18-64 to the population of that age from the March 1998 CPS.

**The Model for the Poverty Ratio of People Age 65 and over**

The model of 1997 state poverty ratios for people age 65 and over is slightly different from those above. We have more appropriate predictors because we can separate the tax exemptions for people age 65 years and over and because we have data from the Supplemental Security Income program for people this age. In addition, the poverty rate for people age 65 and over from the prior census is a better predictor for this age group than the regression residuals we have employed for other age groups. The predictors of 1997 state poverty rates for people age 65 and over are:

- an intercept term;
- the 1997 "tax return poverty rate" for people age 65 years and over -- the number of exemptions age 65 years and over which are on returns for which the adjusted gross income falls below the official poverty threshold for a family of the size implied by the number of exemptions on the return divided by the population for that age group;
- the 1997 "nonfiler rate" for people age 65 and over -- the difference between the estimated population age 65 and over and the number of exemptions age 65 and over, expressed as a percentage of the population age 65 and over;
- the 1997 Supplemental Security Income recipiency rate -- the number of SSI recipients age 65 years and over divided by the population of that age; and
- the poverty ratio for people age 65 years and over from the 1990 census.

For further information on these variables, go to Information about Data Inputs.

The dependent variable is the CPS estimate of 1997 state poverty rates for people age 65 and over. We average the predicted rates with the CPS direct sample estimates using Bayesian techniques, and transform the results into estimated numbers of poor by multiplying them by the appropriate estimate of each state's noninstitutional population. Finally, we ratio-adjust the estimated numbers for each state to the CPS national estimate.

**The Model for the Total Number of Poor People**

We derived the estimate of the total number of poor people in a state by summing the separate model-based estimates of the number of poor people by age (not limited to related children). The age groups with separate models were 1) people under 5 years of age, 2) people age 5 to 17 years, 3) people age 18 to 64 years, and 4) people age 65 years and over. Summing state-level estimates from separate models for these groups produces superior estimates of the total relative to a single state-level model for the total number of poor.

**The Model For Median Household Income**

The regression model for the 1997 median household income for states has the following predictor variables:

- the 1989 median household income derived from the 1990 census; and
- the 1989 median household income derived from the 1990 census adjusted by the ratio of the median 1997 adjusted gross income to the median 1989 adjusted gross income from tax returns.

The dependent variable is the direct estimate of median household income in 1997 from the March 1998 CPS. We average the model estimates with the direct sample estimates from the March CPS using Bayesian techniques. We do not ratio-adjust the state medians to the CPS estimate of the national median. Unlike estimates of poverty ratios, estimates of state-level medians cannot be averaged to get a national median.

Source: U.S. Census Bureau | Small Area Income and Poverty Estimates |
Last Revised:
April 29, 2013