- We estimate a regression model which predicts the number of poor persons using three-year averages of county-level observations from the March Current Population Survey (CPS) as the dependent variable and administrative record and census data as the predictors. Although we use only the approximately 1,200 counties with CPS sample cases to estimate the equation, we make regression "predictions" for all 3,141 counties.
- The model is multiplicative; that is, we model the number of poor as the product of a series of predictors which are numbers (not rates) and have unknown errors. When estimating the coefficients in the model, we take logarithms of the dependent and all independent variables. While we may omit reference to logs, all variables in the county regression models for numbers of poor are logarithmic.
- The estimates for different counties are of different reliability because of the size of the CPS sample in them. Our estimates take this factor into account.
- To use the information contained in the direct estimates for the 1,200 counties in the CPS we combine the regression predictions with these direct estimates using Empirical Bayes (or "shrinkage") techniques. The Empirical Bayes (EB) techniques weight the contribution of the two components (regression and direct estimates) based on their relative precision.
- We control the estimates for the counties of a given state to sum to the independently derived state estimate (which in turn sum to the official national estimate).
- We provide a confidence interval, which represents uncertainty from both sampling and from modeling, for each estimate.

*Using counties in the CPS sample.* Our use of the CPS implicitly
assumes that the counties in the survey sample are representative of those
not selected, but this need not be the case. The CPS sample is designed
to represent each state's population and only incidentally represents counties.
The characteristics of some counties guarantee that they are included,
e.g., most counties in large metropolitan areas and counties with large
populations. More generally, while all counties have a nonzero probability
of being included in the sample, some have higher probabilities than others.
Further, the probability of selecting a county may be related to its
income and poverty level. On the other hand, comparison of regression equations
based on 1990 census data for counties in the CPS sample and equations based on all
counties indicate remarkably similar results, providing some assurance that the
CPS counties are largely representative of all counties.

The survey weights used in estimation at the national level are not appropriate for county-level estimates. The CPS sample design selects some primary sampling units (usually a county or group of counties) to represent a set of counties in the same stratum. The sum of the weights for sample households from such a county estimates the total population of the entire set of counties it represents. Because we want each county in the CPS sample to stand for itself, we have adjusted the weights to make each county self-representing.

*Estimation of the model equation.* CPS sampling variances are
not constant over all counties. We avoid giving observations with larger
variances (a great deal of uncertainty) the same influence on the regression
as observations with smaller variances (less uncertainty) by, in effect,
weighting each observation by the inverse of its uncertainty. Representing
this uncertainty requires recognizing that it arises from two sources:

- uncertainty about where the estimates lie relative to the true values for each county (sampling error), and
- uncertainty about where the true county values lie with respect to the regression surface (lack of fit).

To estimate the lack-of-fit component, we estimate our model using the 1990 census data and assume that the lack-of-fit component of residual variance is the same when the same model is fit to the CPS and to the census. Since we have separate estimates of sampling variance for each observation in the 1990 census, we use them to estimate the unknown lack-of-fit component with a maximum likelihood procedure (see "Appendix C: Accuracy of the Data" in 1990 Census, STF3 documentation) [PDF 93k].

Next we fit a regression equation to the CPS data. We assume the sampling variance of the log of the number of poor is inversely proportional to the sample size (in households) and the lack-of-fit variance is the same as that estimated in the census regression. We estimate the CPS regression parameters and the two components of CPS variance with a maximum likelihood procedure.

*Combining model and direct survey estimates.* Final estimates
are weighted averages of the model predictions and the direct CPS estimates,
where they exist. The two weights for each county add to 1.0, and we compute the weight
on the model prediction as the sampling variance divided by
the total variance (sampling plus lack-of-fit) of the direct estimate.
With this technique, the larger the sampling variance of the direct estimate,
the smaller its contribution and the larger the contribution from the prediction
model. These weights are commonly referred to as "shrinkage weights,"
and the final estimates as "shrinkage" or "Empirical Bayes"
estimates. For counties not in the CPS sample, the weight on the
model's predictions is one and the weight on the direct survey estimate
is zero.

*Controlling to State Estimates.*The last steps in the production process are transforming the county estimates
from the log scale to estimates of numbers and controlling them to the
independently derived state
estimates. We make a simple ratio adjustment
to the county-level estimates to ensure
that they add to the state totals. We control model-based estimates at the state level
to the national level direct estimates derived from the March
1999 CPS. We adjust the estimated standard errors of the county estimates
to reflect this additional level of control.

We do not control estimates of county median household income to the state medians because the estimation model does not produce the entire household income distribution, which would be required to do so.

*Standard Errors and Confidence Intervals.* One goal
of our small area estimation work is providing estimates of the uncertainty
surrounding the estimates of the numbers of poor. The census and model-based
estimates shown in the tables are accompanied by their 90-percent confidence
intervals. These intervals were constructed from estimated standard errors.

For the model-based estimates, the standard error depends mainly on the uncertainty about the model and the CPS sampling variance. While the variance of the shrinkage weights could also be a significant component of uncertainty about our estimates (if sizeable and ignored we would be underestimating the standard errors), our research indicates that its contribution is negligible.

For the census, we derive the standard errors from a set of generalized variance functions that reflect the nature of the census sample design for the long form questionnaire (for further information, see Quantifying Uncertainty in State and County Estimates).

The model is multiplicative; that is, we model the number of poor as the product of a series of predictors that are numbers (not rates), and we model the unknown errors. To estimate the coefficients in the model, we take logarithms of the dependent and all independent variables. Our choice of a multiplicative model is motivated in part by the fact that the distribution of the number of poor has a huge range -- from zero in some counties to more than a million in the largest county (with a mean of 10,000), based on the 1990 census -- and the distribution is highly skewed. Taking the logarithm of all variables makes their distributions more centered and symmetrical and has the effect of diminishing the otherwise inordinate influence of large counties on the coefficient estimates. Another advantage of a multiplicative model is that it makes it plausible to maintain that the (unobserved) errors for every county, no matter how large or small, are drawn from the same distribution.

The predictor variables in the regression model used to estimate the total number of poor people by county for income year 1998 are:

- the log of the number of 1998 tax return exemptions (all ages) on returns whose adjusted gross income falls below the official poverty threshold for a family of the size implied by the number of exemptions on the form;
- the log of number of food stamp recipients in July 1998;
- the log of the estimated total resident population as of July 1, 1999;
- the log of total number of 1998 tax return exemptions; and
- the log of the 1990 census estimate of the total number of poor.

For further information on these variables see Information about Data Inputs.

The dependent variable is the log of the total number of poor in each county as measured by the three-year average of values from the March CPS for 1998, 1999 and 2000. We combine the regression predictions, in the log scale, with the logs of the direct CPS sample estimates, and then transform the results into estimates of the numbers of poor. Finally, we control the estimates to the independent estimates of state totals.

**The Model for the Number of Related Children Ages 5 to 17
in Families in Poverty**

The estimation model for related children age 5 to 17 in poverty parallels that for all people in poverty in structure. There are five predictor variables:

- the log of the number of 1998 child exemptions claimed on tax returns whose adjusted gross income falls below the official poverty threshold for a family of the size implied by the number of exemptions on the form;
- the log of the number of food stamp recipients in July 1998;
- the log of the estimated resident population under age 18 as of July 1, 1999;
- the log of the total number of child exemptions indicated on 1998 tax returns; and
- the log of the 1990 census estimate of the number of poor related children age 5 to 17.

For further information on these variables see Information about Data Inputs.

The dependent variable is the log of the number of poor related children age 5 to 17 in each county as measured by the three-year weighted average of the March CPS for 1998, 1999 and 2000. We combine the regression predictions, in the log scale, with the logs of the direct CPS sample estimates, and then transform the results into estimates of the numbers of poor. Finally, we control the estimates to the independent estimates of state totals.

**The Model for the Number of Poor People Under Age 18**

The estimation model for poor people under age 18 in poverty is quite similar. There are five predictor variables:

- the log of the number of 1998 child exemptions indicated on tax returns whose adjusted gross income falls below the official poverty threshold for a family of the size implied by the number of exemptions on the form;
- the log of the number of food stamp recipients in July 1998;
- the log of the estimated resident population under age 18 as of July 1, 1999;
- the log of the total number of child exemptions indicated on 1998 tax returns; and
- the log of the 1990 census estimate of the number of poor persons under age 18.

For further information on these variables see Information about Data Inputs.

The dependent variable is the log of the number of poor people under age 18 in each county as measured by the three-year weighted average of the March CPS for 1998, 1999 and 2000. We combine the regression predictions, in the log scale, with the logs of the direct CPS sample estimates, and then transform the results into estimates of the numbers of poor. Finally, we control the estimates to the independent estimates of state totals.

**The Model for Median Household Income**

The predictor variables in the regression model used to generate the estimates for median 1998 household income by county are:

- the median adjusted gross income from 1998 tax returns;
- the ratio of the number of dependent ("zero exemption") 1998 tax returns -- returns representing people claimed as dependents on other returns -- to the total number of returns;
- an interaction term between a dummy variable equal to one if the county is in Alaska (and zero otherwise) and the ratio of the number of dependent 1998 tax returns to the total number of returns.
- the log of the proportion of the 1998 Bureau of Economic Analysis (BEA) estimate of total personal income derived from government transfers;
- the 1990 census estimate of median household income;
- the ratio of the 1998 BEA estimate of per capita total personal income to the 1989 estimate; and
- the product of the 1990 census median household income and the ratio of 1998 to 1989 BEA per capita total personal income.

For further information on these variables see Information about Data Inputs.

The dependent variable is the county median household income as measured
by the three-year average of the March CPS for 1998, 1999 and 2000 (income for years
1997, 1998 and 1999, respectively). We adjusted the March 1998 and 2000 CPS to express
incomes in 1998 dollars before we computed the median incomes.

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