Small-Area Estimates of School-Age Children in Poverty Interim Report 2, Evaluation of Revised 1993 County Estimates for Title I Allocations (1998) / Chapter Skim
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APPENDIX A. MODELS FOR COUNTY AND STATE POVERTY ESTIMATES
APPENDIX A. MODELS FOR COUNTY AND STATE POVERTY ESTIMATES
Pages 93-108
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Appendices
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From page 95... ...
The same model forms can be used for poverty statistics for other age groups, with appropriately defined dependent and regression variables. NOTATION The following notation is used in the estimation program: · Yin = CPS 5-17 poverty estimate for county i in year t; · Ceni = previous census estimate for county i (where necessary, a specific census is distinguished by writing Cen90i or Cen80i)
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N(0, ~2 ~ and independent of each other.) The basic regression variables xit are defined below.
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and (2) are thus the regression parameters A, A, it, and I; the common model error variance (S2; and the sampling error variance parameter vet Decennial census sampling error variances for estimates of number of poor are available from published formulas (generalized variances)
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From page 98... ...
The CPS poverty universe, and the number of poor related children aged 5-17, are estimated from CPS data for each year using CPS weights modified to make each county "self-representing." For counties with a CPS sample in only 1 or 2 of the 3 years, the values for only that year, or for the 2-year average corresponding to (6) , are used.
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and (10) to be estimated are thus the regression parameter vectors ~ and it; the common model error variance ~u2; the model error correlation p; and the sampling error variance parameter vet Note that the bivariate model form differs from the SAIPE model form in that it does not include the previous census data as a regression variable, and it also allows the model errors to be correlated.
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always denotes Cen90i. (The bivariate model approach can be applied to jointly model 1990 CPS and 1990 census data, but this is a different exercise, since the resulting smoothed estimates of Y
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are needed to convert smoothed poverty rate estimates to estimates of the number of poor children. For some counties with very small CPS sample sizes there may be no related children aged 5-17 observed in the sample.
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To assume that the CPS sampling errors of direct poverty rate estimates have variance of the form ve I nit is inconsistent with making the same assumption for CPS direct estimates of log number poor or log poverty rate. Simple Taylor series approximations suggest that if ve I nit is the appropriate variance for poverty rate estimates, then the sampling error variance for log poverty rates will depend on the underlying true poverty rate p, and vice versa.
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and (2~: the census data appearing on the right-hand side of the equations are analogous to the other regression variables defined as log number poor children 5-17, whereas Cen90i appearing on the left-hand side is the log census poverty rate. With the data thus defined, the model fitting proceeds in the same fashion as for the other models discussed.
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be vectors containing the county CPS and census data to be used for model fitting, and let X~ and X~g be the corresponding matrices of regression variables for their respective equations. The SAIPE model form given by (1)
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, one also needs to allow for the correlations between ut and z when estimating the regression parameters (p, 71~- This can be done by applying generalized least squares to (18~. In fact, it is simpler to structure the equations for the bivariate model so that the CPS and census data are paired off (for those counties with CPS data available for model fitting)
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The two different model fitting approaches were adopted because some analysts use SAS and others use Splus and because the SAS code was developed for the original SAIPE model and could not be used to fit models of bivariate form, necessitating development of a second program. Generalization of the Splus bivariate model software is a recent development, and there has not been time to make extensive comparisons of the two programs for models they can both fit.
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For models with fixed state effects, smoothed estimates and their variances are obtained from expressions analogous to (19)
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Prediction error variances in these cases could be taken to be those for the smoothed poverty rates multiplied by the square of the population estimates, though this ignores error in the 5-17 population estimates. Formal measures (variances)
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