Small-Area Estimation With State–Space Models Subject to Benchmark Constraints

Abstract

This article shows how to benchmark small-area estimators, produced by fitting separate state–space models within the areas, to aggregates of the survey direct estimators within a group of areas. State–space models are used by the U.S. Bureau of Labor Statistics (BLS) for the production of all of the monthly employment and unemployment estimates in census divisions and the states. Computation of the benchmarked estimators and their variances is accomplished by incorporating the benchmark constraints within a joint model for the direct estimators in the different areas, which requires the development of a new filtering algorithm for state–space models with correlated measurement errors. The new filter coincides with the familiar Kalman filter when the measurement errors are uncorrelated. The properties and implications of the use of the benchmarked estimators are discussed and illustrated using BLS unemployment series. The problem of small-area estimation is how to produce reliable estimates of area (domain) characteristics and compute their variances when the sample sizes within the areas are too small to warrant the use of traditional direct survey estimates. This problem is commonly handled by borrowing strength from either neighboring areas and/or from previous surveys, using appropriate cross-sectional/time series models. To protect against possible model breakdowns and for consistency in publication, the area model–dependent estimates often must be benchmarked to an estimate for a group of the areas, which it is sufficiently accurate. The latter estimate is a weighted sum of the direct survey estimates in the various areas, so that the benchmarking process defines another way of borrowing strength across the areas.