Appendix E

Composite Estimation

This appendix briefly presents the simplest composite estimator for a rotating panel design. There is an extensive literature on composite estimation to which the reader can turn for descriptions of more elaborate approaches.

This discussion assumes an annual data collection. Let image denote a parameter to be estimated in year t, and let image denote the “usual unbiased estimate” of image based only on data from time t. Note that it is also possible to compute an estimate image based only on the overlapping (or “matched”) sample, i.e., using only the sample that is being interviewed for the second time (or more). Since values exist from the previous year for the matched sample, it is possible to compute image and so image is an estimate of the change between year t and year t–1 based on the same sample units. By adding this estimate of change to last year’s estimate of image a second estimate of image, namely, image is obtained. (In words, the estimate for this year is equal to the estimate for last year plus the estimated year-to-year change.) Since there are now two estimates of the same quantity image, it is natural to combine them to obtain an improved estimate. Thus the so-called composite estimate of image is image a weighted average of two estimates (assuming α is between 0 and 1). Because image is a better estimate than image then in practice, the following composite estimate is used: image.

The value of α is chosen to minimize the variance of image.



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Appendix E Composite Estimation This appendix briefly presents the simplest composite estimator for a rotating panel design. There is an extensive literature on composite es- timation to which the reader can turn for descriptions of more elaborate approaches. This discussion assumes an annual data collection. Let θt denote a ˆ parameter to be estimated in year t, and let θ t denote the “usual unbiased estimate” of θt based only on data from time t. Note that it is also possible ˆ to compute an estimate θ t ,m based only on the overlapping (or “matched”) sample, i.e., using only the sample that is being interviewed for the second time (or more). Since values exist from the previous year for the matched ˆ ˆ ˆ sample, it is possible to compute θ t –1,m , and so θ t ,m – θ t −1,m is an estimate of the change between year t and year t –1 based on the same sample units. By adding this estimate of change to last year’s estimate of θt–1, a second ˆ ˆ ˆ estimate of θt, namely, θ t −1 + θ t ,m – θ t −1,m is obtained. (In words, the estimate for this year is equal to the estimate for last year plus the estimated year-to- year change.) Since there are now two estimates of the same quantity θt, it is natural to combine them to obtain an improved estimate. Thus the so-called ˆ ˆ ˆ ˆ ˆ composite estimate of θt is θ tc = αθ t + (1 – α )(θ t −1 + θ t ,m – θ t −1,m ), a weighted ˆ average of two estimates (assuming a is between 0 and 1). Because θ tc−1 is a ˆ , then in practice, the following composite estimate better estimate than θ t –1 ˆ ˆ ˆ ˆ ˆ is used: θtc = αθ t + (1 – α )(θ tc−1 + θ t ,m – θ t −1,m ). ˆ The value of a is chosen to minimize the variance of θ tc . 105

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