Time Series Methods and Repeated Sample Surveys

When applied to a sequence of repeated surveys, the traditional sample survey estimators of means or totals for one time period only, fail to take advantage of any time series structure. Such structure may result from correlation between successive responses for resampled individuals, or from time series properties in the parameters of interest. Historically, the initial published papers on time series improvement of repeated sample survey estimates allowed only the first possibility, treating the sum over the population of the individual responses as fixed; individual responses were seen as having stochastic properties only with respect to the sampling scheme. The alternative and later development allowed that both individual responses and their sum have stochastic properties with respect to a superpopulation from which the population of individual responses are drawn. Superpopulations allowed the application of mainstream time series techniques, including signal extraction and stochastic least squares, to repeated sample survey data. These developments in their historical perspective are the topic of Chapter 1. Superpopulation models may also be applied to sample surveys from a single time period, and superpopulation and design properties of the one period linear non-homogeneous sample survey estimator form the topic of Chapter 2; this estimator is sufficiently general to subsume almost all single period non-informative sample survey estimators, and Chapter 2 allows systematisation of a wide range of previously disparate results. This linear estimator may also be extended beyond one time period to include the known estimators for repeated surveys, and this topic, together with a consideration of the effects of data agqregation on non-stochastic and stochastic least squares, is the subject of Chapter 3. Given the central role of the general linear model, and the time series nature of repeated surveys, projection and parameter updating formulae for linear models should form an integral part of repeated survey analysis. The correlation of sample survey errors however, invalidates the formulae appropriate to the known iid error case, and Chapters 4 and 5 develop the general formulae to allow correlated error structure. Chapter 4 considers parameter vectors of fixed length, as for example, for polynomial models, and provides formulae for estimating the length of the parameter vector, and for calculating independent recursive residuals and cusums when further data are added to the model. Chapter 5 considers updating and projection formulae in a wider context, and allows that the parameter vector may be stochastic or non-stochastic and that its length may increase with additional data; it consequently provides a general extension of the Kalman filter to the case of coloured noise over time. The paucity of suitable data has limited data analysis to that contained in Chapter 6, where a simulation study and an analysis of medical data gauge the efficacy of polynomial models in time with multiple observations per time point and autocorrelated errors. The formulae of Chapter 4 allow testing for the constancy of the regression relationships over time. The appendix details SAS computer programs for fitting the polynomial models of Chapter 6.