Estimation of the variance for different estimators of the change over time for overlapping samples

This work was inspired by the growing need to have a measure of the accuracy of the estimates produced within the short-term statistics in the Official Statistics. In particular, the aim of the work is to illustrate the methodology for the computation of the variance for the estimators currently used in the service turnover survey carried on by the Italian National Institute of Statistics, for the quarterly turnover growth rate estimation. While the calculation of the variance of the estimates produced for a given instant of time is now a good practice (also through the development of software packages), the same does not happen for the variation of two quantities over time. An estimator of variance must take into account of both the estimator and the sampling design (Wolter, K.M. (1985)). The greatest difficulty is that for many surveys, the samples for producing estimates in two different time are not independent each other, due to the rotation operations of the sample. In particular for business surveys, in order to take into account the birth-mortality of units in the population and changes in stratification variables (such as size category and type of economic activity), the sample is updated, and a part of the units is replaced with others. Moreover, many indicators are non-linear function of linear estimators (e.g. simple ratio, difference of ratios), therefore, to calculate their variance a first-order Taylor approximation can be used. Alternatively, balanced repeated replication (BRR) can be used. My methodological contribution is not only to suggest how to assess the variance of possible estimators of the turnover variation over time, but also to compare such estimators with respect to their variance to identify the best one. The performance of these estimators is assessed by a simulation study, which also has the aim of exploring under which conditions it is better to use all the observations or only the overlapping observations. The change estimators and the corresponding estimators of the variance are defined at stratum and estimation domain level and take into account the use of a stratified sampling design and the updating of the sample due to a replacement of some units and to a dynamic stratification of the population.