1. Introduction
Paul Knottnerus
In many countries a monthly business survey is held for the major Standard Industrial Classification (SIC) codes to estimate the level of the monthly turnover and the change in that level compared to a month or a year ago. When repeatedly sampling a population, a complicating factor is that there are various methods for estimating the (relative) change from a panel with different outcomes especially when the samples on different occasions are not completely overlapping.
Kish (1965), Tam (1984), Laniel (1987), Hidiroglou, Särndal and Binder (1995), Nordberg (2000), Berger (2004), Qualité and Tillé (2008), Wood (2008) and Knottnerus and Van Delden (2012) examined various estimators for the parameter of change in different situations. The main aim of this paper is to derive estimators for a relative change as well as the corresponding population totals that are in line with each other and that have minimum variance property. The derivation of the aligned composite estimators is based on the general restriction (GR) estimator of Knottnerus (2003). Composite estimators for totals and (absolute) changes are also proposed by Särndal, Swensson and Wretman (1992, pages 370-378) but in separate steps. Moreover, this paper focuses on estimators for growth rates because: (i) users of figures from business surveys for a specific SIC code often are more interested in growth rates than in absolute changes, (ii) in practice there might be model-assisted reasons to look at growth rates (auxiliary variables in regression models often explain the different growth rates of the units rather than their different levels), and (iii) growth rates are needed for making an overall index for the (monthly) turnover for each of the major SIC codes. For instance, Smith, Pont and Jones (2003) describe the method of matched pairs to measure a change from month to month, using responses that are common to both periods. The authors use this method for deriving the monthly retail sales index (RSI).
The outline of the paper is as follows. Section 2 briefly describes two methods for estimating a growth rate of the total turnover for enterprises with a certain SIC code. Two examples illustrate the possibly substantial differences between the two approaches. Section 3 discusses the question of which estimation method is to be preferred and explains as to why the difference between the variances of both estimators might be so large. For various situations Section 4 and Section 5 propose an optimal composite estimator. Section 6 discusses some extensions of the aligned composite (AC) estimator for growth rates and totals. Section 7 summarizes the main conclusions and issues to be further investigated.
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