Growth Rates Preservation (GRP) temporal benchmarking: Drawbacks and alternative solutions
Section 1. Introduction
Benchmarking monthly and quarterly series to annual data is a common practice in many National Statistical Institutes. For example, each year Statistics Netherlands aligns 12 quarterly Supply and Use Tables with the three most recent annual accounts (Eurostat, 2013, Annex 8C).
The benchmarking problem arises when time series data for the same target variable are measured at different frequencies with different levels of accuracy. One might expect that a temporal aggregation relationship between these time series is fulfilled, e.g., that four quarterly values add up to one annual value, but because of differences in data sources and processing methods, this is often not the case. Benchmarking is the process to remove such discrepancies. In this process the preliminary values are adjusted to achieve mathematical consistency between low-frequency (e.g., annual) and high-frequency (e.g., quarterly or monthly) time series.
There are two main principles of benchmarking. Firstly, low-frequency benchmarks are fixed, because these data sources describe levels and long-term trends better than high-frequency sources. Secondly, short-term movements of high-frequency time series are preserved as much as possible, as these data sources provide the only information on short-term movements.
Several benchmarking methods are available in the literature. These methods differ in the way short-term movements of high-frequency series are defined. A distinction can be made between multiplicative and additive methods. Multiplicative methods try to preserve relative changes of preliminary high-frequency time series, while additive methods aim to preserve changes in absolute terms. In this paper the focus will be solely on multiplicative variants.
Two well-known multiplicative methods are Denton Proportionate First Differences (PFD), by Denton (1971), and Growth Rates Preservation (GRP) by Causey and Trager (1981; see also Trager, 1982 and Bozik and Otto, 1988).
In the literature it is generally agreed that GRP is grounded on the strongest theoretical foundation (Bloem, Dippelsman and Maehle, 2001, page 100). It explicitly preserves the period-to-period rates of change of the preliminary series. However, Denton PFD is more popularly used, because it is technically easier to apply. Mathematically, the Denton method deals with a standard linearly constrained quadratic optimization problem, while GRP solves a more difficult linearly constrained nonlinear problem that can be efficiently solved by an interior-point-algorithm (Di Fonzo and Marini, 2015).
From a number of simulation studies it is known that Denton PFD and GRP lead to similar or close to similar results for the large majority of cases (Harvill Hood, 2005; Titova, Findley and Monsell, 2010; Di Fonzo and Marini, 2012 and Daalmans and Di Fonzo, 2014). Therefore Denton PFD can be used as an approximation of GRP.
The aim of this paper is to demonstrate that GRP suffers from drawbacks that are, to the best of our knowledge, not described in the literature. A first drawback is that it matters whether benchmarking is applied “forward” or “backward” in time. In this context, we will present a link with the time reversibility property from index number theory. A second drawback is that undesirable results may be obtained due to singularities in the GRP objective function.
A second aim of this paper is to present alternative benchmarking methods that do satisfy time reversibility. This paper may be valuable for practitioners who apply or consider to apply benchmarking techniques.
First, in Section 2, we will give a formal description of the Denton PFD and GRP benchmarking methods. Section 3 describes the drawbacks of the GRP method. In Section 4 two new benchmarking methods are proposed that can be used as an alternative for GRP. Results of an illustrative application to real-life data are given in Section 5. Finally, Section 6 concludes this paper.
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