Growth Rates Preservation (GRP) temporal benchmarking: Drawbacks and alternative solutions
Section 2. Temporal benchmarking methods

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This section explains the Denton PFD and GRP benchmarking procedures. Because temporal aggregation constraints are the same for Denton PFD and GRP, these are described first. Thereafter, the Denton PFD and GRP benchmarking procedures are explained.

We focus on univariate variants of these methods, in which temporal consistency is the main constraint of interest. The observations that are presented in the remainder of this paper are however also valid for the multivariate case, in which multiple time-series are reconciled simultaneously and additional constraints between time-series apply (see Di Fonzo and Marini, 2011 and Bikker, Daalmans and Mushkudiani, 2013).

2.1  General notation and temporal constraints

In general, temporal aggregation constraints can be expressed as a linear system of equalities $Ax=b,$ where $x$ is the target vector of high-frequency values, $b$ is a vector of low-frequency values, and $A$ is a temporal aggregation matrix converting high- into low-frequency values.

The specific form of these constraints depends on the nature of the variables involved. For flow variables, a sum of subannual values, e.g., four quarterly values, usually needs to be the same as one annual value. For stock variables, one of the subannual values, usually the first or the last, needs to be the same as the relevant annual value. For example, for quarterly/annual flow variables, assuming for the sake of simplicity that the available time span begins on the first quarter of the first year and ends on the fourth quarter of the last observed year, it is

$$A=\left[\begin{array}{ccccccccccccc}1& 1& 1& 1& 0& 0& 0& 0& \cdots & 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 1& 1& \cdots & 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0& 0& 0& 0& 0& \cdots & 1& 1& 1& 1\end{array}\right].$$

Denoting by $p$ a vector of preliminary values, in general it is $Ap\ne b,$ otherwise no adjustment would be needed. We look for a vector of benchmarked estimates $x^{*},$ a particular outcome for $x,$ which should be “as close as possible” to the preliminary values and that satisfies $A{x}^{*}=b.$

Not all sub annual periods need to be covered by a benchmark. Thus, the number of rows in $A$ may be smaller than the total number of annual periods, see e.g., Dagum and Cholette (2006) for more details.

In a benchmarking operation, characteristics of the original series $p$  should be considered. For example, in an economic time series framework, the preservation of the temporal dynamics (however defined) of the preliminary series is often a major interest of the practitioner.

2.2  Growth Rates Preservation (GRP) and Denton PFD

This section gives a formal description of GRP and Denton PFD.

Causey and Trager (1981; see also Monsour and Trager, 1979 and Trager, 1982) obtain the benchmarked values $x_t^{*}\text{},t=1,\dots ,n$  as a solution to the following optimization problem:

$$\underset{x_t}{\text{min}}{f}_{\text{F}}^{\mathrm{GRP}}\left(x\right)\text{subject to}Ax=b,\text{where}{f}_{\text{F}}^{\text{GRP}}\left(x\right)={\displaystyle \sum }_{t=2}^n{\left(\frac{x_t}{x_{t-1}}-\frac{p_t}{p_{t-1}}\right)}^2\text{}.(2.1)$$

The GRP criterion to be minimized, $f_{\text{F}}^{\text{GRP}}\left(x\right),$ explicitly relates to growth rates: it minimizes the sum of squared differences between growth rates of preliminary and benchmarked values. The subscript “F” in the minimization function stands for “Forward”, later in this paper a “Backward” minimization function will be defined.

Denton (1971) proposed a benchmarking procedure grounded on the Proportionate First Differences (PFD) between target and original series. Cholette (1984) slightly modified the result of Denton, in order to correctly deal with the starting conditions of the problem. The PFD benchmarked estimates are thus obtained as the solution to the constrained quadratic minimization problem

$$\underset{x_t}{\min }{f}_{\text{F}}^{\text{PFD}}\left(x\right) \text{subject to}Ax=b,\text{where}{f}_{\text{F}}^{\text{PFD}}\left(x\right)={\displaystyle \sum }_{t=2}^n{\left(\frac{x_t}{p_t}-\frac{x_{t-1}}{p_{t-1}}\right)}^2.(2.2)$$

The Denton PFD criterion to be minimized, $f_{\text{F}}^{\text{PFD}}\left(x\right),$ is a sum of squared linear terms, which is easier to deal with than the nonlinear GRP objective function.

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