Growth Rates Preservation (GRP) temporal benchmarking: Drawbacks and alternative solutions
Section 4. Alternative benchmarking techniques

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In Section 3 we identified two problems with GRP methods. In this section we consider two alternative benchmarking techniques that solve the time irreversibility property.

4.1  Simultaneous growth rate preservation

Here, we propose two alternative objective functions for GRP. The first is a “time symmetric” variant of GRP, defined by

$$f_{\text{S}}^{\text{GRP}}\left(x\right)=\frac{1}{2}{\displaystyle \sum }_{t=2}^n{\left(\frac{x_t}{x_{t-1}}-\frac{p_t}{p_{t-1}}\right)}^2+\frac{1}{2}{\displaystyle \sum }_{t=2}^n{\left(\frac{x_{t-1}}{x_t}-\frac{p_{t-1}}{p_t}\right)}^2,(4.1)$$

where subscript “S” stands for “simultaneous”. The method will be called GRPS in the remainder of this paper. The GRPS objective function both preserves forward and backward growth rates. As far as the authors know this method has not been mentioned elsewhere in the literature. It can be easily seen that GRPS satisfies time reversibility: interchanging $t$ and $t-1$ does not alter the objective function.

However, the second problem in Section 3 (singularity of objective function) is not considered. One of the consequences, negative benchmarked values, can be avoided by imposing lower bounds of zero on the benchmarked values. This can be done by including inequality constraints to an optimization problem, which is a well-established technique (e.g., Nocedal and Wright, 2006). The other problems related with singularity can however still occur.

4.2  Logarithmic growth rate preservation

Another “time symmetric” variant of GRP is given by the logarithmic form:

$$f_{\text{L}}^{\mathrm{GRP}}\left(x\right)={\displaystyle \sum }_{t=2}^n{\left[\text{log}\left(\frac{x_t}{x_{t-1} }\right)-\text{log}\left(\frac{p_t}{p_{t-1}}\right)\right]}^2.(4.2)$$

This function was firstly considered by Helfand, Monsour and Trager (1977). It is immediately verified that function (4.2) satisfies the time reversal property as well. The objective function can be considered the logarithmic version of GRP and equally well as the logarithmic version of Denton PFD. It will be denoted GRPL in the remainder of this paper, where “L” stands for “logarithmic”.

Note that (4.2) can be used for strictly positive preliminary values only, and that it produces benchmarked values that are larger than zero as well. This does not seem an important limitation, as Section 3 already mentioned that growth rate preservation can be considered inappropriate for problems with positive and negative values. Nevertheless, a potential solution for time series with negative values is to add a sufficiently large constant to the series prior to benchmarking and subtract that constant from the benchmarked series. A potential drawback of this solution is that adding a constant distorts initial growth-rates. Thus, it is unclear whether preliminary growth rates are actually preserved. Further research is necessary to better understand the implications of this solution.

Although GRPL necessarily produces positive values, other problems in Section 3.2, related to a singular objective function can still occur.

4.3  Comparison

When comparing GRPS and GRPL, it can be expected that GRPL behaves more like Denton PFD. Below we will give two reasons for this.

Firstly, because of the asymptotic properties of the log function, the problem that close-to-zero values are avoided is less severe for GRPL than for GRPS. Close-to-zero values are associated with large adjustments of growth rates. Very large adjustments of growth rates are penalised less in GRPL than in GRPS, since GRPS’s objective function grows faster when corrections are large.

Secondly, the first-order Taylor linearization of GRPL’s objective function corresponds to Denton PFD’s function, whereas the approximation of GRPS leads to a different result. The linearization of the squared terms of the objective function in the preliminary values are given by $\left(\frac{x_t}{p_t }\right)-\left(\frac{x_{t-1}}{p_{t-1}}\right)$ and $\left(\frac{p_t}{P_{t-1}}+\frac{p_{t-1}}{p_t}\right)\left\{\left(\frac{x_t}{p_t }\right)-\left(\frac{x_{t-1}}{p_{t-1}}\right)\right\}$ for GRPL and GRPS respectively.

4.4  Example

In order to explore the properties of GRPL and GRPS, we will consider the example of Subsection 3.3 again. Figure 4.1 compares results of the symmetric $f_{\text{S}}^{\text{GRP}}\text{},$ $f_{\text{L}}^{\text{GRP}}$ and $f^{\text{PFD}}\left(x\right)$ methods.

Firstly, it can be seen that the peaks and troughs occur at the same periods for all time symmetric methods.

Secondly, some of the drawbacks related to the singularity of the objective function still occur. When compared to Denton PFD, GRP methods tend to avoid close-to zero values, move away relatively slowly from low values (in both directions) and lead to irregular large peaks.

Thirdly, in accordance to Subsection 3.3, GRPL resembles Denton PFD more than GRPS, which follows from the slightly lower peaks of GRPL.

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