Benchmarking and Movement Preservation: Evidences from Real-Life and Simulated Series

Abstract

The benchmarking problem arises when time series data for the same target variable are measured at different frequencies with different level of accuracy, and there is the need to remove discrepancies between annual benchmarks and corresponding sums of the sub-annual values. Two widely used benchmarking procedures are the modified Denton Proportionate First Differences (PFD) and the Causey and Trager Growth Rates Preservation (GRP) techniques. In the literature it is often claimed that the PFD procedure produces results very close to those obtained through the GRP procedure. In this chapter we study the conditions under which this result holds, by looking at an artificial and a real-life economic series, and by means of a simulation exercise.

The views expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board, or its management

Appendix: Non-singularity of the Coefficient Matrix of System (45.2)

Luenberger [9, p. 424] shows that a unique solution to the problem

$$\displaystyle{\min _{\mathbf{y}}\mathbf{y}\prime\mathbf{Qy}\qquad \mbox{ subject to}\quad \mathbf{Cy} = \mathbf{Y}}$$

exists if the matrix C is of full rank, and the matrix Q is positive definite on the null space of matrix C: $\mathcal{N}(\mathbf{C}) =\{ \mathbf{x} \in {\mathbb{R}}^{n} : \mathbf{Cx} = \mathbf{0}\}$.

Let us consider the matrix $\mathbf{Q} ={ \mathbf{P}}^{-1}\boldsymbol{\Delta }^{\prime}_{n}\boldsymbol{\Delta }_{n}{\mathbf{P}}^{-1}$, and let the vector y belong to $\mathcal{N}$(C). We assume p _t ≠0, t = 1, …, n (otherwise the objective function is not defined), and Cp≠Y, which corresponds to exclude the trivial solution y ^∗ = p, valid when there is no benchmarking problem. Given that

$$\displaystyle{\mathbf{y}\prime\mathbf{Qy} =\sum _{ t=2}^{n}{\left (\frac{y_{t}} {p_{t}} -\frac{y_{t-1}} {p_{t-1}}\right )}^{2},}$$

it is immediately recognized that the expression above is strictly positive, which means that matrix $\mathbf{Q} ={ \mathbf{P}}^{-1}\boldsymbol{\Delta }^{\prime}_{n}\boldsymbol{\Delta }_{n}{\mathbf{P}}^{-1}$ is positive definite on the null space spanned by the columns of matrix C, and thus the coefficient matrix of system (45.2) is nonsingular.