On the Euro method

Abstract

This paper critically examines the Euro method usage for the purposes of updating supply and use tables (SUTs) and/or input–output tables. Its known restricted applicability to only unnecessarily aggregated and symmetric SUTs (and not their underlying rectangular versions) is already an issue of concern. However, by studying analytically the nature of Euro's adjustments of the SUT elements and empirically assessing some of its underlying assumptions, including newly revealed ones, it is concluded that the Euro method is a largely ad hoc updating procedure. Its recently claimed superiority over the generalized RAS approach (GRAS, or SUT-RAS) in the absence of industry output is challenged. It is shown that applying the standard GRAS with exogenously given estimates of industry outputs under such restricted data-availability environment still outperforms the Euro method.

Keywords:

• Updating supply and use tables
• non-survey techniques
• Euro method
• GRAS method

JEL Classifications:

• C02
• C61
• C80
• D57

Acknowledgments

The author is grateful to Ronald Miller, Jan Oosterhaven, Iñaki Arto, Michael Lahr, Manfred Lenzen and three anonymous referees for useful comments and suggestions, but is solely responsible for the views expressed in this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Since SUTs are more fundamental data than IOTs (the latter are obtained from SUTs under different transformation assumptions), this study focuses on the SUT-Euro method only, which is simply referred to as the Euro method. Nevertheless, all the results of this paper are also valid with respect to the traditional Euro method. It should be noted that the first complete mathematical presentation of the SUT-Euro method as proposed and demonstrated in Beutel (2008) appeared in Temurshoev et al. (2011).

2 See Lahr and De Mesnard (2004) for details of the RAS-type methods, including its history. The most flexible RAS-type framework is the so-called KRAS (K for Konfliktfreies) of Lenzen et al. (2009), which generalizes the GRAS method to: (i) incorporate constraints on arbitrary subsets of matrix elements, including cases of constraints' coefficients being different from 1 or −1, (ii) include reliability of the initial estimate and the external constraints, and (iii) find a compromise solution between conflicting constraints.

3 Matrices are in bold, capitals; vectors in bold, lower-case; and scalars in italicized, lower-case letters. Vectors are columns by definition, row vectors are obtained by transposition, indicated by a prime. $\widehat{\mathbf{\text{x}}}$ denotes a diagonal matrix with the entries of $\mathbf{\text{x}}$ vector on its main diagonal and zeros elsewhere. The symbol ⊗ indicates Hadamard product or the element-wise multiplication of matrices. A vector of ones of appropriate dimension is denoted by $\text{ı}$. Finally, subscript zero indicates that the variable in question refers to the base (or benchmark) year, whose value is thus known/observed.

4 This approach is analytically easier to comprehend as compared to the usual (largely) verbal description of the method, including the use of illustrations of hypothetically worked examples and flowcharts.

5 One can ‘generalize’ the framework by incorporating TLS matrices ${\mathbf{\text{N}}}_u$ and ${\mathbf{\text{N}}}_y$ instead of, respectively, their column totals of ${\mathbf{\text{n}}}_u^{\prime }$ and ${\mathbf{\text{n}}}_y^{\prime }$ within the Use table. In such a setting, there is a (much) higher chance of encountering cases when the signs of product-level TLS totals, $[{\mathbf{\text{N}}}_u,{\mathbf{\text{N}}}_y]\text{ı}$, switch across periods.

6 The expression in Proposition 2.1 also reveals the following relation between the preliminary and final estimate of gross output at each iteration: ${\mathbf{\text{x}}}_t=({\widehat{\mathbf{\text{r}}}}_t^v{\widehat{\mathbf{\text{v}}}}_0{)}^{-1}{\widehat{\mathbf{\text{v}}}}_t{\mathbf{\text{x}}}_t^{\ast }$.

7 Sometimes when a reference is made to the ultimate (i.e. after convergence) multipliers and variables, their iteration identifier t is dropped for convenience.

8 A point of clarification with respect to the use of the geometric mean option is in order here. For negative entries of SUTs (such as changes in inventories or net taxes on products), using the standard geometric mean formula is insufficient, if one opts to use the known algorithm of the Euro method, (1.1)–(1.15). Then to keep the negative sign, the latter has to be added manually. Thus, in practice equation (1.4) with geometric mean option needs to be implemented as: ${\mathbf{\text{Y}}}_t=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\mathbf{\text{Y}}}_0\right)\otimes \sqrt{\left({\widehat{\mathbf{\text{r}}}}_t{\mathbf{\text{Y}}}_0\right)\otimes \left({\mathbf{\text{Y}}}_0{\widehat{\mathbf{\text{r}}}}_t^y\right)}$. However, such sign adjustment is not necessary with the alternative formulation of the Euro method presented in Proposition 2.2, provided that all the multipliers are positive.

9 In this respect, the generality and flexibility of the Harthoorn and van Dalen (1987, HvD) method can be seen from the expression of its optimal cell-specific factor as $a_{ij}=1+g_{ij}({\lambda }_i+{\tau }_j)/u_{0,ij}$. Here, $a_{ij}$ minimizes the objective function $\sum _i\sum _j(f_{ij}{u}_{0,ij}-u_{0,ij}{)}^2/g_{ij}$ subject to the row and column sums constraints of the new Use table, ${\lambda }_i$ and ${\tau }_j$ are, respectively, the Lagrange multipliers of these constraints, and $g_{ij}$ is the exogenously specified relative confidence of the benchmark element $u_{0,ij}$. Different specifications of $g_{ij}$ are possible, such as $g_{ij}=1$ for all i and j, $g_{ij}=|u_{0,ij}|$ and $g_{ij}=(u_{0,ij}{)}^2$. Note that multipliers ${\lambda }_i$ and ${\tau }_j$ are also a function of $g_{ij}$. From these three specifications (one might try other justifiable choices), the empirical evaluations of updating SUTs by Temurshoev et al. (2011) showed that the choice $g_{ij}=|u_{0,ij}|$ resulted in the best-performing HvD variant.

10 From their empirical assessments, Valderas-Jaramillo et al. (2019) conclude that ‘the geometric version of the SUT-EURO method usually performs better than its arithmetic version’ (p. 442).

11 This applies to the intermediate Use table column multipliers, but not imports row multipliers which by construction are proportional to the OMF-to-benchmark GVA ratio, (3(3) ${\mathbf{\text{r}}}_t^m={\mu }_t{\widehat{\mathbf{\text{v}}}}_0^{-1}\mathbf{\text{v}}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{\mu }_t=c_t^m{c}_{t-1}^m\cdots c_2^m>0.$(3) ). For example, the iteration t imported intermediate Use matrix in (6.5(6.5) $\begin{array}{rl}{\mathbf{U}}^m& =\sqrt{\mu }\sqrt{{\hat{\mathbf{v}}}_0^{-1}\hat{\mathbf{v}}}{\mathbf{U}}_0^m{\hat{\mathbf{v}}}_0^{-1}\hat{\mathbf{v}}(\sqrt{{\hat{\mathbf{r}}}^v}{)}^{-1}={\hat{\mathbf{r}}}_m{\mathbf{U}}_0^m{\hat{\mathbf{s}}}_u\end{array}$(6.5) ) with explicit t can be written as ${\mathbf{\text{U}}}_t^m=\sqrt{{\mu }_t}\sqrt{{\widehat{\mathbf{\text{v}}}}_0^{-1}\widehat{\mathbf{\text{v}}}}{\mathbf{\text{U}}}_0^m{\widehat{\mathbf{\text{v}}}}_0^{-1}{\widehat{\mathbf{\text{v}}}}_t(\sqrt{{\widehat{\mathbf{\text{r}}}}_t^v}{)}^{-1}$.

12 These last four expressions, but without square root, would also capture – within the corresponding arithmetic mean terms – the effects of substitution and price indexes under the Euro-A approach. E.g. the term $0.5(1+r_i^v/r_j^v)$ in (5.4(5.4) $\begin{array}{rl}{\mathbf{U}}^d& ={\mathbf{A}}_U^d\otimes {\mathbf{U}}_0^d\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}({\mathbf{A}}_U^d{)}_{ij}=\frac{1}{2}(1+\frac{r_i^v}{r_j^v})\frac{v_j}{v_{0,j}}\end{array}$(5.4) ) may be interpreted as the relative price index relevant to sector j's purchases of domestic intermediates per unit of its output. The two terms in this average include the base-year unitary relative price of products i to j and the corresponding relative price for the forecast year $r_i^v/r_j^v$.

13 The Euro-A counterpart of this expression is $u_{ij}^d=0.5(r_i^v+r_j^v)u_{0,ij}^d{r}_j^x$ as follows from (5.4(5.4) $\begin{array}{rl}{\mathbf{U}}^d& ={\mathbf{A}}_U^d\otimes {\mathbf{U}}_0^d\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}({\mathbf{A}}_U^d{)}_{ij}=\frac{1}{2}(1+\frac{r_i^v}{r_j^v})\frac{v_j}{v_{0,j}}\end{array}$(5.4) ).

14 Since this general biproportion holds true for updating domestic final demand matrix in (6.1(6.1) $\begin{array}{rl}{\mathbf{Y}}^d& =\sqrt{{\hat{\mathbf{r}}}^v}{\mathbf{Y}}_0^d\sqrt{{\hat{\mathbf{r}}}^y}={\hat{\mathbf{r}}}_v{\mathbf{Y}}_0^d{\hat{\mathbf{s}}}_y\end{array}$(6.1) ), then one can expect that the Euro-G estimate of ${\mathbf{\text{Y}}}^d$ (especially, for categories with non-negative entries) will have lower error compared to the estimates of other components of the Use table. This is indeed what empirical results show.

15 As follows from (5.5(5.5) $\begin{array}{rl}{\mathbf{U}}^m& ={\mathbf{A}}_U^m\otimes {\mathbf{U}}_0^m\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}({\mathbf{A}}_U^m{)}_{ij}=\frac{1}{2}(1+\frac{\mu v_i}{v_{0,i}{r}_j^v})\frac{v_j}{v_{0,j}}\end{array}$(5.5) ), the Euro-A's counterpart of (8.2(8.2) $\begin{array}{rl}{u}_{ii}^m& =\sqrt{\mu }{\left(\frac{v_i}{v_{0,i}}\right)}^{\frac{3}{2}}{\left(r_i^v\right)}^{-\frac{1}{2}}{u}_{0,ii}^m.\end{array}$(8.2) ) takes the form : $u_{ii}^m=\frac{1}{2}[v_i/v_{0,i}+(\mu /r_i^v\left)\right(v_i/v_{0,i}{)}^2]u_{0,ii}^m$. Recall that the same multipliers (e.g. μ or $r_i^v$) differ in value across the Euro-A and Euro-G variants.

16 The last expression in quotes is borrowed from Kop Jansen (1994, p. 64) when the author discusses how Thijs ten Raa disliked the ‘rather odd-looking coefficients’ of the West's (1986) formulas of multiplier bias and variance, suggesting ‘more natural formulas’.

17 See De Mesnard (2011) on economic implications of negatives in the inverse of ${\mathbf{\text{D}}}_0$.

18 The practical advantage of the SUT-RAS method as presented in Temurshoev and Timmer (2011) is or might be that the main components of SUTs are used individually and there is no need to put all of them within one integrated framework such as ${\mathbf{\text{X}}}_0$ and its row and column sums as in (13(13) ${\mathbf{\text{X}}}_0=\left[\begin{array}{cccc}-{\mathbf{\text{S}}}_0^d& {\mathbf{0}}_p& {\mathbf{\text{U}}}_0^d& {\mathbf{\text{Y}}}_0^d\\ {\mathbf{O}}_{p\times s}& -{\mathbf{\text{m}}}_0& {\mathbf{\text{U}}}_0^m& {\mathbf{\text{Y}}}_0^m\\ {\mathbf{0}}_s^{\prime }& 0& {\mathbf{\text{n}}}_{u0}^{\prime }& {\mathbf{\text{n}}}_{y0}^{\prime }\end{array}\right],{\mathbf{\text{u}}}_{\mathrm{G}\mathrm{R}\mathrm{A}\mathrm{S}}=\left[\begin{array}{c}{\mathbf{0}}_{2p}\\ n\end{array}\right]\mathrm{a}\mathrm{n}\mathrm{d}{\mathbf{\text{v}}}_{\mathrm{G}\mathrm{R}\mathrm{A}\mathrm{S}}=\left[\begin{array}{c}-\mathbf{\text{x}}\\ -m\\ \mathbf{\text{x}}-\mathbf{\text{v}}\\ \mathbf{\text{y}}\end{array}\right].$(13) ), including changing the signs of certain components of SUTs.

19 We use only WAPE because: (a) WAPE is often equivalent (in terms of rankings of matrix goodness-of-fit statistics) to the information-based psi statistic $\hat{\psi }$ widely used in geographical literature and to the simpler standardized absolute error (SAE) indicator (see Voas Williamson, 2001; Temurshoev et al., 2011), (b) only one statistic is sufficient for the purposes of this paper, and (c) space consideration.

20 This is especially the case for the 2010–2015 projection where most components of SUTs are estimated better under Euro-G than GRAS-1. The Euro method, however, is often used for updating missing SUTs in-between two available benchmark tables, for which purposes GRAS-1 seems to be largely superior as follows from the results in Table 4. When there are two benchmark SUTs available and there is a missing SUT in-between, there are other simple options of obtaining the estimate of $\mathbf{\text{x}}$ in its absence that use the relevant benchmark information. Assume one needs to project SUTs for k years between the start and ending periods of available SUTs. Then for year $t\in (0,k+1)$ with missing SUTs take ${\widetilde{\mathbf{\text{x}}}}_t=\left(\frac{k+1-t}{k+1}{\widehat{\mathbf{\text{x}}}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}{\widehat{\mathbf{\text{v}}}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}^{-1}+\frac{t}{k+1}{\widehat{\mathbf{\text{x}}}}_{\mathrm{e}\mathrm{n}\mathrm{d}}{\widehat{\mathbf{\text{v}}}}_{\mathrm{e}\mathrm{n}\mathrm{d}}^{-1}\right){\mathbf{\text{v}}}_t\mathrm{o}\mathrm{r}{\widetilde{\mathbf{\text{x}}}}_t=\frac{k+1-t}{k+1}{\mathbf{\text{x}}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}+\frac{t}{k+1}{\mathbf{\text{x}}}_{\mathrm{e}\mathrm{n}\mathrm{d}},$ where subscript start and end refer to the two available benchmark SUTs, considered as starting and ending period of SUTs interpolation. Such a time-weighted estimate roughly accounts for the trend of, respectively, industry output-to-GVA ratio and industry output movements under a ‘normal development’ scenario assumption. In fact, one could apply the same time-weighting procedure to the GRAS benchmark matrix ${\mathbf{\text{X}}}_0$ as well in order to, at least partially, account for possible structural change that affects the structure and denseness of SUT components. (The idea of time-weighting for SUT update purposes goes back at least to Temurshoev et al. (2011, see footnote 8, p. 94)).

21 All these advantages are also mentioned in Beutel (2002).