The multidimensional nD?GRAS method: Applications for the projection of multiregional input–output frameworks and valuation matrices

We present a multidimensional generalization of the GRAS method (nD-GRAS) for estimation multiple matrices in an integrated framework. The potential applications this regional and multi-regional input–output analyses based on national/regional accounts frameworks are many. provide two real applications, 3D-GRAS that estimates use table at basic prices jointly with valuation Denmark; 4D-GRAS estimating intercountry tables OECD data. show higher dimensional methods more consistent accurate than those lower number dimensions. analytical closed-form solution RAS-like algorithm easy operationalization. En este artículo se presenta una generalización del método para la estimación de múltiples en un marco integrado. Las aplicaciones potenciales los análisis input-output regionales y multirregionales basados marcos cuentas nacionales o son numerosas. Se incluyen dos reales, que estima tabla uso precios básicos conjuntamente con valoración Dinamarca; estimar tablas entre países datos OCDE. demuestra métodos mayores dimensiones proporcionan estimaciones más consistentes precisas aquellos menor número dimensiones. Para fácil operacionalización, proporciona solución analítica forma cerrada el algoritmo tipo RAS. ??????????????????????????????GRAS?(n D-GRAS)??????????????????????????????????????????????????????????????????????????????????????????????????????3 D-GRAS???OECD?????????????????????????4 D-GRAS??????????????????????GRAS???????GRAS????????????????????????????????????????????????????RAS????????????? Multiple variations biproportional techniques have been applied to field analysis since Leontief's (1941) pioneering work, which he used technique identify sources inter-temporal change cells series (Lahr & Mesnard, 2004). This also includes –broadly speaking– RAS-family (Bacharach, 1965, 1970) their extensions. A recent summary provides good overview large compilation these can be found Chapter 18 UN Handbook supply, extensions (United Nations, 2018). idea stems from fact often practical situations rather simply imposing constraints summing all elements matrix row-wise or column-wise (as standard GRAS), it is necessary rearrange representing multiregional information into arrays larger dimension dimensions array. For instance, frameworks, where national IOTs split using bilateral exports imports, may corresponding imports might serve as balancing IOT. Another situation requires matrices, is, trade transport margins (TTM), taxes less subsidies products (TLS) make them purchasers´ prices. estimate each independently method, but result TTM, TLS would equal initial only by chance. main contribution paper derivation framework arbitrary handle problems accessible way. bi-dimensional case (2D-GRAS according our terminology). As will described next section, addressed nD-GRAS embedded solved within KRAS framework; nonetheless, approach easily made operational algorithm, among other differences. Finally yet importantly, another projections dimensions, apart global coherence estimations, lead better performance greater accuracy independent general, 3D slices resulting yield fit results we could achieve 4D see statement holds 2D projections. organized follows. section frames theoretical background work latest related literature. Section 3 introduces how several important approach. 4 contains theory about nD-GRAS, set-up optimization algorithm. 5 explores some conditions regarding feasibility issues convergence. 6 examples methods. Finally, concluding findings contributions. RAS fall under category what known fields iterative proportional fitting procedures (IPFP). In context, Deming Stephan (1940) similar contingency tables.2 find early literature Sheleikhovskii (Bregman, 1967) Kruithof Generalization three-dimensional was done (1943),3 Darroch (1962) Ireland Kullback (1968). techniques, implementation references documentation R-package “mipfp” developed Barthelemy et al. (2018). concerning Tilanus (1976) first introduced algorithmic way, generalizing four Oosterhaven (1986) interregional system must add up figure. followed (MR-GRAS) Temursho, Oosterhaven, Cardenete (2020). Both constitute additional constraint sort third dimension. MR-GRAS, set (different typical sum constraints) same objective function method. MR-GRAS linear restrictions operates across any non-overlapping subsets IOT given total. (2020) show, adapted updating inter-national/regional SUTs, blocks has extensively creation baseline scenario called PIRAMID4 2018 Global Energy Climate Outlook (Rey Los Santos al., 2018; Cardenete, 2020) projection context future years, ensuring consistency National Accounts. principle, approaches not because authors over hence, taken account such. Nonetheless, introducing restrictions, produce solution. terms Cardenete's approach, possible formulate array matrix, thus making correspondent. However, rightly mention, accounting sets included Indeed, general so describe geometrical intuitive makes efficient straightforward, regardless without requiring aggregation matrices. Eurostat's FIGARO5 project (Remond-Tiedrez Rueda-Cantuche, 2019) being profusely construction European inter-country (SUIOTs). Other Gilchrist St. Louis (1999, 2004) Lenzen (2006). TRAS papers sums. “cRAS” proposed (2006) generalizes constraints. papers, (1976), resolution presented way obtaining estimated meets desired mathematical proof. purchasers inputs Dalgaard Gysting (2004). opinion, does algorithms, there constraint, such cell-wise purchasers' table. Instead, GDP output side demand respecting outputs provided supply target derived individually adjustment balancing, them. continues until balance achieved. Holý Šafr (DRAS) reformulates problem purely deals non-negative arrays. (2020), (1976). prove cross-entropy model generalization. DRAS particular when no negative exist matrix. They apply regional, quarterly domestic/imported (industry-by-industry) Czech Republic. Their addition dimension, totals different layers (either disaggregation, disaggregation split) allows overall structure. application Isard's shows decisive development continuation (2009). vectorization formulation unidimensional set-up. embedded, arbitrarily sized shaped either unity non-unity coefficients, generalized formulation. incorporates features reliability supplied autonomous management conflicts external data inconsistent core deal problems. Geschke (2011) tool custom processing language (AISHA) described. performs tables. describes representation vectorized one-to-one correspondence vector. methodology implemented (2013) EORA MRIO database, eight-tiered hierarchy used. actually imply eight-dimension 8D-GRAS work. That even though 8D-array, do eight together represent aggregations identical Additional include possibility functions function. Among them, quadratic programming barrier penalty (Geschke 2011; 2012). addition, solvers prepared large-scale sparse including parallelization Cimmino algorithms 2019; Lenzen, Geschke, 2014) available. Besides, dealing tailored sectoral defined subset fixed classification (Lenzen, 2014). methodological proposal, deriving paper, goes one step further virtual laboratories (2012, 2017), (2014) Hadjikakou (2017). laboratories, researchers assemble own versions collaborative research environment cloud-computing platforms enabling multitude carbon, water, ecological footprints, life-cycle assessments trend key driver analyses. Undoubtedly, framework, long feasible exists, function, coefficients conflicting information. applies, rest mentioned Even advantages extended labs proposals obvious, aspects account. proposal advantage having solution, cost coefficients. Finding involves providing non-conflicting exogenous well-posed prior associated enables variety IT hardware. On hand, handled process, clearing information, size involved regions, entry users. usually leads very especially demanding computing performance. operationalization methods, automation “data mining, processing, reclassification much possible” (Lenzen 2012, p. 8376). Optimization address kind covered nD-GRAS. Jackson Murray (2004) article excellent review bridging between algorithms. proliferation RAS-based already article, cause impression sacrifice simplicity capability (KRAS probably example this). Although seem complex, existence its with. efficiently widely available free software programs like R. Alternatively, complex numerical require commercial need high-performance software. true setting try weighting patterns, penalties, functions. infeasibilities, process approaches, reason non-convergence easier trace compared underlying routines (see 2020). It inspect qualitative relationship targets 5) checking paths factors process. Before illustrate Figure 1 frame methodology. Going simpler start examples. fertile ground applications. Let us assume row (products industries) arranged countries/regions applies column-wise. if think regions. (i.e. domestic part voided), rows denote destination partner/user, columns, country origin. schematically represents, international (in diagonal block—shaded light grey—set zero). If concentrate attention on, say, trading partner (region) (shaded 1), description region k's imported element , product user coming (the block zero implies every ). column represented array, portrayed upper right-hand 1. aggregate origin (i.e., ) import dark grey depict graphically . Many representation. users set-up). added (e.g., implying total), then becomes balancing. vectors known, adding one. solve straightforwardly consisting 2, prices, margin target. territorial problem, therefore, industries.6 consider time situation, aggregated annual table, temporal quarterly). breakdown total figures Šafr, When performing interpret three optimizations simultaneously: layer accounted independently; contrary, constraint—cell-wise plus values—ensures connecting single simultaneously. illustrated 6. always setup 1, whole multicountry constitutes table). increase, define aggregator meet inclusion new multipliers cell once specific cannot again others). intuition feature paper. First, note that, mentioned, expressed hide sub-indexes. Hence, should sub-index represents difference essentially Conditioning slice full perform selecting time). At end, necessarily exactly takes place Recalling previous marginal 2D-GRAS independently. adds and, see, positive impact quality estimations. knowledge values, although assumed they were values missing, skips natural implement columns and/or unknown. executed leaving vector free. -dimension technique. sake simplicity, understand point view, initially expose tridimensional (3D-GRAS), generalize setup, consists elements: (prior constraints), usual, necessary, ; take along, bidimensional slice, matrix) required. sequentially, initiating iteration repeats steps iteration. flow diagram 3, number, concept biproportion De 1994) proportion, figures. 4, (assumed example) updated factor, multiplication convergence depicted procedure till tolerance R, operationalize. R scripts upon request. sequence step. guaranteed well exists. such, (1) strictly convex convex. All functions, hence Lagrangian (2) (strictly) converge characteristics constraints, Chiang (1984), unique.9 One solutions simple controlling fixing infeasibilities. result, highlight. These dimensions; otherwise, infeasible. Bacharach (1970) proved10 condition remains valid generalization, otherwise. Second, null issue zeros reduce degrees freedom zeroes happen sub-arrays, identified (1970)11 sub-array. (1970)12 n