We present a multidimensional generalization of the GRAS method (nD-GRAS) for the estimation of multiple matrices in an integrated framework. The potential applications of this method in regional and multi-regional input–output analyses based on national/regional accounts frameworks are many. We provide two real applications, a 3D-GRAS that estimates a use table at basic prices jointly with valuation matrices for Denmark; and a 4D-GRAS for estimating intercountry input–output tables with OECD data. We show that higher dimensional GRAS methods provide more consistent and accurate estimates than those with lower number of dimensions. We provide the analytical closed-form solution and the RAS-like algorithm for an easy operationalization.

中文翻译：

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多维 nD-GRAS 方法：多区域投入产出框架和估值矩阵投影的应用

我们提出了 GRAS 方法 (nD-GRAS) 的多维推广，用于在集成框架中估计多个矩阵。该方法在基于国家/区域账户框架的区域和多区域投入产出分析中的潜在应用很多。我们提供了两个实际应用程序，一个 3D-GRAS，它以基本价格与丹麦的估值矩阵一起估计使用表；以及 4D-GRAS，用于使用经合组织数据估算国家间投入产出表。我们表明，与具有较少维数的 GRAS 方法相比，高维 GRAS 方法提供了更一致和准确的估计。我们提供分析封闭形式的解决方案和类似 RAS 的算法，以便于操作。