Extensive research highlights the importance of modeling CSD through multiplicative interactive effects (IE). Two leading methodologies—the principal component (PC) estimation method by Bai (2009) and the common correlated effects (CCE) estimator by Pesaran (2006)—have been widely adopted and expanded upon. Recent empirical needs have driven the development of econometric techniques suitable for multi-dimensional panel datasets, which arise in contexts like trade flows, employer employee matches, and social networks. Building on the LM procedure introduced by Kapetanios et al 2023 for two-dimensional panels, this study extends the test to a three-dimensional (3D) framework where units i and j  are observed across time t. We propose a 3D version of the LM test to validate the conditional independence of factor loadings and regressors. Additionally, we introduce unit-specific LM tests to identify subsets of data where loadings remain conditionally independent. This allows for valid estimation without fully accounting for CSD, offering a significant advantage over two-dimensional approaches by isolating and excluding correlated units. This innovation underscores the potential of multi-dimensional datasets in overcoming the complexities of CSD in econometric analysis.

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