Identification of Multivariate Measurement Error Models

Multivariate continuous latent variables arise in numerous empirical settings in economics, psychology, marketing, epidemiology, and the social sciences. Examples include multidimensional skills, cognitive factors, latent preferences, health indices, productivity components, and risk attitudes. In practice, such latent constructs are rarely observed directly; instead, researchers rely on multiple imperfect measurements that contain potentially correlated forms of noise. The resulting measurement-error problem is especially severe when the latent variable is multidimensional: each measurement typically captures only a low-dimensional projection of the latent vector, and the noise may exhibit dependence or heterogeneity across measurement channels. As a result, classical approaches to continuous measurement error, which often rely on injectivity, offer limited guidance. This paper develops a identification strategy tailored specifically to multivariate continuous latent variables measured with noise. The key innovation is to combine tools from multi-linear algebra--specifically the uniqueness properties of so-called CP tensor decompositions-- with the multivariate extension of Kotlarski's identity, a powerful deconvolution result based on characteristic functions. The starting point is the observation that third-order cross-moments of three separate measurements form a three-way moment tensor whose CP decomposition is governed by the latent factor loading matrices. By invoking Kruskal's theorem (Kruskal, 1977a), I show that these loadings are generically identifiable even when each measurement matrix is rank-deficient and--even more surprisingly--when the stack of all measurement matrices is non-injective.