Abstract

In this talk, we consider multivariate regression with hidden variables, Y = ΘTX + BTZ + E, where Y is a m-dimensional response vector, X is a p-dimensional vector of observable features, Z represents a K-dimensional vector of unobserved hid-den variables, possibly correlated with X, and E is an independent error. The number of hidden variables K is unknown and both m and p are allowed (but not required) to grow with the sample size n. We address several fundamental challenges of this problem, (1) parameter identification, (2) estimation methods/non-asymptotic analysis, (3) asymptotic inference, and (4) generalizations to heteroscedastic errors and GLM. This talk is based on a sequence of works primarily with my students. Related pa-pers include https://arxiv.org/abs/2003.13844, https://arxiv.org/pdf/2201.08003, https://arxiv.org/abs/2509.00196.