Latent Variable Modelling by Supervised Diffusion

Abstract

This paper proposes a new methodological framework for estimating inferential models with latent variables. It also introduces a new latent variable regression model called LARX: an extension of the ubiquitous autoregressive model with exogenous inputs (ARX) in which any or all input variables can be latent. In deriving the LARX model, a minor contribution is also made to the field of matrix calculus: A new matrix operator is defined and applied to solve a class of Lagrangian optimisation problems with interactions between multiple coefficient vectors subject to case-by-case constraints. In the empirical section, the LARX model is used to re-examine the relationship between stock market performance and real economic activity in the United States. The LARX model attains an out-of-sample R-squared of up to 79.7% compared to 50.3% for the baseline OLS approach. It also reveals new information about the underlying drivers of the relationship between stock returns and economic growth, including the predictive power of sector rotations.