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Probabilistic Targeted Factor Analysis

Miguel C. Herculano, Santiago Montoya-Blandón

TL;DR

Probabilistic Targeted Factor Analysis (PTFA) provides a likelihood-based framework for extracting latent factors that are explicitly targeted to economically meaningful outcomes. By modeling $\mathbf{x}=\mathbf{P}\mathbf{f}+\mathbf{e}_x$ and $\mathbf{y}=\mathbf{Q}\mathbf{f}+\mathbf{e}_y$ with $\mathbf{f}\sim\mathcal{N}(\mathbf{0},\mathbf{V}_F)$, PTFA yields a Gaussian posterior $\mathbf{f}|\mathbf{x},\mathbf{y}$ and employs a fast EM algorithm to estimate the loadings and noise variances, enabling uncertainty quantification. The framework naturally extends to incomplete data, mixed-frequency sampling, stochastic volatility, and dynamic factor behavior, and it is demonstrated to improve the recovery and forecasting of targeted factors in simulations and three economic/financial applications: targeted financial conditions indices, macroeconomic forecasting, and equity premium prediction. Simulation results and empirical exercises show PTFA outperforms traditional PLS and PCA approaches, particularly under high noise or missing data, and the method comes with an open-source implementation to facilitate adoption. The probabilistic formulation also opens avenues for Bayesian or variational extensions and richer prior structures while maintaining computational efficiency.

Abstract

We develop Probabilistic Targeted Factor Analysis (PTFA), a likelihood-based framework for constructing latent factors that are explicitly targeted to variables of economic interest. PTFA provides a probabilistic foundation for Partial Least Squares, allowing supervised factor extraction under uncertainty. The model is estimated via a fast expectation maximization algorithm and naturally accommodates missing data, mixed-frequency observations, stochastic volatility, and factor dynamics. Simulation evidence shows that PTFA improves recovery of economically relevant latent factors relative to standard PLS, particularly in noisy environments. Applications to financial conditions indices, macroeconomic forecasting, and equity premium prediction illustrate the measurement and forecasting gains delivered by targeted probabilistic factor extraction.

Probabilistic Targeted Factor Analysis

TL;DR

Probabilistic Targeted Factor Analysis (PTFA) provides a likelihood-based framework for extracting latent factors that are explicitly targeted to economically meaningful outcomes. By modeling and with , PTFA yields a Gaussian posterior and employs a fast EM algorithm to estimate the loadings and noise variances, enabling uncertainty quantification. The framework naturally extends to incomplete data, mixed-frequency sampling, stochastic volatility, and dynamic factor behavior, and it is demonstrated to improve the recovery and forecasting of targeted factors in simulations and three economic/financial applications: targeted financial conditions indices, macroeconomic forecasting, and equity premium prediction. Simulation results and empirical exercises show PTFA outperforms traditional PLS and PCA approaches, particularly under high noise or missing data, and the method comes with an open-source implementation to facilitate adoption. The probabilistic formulation also opens avenues for Bayesian or variational extensions and richer prior structures while maintaining computational efficiency.

Abstract

We develop Probabilistic Targeted Factor Analysis (PTFA), a likelihood-based framework for constructing latent factors that are explicitly targeted to variables of economic interest. PTFA provides a probabilistic foundation for Partial Least Squares, allowing supervised factor extraction under uncertainty. The model is estimated via a fast expectation maximization algorithm and naturally accommodates missing data, mixed-frequency observations, stochastic volatility, and factor dynamics. Simulation evidence shows that PTFA improves recovery of economically relevant latent factors relative to standard PLS, particularly in noisy environments. Applications to financial conditions indices, macroeconomic forecasting, and equity premium prediction illustrate the measurement and forecasting gains delivered by targeted probabilistic factor extraction.

Paper Structure

This paper contains 25 sections, 37 equations, 16 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Methodological Framework
  3. Review: PLS regression
  4. Probabilistic Targeted Factor Analysis
  5. Extensions
  6. Incomplete Data
  7. Missing at Random
  8. Mixed-Frequency Data
  9. Stochastic Volatility
  10. Factor Dynamics
  11. Simulation Exercises
  12. Empirical Applications
  13. Targeted Financial Conditions Indices
  14. Macroeconomic Forecasting
  15. Predicting the Equity Premium
  16. ...and 10 more sections

Figures (16)

  • Figure 1: Comparison of PLS and PTFA on a single realization of simulated data with independent Gaussian errors \ref{['DGP:Simple']}
  • Figure 2: Comparison of PLS and PTFA on a single realization of simulated data with correlated Gaussian errors \ref{['DGP:System']}
  • Figure 3: Comparison of PLS and PTFA on a single realization of simulated data with heavy-tailed non-Gaussian errors \ref{['DGP:NonGaussian']}
  • Figure 4: Comparison of the distributions of average $R^2$ statistics between PLS and PTFA across 1000 replications of \ref{['DGP:Simple']}
  • Figure 5: Comparison of the median $R^2$ statistics for PLS (a) and PTFA (b) across 1000 replications from \ref{['DGP:Simple']}, varying noise in features ($\sigma_x$) and targets ($\sigma_y$)
  • ...and 11 more figures