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Time-varying Factor Augmented Vector Autoregression with Grouped Sparse Autoencoder

Yiyong Luo, Brooks Paige, Jim Griffin

TL;DR

The paper tackles nonlinear and time-varying dynamics in macro FAVAR by proposing a Grouped Sparse autoencoder (GS-AE) with a Spike-and-Slab Lasso prior to achieve semi-identifiability and interpretability across economic groups, coupled with a time-varying parameter VAR (TVP-VAR) to capture evolving dynamics. The GS-AE uses group-specific sparsity to identify factors up to element-wise transformations, enabling clearer economic interpretation, while the TVP-VAR allows factor relationships to drift over time. Empirical results on 168 quarterly US macro series show improved interpretability and forecasting performance when combining GS-AE with TVP-VAR, and impulse response analyses reveal time-varying monetary policy effects with higher uncertainty during recessions. The framework offers a parsimonious, interpretable, and flexible approach to nonlinear and nonstationary macro modeling with practical forecasting and policy-analytic benefits.

Abstract

Recent economic events, including the global financial crisis and COVID-19 pandemic, have exposed limitations in linear Factor Augmented Vector Autoregressive (FAVAR) models for forecasting and structural analysis. Nonlinear dimension techniques, particularly autoencoders, have emerged as promising alternatives in a FAVAR framework, but challenges remain in identifiability, interpretability, and integration with traditional nonlinear time series methods. We address these challenges through two contributions. First, we introduce a Grouped Sparse autoencoder that employs the Spike-and-Slab Lasso prior, with parameters under this prior being shared across variables of the same economic category, thereby achieving semi-identifiability and enhancing model interpretability. Second, we incorporate time-varying parameters into the VAR component to better capture evolving economic dynamics. Our empirical application to the US economy demonstrates that the Grouped Sparse autoencoder produces more interpretable factors through its parsimonious structure; and its combination with time-varying parameter VAR shows superior performance in both point and density forecasting. Impulse response analysis reveals that monetary policy shocks during recessions generate more moderate responses with higher uncertainty compared to expansionary periods.

Time-varying Factor Augmented Vector Autoregression with Grouped Sparse Autoencoder

TL;DR

The paper tackles nonlinear and time-varying dynamics in macro FAVAR by proposing a Grouped Sparse autoencoder (GS-AE) with a Spike-and-Slab Lasso prior to achieve semi-identifiability and interpretability across economic groups, coupled with a time-varying parameter VAR (TVP-VAR) to capture evolving dynamics. The GS-AE uses group-specific sparsity to identify factors up to element-wise transformations, enabling clearer economic interpretation, while the TVP-VAR allows factor relationships to drift over time. Empirical results on 168 quarterly US macro series show improved interpretability and forecasting performance when combining GS-AE with TVP-VAR, and impulse response analyses reveal time-varying monetary policy effects with higher uncertainty during recessions. The framework offers a parsimonious, interpretable, and flexible approach to nonlinear and nonstationary macro modeling with practical forecasting and policy-analytic benefits.

Abstract

Recent economic events, including the global financial crisis and COVID-19 pandemic, have exposed limitations in linear Factor Augmented Vector Autoregressive (FAVAR) models for forecasting and structural analysis. Nonlinear dimension techniques, particularly autoencoders, have emerged as promising alternatives in a FAVAR framework, but challenges remain in identifiability, interpretability, and integration with traditional nonlinear time series methods. We address these challenges through two contributions. First, we introduce a Grouped Sparse autoencoder that employs the Spike-and-Slab Lasso prior, with parameters under this prior being shared across variables of the same economic category, thereby achieving semi-identifiability and enhancing model interpretability. Second, we incorporate time-varying parameters into the VAR component to better capture evolving economic dynamics. Our empirical application to the US economy demonstrates that the Grouped Sparse autoencoder produces more interpretable factors through its parsimonious structure; and its combination with time-varying parameter VAR shows superior performance in both point and density forecasting. Impulse response analysis reveals that monetary policy shocks during recessions generate more moderate responses with higher uncertainty compared to expansionary periods.

Paper Structure

This paper contains 25 sections, 1 theorem, 14 equations, 19 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. FAVAR
  4. Autoencoder and its Econometric Applications
  5. Challenges of Standard Autoencoder and Solutions
  6. Methodology
  7. Grouped Sparse Autoencoder
  8. TVP-FAVAR
  9. Estimation
  10. Training the Deep Learning Model
  11. Bayesian Inference
  12. Empirical Results
  13. Data and Implementation Detail
  14. Analysis of Latent Factors
  15. Forecasting Performance
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose the following assumptions hold: If we have two sets of decoder parameters and factors: $\{\boldsymbol{\theta}, \boldsymbol{B}, \boldsymbol{f}_t\}$ and $\{\boldsymbol{\theta}^*, \boldsymbol{B}^*, \boldsymbol{f}^*_t\}$, which yield the same reconstructions of $\hat{\boldsymbol{x}}_t$, for $t=1,\dots, T$, then the recovery of $f_{t,k

Figures (19)

  • Figure 1: The first factor extracted from the PCA and non-linear GS autoencoder (top panel), and variables with the 15 highest correlation magnitudes with the corresponding factors (bottom panel). The time series are standardized to have zero mean and variance one. The grey bands highlight the recession periods.
  • Figure 2: The second factor extracted from the PCA and non-linear GS autoencoder (top panel), and variables with the 15 highest correlation magnitudes with the corresponding factors (bottom panel). The time series are standardized to have zero mean and variance one. The grey bands highlight the recession periods.
  • Figure 3: The third factor extracted from the PCA and non-linear GS autoencoder (top panel), and variables with the 15 highest correlation magnitudes with the corresponding factors (bottom panel). The time series are standardized to have zero mean and variance one. The grey bands highlight the recession periods.
  • Figure 4: The fourth factor extracted from the PCA and non-linear GS autoencoder (top panel), and variables with the 15 highest correlation magnitudes with the corresponding factors (bottom panel). The time series are standardized to have zero mean and variance one. The grey bands highlight the recession periods.
  • Figure 5: The fifth factor extracted from the PCA and non-linear GS autoencoder (top panel), and variables with the 15 highest correlation magnitudes with the corresponding factors (bottom panel). The time series are standardized to have zero mean and variance one. The grey bands highlight the recession periods.
  • ...and 14 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • proof
  • proof