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Diffusion Index Forecasting with Tensor Data

Bin Chen, Yuefeng Han, Qiyang Yu

TL;DR

This paper develops a diffusion-index forecasting framework for tensor data using CP low-rank factor models and the CC-ISO estimator to extract latent factors without vectorizing the tensor. It provides asymptotic theory for factor estimation, diffusion-index inference, and robust covariance estimation, with a high-dimensional extension via MS-FASR that combines tensor factors with sparse regression on many predictors. Simulation studies confirm factor consistency, valid prediction intervals, and strong performance of CP-based methods, while empirical application to US trade flows demonstrates forecast gains over standard benchmarks. The approach offers a principled way to exploit multidimensional data in macro forecasting and invites further exploration of multi-source tensor analytics in economics.

Abstract

In this paper, we consider diffusion index forecasting with both tensor and non-tensor predictors, where the tensor structure is preserved with a Canonical Polyadic (CP) tensor factor model. When the number of non-tensor predictors is small, we study the asymptotic properties of the least squares estimator in this tensor factor-augmented regression, allowing for factors with different strengths. We derive an analytical formula for prediction intervals that accounts for the estimation uncertainty of the latent factors. In addition, we propose a novel thresholding estimator for the high-dimensional covariance matrix that is robust to cross-sectional dependence. When the number of non-tensor predictors exceeds or diverges with the sample size, we introduce a multi-source factor-augmented sparse regression model and establish the consistency of the corresponding penalized estimator. Simulation studies validate our theoretical results and an empirical application to U.S. trade flows demonstrates the advantages of our approach over other popular methods in the literature.

Diffusion Index Forecasting with Tensor Data

TL;DR

This paper develops a diffusion-index forecasting framework for tensor data using CP low-rank factor models and the CC-ISO estimator to extract latent factors without vectorizing the tensor. It provides asymptotic theory for factor estimation, diffusion-index inference, and robust covariance estimation, with a high-dimensional extension via MS-FASR that combines tensor factors with sparse regression on many predictors. Simulation studies confirm factor consistency, valid prediction intervals, and strong performance of CP-based methods, while empirical application to US trade flows demonstrates forecast gains over standard benchmarks. The approach offers a principled way to exploit multidimensional data in macro forecasting and invites further exploration of multi-source tensor analytics in economics.

Abstract

In this paper, we consider diffusion index forecasting with both tensor and non-tensor predictors, where the tensor structure is preserved with a Canonical Polyadic (CP) tensor factor model. When the number of non-tensor predictors is small, we study the asymptotic properties of the least squares estimator in this tensor factor-augmented regression, allowing for factors with different strengths. We derive an analytical formula for prediction intervals that accounts for the estimation uncertainty of the latent factors. In addition, we propose a novel thresholding estimator for the high-dimensional covariance matrix that is robust to cross-sectional dependence. When the number of non-tensor predictors exceeds or diverges with the sample size, we introduce a multi-source factor-augmented sparse regression model and establish the consistency of the corresponding penalized estimator. Simulation studies validate our theoretical results and an empirical application to U.S. trade flows demonstrates the advantages of our approach over other popular methods in the literature.

Paper Structure

This paper contains 27 sections, 20 theorems, 284 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Model and Estimation
  4. Asymptotic Properties
  5. Estimation of Factors
  6. Inference for Diffusion Index Model
  7. Covariance matrix estimation of factor process by thresholding
  8. Multi-Source Factor-Augmented Sparse Regression
  9. Simulation
  10. Factor Estimator Consistency
  11. Factor Estimator Distribution
  12. Prediction Interval
  13. Multi-Source Factor-Augmented Sparse Regression
  14. Empirical Application
  15. Data and sample
  16. ...and 12 more sections

Key Result

Theorem 3.1

Suppose Assumptions asmp:error-asmp:weakfactor hold. Assume that $\max_{k\le K}\| A_k^\top A_k - I_r\|_2<1$ and $T\le C\exp\left(d_{\max} \right)$ for some constant $C$. Suppose that the initial estimation error bounds satisfy the condition: where $C_{1,K}$ is some constant depending on $K$ only. Then the estimated tensor factors satisfy where $H$ is defined in eqn:rotation_matrix.

Figures (13)

  • Figure 1: Boxplots of log estimation errors of $\widehat{f}_T$.
  • Figure 2: Sample distribution of $\widehat{\Sigma}_{Be}^{-1/2}(\widehat{\widetilde{f}}_t - \widetilde{f}_t)$ over 2000 repetitions. The orange line is the pdf of standard normal.
  • Figure 3: Estimation error of $\beta_0$ and prediction error of $y_{T+h|T}$ over 1000 repetitions under strong and weak factor setting.
  • Figure 4: Heatmap of the absolute value of correlation between the common factors and the monthly variation in bilateral tradeflow among selected countries. The saturation represents the correlation strength, with high saturation indicating a stronger correlation.
  • Figure 5: Plot of $\gamma_q$ for $d_k = 10$ and $\tau = 0.5$.
  • ...and 8 more figures

Theorems & Definitions (44)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Remark 3.4
  • ...and 34 more