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Weight-calibrated estimation for factor models of high-dimensional time series

Xinghao Qiao, Zihan Wang, Qiwei Yao, Bo Zhang

TL;DR

The paper develops a weight-calibrated autocovariance estimator for high-dimensional factor models of time series by embedding a reduced-rank autoregression in a projected subspace. By introducing a projection-based weight matrix ${\widehat{\bf W}}={\bf Q}( {\bf Q}^T \widehat{\boldsymbol\Omega}_y {\bf Q})^{-1}{\bf Q}^T$ and aggregating information across lags in ${\widehat{\bf M}}$, the authors achieve improved separation of strong factors from weak factors and idiosyncratic noise, even when idiosyncratic components are serially correlated. They develop a ratio-based procedure to estimate the number of strong factors $r_0$, establish theoretical results under uniform and heterogeneous factor strengths, and demonstrate superior finite-sample performance through simulations and real-data analyses of S&P 500 daily returns and US macroeconomic data. The work advances the understanding of covariance versus autocovariance-based approaches and lays groundwork for extensions to matrix and tensor factor models, with practical implications for reliable factor recovery and forecasting in high dimensions.

Abstract

The factor modeling for high-dimensional time series is powerful in discovering latent common components for dimension reduction and information extraction. Most available estimation methods can be divided into two categories: the covariance-based under asymptotically-identifiable assumption and the autocovariance-based with white idiosyncratic noise. This paper follows the autocovariance-based framework and develops a novel weight-calibrated method to improve the estimation performance. It adopts a linear projection to tackle high-dimensionality, and employs a reduced-rank autoregression formulation. The asymptotic theory of the proposed method is established, relaxing the assumption on white noise. Additionally, we make the first attempt in the literature by providing a systematic theoretical comparison among the covariance-based, the standard autocovariance-based, and our proposed weight-calibrated autocovariance-based methods in the presence of factors with different strengths. Extensive simulations are conducted to showcase the superior finite-sample performance of our proposed method, as well as to validate the newly established theory. The superiority of our proposal is further illustrated through the analysis of one financial and one macroeconomic data sets.

Weight-calibrated estimation for factor models of high-dimensional time series

TL;DR

The paper develops a weight-calibrated autocovariance estimator for high-dimensional factor models of time series by embedding a reduced-rank autoregression in a projected subspace. By introducing a projection-based weight matrix and aggregating information across lags in , the authors achieve improved separation of strong factors from weak factors and idiosyncratic noise, even when idiosyncratic components are serially correlated. They develop a ratio-based procedure to estimate the number of strong factors , establish theoretical results under uniform and heterogeneous factor strengths, and demonstrate superior finite-sample performance through simulations and real-data analyses of S&P 500 daily returns and US macroeconomic data. The work advances the understanding of covariance versus autocovariance-based approaches and lays groundwork for extensions to matrix and tensor factor models, with practical implications for reliable factor recovery and forecasting in high dimensions.

Abstract

The factor modeling for high-dimensional time series is powerful in discovering latent common components for dimension reduction and information extraction. Most available estimation methods can be divided into two categories: the covariance-based under asymptotically-identifiable assumption and the autocovariance-based with white idiosyncratic noise. This paper follows the autocovariance-based framework and develops a novel weight-calibrated method to improve the estimation performance. It adopts a linear projection to tackle high-dimensionality, and employs a reduced-rank autoregression formulation. The asymptotic theory of the proposed method is established, relaxing the assumption on white noise. Additionally, we make the first attempt in the literature by providing a systematic theoretical comparison among the covariance-based, the standard autocovariance-based, and our proposed weight-calibrated autocovariance-based methods in the presence of factors with different strengths. Extensive simulations are conducted to showcase the superior finite-sample performance of our proposed method, as well as to validate the newly established theory. The superiority of our proposal is further illustrated through the analysis of one financial and one macroeconomic data sets.
Paper Structure (32 sections, 18 theorems, 105 equations, 2 figures, 5 tables)

This paper contains 32 sections, 18 theorems, 105 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Reduced-rank autoregression formulation
  4. Determining the tuning parameter
  5. Theoretical results with uniform factor strength
  6. Theoretical results with different factor strengths
  7. Covariance-based method
  8. Standard autocovariance-based method
  9. Weight-calibrated autocovariance-based method
  10. Summary of theoretical results
  11. Simulations
  12. The model with uniform factor strength
  13. The model with different factor strengths
  14. Real data analysis
  15. Daily returns for S&P 500 stocks
  16. ...and 17 more sections

Key Result

Theorem 1

Let Conditions cond.A--cond.auto hold. For each $k\in[m],$ the following assertions hold: (i) $\lambda_{k1} \asymp \lambda_{kr_0} \asymp p^{\delta_0}$ with probability tending to 1, and $\lambda_{k,r_0+1}=O_p(n^{-1}p);$ (ii) $\|\boldsymbol \Phi_{k,A}\boldsymbol \Phi_{k,A}^{{ \mathrm{ T} }}-{\bf A}{

Figures (2)

  • Figure 1: Plots of ratios of adjacent cumulative weighted eigenvalues in the first and second steps for Auto and WAuto. With such ratio proposed in (\ref{['eq.deter_r']}) for WAuto, $\widetilde{R}_j,R_j^*$ and $\widetilde{R}_j^*$ can be defined analogously using $\{\mu_{kj}\}$, $\{\lambda_{kj}^*\}$ and $\{\mu_{kj}^*\}$, respectively, where $\{\lambda_{kj}^*\}$ and $\{\mu_{kj}^*\}$ represent the corresponding eigenvalues obtained in the second step when applying WAuto or Auto to $\{{\mathbf y}_t^*\}_{t=1}^{n}$.
  • Figure 2: The left and right heatmaps display the varimax-rotated loadings of the first $(\tilde{r}_0 + \tilde{r}_1)$ and $(\hat{r}_0 + \hat{r}_1)$ identified factors for Auto and WAuto, respectively.

Theorems & Definitions (36)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Remark 5
  • Remark 6
  • ...and 26 more