Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-Dimensional Knockoffs Inference for Time Series Data

Chien-Ming Chi, Yingying Fan, Ching-Kang Ing, Jinchi Lv

TL;DR

This work tackles high-dimensional variable selection for time series by introducing Time Series Knockoffs Inference (TSKI), a framework that combines subsampling, robust knockoffs, and e-value aggregation to achieve false discovery rate control under serial dependence. By relaxing the model-X assumptions of i.i.d. data and known covariate distributions, TSKI extends knockoffs to time series and provides theoretical guarantees of asymptotic FDR control under $\beta$-mixing, along with power analyses in generalized linear time series models. The authors demonstrate finite-sample performance through simulations on SETARX-type models and apply the method to inflation data, showing practical interpretability and robust performance when sample sizes are limited. Overall, TSKI offers a principled, scalable approach for interpretable forecasting in time series with strong theoretical underpinnings and empirical validity.

Abstract

We make some initial attempt to establish the theoretical and methodological foundation for the model-X knockoffs inference for time series data. We suggest the method of time series knockoffs inference (TSKI) by exploiting the ideas of subsampling and e-values to address the difficulty caused by the serial dependence. We also generalize the robust knockoffs inference in Barber, Candès, and Samworth to the time series setting to relax the assumption of known covariate distribution required by model-X knockoffs, since such an assumption is overly stringent for time series data. We establish sufficient conditions under which TSKI achieves the asymptotic false discovery rate (FDR) control. Our technical analysis reveals the effects of serial dependence and unknown covariate distribution on the FDR control. We conduct a power analysis of TSKI using the Lasso coefficient difference knockoff statistic under the generalized linear time series models. The finite-sample performance of TSKI is illustrated with several simulation examples and an economic inflation study.

High-Dimensional Knockoffs Inference for Time Series Data

TL;DR

This work tackles high-dimensional variable selection for time series by introducing Time Series Knockoffs Inference (TSKI), a framework that combines subsampling, robust knockoffs, and e-value aggregation to achieve false discovery rate control under serial dependence. By relaxing the model-X assumptions of i.i.d. data and known covariate distributions, TSKI extends knockoffs to time series and provides theoretical guarantees of asymptotic FDR control under -mixing, along with power analyses in generalized linear time series models. The authors demonstrate finite-sample performance through simulations on SETARX-type models and apply the method to inflation data, showing practical interpretability and robust performance when sample sizes are limited. Overall, TSKI offers a principled, scalable approach for interpretable forecasting in time series with strong theoretical underpinnings and empirical validity.

Abstract

We make some initial attempt to establish the theoretical and methodological foundation for the model-X knockoffs inference for time series data. We suggest the method of time series knockoffs inference (TSKI) by exploiting the ideas of subsampling and e-values to address the difficulty caused by the serial dependence. We also generalize the robust knockoffs inference in Barber, Candès, and Samworth to the time series setting to relax the assumption of known covariate distribution required by model-X knockoffs, since such an assumption is overly stringent for time series data. We establish sufficient conditions under which TSKI achieves the asymptotic false discovery rate (FDR) control. Our technical analysis reveals the effects of serial dependence and unknown covariate distribution on the FDR control. We conduct a power analysis of TSKI using the Lasso coefficient difference knockoff statistic under the generalized linear time series models. The finite-sample performance of TSKI is illustrated with several simulation examples and an economic inflation study.
Paper Structure (44 sections, 14 theorems, 187 equations, 6 figures, 6 tables, 2 algorithms)

This paper contains 44 sections, 14 theorems, 187 equations, 6 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Robust time series knockoffs inference with TSKI
  5. A brief review of the model-X knockoffs framework
  6. Outline of the TSKI framework
  7. FDR control by TSKI
  8. Stationary processes satisfying Condition \ref{['ge4']}
  9. Power analysis under generalized linear time series models
  10. Simulation studies
  11. Simulation settings
  12. Empirical performance of TSKI
  13. Real data application
  14. Appendix of Section \ref{['Sec2.new']}
  15. Example for Theorem \ref{['theorem1.new']}
  16. ...and 29 more sections

Key Result

Theorem 1

Let $\widehat{S}$ be the set of variables selected by TSKI with Algorithm Algorithm1. Then under Conditions knockoff.generator.3--knockoff.generator.4 and the assumption of positive $T^{k}$'s in tau.2, we have where $0<\tau^*<1$ is the target FDR level and for each $1\le k \le q+1$ and $1\le j\le p$,

Figures (6)

  • Figure 1: The U.S. inflation from May 2013 to January 2023.
  • Figure 2: A graphical representation depicting $Y_{t-1}$ on the $x$-axis and $Y_{t}$ on the $y$-axis is provided under Model \ref{['SETAR1']} with $(\eta, \iota) = (0.2, 5)$.
  • Figure 3: The left panel displays the averages of "having any selections" indicators over 100 repetitions, where the indicator at each rolling window is one if and only if any covariates are selected, and the $x$-axis indicates the ending time of each rolling window. The right panel shows the analogous results, but the indicator is one at each covariate index if that covariate is selected at any rolling window. The first 127 covariates are current time covariates, and the $128$th to $254$th covariates are one-month lag covariates in the AR(2) model. Covariates measuring similar economic values are clustered closer (see mccracken2016fred for detailed definitions of these covariates). The selection method here is the TSKI-LCD without subsampling ($q=0$).
  • Figure 4: These two panels are analogous to those in Figure \ref{['fig:frequency2']} but with $q=1$ for the TSKI-LCD procedure.
  • Figure 5: The black curves in the three panels are the inflation series at time $t+1$. The red curve in panel (a) is the number of new orders for consumer goods at time $t$, the red curve in panel (b) indicates the U.S./Canada exchange rate at time $t$, and the red curve in panel (c) is the U.S. initial claims for unemployment benefits at time $t$. All curves here are standardized and adjusted for visual comparison, and hence the values of these time series are not reported on the $y$-axis.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 1
  • Definition 2: Knockoff generator
  • Example 1: Lasso coefficient difference (LCD)
  • Example 2: Random forests mean decrease accuracy (MDA)
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Example 3: Autoregressive models with exogenous variables (ARX)
  • Proposition 1
  • Theorem 2
  • ...and 18 more