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Double Machine Learning for Time Series

Milos Ciganovic, Federico D'Amario, Massimiliano Tancioni

TL;DR

A calibration rule targeting a "Goldilocks zone", a region of tuning parameters that delivers stable, partialled-out signals and reduced small-sample bias is proposed, and is applied to residualized Local Projections to estimate the dynamic effects of a rise in Tier 1 regulatory capital.

Abstract

We modify the Double Machine Learning estimator to broaden its applicability to macroeconomic time-series settings. A deterministic cross-fitting step, termed Reverse Cross-Fitting, leverages the time-reversibility of stationary series to improve sample utilization and efficiency. We detail and prove the conditions under which the estimator is asymptotically valid. We then demonstrate, through simulations, that its performance remains valid in realistic finite samples and is robust to model misspecification and violations of assumptions, such as heteroskedasticity. In high dimensions, predictive metrics for tuning nuisance learners do not generally minimize bias in the causal score. We propose a calibration rule targeting a "Goldilocks zone", a region of tuning parameters that delivers stable, partialled-out signals and reduced small-sample bias. Finally, we apply our procedure to residualized Local Projections to estimate the dynamic effects of a rise in Tier 1 regulatory capital. The results underscore the usefulness of the methodology for inference in macroeconomic applications.

Double Machine Learning for Time Series

TL;DR

A calibration rule targeting a "Goldilocks zone", a region of tuning parameters that delivers stable, partialled-out signals and reduced small-sample bias is proposed, and is applied to residualized Local Projections to estimate the dynamic effects of a rise in Tier 1 regulatory capital.

Abstract

We modify the Double Machine Learning estimator to broaden its applicability to macroeconomic time-series settings. A deterministic cross-fitting step, termed Reverse Cross-Fitting, leverages the time-reversibility of stationary series to improve sample utilization and efficiency. We detail and prove the conditions under which the estimator is asymptotically valid. We then demonstrate, through simulations, that its performance remains valid in realistic finite samples and is robust to model misspecification and violations of assumptions, such as heteroskedasticity. In high dimensions, predictive metrics for tuning nuisance learners do not generally minimize bias in the causal score. We propose a calibration rule targeting a "Goldilocks zone", a region of tuning parameters that delivers stable, partialled-out signals and reduced small-sample bias. Finally, we apply our procedure to residualized Local Projections to estimate the dynamic effects of a rise in Tier 1 regulatory capital. The results underscore the usefulness of the methodology for inference in macroeconomic applications.
Paper Structure (24 sections, 3 theorems, 68 equations, 2 figures, 3 tables)

This paper contains 24 sections, 3 theorems, 68 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Inference with Reverse Cross-fitting
  3. The mechanics of time-Reversed Cross-Fitting
  4. The RCF-DML estimator
  5. Assumptions for valid inference
  6. Asymptotic properties
  7. Stability-based tuning of nuisance parameters
  8. Simulations
  9. Application
  10. Data and Methodology
  11. Results
  12. Conclusions
  13. Step 1 (Orthogonal expansion with $L_2$-quadratic remainder).
  14. Step 2 (Decomposition into conditional mean and centered part).
  15. Step 2.1 (Term A is small).
  16. ...and 9 more sections

Key Result

Lemma 2.1

For each fold $k$, define the block-average discrepancy Under Assumptions ass:smooth--ass:rate, $R_k=o_p(T^{-1/2})$ uniformly in $k$. Consequently, and the average over $k$ also satisfies

Figures (2)

  • Figure 1: Example of Reverse Cross-Fitting using five folds. The Blue area represents the whole sample. Red blocks are "Main" observations. Green blocks are "Quasi-complementary" observations. White blocks represent left-out observations, and the direction of the arrows indicates the direction of the estimate.
  • Figure 2: Cumulative Impulse response functions to a regulatory capital shock obtained via RCF-DML LPs.The dark (light) blue shaded areas denote 95% (90%) cumulative confidence intervals computed via HAC.

Theorems & Definitions (8)

  • Remark 2.1: On non-independence
  • Remark 2.2: Choice of $K$
  • Lemma 2.1: Orthogonality remainder under dependence
  • Lemma 2.2: Per-fold OLS linearization
  • Theorem 2.1: Fold-average RCF attains oracle asymptotic variance
  • proof
  • proof
  • proof