Double Machine Learning for Time Series

Double Machine Learning for Time Series

Introduction

The recently developed Double/Debiased Machine Learning (DML) estimator (Chernozhukov2018) provides a valid alternative to standard causal inference methods for microeconomic data (ahrens2025introduction). Building on robinson1988root's partially linear regression (PLR) model, it identifies a low-dimensional causal parameter while controlling for potentially high-dimensional, nonlinear confounders. These nuisance functions, though not of direct interest, must be accurately approximated using modern Machine Learning (ML) methods to ensure valid inference. DML aims to reduce the target parameter estimate's sensitivity to approximation error in these functions. Robustness is achieved through Neyman orthogonalization, which removes regularization bias, and randomized cross-fitting (CF), which alternately assigns folds as training and test samples, mitigating overfitting and improving efficiency in i.i.d. settings.

This paper examines the applicability of DML to macroeconomic time series, which are typically short, strongly dependent, and highly endogenous. Such features can create high-dimensional settings where the parameter space may exceed the sample size. Standard DML, designed for independent observations, cannot yield valid inference in this context, as randomized cross-fitting would violate the series's sequential structure.

By modifying specific features of DML, we establish its validity for time-dependent data. The adapted estimator has two key components. First, a novel Reverse Cross-Fitting (RCF) procedure leverages the time-reversibility of stationary Gaussian processes to improve sample usage and enhance efficiency in realistic time-dependent samples. Second, we introduce a fold-specific, stability-based calibration for nuisance learners, replacing standard predictive tuning. This criterion identifies a “Goldilocks zone” of tuning parameters in which the fold-specific RMSE varies minimally. Consistent with Neyman orthogonality, estimates within this stability region achieve minimal bias, a property that may fail when using predictive metrics in high-dimensional control settings (mcgrath2025optimal).

To parallel the plain DML framework, we express the time-series inference problem using the PLR model of robinson1988root, as in Chernozhukov2018. We aim for root-$T$-consistent estimation and valid inference for a low-dimensional causal parameter $\theta_0$ in the presence of high-dimensional controls and nuisance functions $\eta_0 = (m_0, g_0)$:

where $y_t$ is the outcome, $d_t$ a scalar policy variable, $X_t = (X_1,\ldots, X_p)$ the vector of controls, and $\epsilon_t$ and $\xi_t$ disturbances, for $t = 1,\ldots,T$.

belloni2014high propose the post-double-selection procedure, which substantially reduces regularization bias by applying Lasso to both the reduced-form of (ref) and (ref) and regressing $y_t$ on $d_t$ and the union of selected controls.\footnote{Naively applying selection methods to (ref) may discard controls with coefficients “moderately close to zero,” inducing omitted-variable (regularization) bias (belloni2014high)}

Chernozhukov2018 extend this approach to settings with highly complex, nonlinear nuisance functions via DML, allowing valid inference with general ML learners. Relative to post-double-selection, DML introduces three core elements: (i) Neyman orthogonalization, which removes regularization bias by forming a residualized outcome and policy equation from the reduced-form outcome equation $y_t = g^r_0(X_t) + \chi_t$, with $g^r_0 \neq g_0$), and the policy equation (ref)

so that applying OLS to residuals accommodates unspecific ML estimators for $g^r_0$ and $m_0$; (ii) sample splitting, which separates data used to learn nuisance functions from data used to estimate $\theta_0$ and mitigates overfitting; and (iii) cross-fitting (CF), which alternates the roles of training and test sets to restore efficiency and full sample usage.

While the first two features transfer naturally to time-series applications, standard CF does not, as it relies on independence across folds. Random splitting would violate the temporal dependence structure, motivating the need for a time-series–adapted CF procedure.

To handle temporal dependence in microeconomic DML applications, semenova2023inference propose the Neighbours-Left-Out (NLO) CF procedure for weakly dependent panel data. NLO exploits the mixing properties of the series by partitioning data into blocks and omitting neighboring observations to create quasi-independent folds. For typical macroeconomic time series, achieving approximate independence requires substantial truncation, reducing sample efficiency. We introduce RCF to address this limitation.

We structure the analysis in three steps. We begin by detailing the time-series RCF-DML estimator and outlining the conditions for achieving asymptotic minimum-bias properties and efficiency with generic ML learners. A key focus is on conditional stability, a time-series–specific requirement preventing block dependence from reintroducing bias, akin to a “no data leakage” condition. Under this and standard DML conditions, the estimator is asymptotically consistent, with variance converging to an absolutely summable long-run variance, which can be consistently estimated using HAC corrections.

Next, we conduct extensive simulations based on approximately sparse (giannone2021economic) structural VAR and PLR DGPs, leveraging a recursivity assumption for the contemporaneous structure.\footnote{For tractability, nuisance functions are estimated using linear ML methods. The DML framework also accommodates nonlinear learners such as random forests, boosting, neural networks, and ensembles.} The results indicate that RCF-DML delivers root-$T$-consistent estimates with coverage levels approaching nominal rates even under model mis-specification.\footnote{Model mis-specification is introduced by considering all the variables as contemporaneous, even if the true DGP is recursive.} In realistic finite samples, the RCF strategy outperforms NLO, especially in scenarios with strong persistence.

The finite sample simulations also reveal challenges in applying RCF-DML to real data. Standard predictive tuning, common in forecasting and causal ML, does not minimize bias, particularly as dimensionality increases relative to sample size. In these regimes, the divergence between minimum-bias MSE and predictive MSE increases.\footnote{This divergence echoes robinson1988root's findings on allowable kernel complexity in its prototypical PLR. The issue is independent of the CF design and not attributable to failures in relaxing Donsker conditions.} This result is consistent with recent findings (celentano2023challenges, mcclean2024double, mcgrath2025optimal). To address this issue, we propose a stability-based tuning criterion that targets a Goldilocks zone of tuning parameters, producing minimal fold-specific RMSE variation. Importantly, RMSE itself need not be minimal across the tested calibrations. This logic parallels sufficient-variability requirements in IV identification (jaeger1995problems) applied to ML-residualized second-stage regressions (athey2019estimating).

Notably, RCF-DML remains low-bias even when the recursivity assumption is violated, inducing mis-specification in contemporaneous controls. Moreover, by incorporating GARCH dynamics into the SVAR DGP, we demonstrate robustness to severe non-normality, which violates time reversibility. In small samples, the bias advantages of RCF-DML outweigh efficiency losses relative to time-directional CF strategies.

Additionally, we show that, in a dynamic setting, nuisance-function residuals can be utilized to construct unbiased structural impulse responses via residualized Local Projections (DML-LPs). Under conditional unconfoundedness, supported by high-dimensional conditioning, the time-series-adapted DML estimator provides a new framework for identifying macroeconomic shocks and tracing their transmission.

Finally, we apply our method to estimate the dynamic effects of prudential capital shocks, a context well-suited for RCF-DML, given the short time span of regulatory data. The results align closely with consensus findings from established identification strategies.

Our analysis contributes to the growing literature on ML-assisted causal inference (athey2016recursive, athey2019generalized, Chernozhukov2018). Within the DML framework, we relate to recent work on cross-sectional and panel data (ahrens2025model, chang2020double, chernozhukov2022automatic, chernozhukov2022locally, dube2020monopsony, semenova2021debiased, semenova2023inference). In macroeconomic applications, our approach complements adamek2024local, who use desparsified Lasso for high-dimensional time-series LPs, and agostini2024vaccination, who apply dynamic DML-type residualization to panel-data LPs. Regarding learner calibration, our results echo recent work highlighting that optimal tuning for DML inference differs from predictive optimality. mcclean2024double advocate undersmoothing to improve convergence, while mcgrath2025optimal demonstrate that in high-dimensional (inconsistency) regimes, minimum bias is achieved by minimizing the asymptotic variance of the target functional.

The remainder of the paper is structured as follows. Section (ref) introduces RCF and the notion of time reversibility, proving that, under suitable conditions, RCF-DML is asymptotically consistent and efficient. Section (ref) introduces the Goldilocks zone tuning strategy and explains its role in bias control for high-dimensional time series. Section (ref) presents the simulation results. Section (ref) provides an empirical application. Section (ref) concludes.

Inference with Reverse Cross-fitting

Cross-fitting restores efficiency after sample splitting by alternating the roles of training and test sets. In i.i.d. settings, this is achieved through random partitioning. However, random splitting is infeasible for time series because it breaks the temporal dependence structure.

We propose the RCF method as an alternative CF strategy for time series, tailored to macroeconomic data. Rather than enforcing independence across folds, RCF exploits the time-reversibility of stationary processes to construct auxiliary and main samples that preserve the temporal structure while maximizing sample usage. This design avoids the truncation costs of NLO and provides an efficiency-preserving cross-fitting scheme.

A stochastic process $X_t$ is time-reversible if, for any window length $D$, the joint distributions of $X_t = \{X_t,X_{t+1},\ldots, X_{t+D}\}$ and of its time-reversed version, $X'_t = \{X_{t+D},X_{t+D-1},\ldots, X_t\}$ are identical for all $t$. All stationary Gaussian processes, and nonlinear transformations of them, are time-reversible because their joint distributions depend only on the symmetric covariance function (weiss1975time,hallin1988time).\footnote{The PLR model allows nonlinear nuisance functions; under time reversibility, reversing the series does not affect the model parameters. cheng1999miscellanea provides necessary and sufficient conditions for the reversibility of stationary linear processes without requiring higher-order moments, and tong2005time extends these results to the multivariate case, covering many non-Gaussian stationary linear processes. Time reversibility is statistically testable. For zero-mean series, Ramsey’s test (rothman1990characterization) exploits the symmetry of differences of reversible processes by comparing sample bi-covariances at lag ($\tau$), i.e., $\gamma(\tau) = \{X^2_tX_{t-\tau}\}-\{X^2_{t-\tau}X_t\}$. martinez2018detection propose the TiROP test, which relies on ordinal patterns and makes no moment assumptions.}

The mechanics of time-Reversed Cross-Fitting

Consider a partition of the set $\{1, \ldots, T\}$ into adjacent blocks $\{\mathcal{B}_k\}_{k=1}^{K}$, each of equal length (i.e. $T_k = T_{\text{block}} = \left\lfloor \frac{T}{K} \right\rfloor$). For any $k$, define the left and right index sets: \[

\] and the corresponding left- and right-“complements’’ of $\mathcal{B}_k$ as: \[

\] Let the associated data blocks be: \[

\]

Consider $K = 5$, defining corresponding blocks $\{\mathcal{B}_k\}_{k=1}^{5}$ of adjacent time stamps $t \in \{1, \ldots, T\}$, each of size $T_{\text{block}}$, so that $T = KT_{\text{block}}$. The partition defines five non-overlapping blocks $\mathcal{B}_k$, considered as main sets in the five-fold-RCF, and left- and right-complementary blocks, which are auxiliary sets: $\mathcal{B}^{L}_{1} = \varnothing$, $\mathcal{B}^{R}_{1} = \{\mathcal{B}_k\}_{k=2}^{5}$; $\mathcal{B}^{L}_{2} = \mathcal{B}_1$, $\mathcal{B}^{R}_{2} = \{\mathcal{B}_k\}_{k=3}^{5}$; $\mathcal{B}^{L}_{3} = \{\mathcal{B}_k\}_{k=1}^{2}$, $\mathcal{B}^{R}_{3} = \{\mathcal{B}_k\}_{k=4}^{5}$; $\mathcal{B}^{L}_{4} = \{\mathcal{B}_k\}_{k=1}^{3}$, $\mathcal{B}^{R}_{4} = \mathcal{B}_5$; $\mathcal{B}^{L}_{5} = \{\mathcal{B}_k\}_{k=1}^{4}$, $\mathcal{B}^{R}_{5} = \varnothing$. Nuisance functions are therefore estimated in time-reversed mode in the first two folds, in forward mode in the last two, and in both directions in the central fold.

In typical macroeconomic samples, auxiliary blocks adjacent to the main block, e.g., the second-left or fourth-right block, may contain too few observations. Including such blocks leads to nuisance estimates dominated by very short segments. In such cases, it is preferable to omit these undersized blocks from nuisance estimation. RCF addresses this by adopting an auxiliary-set rule, dropping the smaller side of the auxiliary sample: $D_{\text{aux},k} = D_{L,k}$ if $|D_{L,k}| < |D_{R,k}|$, $D_{\text{aux},k} = D_{R,k}$ if $|D_{R,k}| < |D_{L,k}|$, and $D_{\text{aux},k} = D_{L,k} \cup D_{R,k}$ if $|D_{L,k}| = |D_{R,k}|$. For any $K\ge4$, the auxiliary sample used for fold $k$ is therefore only left (backward) or only right (forward) if one side is smaller, both sides when $k$ is central (odd $K$-fold CF), and the auxiliary blocks have equal size.

Figure (ref) illustrates this five-fold Reverse Cross-Fitting scheme.

To generalize across sample sizes, consider each fold as containing a single observation. RCF attains higher, equal, or lower usage than NLO when $K < 11$, $K = 11$, and $K > 11$, respectively. Under NLO, usage exceeds $70\%$ for $K\ge10$. Using many folds in macroeconomic time series reduces the size of the left-out blocks and may violate weak dependence assumptions. By providing higher usage for all $K<11$, RCF offers a practical advantage in small samples.

The RCF-DML estimator

\sloppy Let $(W_t)_{t=1}^T$ be a covariance-stationary, ergodic time series on $(\Omega,\mathcal F,\mathbb P)$. DML targets a finite-dimensional parameter $\theta_0\in \Theta\subset\mathbb{R}^d$ while controlling for (possibly high-dimensional) nuisance objects $\eta_0$. Let $\psi:\mathcal W\times \Theta\times \mathcal H\to\mathbb{R}^d$ be a score (moment) function satisfying Neyman orthogonality, i.e., the local invariance of $\theta_0$ to errors in the approximation of $\eta_0$:

and let $A := \mathbb{E}[\partial_\theta \psi(W_t;\theta_0,\eta_0)]$ be nonsingular. Partition the index set $\{1,\dots,T\}$ into $K$ adjacent, non-overlapping blocks $\{B_k\}_{k=1}^K$, each of length $|B_k|=\lfloor T/K\rfloor$ up to $O(1)$ boundary effects. For each main block $B_k$, define its left and right complements \[ B^L_k := \bigcup_{i< k} B_i, \qquad B^R_k := \bigcup_{i>k} B_i. \]

RCF trains the nuisance function on data lying outside the main block. If $k < \lfloor K/2\rfloor$, training is performed on $B^R_k$ (using time-reversed future data). If $k > \lfloor K/2\rfloor$, training is performed on $B^L_k$ (using past data). If $k = \lfloor K/2\rfloor$ and $K$ is odd, both $B^L_k$ and $B^R_k$ are used. This scheme guarantees strict separation between training and main blocks.

Within $B_k$, use the nuisance estimates $\hat\eta^{(k)}$ to construct residualized outcome and policy series, $(\chi_{t,k},\xi_{t,k})_{t\in B_k}$, and compute the residual-on-residual OLS slope \[ \hat\theta_k := \arg\min_{\theta\in\Theta} \frac{1}{|B_k|} \sum_{t\in B_k}(\chi_{t,k}-\theta\,\xi_{t,k})^2. \] Equivalently, the fold-specific empirical moment condition satisfies: \[ \frac{1}{|B_k|}\sum_{t\in B_k} \psi(W_t;\hat\theta_k,\hat\eta^{(k)}) = 0. \]

The RCF-DML estimator averages the $K$ fold-specific estimates:

Assumptions for valid inference

Define the oracle score $\psi_t^\star := \psi(W_t;\theta_0,\eta_0)$, the lag-$h$ autocovariance $\Gamma(h):=\operatorname{Cov}(\psi_t^\star,\psi_{t-h}^\star)$, and the long-run variance

where the matrix $\Sigma$ is the time-series analogue of $\operatorname{Var}(\psi_t^\star)$ in the i.i.d.\ case.

The RCF–DML estimator does not assume independence between auxiliary and main blocks. The following assumptions are required for valid inference.

Assumption (ref) ensures the oracle score has finite variance and that serial dependence is weak enough for a well-defined long-run variance. This assumption guarantees that the long-run variance exists and can be estimated with HAC methods. It excludes explosive or infinite-variance processes.

Assumption (ref) is the functional central limit theorem (FCLT): the partial-sum process of the oracle scores $\psi_t^\star=\psi(W_t;\theta_0,\eta_0)$, normalized by $\sqrt{T}$, converges weakly in the space $D([0,1];\mathbb{R}^d)$ to a multivariate Brownian motion with covariance kernel $(u\wedge v)\Sigma$.\footnote{Here $u,v\in[0,1]$ denote rescaled time indices, representing fractions of the sample (e.g.\ $u=0.5$ corresponds to the first half of the series), and $(u\wedge v)\Sigma$ is the covariance kernel of the limiting Brownian motion: the variance grows linearly with time $(u)$, while $\Sigma$ captures the contemporaneous dependence across the $d$ components of the score vector.} This FCLT yields the asymptotic normality of the RCF–DML estimator, which considers time-dependent stationary series (see, e.g., Billingsley1968, Hansen1982, Herrndorf1984). Sufficient conditions for Assumption (ref) include the Maxwell–Woodroofe projective criterion (MaxwellWoodroofe2000) and, more generally, summable $\alpha$-mixing.\footnote{The Maxwell–Woodroofe condition guarantees the functional central limit theorem for a broad class of stationary sequences, and summable $\alpha$-mixing implies the Maxwell–Woodroofe condition. Time-reversible processes fall within this class, as their stationarity and covariance symmetry ensure that the criterion is automatically satisfied.} Using this framework, in the supplementary appendix, we verify Assumption (ref) for the DGPs considered in the paper.

Assumption (ref) guarantees smooth, linearizable dependence of the score on $(\theta,\eta)$ and enforces Neyman orthogonality so that nuisance errors enter only at second order. Part (i) thus establishes that, if the nuisance is estimated sufficiently well, the approximation errors are asymptotically negligible at the $\sqrt{T}$ scale. Part (ii) extends weak-dependence and continuity requirements to the score derivatives, allowing for valid Jacobian and variance calculations under serial correlation. Part (iii) ensures that score derivatives change continuously with the nuisance parameter, such that $L_2$ convergence of $\hat\eta^{(k)}$ translates to uniform convergence of the Jacobian matrix across folds.\footnote{Notably, the summability of autocovariances parallels the HAC conditions in Andrews1991 and holds automatically for mixing processes (Hansen1982).}

Assumption (ref) imposes the usual $L_2$-consistency rate ($o_p(T^{-1/4})$) and uniform boundedness of fold-specific nuisance estimators. Moreover, it adds a conditional stability requirement: the block-average conditional plug-in bias, given the training data to obtain $\hat\eta^{(k)}$, is negligible at the $\sqrt{T}$ scale, even though the auxiliary and main blocks belong to the same time-dependent series. This condition replaces the usual fold-independence assumption and is the sole RCF-specific condition that handles dependence between auxiliary and main blocks.

Intuitively, this means that training on auxiliary blocks, which are non-overlapping but adjacent to the main block, does not induce systematic bias in the scores computed on the main block. This condition is the time series analogue of the no-data-leakage principle in statistical learning. As long as each main block is excluded from the training set and the learner generalizes in $L_2$, the condition is mild. We verify this requirement for the DGPs considered in the paper in the supplementary appendix.

Asymptotic properties

This subsection derives the asymptotic distribution of the RCF–DML estimator under time dependence. We show that the estimator admits an asymptotic linear representation with the oracle score, is $\sqrt{T}$-consistent and asymptotically normal, and that a HAC estimator of the long-run variance is consistent. These results imply valid inference without fold independence or temporal gaps between training and main blocks. We provide complete proofs of the results in Appendix A.

Although the main blocks $B_1,\dots, B_K$ are disjoint, serial dependence in $\{W_t\}$ and overlapping auxiliary samples induce cross-fold covariance. The fold-specific estimators \[ \hat\theta_k \;\approx\; \theta_0 + A^{-1}\frac{1}{|B_k|}\sum_{t\in B_k}\psi(W_t;\theta_0,\eta_0), \] are not independent across $k$, thus their average $\hat\theta={K}^{-1}\sum_{k=1}^K\hat\theta_k$, accumulates these dependencies into the long-run variance which determines the asymptotic covariance of $\hat\theta$. Consequently, valid inference requires estimating $\Sigma$ in a way that captures cross-fold serial dependence. We do this via an HAC estimator built from the stacked, time-ordered, cross-fitted score series $\{\hat s_t\}_{t=1}^T$ rather than by averaging per-fold variances. A full description of the variance estimator appears in the supplementary appendix.

Stability-based tuning of nuisance parameters

The validity of DML relies on Neyman orthogonality, which requires the second-stage score to be locally insensitive to small errors in nuisance estimation (Chernozhukov2018). This robustness holds only when nuisance learners satisfy the appropriate complexity conditions. Conditions, equivalently, when they achieve suitable $L_2$-consistency.

In high-dimensional settings, tuning hyperparameters solely for predictive accuracy may violate these requirements. As shown by mcgrath2025optimal, prediction-optimal tuning can lead to nuisance estimates that undermine valid inference, even in cross-sectional designs. In practice, nuisance models cannot be too simple (leaving residual confounding) or too complex (absorbing treatment variation or overfitting noise).

This balance is often described as the Goldilocks zone of model complexity (fort2019goldilocks, snyder2024goldilocks), where $\hat{m_0}(X_t)$ removes confounding while preserving sufficient policy variation for identification, and $\hat{g_0}^r(X_t)$ denoises without attenuating policy-induced signals.

We operationalize this idea by targeting regions of the hyperparameter space that balance predictive accuracy and local stability, rather than selecting the hyperparameter that minimizes predictive error. Choosing within these stability regions avoids overfitting to sharp local minima and yields nuisance estimates that support more reliable second-stage inference. We therefore refer to our tuning method as identifying the Goldilocks zone of the time-series DML estimator.

Let $\Lambda = \{\lambda_1, \lambda_2, \dots, \lambda_M\}$ denote an ordered, equally spaced, grid of scalar hyperparameters (e.g., Lasso regularization penalties), sorted such that $\lambda_i < \lambda_{i+1}$. For a given fold $k$, let $\hat{\eta}^{(k)}(\lambda_i)$ be the nuisance function estimated on the auxiliary sample using parameter $\lambda_i$. Let $\mathcal{R}(\lambda_i)$ denote the RMSE of this estimate evaluated on the main block $B_k$. We consider a moving window of size $S$ consisting of adjacent hyperparameters. For the $j$-th window, denoted as $\mathcal{W}_j$, the set of indices is defined as $\{j, j+1, \dots, j+S-1\}$ for $j = 1, \dots, M-S+1$.

For each window $\mathcal{W}_j$, we compute two component metrics: local variability and local predictive performance. Local variability is defined as

Intuitively, the local performance metric in each window will be given by $\bar{\mathcal{R}}_j$.

To ensure commensurability between stability and performance, we apply min–max normalizations $\tilde{V}_j = (V_j - \min V_j)/(\max V_j - \min V_j)$ and $\tilde{\bar{\mathcal{R}}}_j = (\bar{\mathcal{R}}_j - \min \bar{\mathcal{R}}_j)/(\max \bar{\mathcal{R}}_j - \min \bar{\mathcal{R}}_j)$ across all valid windows $j$.

Each score $\mathcal{S}_j$ then combines the normalized metrics:

The Goldilocks zone, $\mathcal{W}^*$, is the window associated with the minimum score, balancing low predictive error with structural stability:

Finally, the optimal hyperparameter ${\lambda^\star}$ is the minimizer of the predictive error within the Goldilocks zone:

This rule ensures that the chosen regularization parameter attains high predictive performance while lying in a region robust to small perturbations. In our implementation, we set the window size to $S=3$.

Simulations

In this section, we present simulation evidence on the performance of RCF-DML using a contemporaneously recursive SVAR. The data-generating process (DGP) is given by

where $Y_t \in \mathbb{R}^n$ and $P$ is a lower--triangular Cholesky factor encoding contemporaneous impacts. The autoregressive matrix $\Phi_1$ induces approximate sparse local dependence via asymmetric, band-limited decay. We consider $p = 100$ covariance stationary variables and conduct 10{,}000 Monte Carlo simulations. Details on this and alternative DGPs are provided in Supplement S1.

Large sample, low complexity.

We first evaluate the large-sample performance of the RCF-DML estimator by fixing the sample size at $T = 1{,}000$, which provides an approximation to the asymptotic regime. We use RMSE-tinned Lasso for nuisances' approximations.\footnote{Because the SVAR design is low-complexity relative to the sample size, tuning the Lasso penalty via the Goldilocks zone offers no improvement, such that the optimal parameter $\lambda^\star$ is selected using the standard predictive criterion.}

The results in Table (ref) show that the RCF-DML estimator attains nominal coverage and exhibits asymptotic consistency, with percentage bias remaining close to 1.5%. The correct coverage and small bias indicate that the FCLT operates as expected, yielding asymptotic normality. Moreover, the estimator performance is stable across the range of fold choices. Taken together, these findings confirm the analytical properties established for the RCF-DML procedure and, importantly, validate the HAC variance estimation strategy adopted in the asymptotic analysis.

Small samples, high complexity.

We next document the bias-reduction gains achieved by the Goldilocks zone (GZ) tuning rule in finite samples relative to the standard predictive metric (RMSE), and compare the performance of the RCF-DML estimator with the alternative NLO cross-fitting strategy. We consider sample sizes $T \in \{50, 100, 200\}$ and different values of $K$. Using the complexity measure $c=p/T$ (mcgrath2025optimal), this design spans a range of dimensionality regimes, from high-dimensional settings ($c=2$ and $c=1$) to moderately complex ones ($c=0.5$).

The results depicted in Table (ref) show clear bias-reduction benefits of the stability-based tuning rule (RCF-GZ), particularly in small samples where high dimensionality amplifies overfitting risk. In these settings, on average, the RCF-GZ reduces bias by approximately $35\%$ relative to RCF-RMSE. In addition, the bias decreases with larger values of $K$ under both tuning methods, though GZ consistently yields the more substantial improvement. The estimator's coverage is largely insensitive to the choice of the tuning metric. Coverage remains close to the nominal rate under both the GZ and RMSE criteria, and displays substantial stability across different values of $K$.

Relative to the NLO CF estimator (NLO-RMSE), RCF-RMSE reduces bias by about 10% on average, with larger gains in smaller samples. RCF-GZ delivers over a 40% reduction. Even for $K \ge 10$, when NLO-RMSE remains root-$T$ consistent, the bias under NLO-RMSE is still more than 35% higher than under RCF-GZ. To further demonstrate that the GZ metric is not specific to our CF strategy, the supplementary appendix reports parallel simulations using the NLO-GZ procedure. Even under that CF design, the proposed metric improves the estimator's accuracy relative to standard RMSE-based tuning, without compromising coverage.\footnote{Although the NLO design aims to produce approximately independent folds, we estimate its standard errors using the same HAC procedure as for RCF-DML to account for possible residual serial dependence.}

Overall, the results show that RCF can improve on NLO in small samples with high serial dependence. In such cases, the trade-off between sample usage and near-independence becomes binding: for NLO, increasing $K$ leaves too little data in each left-out block to satisfy the independence condition. As expected, these issues diminish as $T$ grows.

Violating the normality assumption: GARCH heteroskedasticity

We consider the case in which the time-reversibility condition is violated by introducing GARCH heteroskedasticity. The specific GARCH structure is detailed, together with simulation results, in the supplementary appendix.

Compared with the homoskedastic benchmark, both RCF and NLO-DML exhibit higher bias. However, RCF-DML coverage remains close to its nominal rate. Notably, the RCF-DML estimator remains approximately root-$T$ consistent across different values of $K$. Compared with the time-directional NLO design, the gains from increased sample usage and estimator efficiency under RCF outweigh the losses induced by heteroskedasticity and non-invertibility. Overall, deviations from homoskedasticity moderately affect the accuracy of the time-series DML estimator but do not impair its inferential properties.

Double Machine Learning for Local Projections.

We show that the RCF-DML methodology can be used to estimate dynamic causal effects. In this case, the ground-truth target is the recursive solution of the SVAR. The IRF of the outcome variable to the shock of interest is estimated using the LP adaptation\footnote{chernozhukov2021causal and agostini2024vaccination provide early applications of DML to LP settings.} of the DML-PLR model defined in equations (ref) and (ref), namely:

for horizons $h=0,1,...,H$.

The DML-LP implementation requires re-estimating, for every horizon, the fold-specific nuisance functions of the reduced-form outcome equation. The resulting dynamic residualized equation is:

where $\hat{\chi}_{t+h} = y_{t+h} - \hat{g_h^r}(X_t)$, $\hat{\xi}_t = d_t - \hat{m_0}(X_t)$, and $g_h^r \neq g_h$ denotes the reduced-form mapping for the outcome equation employed in the residualization step.

The finite sample properties of the RCF-DML-LP estimator, considering eight horizons, are reported in the supplementary appendix. The bias remains very small and converges to zero at longer horizons, where the true IRF decays. Coverage remains close to its nominal level, including the high-dimensional regime with the smallest sample size.

Application

We apply the time-series RCF-DML LP estimator developed above to assess the dynamic effects of prudential capital shocks on Italian GDP, private nonfinancial corporations (PNFC) lending, PNFC spreads, and the components of the Tier 1 capital ratio. Because prudential tools were introduced only recently, the available time series are short. This data feature makes the RCF approach suitable for inference in this setting.

Since its introduction in 1988, the Basel I framework has established minimum capital requirements to strengthen banking system stability and laid the foundation for subsequent Basel Accords. In the aftermath of the 2008 financial crisis, financial institutions exhibited significant fragility (cited in admati2014bankers), leading regulators to reach consensus on the need to strengthen bank capital and liquidity requirements. The Basel Committee on Banking Supervision (BCBS) introduced a broad set of reforms on September 12, 2010, aimed at enhancing the resilience of the global financial system (basel2010assessment). The Basel framework is expected to curb banks’ risk-taking incentives (holmstrom1997financial, kim1988risk) and lower default probabilities following adverse shocks (dewatripont1994theory, Gambacorta2004Does). However, policymakers and industry participants have raised concerns that these regulations may entail sizable costs for banks and the real economy (fbf2010reforming, iif2011cumulative).

A growing empirical literature evaluates the macroeconomic consequences of bank capital shocks. Empirical studies, using various identification schemes and alternative capital measures, largely converge on the same conclusions: prudential capital shocks reduce economic activity in the short run, contract corporate lending, and increase lending spreads. Table (ref) summarizes the main dynamic estimates of the effects of prudential capital shocks on GDP, PNFC lending, and interest-rate spreads from leading empirical contributions.

Data and Methodology

We construct our dataset using official sources and following several related studies (conti2023bank,kanngiesser2020macroeconomic,meeks2017capital,mesonnier2017macroeconomic,noss2016estimating). These works guide the identification of variables that act as transmission channels through which capital requirements influence banks’ capital ratios and the broader economy.

We consider an extensive set of controls to isolate the causal effect of regulatory capital on GDP.\footnote{We provide in the supplementary appendix the lists of all variables used and their sources.} To avoid conditioning on post-treatment mediators, which would block part of the causal transmission mechanism (pearl2009causal), transmission-channel variables enter the model only lagged.\footnote{We include three lags of the control variables, the outcome, and the policy variable.} The policy variable is the Tier 1 Capital to Risk-Weighted Assets, sourced from the IMF.\footnote{https://data.imf.org/?sk=51B096FA-2CD2-40C2-8D09-0699CC1764DA. The IMF reports half-quarterly data for Italy from 2005Q2. To construct a consistent quarterly series, we interpolate the IMF data. Validation exercises confirm that the interpolation preserves the structure of the underlying data.} We test each outcome, the policy, and control series for stationarity and apply log-differences to variables in levels and differences to ratios and interest rates.\footnote{All series are seasonally adjusted using X-13ARIMA-SEATS (monsell2007x).}

The dynamic analysis considers the DML-LP model defined by equations (ref), (ref), and (ref) together with the residualized regression in equation (ref). We approximate the nuisance functions using Lasso. We fix the number of folds to $K = 8$,\footnote{With eight CF folds, and considering $H=8$, we maximize sample usage.} and tune the fold-specific shrinkage intensity parameters according to the Goldilocks zone target over a grid of \(10{.}000\) values. The predictive equations and residuals are fitted on the auxiliary sample and evaluated on the main sample. In practice, the orthogonalized residuals are computed using the “Best’’ estimator among all candidates within the Goldilocks zone.

We focus on four main channels through which prudential capital shocks affect real GDP. These are Tier 1 capital, risk-weighted assets of Italian banks, the real value of banks’ new loans to PNFCs, and the PNFC interest-rate spread, computed as the difference between the lending rate on new loans and the three-month Euribor (conti2023bank).

Results

The shock leads to an increase in the PNFC spread, peaking at about three basis points after three quarters. In turn, the reduction in corporate lending, which carries relatively higher regulatory risk weights, is consistent with banks prioritizing the downsizing of higher risk-weighted assets to meet tighter capital-ratio requirements.

A reduction in credit supply generates adverse short-run macroeconomic effects that gradually fade in the medium term. Real GDP contracts by almost 0.13 percent after four quarters.

The method successfully recovers the expected transmission patterns of prudential capital shocks and aligns closely with established empirical evidence summarized in Table (ref). The supplementary appendix reports the corresponding impulse responses obtained using the RMSE-based calibration of the nuisance functions.

These results further highlight the validity of the novel metric proposed in this paper, particularly for addressing the real GDP response. For the latter, RMSE tuning tends to over-denoise the series and attenuate the policy signal, leading to a non-significant response, contrasting with the existing literature.

Conclusions

This paper develops a Double Machine Learning estimator for macroeconomic time series, where short samples, persistence, and endogeneity limit the effectiveness of standard cross-fitting and predictive tuning.

We contribute two methodological innovations. First, Reverse Cross-Fitting, exploiting the time-reversibility of stationary processes to construct folds that increase sample usage while preserving the asymptotic independence conditions for orthogonal scores. Second, the Goldilocks zone tuning rule is a stability-based criterion that targets hyperparameter regions that jointly deliver predictive performance and local stability. Together, these features improve estimation accuracy in short and dependent samples while maintaining the formal inferential guarantees of Double Machine Learning.

\sloppy We prove that the estimator admits an asymptotic linear representation, is $\sqrt{T}$-consistent and asymptotically normal, and that HAC estimators consistently recover the long-run variance.

\sloppy Monte Carlo experiments confirm these theoretical results. In approximately sparse time-series designs, the estimator with Reverse Cross-Fitting yields lower bias and more stable coverage than existing cross-fitting schemes, particularly in short samples with strong dependence. The Goldilocks tuning rule further improves accuracy and enhances the performance of other cross-fitted DML variants. Deviations from time-reversibility, such as GARCH volatility, reduce accuracy but do not undermine inferential validity.

We also extend the framework to dynamic causal analysis using Local Projections with DML residualization. The results display limited bias and correct coverage under the model, even under high-dimensional controls.

An empirical application to prudential capital shocks in Italy illustrates the method's practical relevance. The DML estimator with Reverse Cross-Fitting yields economically plausible impulse responses aligned with established evidence.

Our approach offers an alternative, rather than a universally superior, adaptation of Double Machine Learning to serially dependent data. Theoretical and simulation results characterize the settings in which Reverse Cross-Fitting is most advantageous.

Overall, the paper provides a feasible and theoretically grounded framework for deploying orthogonal-score methods in macroeconomic time series, a context in which standard DML is unreliable and structural uncertainty is substantial.

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