Localized LOO Frequency Domain CV (FDCV)

• The paper's main contribution is integrating leave-one-out strategies with frequency domain localization to improve the reliability of spectral estimation and HAC inference.
• FDCV systematically tunes both model structure and smoothing parameters by minimizing a localized leave-one-out log-likelihood, effectively addressing issues of autocorrelation and heteroskedasticity.
• The approach employs a prewhitened kernel estimator with the Burg algorithm to ensure stable, lower-bias spectral density estimates for more accurate long-run variance estimation.

Localized Leave-One-Out Frequency Domain Cross-Validation (FDCV) is a class of cross-validation techniques tailored for reliable evaluation and tuning of prediction rules and spectral estimators where the focus is localized to specific frequency components—typically low frequencies where estimation of the long-run variance or spectral density at zero is critical. By integrating leave-one-out strategies with frequency-domain localization, FDCV addresses both statistical and computational challenges in robust, heteroskedasticity and autocorrelation consistent (HAC) inference, model assessment, and regularization parameter selection, especially in time series, regression, and high-dimensional settings.

1. Frequency Domain Localization and Leave-One-Out Principle

FDCV adapts the leave-one-out approach to the frequency domain, systematically excluding individual or blocks of frequencies when computing spectral density or related estimators. The canonical FDCV procedure operates as follows:

• For each Fourier frequency $\omega_j$, a leave-one-out estimator $f^{(-j)}(\omega_j)$ is constructed, omitting the contribution at frequency $\omega_j$—often by replacing the Fourier coefficient at $\omega_j$ with a local average (e.g., of its nearby values) to avoid information leakage and ensure invariance properties (Xu et al., 2021, Li et al., 27 Sep 2025).
• The FDCV criterion aggregates prediction errors or likelihood contributions only over a select band of frequencies near zero: $$\mathrm{FDCV}(f, c) = \frac{1}{N_c} \sum_{j=1}^{N_c} Q\big(I(\omega_j), f^{(-j)}(\omega_j)\big)$$ where $I(\omega_j)$ is the periodogram, $f^{(-j)}$ is the leave-one-out estimator, $Q(\cdot)$ is a local prediction error or log-likelihood loss, and $N_c \approx \left\lfloor (n/2)^c \right\rfloor$ for some $c \in (0,1)$ so that, asymptotically, only the lowest frequencies are included (Xu et al., 2021, Li et al., 27 Sep 2025).

The leave-one-out mechanism, localized to frequencies critical for application (most typically, the vicinity of $\omega=0$ for HAC estimation), has the effect of targeting predictive accuracy where inference is most sensitive and theoretical guarantees of estimator performance are most relevant.

2. Simultaneous Model and Tuning Parameter Selection

A central innovation of FDCV is its ability to tune not just smoothing bandwidths, but also model structure parameters such as the order of prewhitening filters:

• In the context of prewhitened kernel-based HAC variance estimation, FDCV is used to select both (a) the order $q$ of a prewhitening VAR (vector autoregression) fitted to the series of interest and (b) the bandwidth $m$ of the smoothing kernel (Li et al., 27 Sep 2025).
• The multivariate localized leave-one-out log-likelihood function is minimized over candidate $(q, m)$:

$$\mathrm{CVLL}_c(q, m) = \sum_{j=1}^{N_c} \left\{ \log\det\big[ \hat f_{-j}(\omega_j; q, m) \big] + \mathrm{tr}\big[ I(\omega_j) \hat f_{-j}^{-1}(\omega_j; q, m) \big] \right\}$$

where $\hat f_{-j}$ is the leave-one-out estimator with parameters $(q,m)$.

• This enables fully data-driven selection of all structural tuning parameters, adapting to both the persistence and the smoothness of the underlying process (Xu et al., 2021, Li et al., 27 Sep 2025).

This two-dimensional tuning is critical for robust long-run variance estimation: conventional methods without frequency localization or prewhitening can underperform or misrepresent uncertainty, especially when the spectral peak is sharp or processes are highly persistent.

3. Prewhitened Kernel Estimators and Burg Method Implementation

The effectiveness of FDCV for HAC estimation rests heavily on the underlying prewhitened kernel estimator:

• The prewhitening step fits a VAR($q$) to a series such as $V_t = u_t X_t$ (regression residual times predictor), yielding residuals $\tilde V_t$ assumed to be closer to white noise (Li et al., 27 Sep 2025).
• Smoothing is performed on $\{\tilde V_t\}$ with a kernel $k(r/m)$ to produce an estimate $\widehat{S}_{\tilde V}$. "Recoloring" is accomplished via inversion of the VAR coefficient matrix $A$: $\widehat{S}_V = \Phi \widehat{S}_{\tilde V} \Phi'$ where $\Phi = (I - \sum_{i=1}^q \widehat{A}_i)^{-1}$.
• The paper (Li et al., 27 Sep 2025) finds that VAR estimation via the Burg algorithm is preferable to OLS: Burg estimation guarantees that the filter is stationary by construction (all roots strictly within the unit circle), eliminating the need for ad hoc eigen adjustments that can introduce bias when regressors have nonzero mean.

This construction produces HAC estimators that are both consistent and exhibit reduced mean squared error compared to traditional alternatives, particularly in the presence of strong autocorrelation or structural mean shifts.

4. Flaws of Eigen Adjustment and Empirical Validation

Prior methods such as the AM-PW estimator [Andrews and Monahan, 1992] employ eigen value truncation for the VAR coefficients, intending to prevent instability when inverting $(I - A)$. The FDCV approach exposes two critical flaws:

• When regressors have nonzero mean, the singular values used to trigger eigen adjustment can become inflated, causing the adjustment to activate needlessly. This adjustment distorts the filter and can lead to substantial overestimation of standard error or confidence intervals, with distortions as much as 56% depending on mean shift and sample size (Li et al., 27 Sep 2025).
• The adjustment procedure lacks scale invariance; simply rescaling data may trigger or avoid the adjustment unpredictably.

Extensive Monte Carlo simulations and empirical applications (including regression analyses of unemployment vs. GDP and purchasing power parity data) show that:

• The FDCV-Burg estimator avoids the pitfalls of eigen adjustment, producing lower bias and more stable coverage rates, especially in cases with strong autocorrelation or nonzero regressor mean.
• Standard errors and coverage rates from the FDCV estimator remain appropriate, whereas eigen-adjusted estimators can be heavily upwardly biased or deliver intervals that are too conservative.

5. Mathematical and Algorithmic Formulation

The FDCV methodology involves several algorithmic and mathematical steps directly reflected in the cited source:

• Compute the (residual-based) product series $V_t$.
• Estimate the prewhitening VAR via the Burg method.
• Obtain prewhitened residuals $\tilde V_t$; compute leave-one-out DFTs $J_{-j}(\omega_k)$ for each frequency of interest, using substitution or local averaging (e.g., $$J^{(-j)}(\omega_k) = (J(\omega_{j-1}) + J(\omega_{j+1}))/2$$ for $k=j$ or $k=n-j$).
• Form leave-one-out estimators $\widehat{f}_{-j}(\omega_j; q, m)$ and periodograms $I(\omega_j)$.
• Minimize the localized CVLL criterion over $(q, m)$.
• Recolor and aggregate the selected spectral estimate to recover the HAC covariance estimate for inference.

This approach is both theoretically justified and numerically efficient, especially as the leave-one-out computations are restricted to a small band near frequency zero.

6. Implications and Scope of Applicability

FDCV provides a principled and robust route for automated tuning in HAC and related spectral inference contexts, including:

• Model selection in time series regression, where optimal prewhitening and smoothing are crucial for inference under heteroskedasticity and autocorrelation.
• Economic applications (e.g., Phillips–Perron unit root testing, KPSS stationarity tests) where accurate long-run variance and its uncertainty quantification are fundamental.
• High-dimensional series with possibly nonzero mean regressors, where prior eigen adjustment-based correction methods may actually harm statistical inference (Li et al., 27 Sep 2025).

The localization and leave-one-out machinery can theoretically be extended to other model structures or regimes (e.g., graph estimation in the frequency domain, or regularized models with frequency-localized risk), provided similar principles regarding leave-one-out constructions and critical frequency localization are obeyed.

7. Limitations and Future Directions

While the FDCV procedure as constructed addresses many pitfalls of prior HAC estimation approaches, several open challenges remain:

• Rigorous analysis of finite-sample behavior and robustness to structural model misspecification, particularly in the presence of weak identification or pronounced deviations from stationarity.
• Extension to multivariate and high-dimensional settings, including models with contemporaneous dependence among innovations or cross-series.
• Integration with advanced block or group cross-validation strategies to further mitigate overfitting or bias in settings with strong local (temporal or spatial) dependence structures (Wood, 25 Apr 2024).
• Consideration of distributional bias in localized frequency holds, as new work on rebalancing cross-validation identifies the danger of shifted sample means under leave-one-out deletions (Austin et al., 3 Jun 2024).

The FDCV framework as summarized here offers a robust, efficient, and practical route to improved uncertainty quantification and model assessment in the frequency domain, with empirical and theoretical support for its central innovations (Xu et al., 2021, Li et al., 27 Sep 2025).