Consistent Variable Selection for GARCH-X Models

1. Problem Statement

Financial volatility modeling often relies on GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models. While standard GARCH models capture endogenous volatility dynamics (past shocks and past volatility), empirical evidence suggests that external (exogenous) factors—such as macroeconomic indicators, news intensity, or commodity prices—significantly influence volatility.

The GARCH-X model extends the standard framework to include these exogenous covariates. However, a critical challenge arises when the number of potential covariates is large:

• Model Selection Difficulty: Existing methods primarily focus on selecting the order of the model (e.g., determining $p$ and $q$ in GARCH($p,q$)) rather than selecting specific exogenous variables.
• Computational Burden: Exhaustively testing all combinations of covariates using penalized criteria (like BIC or AIC) is computationally prohibitive when the number of covariates is moderate to large.
• Lack of Theoretical Guarantees: There was no established procedure with theoretical consistency results specifically for selecting exogenous covariates in volatility models. The goal is to identify the "true" subset of relevant covariates that drive volatility while excluding irrelevant ones.

2. Methodology

The authors propose a consistent variable selection procedure based on a multiple hypothesis testing framework combined with False Discovery Rate (FDR) control.

A. The Model Framework

The paper utilizes the GARCH($p,q$)-X model:
$$\epsilon_t = \sigma_t w_t$$
$$\sigma_t^2(\theta) = \omega + \sum_{i=1}^p \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^q \beta_j \sigma_{t-j}^2 + \gamma^T X_{t-1}$$
Where $X_{t-1}$ is a vector of $d$ exogenous covariates, and $\gamma = (\gamma_1, \dots, \gamma_d)'$ represents the coefficients to be tested. The objective is to recover the true index set $I_0 = \{k : \gamma_k \neq 0\}$.

B. Hypothesis Testing

For each covariate $k$, the authors formulate a one-sided hypothesis test:

• $H_0^k: \gamma_k = 0$ (Covariate is irrelevant)
• $H_1^k: \gamma_k > 0$ (Covariate is relevant)

They utilize the Quasi-Maximum Likelihood Estimator (QMLE) $\hat{\theta}_n$. Under regularity conditions (established in Francq and Thieu, 2019), the estimator is consistent and asymptotically normal (projected onto the parameter space due to non-negativity constraints).

The test statistic for each covariate is a Wald-type statistic:
$$t_n(k) = \frac{\sqrt{n} e_k^T \hat{\theta}_n}{\sqrt{e_k^T \hat{\Sigma}_n e_k}}$$
Where $\hat{\Sigma}_n$ is the estimated asymptotic covariance matrix. Due to the positivity constraint on parameters, the squared statistic $t_n^2(k)$ follows a chi-bar-square distribution (a mixture of a point mass at 0 and a $\chi^2_1$ distribution).

C. Variable Selection via FDR Control

Instead of using a fixed significance level (which leads to high Type I errors in multiple testing), the authors apply the Benjamini–Yekutieli (2001) procedure to control the False Discovery Rate (FDR).

1. Compute p-values $\pi_k$ for all $d$ covariates.
2. Order the p-values: $\pi_{(1)} \leq \dots \leq \pi_{(d)}$.
3. Determine the largest index $\ell$ such that:
   $$\pi_{(k)} \leq \frac{k}{d} \frac{\alpha_n}{\sum_{l=1}^d l^{-1}}$$
4. Reject the null hypotheses for the first $\ell$ covariates. The selected set $\hat{I}$ consists of the indices of these rejected hypotheses.

D. Consistency Conditions

The method is proven to be consistent (i.e., $P(\hat{I} = I_0) \to 1$ as $n \to \infty$) under specific conditions:

• Condition (C1): The asymptotic variances of the estimators must decrease sufficiently fast relative to the sample size ($\sigma_{kk} < n/\log(n)$).
• Signal Strength: The significance level $\alpha_n$ must decay to 0 at a specific rate ($\alpha_n = o(e^{-cn})$) as the sample size increases, allowing the method to distinguish true signals from noise even if the signal strength diminishes slightly.

3. Key Contributions

1. Theoretical Consistency: The paper provides the first formal proof that a variable selection rule based on FDR control is consistent for GARCH-X models, ensuring the recovery of the true set of covariates asymptotically.
2. Handling Constraints: The methodology explicitly accounts for the non-negativity constraints of GARCH parameters, utilizing the correct chi-bar-square distribution for p-value calculation rather than standard normal approximations.
3. Computational Efficiency: By avoiding exhaustive search over all subsets and using a step-up FDR procedure, the method scales efficiently with the number of covariates.
4. Robustness to Signal Decay: The theoretical framework allows for "diminishing signal strength," meaning the method works even if the effect size of relevant covariates becomes smaller as the sample size grows, provided the decay is not too rapid.

4. Results

A. Monte Carlo Simulations

The authors conducted extensive simulations across four scenarios:

• Scenarios 1 & 2: GARCH(1,1)-X with independent and exponentially decaying correlated covariates.
• Scenarios 3 & 4: GARCH(2,1)-X with independent and correlated covariates.
• Distributions: Errors were generated from Normal, Student's $t$ (5 and 7 degrees of freedom) to test robustness against heavy tails.
• Sample Sizes: $n = 500, 1000, 5000, 10000$.

Key Findings:

• Accuracy: As sample size increased, the method correctly identified relevant covariates with probability approaching 1. For $n=5000$ and $n=10000$, selection rates for true covariates were near 100%, while false selection rates for irrelevant covariates were negligible (< 5%).
• Robustness: The method performed well under heavy-tailed distributions (Student's $t$), though convergence was slightly slower compared to the Normal case.
• Model Selection Criteria: The models selected by the proposed method consistently achieved lower (better) BIC and AIC values compared to the "full model" (all covariates) and the "null model" (no covariates) at larger sample sizes, demonstrating an optimal trade-off between parsimony and fit.

B. Real Data Application: S&P 500 Volatility

The method was applied to daily S&P 500 returns (Jan 2000 – May 2025) using 9 exogenous covariates (NASDAQ, Dow Jones, Crude Oil, Natural Gas, Gold, Palladium, Rice, Steel, Wheat).

• Selection Results: At a 0.05 adjusted significance level, the procedure selected NASDAQ and Crude Oil as the only statistically significant drivers. Steel was marginally significant.
• Model Comparison:
  • The Selected Model (NASDAQ + Crude Oil) had a BIC of 141,721.2 and AIC of 141,689.4.
  • The Full Model (all 9 covariates) had a higher BIC (141,792.3) and AIC (141,715.9), indicating overfitting.
  • The Null Model (no covariates) performed worse than the selected model in terms of AIC, confirming that external factors do add explanatory power.
• Conclusion: The selected model provided a more parsimonious and interpretable explanation of S&P 500 volatility, capturing market sentiment (NASDAQ) and energy/inflation shocks (Oil) without unnecessary complexity.

5. Significance

This paper bridges a critical gap in financial econometrics by providing a rigorous, theoretically grounded, and computationally feasible tool for variable selection in volatility modeling.

• Practical Utility: It allows researchers and practitioners to objectively identify which macroeconomic or financial indicators truly drive market volatility, avoiding the "kitchen sink" approach of including all available data.
• Theoretical Advancement: It extends the application of FDR control from regression contexts to non-linear, heteroskedastic time series models with complex parameter constraints.
• Risk Management: By identifying the true drivers of volatility, the method improves the accuracy of risk management tools (e.g., Value-at-Risk) and asset pricing models, particularly during periods of market stress where exogenous factors are most influential.