Hannan-Quinn Information Criterion

• Hannan-Quinn Information Criterion is a penalized likelihood criterion defined as -2 ln(L) + 2k ln ln(n) that balances model fit and complexity.
• It guarantees strong consistency by using a slowly growing penalty that mitigates both underfitting and overfitting, outperforming AIC in over-parameterized settings.
• HQIC is applied across various models such as time series, linear regression, and signal processing, offering practical benefits in finite and large-sample regimes.

The Hannan-Quinn Information Criterion (HQIC) is a penalized likelihood model selection criterion developed to provide strong consistency in choosing among parametric models, most notably for time series, linear regression, and more general statistical frameworks. It occupies an intermediate position between the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) by employing a penalty that grows more slowly with sample size than BIC but fast enough to guarantee consistency, thereby controlling both underfitting and overfitting risks in asymptotic regimes.

1. Formal Definition and Structure

In the context of maximum likelihood estimation, let $\mathcal{L}(\hat\theta)$ denote the maximized likelihood for a model with $k$ parameters fitted to a dataset of size $n$. The HQIC is defined as

$$\mathrm{HQIC} = -2 \ln \mathcal{L}(\hat\theta) + 2 k \ln\!\bigl(\ln(n)\bigr),$$

where the penalty term $2k \ln\ln(n)$ diverges with $n$—but slowly, as required for consistency (Ponta et al., 2012, Chen et al., 2017, Pas et al., 2014).

For linear regression, HQIC is often written as

$\mathrm{HQIC}(T) = n\log S(T) + |T|\,d_n,$

with $S(T)$ the residual sum of squares for subset $T$ and $d_n=2\log\log n$ (Suzuki, 2010).

2. Theoretical Justification and Consistency

The HQIC was introduced to address deficiencies in non-consistent criteria such as AIC, whose penalty ($2k$) is insufficient to prevent overfitting as $n\to\infty$, and BIC, whose rapidly increasing penalty ($k\ln n$) often produces excessive underfitting in small to moderate samples. Hannan and Quinn established that the minimal penalty function ensuring "strong consistency" (probability converges to 1 of selecting the true model among a finite list as $n\to\infty$) must satisfy

$$p(n)\to\infty,\quad \frac{p(n)}{n}\to 0,\quad\sum_{n=1}^\infty e^{-p(n)/2}<\infty,$$

and that $p(n)=2\ln\ln n$ is the slowest-growing penalty admissible under these constraints (Ponta et al., 2012, Suzuki, 2010). This grows more slowly than $k\ln n$ and ensures that the model selection probability converges almost surely.

In the nested-model setting, HQIC can be equivalently formulated as a likelihood difference threshold: $$\log\left[\frac{p_{\hat\mu_1}(x^n)}{p_{\hat\mu_0}(x^n)}\right] > c\,\log\log n$$ with $c>1$—typically $c=2$—to guarantee strong consistency (Pas et al., 2014).

3. Risk Properties and Comparison with Other Criteria

AIC achieves minimax-optimal parametric risk rates ($O(1/n)$) but is inconsistent: it can select over-parameterized models indefinitely. BIC is strongly consistent, with risk order $O\left(\frac{\log n}{n}\right)$. HQIC achieves a risk rate of $O\left(\frac{\log\log n}{n}\right)$, thereby interpolating between these two: it is consistent and loses only a double-logarithmic factor in worst-case risk relative to AIC (Pas et al., 2014). This balance is uniquely valuable: HQIC's penalty is asymptotically minimal among all consistent criteria, making it robust against overfitting but generally less conservative than BIC in moderate sample regimes.

Table: Penalty Comparison for Common Information Criteria

Criterion	Penalty Term	Consistency	Asymptotic Risk
AIC	$2k$	No	$O(1/n)$
HQIC	$2k\ln\ln n$	Yes (strong)	$O(\log\log n/n)$
BIC	$k\ln n$	Yes (strong)	$O(\log n/n)$

4. Applications Across Model Classes

The HQIC framework extends beyond classical time series or regression. In compound Poisson models for non-stationary financial returns (Ponta et al., 2012), Hawkes process order selection (Chen et al., 2017), and structured covariance estimation in radar signal processing with missing data (Aubry et al., 2021), HQIC consistently offers an effective compromise between overfitting and underfitting.

• For high-frequency financial modeling, HQIC's intermediate penalty achieves accurate model order detection for sample sizes $n \gtrsim 10^3$, especially in the regime where AIC tends to overfit and BIC underfits.
• In linear regression, HQIC is the minimal penalty ensuring strong consistency in subset selection (Suzuki, 2010).
• For radar and array signal processing, HQIC (generalized via EM) provides reliable source number estimation even in missing-data settings, outperforming both AIC and MDL in moderate samples (Aubry et al., 2021).
• For affine-causal time series models (including ARMA, GARCH, APARCH), generalized HQIC with a calibrated penalty maintains strong consistency under weak moment and information conditions (Kamila, 2021).

5. Limit Theory and Empirical Findings

Extensive limit-theoretic analyses confirm that HQIC is strongly consistent in numerous regimes:

• In AR($p$) and linear regression, HQIC with $d_n=2\log\log n$ is the minimal divergent penalty to rule out persistent overfitting as $n\to\infty$ (Suzuki, 2010).
• In time series model selection for processes such as ARMA, GARCH, and their generalizations, HQIC—when calibrated—outperforms BIC in moderate samples, recovering the true model at a higher rate, and remains consistent asymptotically (Kamila, 2021).
• In distinguishing AR(1) with unit roots and various classes of explosive roots, HQIC (like BIC) is consistent for unit roots and various explosive regimes, but not for local-to-unit-root alternatives; this sharply delineates the scope of its strong consistency (Tao et al., 2017).

Monte Carlo simulations consistently show that HQIC provides a high correct-model selection frequency as $n$ grows, converging to $100\%$ in large-sample limits for models where theoretical consistency is guaranteed (Ponta et al., 2012, Chen et al., 2017). In finite samples, HQIC's performance is situated between AIC’s tendency to overfit and BIC's underfitting.

6. Adaptation, Calibration, and Practical Recommendations

Generalized HQIC formulations allow for adaptive calibration of the penalty constant when model complexity or innovation distribution deviates from standard settings (Kamila, 2021). Estimating model-specific constants via slope heuristics or empirical moment estimation is effective for high-dimensional or nonstandard models. In practice:

• For moderate model complexity and sufficient sample size, HQIC is recommended for balancing over- and underfitting risks.
• If candidate models are highly complex or data are sparse, combining HQIC with cross-validation or hold-out techniques is advisable.
• HQIC should not be used in isolation when selection consistency is critical for high-order or complex models under weak information; BIC or cross-validated selection may provide better robustness to underfitting.

7. Extensions and Implications

The HQIC penalty rate is now recognized as a theoretical benchmark for minimal consistent penalization in model selection (Pas et al., 2014, Chen et al., 2017, Kamila, 2021). The switch criterion, a Bayesian-agile alternative, has been shown to essentially match HQIC asymptotically in nested exponential family settings, confirming conjectures by Lauritzen and Cavanaugh and establishing the $2k\log\log n$ penalty as “almost best” for the trade-off between risk and consistency (Pas et al., 2014).

Recent empirical and theoretical work demonstrates that HQIC is robust for contemporary financial, signal processing, and econometric time series applications, with adaptation available for heavy-tailed or nonstandard settings. Limitations exist in near-nonidentifiable or locally subcritical regimes (e.g., local-to-unity alternatives in AR processes), where all consistent ICs fail to detect the alternative, reflecting a fundamental limit rather than a defect of HQIC specifically (Tao et al., 2017).