Quasi-Likelihood Estimation Method

• Quasi-likelihood estimation is a suite of statistical techniques that use surrogate functions to reliably estimate parameters when the true likelihood is unavailable.
• It employs both Gaussian and non-Gaussian QMLE approaches to achieve consistency and efficiency even under heavy-tailed conditions and model misspecification.
• Extensions such as penalized, composite, and adaptive quasi-likelihood techniques enhance variable selection and accommodate high-dimensional or dependent data.

The quasi-likelihood estimation method is a suite of statistical techniques for parameter estimation in stochastic models where the true likelihood is unavailable, analytically intractable, or computationally prohibitive. Rather than maximizing the actual likelihood, quasi-likelihood approaches maximize or otherwise use a surrogate criterion, typically chosen to match certain key properties (such as mean, variance, or tail behavior) of the data-generating process. Quasi-likelihood methods have become foundational in fields such as time series analysis, stochastic processes, spatial statistics, econometrics, and high-frequency financial modeling. Their theoretical and practical significance is reflected in the diversity of extensions, including penalized quasi-likelihood for variable selection, measure-transformed quasi-likelihood for robust inference, and composite forms for high-dimensional, structured, or dependent data.

1. Key Concepts and General Framework

At its core, a quasi-likelihood is a function $Q(\theta; x)$ that plays the formal role of a likelihood in parameter estimation, but may not correspond to the true conditional density of the data. Formally, the quasi-maximum likelihood estimator (QMLE) is defined as

$$\hat{\theta}_n = \mathrm{argmax}_{\theta\in\Theta} Q_n(\theta)$$

where the "quasi-log-likelihood" $Q_n(\theta)$ is constructed from a model-based approximation or a function with certain optimality properties (e.g., unbiasedness, minimal asymptotic variance). In the time series context, the QMLE often relies on Gaussian approximations, but extensions to non-Gaussian and heavy-tailed settings are now well developed (Qi et al., 2010, Todros et al., 2015, Masuda, 2016).

A defining feature is that QMLE can provide consistent and asymptotically normal estimates even under model misspecification, provided the quasi-likelihood satisfies appropriate regularity and identifiability conditions.

2. Quasi-Likelihood in Time Series and Stochastic Processes

In GARCH, diffusion, and Lévy-driven models, the quasi-likelihood approach enables consistent estimation even when the exact innovation or noise distribution is unknown or heavy-tailed.

Gaussian and Non-Gaussian QMLE

• Gaussian QMLE: For example, in GARCH or state-space models, the QMLE based on a Gaussian innovation density is consistent under general moment and mixing conditions (Qi et al., 2010, Schlemm et al., 2012, Höök et al., 2015). The estimator remains robust as long as the first two moments are correctly specified, but may lose efficiency under heavy-tailed innovations due to increased asymptotic variance.
• Non-Gaussian QMLE and Scale Correction: When the innovation is heavy-tailed, using a heavy-tailed quasi-likelihood (e.g., Student's $t$, generalized Gaussian) can dramatically improve efficiency. However, direct use without a scaling correction leads to inconsistency – a phenomenon corrected by the two-step approach in (Qi et al., 2010), where an unknown scale parameter $\eta_f$ is estimated from residuals, yielding the "two-step non-Gaussian QMLE" (2SNG-QMLE) that is both consistent and more efficient.

Quasi-Likelihood in Diffusion Processes

• Numerical Approaches: For discretely observed SDEs, efficient computation of the Gaussian quasi-likelihood is achieved by numerically solving the Kolmogorov-backward equation, enabling unbiased parameter estimation even under random sampling architectures and without the biases suffered by Euler-Maruyama approximations (Höök et al., 2015).
• Nonparametric Estimation: Penalized quasi-likelihood estimation is used for nonparametric inference of the diffusion coefficient, with the maximizer being a natural spline whose degree and knots are determined by the penalization parameter and data (Hamrick et al., 2010). This approach directly enforces smoothness via a roughness penalty and exhibits convergence rates comparable to kernel-based estimators.

Lévy and Stable-Driven Models

In pure-jump SDEs, the small-time distribution is generally non-Gaussian and often stable (heavy-tailed); using a non-Gaussian quasi-likelihood (constructed from the stable law or from a Cauchy approximation in the Student-Lévy case) yields consistent and efficient estimation even in situations where Gaussian QMLE fails (Masuda, 2016, Masuda et al., 2023).

3. Composite, Penalized, and Adaptive Quasi-Likelihood Techniques

Composite Quasi-Likelihood

Composite quasi-likelihood (CQL) methods maximize a sum of lower-dimensional marginal or conditional quasi-likelihood contributions, making high-dimensional or structurally complex estimation feasible (Chu, 2017). This framework supports models with group-specific heterogeneity and spatially dependent errors, and facilitates simultaneous estimation and classification (e.g., latent group membership).

Penalized Quasi-Likelihood for Variable Selection

Penalization (lasso, bridge, or adaptive forms) is integrated into the quasi-likelihood framework to address high-dimensional inference and variable selection (Kinoshita et al., 2019, Ning et al., 2 May 2024). The theory guarantees, under a polynomial-type large deviation inequality, that moments of the penalized estimator converge and that the correct model is selected with probability tending to one.

Adaptive and Partial Quasi-Likelihood

For degenerate or partially observed diffusion processes, "adaptive" quasi-likelihood estimation employs preliminary estimators from the nondegenerate component to refine estimation of the degenerate (latent) part (Gloter et al., 23 Feb 2024). Partial QLA, in turn, handles models with slow-mixing components by conditioning out the problematic part, ensuring limit theorems still hold for the quasi-likelihood estimator (Yoshida, 2017).

4. Extensions: Semi- and Nonparametric, Robust, and Non-Standard Data Domains

Robust and Measure-Transformed Quasi-Likelihood

Robustness to model misspecification, heavy tails, and outliers can be achieved through measure transformation. The measure-transformed GQMLE applies a data-dependent transformation (via a weight function $u(x)$), optimizing sensitivity to higher-order moments and reducing the influence of outliers (Todros et al., 2015).

Semi-Parametric and Pseudo-Variance Quasi-Likelihood

Estimation in observation-driven time series models with nonstandard distributions (e.g., integer-valued or bounded support) can be accomplished by specifying a parametric conditional mean and a pseudo-variance function. Imposing constraints on the pseudo-variance can yield substantial efficiency gains and facilitate specification testing, with asymptotic theory covering both unrestricted and restricted estimators (Armillotta et al., 2023).

Spatial and Marked Point Processes

In spatial statistics, the optimal first-order estimating function for spatial point process intensity is derived via a Fredholm integral equation that incorporates the pair correlation structure; the resulting quasi-likelihood estimator substantially outperforms composite likelihood estimators in clustered or inhomogeneous settings (Guan et al., 2013).

For multivariate marked point processes—including marked Hawkes processes—the QLA framework enables LAN expansions and moment convergence under verifiable ergodicity and stability conditions, supporting rigorous inference in high-frequency event modeling (Clinet, 2020).

5. Practical Implementation and Empirical Evidence

Quasi-likelihood methods are implemented with techniques ranging from direct iterative maximization (including Expectation-Maximization with embedded quasi-likelihood steps (Cheng et al., 9 Dec 2024)) to numerical solutions of penalized spline criteria (Hamrick et al., 2010) and the use of Kalman filtering in state-space models (Schlemm et al., 2012). Specialized algorithms such as the ECME and DC (Difference-of-Convex) programming address computational challenges and non-convexity in high-dimensional spaces (Phillips, 2017, Chu, 2017).

Monte Carlo and real data experiments in the literature demonstrate:

• Improved efficiency in heavy-tailed and non-Gaussian scenarios (GARCH models, financial time series, spatial processes)
• Consistency and robustness under misspecification, heteroskedasticity, and dependent data (dynamic panels, regime-switching SDEs)
• Effective model selection and variable screening when combined with penalization or pseudo-variance restrictions

Performance comparisons systematically show that quasi-likelihood estimators can match or surpass classical MLE, GMM, or method-of-moments estimators in both efficiency and finite-sample accuracy, especially in nonideal, high-frequency, or dependent scenarios.

6. Limitations, Open Problems, and Future Research

While quasi-likelihood approaches are broadly applicable, certain limitations remain:

• The accuracy of the quasi-likelihood depends on the quality of the chosen surrogate function; improper choice may yield inefficient or, in some non-Gaussian settings, inconsistent estimates unless corrected (as with the scale-corrected 2SNG-QMLE).
• Boundary effects, discretization bias, and stability of numerical schemes may affect estimation, particularly in nonparametric or high-frequency diffusion models.
• Asymptotic theory often requires moment or mixing conditions (e.g., strong mixing, ergodicity); handling models with weak dependence, long memory, or extreme heavy tails demands further refinement.
• Extensions to high-dimensional latent variable models, semi/nonparametric models with infinite dimensions, and models with nonstationary regimes are active areas of research.

Ongoing directions include the development of adaptive and robust methods that combine penalized or measure-transformed QLA with data-driven parameter selection, the unification of composite quasi-likelihood for broad classes of structured data, and the refinement of algorithms for real-time or online inference in large-scale dependent systems.

Summary Table: Representative Quasi-Likelihood Estimation Methods

Class of Model	Quasi-Likelihood Approach	Key Innovations / Features
GARCH, GARCH-like (heavy-tails)	2SNG-QMLE, Non-Gaussian QMLE	Scale correction, efficiency in heavy tails
Diffusion / SDE (discrete observations)	Penalized, Splines, Kolmogorov-Backward	Nonparametric, efficient numerical schemes
Lévy-driven SDE, Student-Lévy regression	Stable/Cauchy-based QL, Two-step QLE	Local heavy-tail approximation, thinning
Dynamic Panels / Mixed Models	QMLE, Penalized QL, Composite QL	Robust to heterogeneity, selection, DC alg.
Spatial Point Process	Integral equation-based QL	Incorporates pair correlation structure
Robust Linear/Array Regression	Measure-Transformed QMLE	Weighted moment matching, resilience
Marked Point/Hawkes Processes	QLA for general intensity	LAN/moment convergence, ergodicity
Semi-parametric/Count/Bonded Time Series	Pseudo-variance QMLE, Restriction tests	Efficient, supports model specification
Regime-Switching SDEs (latent Markov)	QL-EM, small-time approximation	EM with Cauchy surrogates for NIG noise

This summary encapsulates the foundational principles, methodological details, and breadth of application of quasi-likelihood estimation, as well as significant advances in estimation accuracy, robustness, and computational tractability that the various forms of QLA deliver across modern statistical and probabilistic modeling domains.