Quasi-Likelihood Functions

• Quasi-likelihood functions are statistical constructs that approximate the full log-likelihood using moment conditions when complete distributional specifications are unavailable.
• They employ regularization techniques such as penalization and spline-based methods to stabilize estimation in nonparametric and high-dimensional contexts.
• Widely applied in econometrics, time series, spatial statistics, and stochastic processes, they offer robust asymptotic properties and computational efficiency.

A quasi-likelihood function is a statistical construct that generalizes the traditional likelihood by approximating or substituting the true probability model with a function that encodes only partial distributional information (typically the mean and variance, or derived estimating equations). In econometrics, time series, spatial statistics, and stochastic process inference, quasi-likelihoods are used when the full likelihood is intractable or unknown, but moment approximations or local expansions allow for efficient estimation, robust inference, or computational feasibility. Their adoption in nonparametric settings, high-dimensional models, Bayesian inference, and penalized estimation contexts is supported by strong asymptotic theories and modern computational strategies.

1. Core Principles and Mathematical Formulation

The quasi-likelihood function $\mathcal{Q}(\theta)$ is constructed to approximate the log-likelihood $\log L(\theta)$ using a working model—often relying only on the first two moments or conditional moment restrictions. In stochastic process models, quasi-likelihoods are frequently based on local Gaussian approximations:

• For a diffusion process $dY_t = b(Y_t) dt + \sigma(Y_t) dW_t$, high-frequency increments are approximated as

$$Y_{t_i} | Y_{t_{i-1}} \approx \mathcal{N}\left(Y_{t_{i-1}}, \sigma^2(Y_{t_{i-1}})(t_i - t_{i-1})\right),$$

leading to a (scaled) quasi-log-likelihood of the form

$$qll(\theta) = \frac{1}{n} \sum_{i=1}^n \left\{ \theta(Y_{t_{i-1}}) - \frac{1}{2} r_{i-1}^2 \exp(2\theta(Y_{t_{i-1}})) \right\},$$

where $r_{i-1}$ are local volatility estimators and $\theta(x) = -\log \sigma(x)$ (Hamrick et al., 2010).

In generalized linear or semiparametric models, the quasi-likelihood is defined so that its first derivative matches the optimal unbiased estimating equation for the mean structure and variance function:

$$\ell_q(\beta; y, X, \psi) = \frac{1}{\psi} \sum_{i=1}^n \int_a^{\mu_i} \frac{y_i - t}{V(t)} dt,$$

where $\psi$ is a dispersion parameter and $V(\cdot)$ is a mean-variance function (Agnoletto et al., 2023).

In nonparametric settings built upon conditional moment restrictions—such as nonparametric instrumental variables models—the quasi-likelihood may take the form of a penalization of squared empirical moment conditions:

$$\mathcal{Q}(g) = \exp\left\{ -\frac{n}{2} \mathbb{E}[\hat{m}(W, g)^2] \right\}.$$

(Kato, 2012).

2. Regularization and Penalization

To achieve well-posedness and prevent overfitting in nonparametric and high-dimensional settings, quasi-likelihoods are often penalized:

$$\text{pqll}(\theta; m, \lambda) = qll(\theta) - \frac{\lambda}{2}\int_{-\infty}^{+\infty} |\theta^{(m)}(z)|^2 dz$$

where $\lambda$ is a tuning parameter and $m$ determines the order of smoothness (e.g., $m=2$ for cubic splines) (Hamrick et al., 2010). Penalized quasi-likelihood estimation naturally regularizes the estimator, shrinks coefficients, or enforces sparsity (for example, by adding a Bridge or Lasso-type penalty in variable selection (Kinoshita et al., 2019)). In the context of model selection and sparsity, such penalization enables consistent identification of zero components, with precise rates governed by large deviation inequalities.

3. Numerical Implementation and Computational Aspects

The structure of the quasi-likelihood objective enables efficient numerical algorithms:

• Spline-based representation: For diffusion and volatility estimation, the penalized maximum quasi-likelihood estimator for the transformed volatility function is characterized as a natural spline of order $2m-1$ with knots at observed data points. The fundamental equations governing the estimator are given by piecewise recursion:

$$\theta_{*}^{(2m-1)}(y) = \frac{(-1)^m}{n\lambda}\left\{ k - \sum_{j=1}^k r_j^2 \exp(2\theta_{*}(y_j)) \right\},\quad y \in [y_k, y_{k+1})$$

(Hamrick et al., 2010).

• Shooting/Root-finding method: The boundary value characterization leads to a shooting algorithm or Newton-type iterative procedures in low dimensions, with boundary constraints enforcing natural spline properties. Quantities such as derivatives of the objective with respect to initial values are computed recursively for efficient convergence (Hamrick et al., 2010).
• Profiling and Marginalization: For models with random effects or high-dimensional latent variables, profiling (i.e., maximizing with respect to nuisance parameters before plugging the maximizer back into the objective) reduces computational dimensionality and leverages the tractable structure of the quasi-likelihood (Delattre et al., 25 Aug 2025).
• Handling data irregularities: In high-frequency settings, data might contain ties or irregular intervals; tie breaking (adding small random perturbations) and accurate handling of sampling interval lengths are crucial for validity (Hamrick et al., 2010).

4. Asymptotic Properties and Theoretical Guarantees

Under regularity and high-frequency asymptotics, quasi-likelihood estimators display the following properties:

• Convergence rates: For penalized estimators, optimal rates are achieved depending on the degree of regularization. With the regularization parameter chosen as $\lambda_n \propto n^{-2m/(2m+1)}$, convergence in $L^2$ is of rate $n^{-m/(2m+1)}$. For example, cubic spline penalization ($m=2$) leads to an empirical slope of about $-0.398$ in RMSE vs. $n$ on log-log scale (Hamrick et al., 2010).
• Asymptotic (mixed) normality and moment convergence: Detailed expansions (such as local asymptotic normality or quadratic approximations of the log-likelihood field) support central limit theorems, with scaling determined by the information matrix. Quasi-likelihood ratio statistics based on adaptive estimators converge in distribution to $\chi^2$-distributions under the null hypothesis, facilitating hypothesis testing (Nishikawa et al., 24 Feb 2025).
• Robustness to model misspecification and nonparametric efficiency: Quasi-likelihood estimators achieve optimal minimax rates in nonparametric settings, and their Bayesian equivalents (quasi-posteriors) satisfy Bernstein–von Mises theorems, with explicit normal approximations in the limit (Kato, 2012).
• Model selection properties: Via polynomial type large deviation inequalities (PLDI), moment convergence and sharp control over type-I and type-II errors in variable selection are established for penalized quasi-likelihood estimators (Kinoshita et al., 2019).

5. Applications and Case Studies

Quasi-likelihood methodologies are applied broadly in stochastic process inference and econometrics, notably:

• Financial econometrics: Inference for diffusion coefficients from LIBOR, exchange rates (USD/EUR, USD/GBP, etc.), and treasury yields demonstrates the flexibility of nonparametric volatility estimation, revealing state-dependent, non-monotonic volatility functions that deviate from standard parametric models (Hamrick et al., 2010).
• Adaptive estimation in jump-diffusions: In multidimensional SDEs with jumps, adaptive quasi-likelihood functions separate inference for continuous (diffusive) and discontinuous (jump) components, improving stability and accuracy in high dimensions. Observed increments are filtered via thresholds to construct the quasi-likelihood for each component, and separate maximization is adopted for each subset of parameters (Nishikawa et al., 24 Feb 2025).
• Mixed-effects models at high frequency: In SDEs with fixed and random effects, profiling the random effect in the diffusion coefficient enables stepwise and computationally efficient inference; explicit quasi-likelihood formulas bypass the need for high-dimensional integration, facilitating high-dimensional or many-individual settings (Delattre et al., 25 Aug 2025).

The table below summarizes trade-offs between approaches:

Quasi-Likelihood Setting	Regularization/Filtering	Main Computational Tool
Nonparametric volatility (diffusion)	m-th derivative penalization	Shooting method for spline ODE
SDEs with jumps	Adaptive thresholding	Separate maximization for diffusion and jump parts
Mixed-effects SDE	Profiling of random effect	Profile likelihood and marginalization (Euler approx.)

6. Limitations and Practical Considerations

While quasi-likelihood methods are highly flexible and general, care must be taken regarding:

• Boundary effects: Spline-based estimators may over-smooth at the boundaries due to natural boundary conditions (Hamrick et al., 2010).
• Choice of regularization: Theoretical rates depend on appropriate scaling of the penalization parameter; over/under-regularization impacts estimator smoothness and bias.
• Computational stability: Newton or shooting methods require good initial guesses and may struggle as data size increases or penalty weight becomes large; thinning and sub-sampling strategies can mitigate these issues (Hamrick et al., 2010).
• Approximation error: In high-frequency and non-Gaussian settings, the validity of the local approximation (e.g., Cauchy vs. true Student–t increments) depends on carefully controlled thinning and attention to small-interval error (Masuda et al., 2023).

7. Significance in Modern Inference

Quasi-likelihood functions provide a foundational framework for likelihood-based inference under model uncertainty, misspecification, and computational intractability. Their seamless integration with nonparametric regularization, adaptive and penalized estimation, and stepwise high-dimensional optimization make them central to contemporary parametric, semiparametric, and nonparametric statistical modeling.

Their rigorously established asymptotic properties (including rates, limit theorems, and moment convergence) and practical flexibility—demonstrated in financial data, spatial modeling, and random-effects inference—ensure they remain a fundamental tool in the analysis of stochastic process and high-frequency data.