Quasi Maximum Likelihood Estimator

• Quasi Maximum Likelihood Estimator is a statistical method that maximizes a pseudo likelihood function even under potential distribution misspecification, ensuring consistency when conditional means are correctly specified.
• It is widely employed in models like GARCH, ARMA, and panel data, where tractable working densities such as Gaussian or Laplacian enhance estimation robustness through scale adjustments.
• Advanced variants such as measure-transformed and penalized QMLE improve efficiency and enable sparsity in high-dimensional settings, addressing non-regular and boundary parameter issues.

The quasi maximum likelihood estimator (QMLE) is a statistical estimation approach wherein a likelihood function is constructed under potentially misspecified distributional assumptions, while retaining consistent estimation of target parameters under relatively mild conditions. The QMLE framework is integral to modern inference across diverse econometric, time series, and machine learning models—particularly for complex or misspecified data-generating processes where true likelihoods are analytically intractable, unknown, or not fully specified.

1. Definition and General Principles

The QMLE generalizes classical maximum likelihood estimation by maximizing a "quasi" (or "pseudo") likelihood function which may not correspond to the true parametric family governing the data. Formally, for a sequence of observations $X_1, ..., X_n$, and parameter vector $\theta \in \Theta$, the QMLE $\hat\theta_n$ solves

$$\hat\theta_n = \arg\max_{\theta\in\Theta} \sum_{t=1}^n \log f(X_t; \theta)$$

where $f(\cdot; \theta)$ is a working (quasi-)density, not necessarily the density of the actual data-generating process. This likelihood-based criterion typically leverages the tractability and properties of well-understood families (such as the Gaussian, Laplace, Poisson, or exponential families), even when the actual error structure deviates from these choices.

Key generic properties of the QMLE include:

• Consistency under correct specification of the mean or conditional mean, even when higher moments or the overall distribution is mis-specified.
• Asymptotic normality under suitable regularity and identifiability conditions, with an asymptotic covariance matrix given in the "sandwich" form $J^{-1} I J^{-1}$, where $J$ is the limit Hessian and $I$ is the limit variance of the score.
• Robustness to certain forms of misspecification, notably in linear models, GARCH, ARMA, panel data, and state space settings.

2. QMLE in Heteroskedastic and Dependent Data Models

In time series models such as GARCH and ARMA, the QMLE is commonly employed with working Gaussian, Student-t, Laplace, or heavy-tailed likelihoods:

• GARCH models: The classical (Gaussian) QMLE for GARCH is robust (consistent) for the volatility parameters under very general innovation distributions, provided the conditional variance is correctly parameterized. However, as shown in "Non-Gaussian Quasi Maximum Likelihood Estimation of GARCH Models" (Qi et al., 2010), simply replacing the Gaussian quasi-likelihood with a heavy-tailed density (non-Gaussian QMLE) can lead to inconsistency unless a crucial scale parameter is estimated; their two-step QMLE (2SNG-QMLE) corrects this bias by:
  1. Fitting the Gaussian QMLE to obtain standardized residuals.
  2. Estimating a scale $\eta_f$ by maximizing The sample-average of $-\log \eta + \log f(\tilde\epsilon_t/\eta)$.
  3. Re-estimating the parameters via a scale-corrected likelihood. This approach significantly improves efficiency under heavy-tailed innovations, while retaining consistency.

• Affine Causal Time Series: In the context of general affine models $X_t = M_\theta(\cdot)\zeta_t + f_\theta(\cdot)$, the Laplacian QMLE (using an $L_1$-based likelihood) outperforms the Gaussian QMLE in both robustness and moment requirements, requiring only finite first moments for consistency and second moments for asymptotic normality (Bardet et al., 2016).

• Double Autoregressive (DAR) models: QMLEs can be tailored to specific working densities. The E-QMLE (Laplace) requires only a fractional moment for consistency, enabling robust inference for processes with very heavy tails (Liu et al., 2020).
• State Space Models and Lévy-Driven CARMA: QMLE, constructed using Kalman innovations in a Gaussian working likelihood, is strongly consistent and asymptotically normal under strong-mixing and spectral identifiability conditions, applicable even when the underlying innovations are non-Gaussian and possibly heavy-tailed (Schlemm et al., 2012). This framework extends to sampled continuous-time MCARMA models and yields small bias and accurate standard errors in multivariate settings.

3. Extensions: Measure-Transformed and Penalized QMLE

• Measure-Transformed QMLE (MT-GQMLE): Robustifies the classical Gaussian QMLE by transforming the data's underlying measure with a chosen weighting function before estimation. This approach allows for resilience to outliers and sensitivity to higher order moments, generalizes the QMLE beyond moment matching, and is still consistent and asymptotically normal (Todros et al., 2015).
• Penalized QMLE and Nonstandard Conditions: QMLE theory is extended to handle non-regular models where parameters may lie on the boundary of the parameter space, or identifiability is violated (Yoshida et al., 2022). Under these settings, adaptive normalization matrices and general local approximations allow for asymptotic mixed-normality and support penalized quasi-likelihood estimation, including selection consistency results for Lasso/Bridge penalties under suitable tradeoffs between penalty strength and parameter convergence rates.

4. QMLE in Dynamic Panels, Factor Models, and Large Systems

• Dynamic Panel Data: In panels, QMLE methods provide strong consistency and asymptotic normality for levels and differenced estimators, with robustness to initial conditions, heteroskedasticity, and misspecification. The ECME algorithm efficiently computes the QMLE even with time-series heteroskedasticity (Phillips, 2017). Monte Carlo results show that QMLE, especially in levels, consistently outperforms system GMM and other methods in bias and root mean squared error.
• Large Approximate Factor Models: In high-dimensional dynamic factor models, QMLE (often implemented via EM and the Kalman smoother) achieves $\sqrt{T}$-consistency for loadings and $\sqrt{n}$-consistency for factors, with asymptotic equivalence to principal components and WLS estimators (Barigozzi et al., 2019, Barigozzi, 2023). Under standard pervasiveness and weak dependence, the asymptotic variance of the QMLE matches the "oracle" OLS/PC variance. As $n \to \infty$, the factor estimation problem exhibits a "blessing of dimensionality": the identification error due to latentness vanishes.
• Cointegrated Continuous-Time State Space Models: For cointegrated MCARMA models observed at low frequency, QMLE estimators exhibit mixed-normal limiting distributions for "long-run" parameters (with $n$-rate or super-consistency) and standard $\sqrt{n}$-normality for "short-run" parameters, with theory adapted to handle the infinite-order dependence and non-standard innovation structure (Fasen-Hartmann et al., 2017).

5. QMLE in Nonparametric and Semi-parametric and Discrete Data Models

• Nonparametric Diffusion Function Estimation: Quasi-likelihood-based nonparametric estimators, such as the maximum penalized quasi-likelihood for continuous-time diffusion models, formulate the estimation as a penalized smoothing (splines) problem. This approach gives rate-optimal estimators compared to conventional kernel methods and ensures smoothness through natural splines (Hamrick et al., 2010).
• Quasi-Concave Density Estimation: QMLE can be interpreted as a regularized constrained shape estimation problem, e.g., log-concave or Renyi entropy–based quasi-likelihood, enforcing log-concavity or quasi-concavity via convexity constraints in the optimization, with a duality to maximum entropy estimators (Koenker et al., 2010).
• Discrete/Count Data and Multivariate Models: In discrete-valued observation-driven settings, QMLE routinely targets conditional mean parameters with robust consistency, though it may lose efficiency if the working variance is misspecified or dependence is ignored. Two-stage MWLSEs generalize this by estimating optimal weights to reduce finite-sample mean squared error (Armillotta, 2023). When only a conditional mean is specified and the full law is unknown, QMLE based on a Gaussian pseudo-likelihood with a parametric pseudo-variance offers a flexible, efficient, and robust alternative (Armillotta et al., 2023).

6. Implementation, Rate and Efficiency Considerations

Model Type	Robustness to Misspecif.	Moment Requirement	Efficiency
Gaussian QMLE (heterosked)	High (volatility models)	4th moment (GARCH)	Poor under heavy-tails
2SNG-QMLE (e.g. GARCH)	High (scale-adjusted)	2nd–3rd moment	Substantial gain with heavy-tails
Laplacian QMLE (affine)	High (heavy-tails/outliers)	1st (consistency); 2nd (normality)	Robust
MT-QMLE	High (outliers, non-Gaussian)	User-tunable (via weights)	Robust, can exceed GQMLE
Penalized QMLE (semi-param)	Varies (depends on penalty)	Varies; boundary regime	May detect sparsity

Key tuning parameters in advanced QMLE frameworks include the choice of working density, scale/covariance adjustments for misspecification, penalty function parameters, and in measure-transformed approaches, the transformation weights; rigorous selection of these via empirical or adaptive criteria (e.g. efficiency minimization over candidate families) is crucial for practical performance (Qi et al., 2010, Todros et al., 2015).

7. Practical Impact and Frontiers

QMLE remains a workhorse for robust estimation in non-Gaussian, dependent, high-dimensional, and semi-parametric models. Its equivalence to principal components and OLS estimation in large factor models, its robustness via adaptive and measure-transformed constructions, and its mathematical tractability in near-boundary and non-regular problems support its dominance in contemporary statistics and econometrics.

Active research targets further efficiency improvement, computational scalability (e.g., via EM/Kalman filtering), adaptation to nonergodic and highly structured models (Yoshida et al., 2022), and systematic approaches to transformation, penalization, and pseudo-variance specification (Armillotta et al., 2023). Key challenges include optimizing efficiency in multivariate and nonparametric settings, extending selection consistency under irregular constraints, and refining inference under vanishing or degenerate innovations.

The QMLE paradigm is essential whenever full likelihood-based inference is impractical, unknown, or too rigid—a principle underpinning much of modern data-driven statistical modeling.