Skip to content

Quasi-maximum likelihood estimation

Fit a working likelihood under misspecification: Gaussian moment targets, the pseudo-true parameter, sandwich uncertainty and dependence-aware inference.

Quasi-maximum likelihood estimation (QMLE) fits parameters by maximising a working likelihood while allowing its probability distribution to be misspecified. In quantitative finance, a Gaussian working likelihood can estimate a return model's conditional mean and variance even when innovations are non-Gaussian—provided those moment models and the conditions for estimation are valid.
The maximisation can be identical to maximum likelihood estimation. What changes is the justification of the target and its uncertainty. “Quasi” does not mean an approximate numerical optimiser, and it does not make an arbitrary likelihood robust to every modelling error.

A Gaussian fit without a Gaussian claim

Take three fictional observations, in arbitrary units: 0,0,30,0,3. Use the same working Gaussian density for every observation, with an unrestricted common mean μ\mu and variance v>0v>0. The summed log-likelihood, dropping a parameter-independent constant, is
ℓn(μ,v)=−n2log⁡v−12v∑i=1n(yi−μ)2.\ell_n(\mu,v)=-\frac{n}{2}\log v-\frac{1}{2v}\sum_{i=1}^n(y_i-\mu)^2.

For fixed vv, maximising this expression minimises the squared residuals, giving μ^=yˉ\hat\mu=\bar y. Differentiating with respect to vv at that mean gives

∂ℓn∂v=−n2v+∑i(yi−yˉ)22v2=0,v^=1n∑i(yi−yˉ)2.\frac{\partial\ell_n}{\partial v} =-\frac{n}{2v}+\frac{\sum_i(y_i-\bar y)^2}{2v^2}=0, \qquad \hat v=\frac{1}{n}\sum_i(y_i-\bar y)^2.

Here yˉ=1\bar y=1, the squared residuals are 1,1,41,1,4, and their sum is 66. Thus

μ^=1,v^=6/3=2.\boxed{\hat\mu=1,\qquad \hat v=6/3=2.}

These are global maximisers: the sample mean minimises the residual sum of squares, and the resulting profile log-likelihood increases up to v=2v=2 and decreases afterwards. Its limits at zero and infinity are both −∞-\infty for this sample.

Dividing by n−1n-1 instead produces s2=6/2=3s^2=6/2=3, the usual unbiased sample variance under iid sampling with finite variance. It does not maximise this working likelihood. Unbiasedness and likelihood maximisation are different objectives; QMLE does not introduce a degrees-of-freedom correction.
Fictional observations 0, 0 and 3 have mean 1 and squared residuals 1, 1 and 4. Holding the mean at 1, the Gaussian log likelihood peaks at variance 2. The n minus 1 variance estimate is 3 and has log likelihood 0.1082 below the maximum.
Fictional observations 0, 0 and 3 have mean 1 and squared residuals 1, 1 and 4. Holding the mean at 1, the Gaussian log likelihood peaks at variance 2. The n minus 1 variance estimate is 3 and has log likelihood 0.1082 below the maximum.
Original fictional example. The curve varies the candidate variance while fixing the mean at 1. It measures fit within the working model; it does not test normality.

Neither the fit nor this three-observation calculation establishes an asymptotic approximation or a credible tail forecast.

Which parameter does it estimate?

For iid data, the population target is the pseudo-true parameter
θ∗∈arg max⁡θ∈ΘEtrue[log⁡f(Y;θ)].\theta^*\in\operatorname*{arg\,max}_{\theta\in\Theta} \mathbb{E}_{\mathrm{true}}[\log f(Y;\theta)].
With compatible true and working densities and a finite, well-defined Kullback–Leibler divergence, this also minimises DKL(g∥fθ)D_{\mathrm{KL}}(g\Vert f_\theta). It is a projection onto the chosen family, not recovery of the entire true distribution. White's consistency analysis requires identification of the target and regularity conditions controlling the objective and parameter space. A score root alone is insufficient.

The unrestricted Gaussian mean/variance case has a useful property. If the true population has mean mm and finite positive variance τ2\tau^2,

E[(Y−μ)2]=τ2+(m−μ)2.\mathbb{E}[(Y-\mu)^2]=\tau^2+(m-\mu)^2.

Substitution into the expected working log-likelihood makes its maximiser (μ∗,v∗)=(m,τ2)(\mu^*,v^*)=(m,\tau^2). Normality is unnecessary for this moment target. This expected-log-likelihood argument also works for discrete data; calling a continuous Gaussian density a finite-KL approximation to a discrete law would require more care about the underlying measures.

The conditional-moment promise in volatility models

For returns with past information Ft−1\mathcal F_{t-1}, write et(θ)=yt−mt(θ)e_t(\theta)=y_t-m_t(\theta) and a positive working conditional variance ht(θ)h_t(\theta). Gaussian QMLE maximises contributions

ℓt(θ)=−12[log⁡ht(θ)+et(θ)2ht(θ)].\ell_t(\theta)=-\frac12\left[\log h_t(\theta)+\frac{e_t(\theta)^2}{h_t(\theta)}\right].

The relevant correctness claim is that some identified θ0\theta_0 satisfies

E[yt∣Ft−1]=mt(θ0),Var(yt∣Ft−1)=ht(θ0).\mathbb E[y_t\mid\mathcal F_{t-1}]=m_t(\theta_0), \qquad \text{Var}(y_t\mid\mathcal F_{t-1})=h_t(\theta_0).
Under these restrictions, the Gaussian score has zero conditional expectation. This is the key mechanism in Bollerslev and Wooldridge's analysis. Consistency still needs identification, appropriate laws of large numbers, and control of the parameter space and recursion. Normal-limit inference additionally needs smoothness, nonsingular curvature and sufficient score moments; finite variance alone does not establish it. Fourth-moment conditions commonly enter Gaussian variance-parameter inference.

Misspecifying the conditional mean or variance can change the target. Robust standard errors cannot repair that change, and correctly estimating two moments does not validate Gaussian tail probabilities.

The ARCH and GARCH lesson connects this objective to a volatility recursion and residual diagnostics. Its preview is public; the full lesson requires membership.

Curvature and score variation are separate

For iid observations, define the column score si=∇θℓi(θ∗)s_i=\nabla_\theta\ell_i(\theta^*) and

A=−E[∇θ2ℓi(θ∗)],B=E[sisi⊤].A=-\mathbb E[\nabla_\theta^2\ell_i(\theta^*)], \qquad B=\mathbb E[s_i s_i^\top].

Under the conditions for asymptotic normality,

n(θ^−θ∗)→dN(0,A−1BA−⊤).\sqrt n(\hat\theta-\theta^*)\xrightarrow{d} \mathcal N(0,A^{-1}BA^{-\top}).

The sandwich variance estimator replaces A,BA,B with consistent estimates and divides this matrix by nn to estimate the covariance of θ^\hat\theta. With sample-average Hessian and score products, that factor is essential.

Correct specification and the usual likelihood regularity conditions give the information equality A=BA=B, reducing the covariance to A−1/nA^{-1}/n. Under misspecification they generally differ. Stanford's misspecification notes, §16.2, derive this distinction. Sandwich standard errors address sampling uncertainty around the target; they do not make the estimator insensitive to extreme observations.

For a stationary dependent score process, the meat instead uses the long-run covariance

BLR=∑k=−∞∞Cov(st,st−k),B_{\mathrm{LR}}=\sum_{k=-\infty}^{\infty}\text{Cov}(s_t,s_{t-k}),
when that sum and the required limit theory exist. Newey–West standard errors provide a HAC approach under suitable dependence, moment and bandwidth conditions. MIT's HAC notes explain why summing every noisy sample autocovariance is insufficient.
A correctly specified conditional-moment model can make the score a martingale difference sequence; its nonzero-lag score covariances then vanish when the needed moments exist. Dependent returns therefore do not automatically require a HAC correction to an already valid conditional QMLE sandwich. The dependence of the scores determines the appropriate meat.

Sources

Last updated
  • quasi maximum likelihood
  • misspecification
  • pseudo true parameter
  • conditional moments
  • sandwich inference
Keep connecting the ideas
← Back to the glossaryBack to top ↑