Quasi-Maximum Likelihood

Ordinary maximum likelihood requires you to specify the exact distribution your data came from, then find the parameter that makes your observed data most probable under that distribution. In practice, quants routinely don't know — or don't want to assume — the exact distribution; return data is neither perfectly normal nor perfectly anything else. Quasi-maximum likelihood (QMLE) is what happens when you maximize a likelihood built from a convenient but possibly wrong distributional assumption anyway — and, remarkably, for a wide class of models the point estimate you get is still correct, even though the assumed distribution was false.

An analogy: a recipe that still works with the wrong seasoning chart

Imagine following a recipe's cooking-time chart that was written assuming a gas oven, but you actually have an electric one with a slightly different heat curve. If the two ovens happen to reach the target internal temperature at a similar rate — even though the reasoning behind the chart (gas heat transfer) doesn't literally apply to your oven — the dish still comes out properly cooked. What breaks is not the final result, but your confidence in how sensitive that result is to small changes: the gas-oven chart's assumptions about your electric oven's variability from batch to batch are wrong, even if its central prediction happens to survive. QMLE is exactly this: use a convenient model (often the normal distribution) to locate the estimate, get lucky that the location is still right, but recognize that the assumed model's confidence-interval machinery — its stated uncertainty — cannot be trusted without a fix.

The mechanism, one symbol at a time

Suppose the true data-generating distribution has some unknown, possibly complicated form, but you maximize the likelihood as though the data were normal, or Poisson, or whatever convenient family you pick — call this the "quasi-likelihood." Under fairly general conditions, the quasi-MLE $\hat\theta_{QMLE}$ still converges to the true parameter $\theta_0$ as long as the mean of the assumed model is correctly specified, even if the variance or higher moments are wrong.

The catch is in the standard errors. Ordinary maximum likelihood theory says the estimator's asymptotic variance is $I(\theta)^{-1}$, the inverse of the Fisher information — a formula that is only valid if the assumed model is exactly true. Under misspecification, the correct asymptotic variance is instead the "sandwich" formula:

$$\text{Var}(\hat\theta_{QMLE}) \approx I(\theta)^{-1}\, J(\theta)\, I(\theta)^{-1}$$

Read this as: the middle piece $J(\theta)$ is the actual variability of the score function under the true (possibly different) distribution, sandwiched between two copies of $I(\theta)^{-1}$, the inverse information matrix computed from the assumed model. In plain English: your point estimate can still land in the right place using the wrong model, but you must use this sandwich formula — not the plain, textbook MLE variance formula — to get a correct standard error, because the assumed model's own bookkeeping of its uncertainty is only self-consistent when the assumption is actually true. When the model is correctly specified, $J(\theta) = I(\theta)$ and the sandwich collapses back to the ordinary $I(\theta)^{-1}$, recovering standard MLE exactly as a special case.

Imagine the curve above is the assumed normal model, but the real data is drawn from a distribution with fatter tails than shown. The peak (mean) of both distributions can coincide, so a location estimate built assuming the normal curve still lands correctly — but the normal curve's stated spread understates the true spread, exactly the gap the sandwich formula is built to correct.

Now picture the same scatter with a few points much further from the fitted line than the rest, on both sides evenly — a fatter-tailed cloud than the correlation slider alone can produce. The fitted line (your point estimate) barely moves, since the extra spread is symmetric, but the honest uncertainty band around that line should widen to reflect it — precisely the correction the sandwich formula supplies and the naive formula misses.

Worked example 1: normal QMLE on non-normal data still finds the mean

Suppose daily returns actually follow a heavily skewed distribution with true mean $\mu_0 = 0.5\%$, but you fit a normal likelihood (i.e., ordinary least squares / sample mean) anyway. The normal log-likelihood's score equation, from the score-function template, reduces to $\sum(x_i - \hat\mu) = 0 \implies \hat\mu = \bar x$ — this derivation never actually used normality anywhere except in the form of the likelihood; it only required the model's mean to be correctly $\mu_0$. Given returns $0.8, -0.3, 1.2, 0.1, -0.3$ (deliberately skewed, not normal-looking), $\hat\mu = \bar x = \frac{0.8-0.3+1.2+0.1-0.3}{5} = \frac{1.5}{5}=0.3\%$ — a perfectly sensible, consistent estimate of the true mean despite the wrong distributional assumption behind the fitting procedure.

Worked example 2: why the naive standard error is wrong

Continue the example. The naive normal-MLE standard error formula would use the sample variance under the normal assumption: sample variance of the five points is $s^2 = 0.402$ (computed the usual way), giving naive $\text{SE}(\hat\mu) = \sqrt{0.402/5} = 0.284$.

But suppose the true distribution is actually heavily skewed with a fatter tail than the sample of 5 happened to reveal — say the true variance of the underlying process is really $0.9$, not $0.402$ (the small sample just didn't happen to show a big outlier this time). The correct, sandwich-based standard error uses the true variability: $\text{SE}_{true}(\hat\mu) = \sqrt{0.9/5} = 0.424$ — about 50% wider than the naive figure of $0.284$. Reporting the naive $0.284$ would understate the true uncertainty in $\hat\mu$ by a meaningful margin, purely because the normal likelihood's internal bookkeeping assumed a variance structure the data didn't actually have.

What this means in practice

• Fitting a GARCH volatility model with a normal likelihood, then applying the model to fat-tailed returns, is textbook QMLE — the volatility parameter estimates remain broadly reliable, but the reported standard errors on those parameters should use robust ("sandwich" or "Huber-White") standard errors, not the default normal-MLE output.
• Any regression run with robust standard errors or heteroskedasticity-consistent standard errors in a statistics package is applying exactly this sandwich correction — it is one of the most common QMLE fixes used in applied econometrics and factor model research.
• The point estimate surviving misspecification is a property of the specific model, not a universal guarantee — it holds for the normal likelihood's mean parameter under quite general conditions, but does not automatically extend to every parameter of every model; each case needs its own check.