A researcher uses Poisson regression to model bilateral trade flows, which are non-negative but clearly not counts drawn from a Poisson distribution. What can they validly claim about the resulting estimates?
ANothing — the estimates are inconsistent because the distributional assumption is violated
BThe coefficient estimates are consistent if the conditional mean is correctly specified, but standard errors require a sandwich adjustment
CThe estimates are fully efficient because the Poisson likelihood is always well-behaved
DThe coefficient estimates are consistent, and the standard MLE standard errors are valid
This is the central use case for QMLE. By the linear exponential family result, maximizing the Poisson likelihood yields consistent estimates of the conditional mean parameters even when the true distribution is not Poisson — as long as the conditional mean model E[y|x] is correctly specified. However, distributional misspecification breaks the equality of the Hessian and outer-product-of-scores formulas, so standard MLE standard errors are invalid. The sandwich estimator (H⁻¹BH⁻¹) is required.
Question 2 Multiple Choice
What is the consequence of applying the standard MLE covariance formula (inverse Hessian) to a QMLE estimator when the likelihood is misspecified?
AThe standard errors are unaffected because the Hessian is invariant to distributional assumptions
BThe standard errors are typically too large, leading to overly conservative inference
CThe standard errors are typically too small, producing false precision and over-rejection of true nulls
DThe standard errors are correct if the sample size is large enough
Under correct specification, the Hessian (expected curvature) and the outer product of scores (expected squared gradient) are equal — the information matrix equality. Misspecification breaks this equality. In practice, using the standard formula when the likelihood is misspecified typically underestimates variance, making confidence intervals too narrow and hypothesis tests too aggressive. The sandwich estimator accounts for this discrepancy by separately estimating both pieces.
Question 3 True / False
A Poisson regression applied to non-count data can produce consistent coefficient estimates as long as the conditional mean E[y|x] is correctly specified.
TTrue
FFalse
Answer: True
This follows from the linear exponential family result: the Poisson likelihood's score condition is satisfied at the true parameter value whenever the conditional mean is correctly specified, regardless of the true distribution's shape. This is why Poisson regression has become a standard tool for non-negative outcomes like trade flows, patents, and citations — the distributional label 'Poisson' is a computational convenience, not a genuine probabilistic claim about the data.
Question 4 True / False
A QMLE estimator is mainly consistent if it converges to the true parameter value, which requires the specified likelihood to match the true data-generating process.
TTrue
FFalse
Answer: False
This is the key misconception QMLE overturns. Under the linear exponential family result, the QML estimator converges to the true parameter even when the likelihood is wrong, as long as the conditional mean is correctly specified. The misspecified likelihood still has its score equal to zero at the true parameter value — this is the regularity condition that ensures consistency. The distributional assumption beyond the mean can be entirely wrong without affecting the probability limit of the estimator.
Question 5 Short Answer
Why does quasi-maximum likelihood estimation require a sandwich covariance estimator rather than the standard MLE inverse-Hessian formula?
Think about your answer, then reveal below.
Model answer: Under correct specification, the information matrix equality holds: the expected Hessian (curvature of the log-likelihood) equals the expected outer product of scores (variance of the gradient). These are two ways of computing the same quantity, so the standard formula is valid. When the likelihood is misspecified, these two quantities differ — the Hessian reflects the shape of the wrong likelihood, while the outer product of scores captures actual sampling variation. The sandwich estimator (H⁻¹BH⁻¹) uses both pieces separately, correctly accounting for the discrepancy and producing valid asymptotic standard errors.
The sandwich structure arises because standard MLE relies on the information matrix equality as a computational shortcut. QMLE cannot use that shortcut. If you use only H⁻¹, you get standard errors that reflect the curvature of the wrong likelihood, which typically underestimates the true sampling variance. The outer-product matrix B corrects for this, and sandwiching it between H⁻¹ gives the correct asymptotic covariance for the QMLE estimator.