Poisson Quasi-Maximum Likelihood Under Misspecification

Why a Poisson Criterion Can Work Without Poisson Outcomes

A count outcome can have a conditional mean that follows an exponential regression model even when its conditional distribution is not Poisson. Poisson quasi-maximum likelihood (QMLE) uses the Poisson log-likelihood criterion to estimate that mean; it does not require the Poisson distribution to be the true data-generating distribution. The key is that the criterion’s expected value is maximized by the correctly specified conditional mean.

That fact is not, by itself, a complete consistency argument. Consistency also requires identification, a suitable convergence result for the sample criterion, and conditions that keep its maximizers near the population maximizer. This tutorial makes those requirements explicit.

Model and criterion: Let \((Y_i,X_i)\), \(i=1,\ldots,n\), be independent and identically distributed observations, where \(Y_i\) is a nonnegative count and \(X_i\) is a \(k\)-dimensional regressor vector. Suppose the conditional mean is correctly specified: \(\mathbb{E}[Y_i\mid X_i]=\mu(X_i;\beta_0)=\exp(X_i'\beta_0)\). For a candidate parameter \(\beta\), the Poisson quasi-log-likelihood contribution, up to a term that does not depend on \(\beta\), is \(q(Y_i,X_i;\beta)=Y_iX_i'\beta-\exp(X_i'\beta)\). The QMLE \(\hat{\beta}\) maximizes the sample average \(Q_n(\beta)=n^{-1}\sum_{i=1}^n q(Y_i,X_i;\beta)\) over a parameter space \(\Theta\).

If \(Y_i\mid X_i\) actually has a Poisson distribution, this criterion is the Poisson log-likelihood apart from the term \(-\log(Y_i!)\), which does not depend on \(\beta\). If the conditional distribution is different, it is still a criterion whose expected value can target the conditional mean. The distinction matters: correct specification of the mean is not a claim that the variance or the full conditional distribution is Poisson.

Conditions for a Consistency Result

A standard sufficient set of conditions gives the estimator a well-defined target and makes the sample criterion approximate its population counterpart. The following version uses a compact parameter space and bounded regressors for clarity. Other sets of assumptions can also establish consistency, but each must provide suitable identification, convergence, and maximizer control.

Sampling and mean: the observations are independent and identically distributed, \(Y_i\geq 0\), and \(\mathbb{E}[Y_i\mid X_i]=\exp(X_i'\beta_0)\).
Parameter space: \(\Theta\) is compact and contains \(\beta_0\).
Regressors and integrability: \(X_i\) is bounded, and the stated conditional mean holds. With bounded \(X_i\) and \(\beta_0\), \(\mathbb{E}[Y_i]=\mathbb{E}[\exp(X_i'\beta_0)]<\infty\).
Identification: for every \(\beta\in\Theta\) with \(\beta\ne\beta_0\), \(P\{X_i'(\beta-\beta_0)\ne 0\}>0\). A sufficient condition is that \(\mathbb{E}[X_iX_i']\) is positive definite.
Optimization: \(\hat{\beta}\) is a maximizer of \(Q_n(\beta)\) over \(\Theta\). More generally, the argument applies to an approximate maximizer whose criterion is within a term converging in probability to zero of the sample supremum.

Compactness and bounded regressors are convenient sufficient conditions, not universal necessities. In less restricted settings, researchers need other assumptions that establish a uniform law of large numbers and prevent maximizing estimates from escaping to parameter values far from \(\beta_0\).

Theorem: Under the conditions above, \(\hat{\beta}\xrightarrow{p}\beta_0\). The result requires both a correctly specified conditional mean and conditions ensuring that the population criterion has a separated maximum and that the sample criterion converges to it uniformly.

Why the Population Criterion Targets the Mean

Define the population criterion \(Q(\beta)=\mathbb{E}[q(Y_i,X_i;\beta)]\). To compare a candidate \(\beta\) with the true parameter \(\beta_0\), let \(d=X_i'(\beta-\beta_0)\). Conditional on \(X_i\), the conditional-mean assumption gives the following difference:

\begin{aligned} \mathbb{E}[q(Y_i,X_i;\beta)-q(Y_i,X_i;\beta_0)\mid X_i] &= \exp(X_i'\beta_0)d -\left[\exp(X_i'\beta)-\exp(X_i'\beta_0)\right]\\ &= -\exp(X_i'\beta_0)\left[\exp(d)-1-d\right]. \end{aligned}

For every real \(d\), \(\exp(d)\geq 1+d\), with equality only at \(d=0\). Thus the conditional difference is nonpositive, and it is strictly negative whenever \(d\ne0\). Taking expectations shows that \(Q(\beta)\leq Q(\beta_0)\). The identification condition ensures that, for any \(\beta\ne\beta_0\), \(d\ne0\) with positive probability. Therefore \(Q(\beta)<Q(\beta_0)\): \(\beta_0\) is the unique population maximizer.

This argument uses the conditional mean, not a Poisson probability mass function. The expected criterion has the correct maximum even if the distribution of \(Y_i\mid X_i\) has a different variance, skewness, or pattern of support, provided the stated mean model and the other conditions hold.

From the Population Maximum to Consistency

A unique population maximum is important, but on a noncompact parameter space it need not be enough for consistency. A population criterion might approach its maximum along parameter values arbitrarily far from \(\beta_0\). Sample criteria that are uniformly close to the population criterion could then have maximizers far away. The compact parameter space in the conditions above rules out this problem and supplies a separated maximum.

More precisely, for any \(\varepsilon>0\), compactness of \(\Theta\), continuity of \(Q\), and the unique maximum at \(\beta_0\) imply a positive gap:

\Delta_\varepsilon =Q(\beta_0)-\sup_{\substack{\beta\in\Theta\\ \|\beta-\beta_0\|\geq\varepsilon}}Q(\beta)>0,

whenever the set over which the supremum is taken is nonempty. The function \(q(Y_i,X_i;\beta)\) is continuous in \(\beta\). Bounded \(X_i\), compact \(\Theta\), and \(\mathbb{E}[Y_i]<\infty\) provide an integrable bound for the criterion and its variation over \(\Theta\); the uniform law of large numbers therefore gives \(\sup_{\beta\in\Theta}|Q_n(\beta)-Q(\beta)|\xrightarrow{p}0\). A continuous sample criterion on compact \(\Theta\) attains its maximum.

To see how these facts imply consistency, consider the event that the uniform difference is less than \(\Delta_\varepsilon/3\). If a maximizer \(\hat{\beta}\) were at least \(\varepsilon\) from \(\beta_0\), then \(Q(\hat{\beta})\leq Q(\beta_0)-\Delta_\varepsilon\). Uniform closeness would imply \(Q_n(\hat{\beta})<Q_n(\beta_0)\), contradicting that \(\hat{\beta}\) maximizes \(Q_n\). The probability of the event tends to one, so \(P(\|\hat{\beta}-\beta_0\|\geq\varepsilon)\to0\) for every \(\varepsilon>0\). This is convergence in probability to \(\beta_0\).

Worked Examples

Worked Example: A Zero Expected Score With Non-Poisson Counts

Suppose a hypothetical population has a binary regressor \(X\in\{0,1\}\). Its conditional means are \(\mathbb{E}[Y\mid X=0]=2\) and \(\mathbb{E}[Y\mid X=1]=5\). Use the log-mean model \(\mathbb{E}[Y\mid X]=\exp(\beta_1+\beta_2X)\), and suppose the conditional distributions need not be Poisson. Find the parameter matching these two means and check the expected score.

The model requires \(\exp(\beta_1)=2\) and \(\exp(\beta_1+\beta_2)=5\). Therefore \(\beta_1=\log 2\) and \(\beta_2=\log(5/2)\). The score contribution for one observation is \(X(Y-\exp(\beta_1+\beta_2X))\), where here \(X\) in the score denotes the regressor vector \((1,X)'\). At the specified parameter, its conditional expectation is \(X(\mathbb{E}[Y\mid X]-\exp(\beta_1+\beta_2X))=0\) for both regressor values.

This verifies a zero expected score at the mean-model parameter. It does not, on its own, prove consistency: the population criterion must identify that parameter, the sample criterion must converge suitably, and its maximizer must be controlled. The conditional distributions could, for example, have variances different from their means without changing this expected-score calculation.

Worked Example: Comparing a Candidate to the True Population Criterion

Continue with the hypothetical binary-regressor model, with \(P(X=0)=P(X=1)=1/2\), conditional means 2 and 5, and true parameter \(\beta_0=(\log 2,\log(5/2))'\). Consider \(\beta=(\log 2+0.2,\log(5/2)-0.1)'\). Calculate the difference \(Q(\beta)-Q(\beta_0)\).

For \(X=0\), the change in the linear predictor is \(d=0.2\), and the mean is 2. For \(X=1\), the change is \(d=0.2-0.1=0.1\), and the mean is 5. Using the population-criterion difference derived above,

\begin{aligned} Q(\beta)-Q(\beta_0) &=-\tfrac12\left\{2[\exp(0.2)-1-0.2] +5[\exp(0.1)-1-0.1]\right\}\\ &\approx-\tfrac12\{2(0.02140)+5(0.005171)\}\\ &\approx-0.03433. \end{aligned}

The candidate has a lower expected criterion by about 0.03433 criterion units per observation. The difference is strictly negative because the candidate changes the linear predictor for both regressor values. This comparison illustrates population identification; it is not a p-value or a sampling uncertainty calculation.

Worked Example: An Intercept-Only QMLE Under a Non-Poisson Distribution

Consider a hypothetical count-generating process with \(P(Y=0)=1/2\) and \(P(Y=3)=1/2\). Its mean is \(\mathbb{E}[Y]=1.5\), while its variance is \(\mathbb{E}[(Y-1.5)^2]=2.25\). The distribution is not Poisson: a Poisson variable with mean 1.5 has variance 1.5. Suppose six hypothetical observations are \(0,3,0,3,3,0\). Compute the intercept-only Poisson QMLE.

With no regressors other than the intercept, the sample criterion is \(Q_n(\beta)=n^{-1}\sum_i[Y_i\beta-\exp(\beta)]\). Its derivative is \(n^{-1}\sum_iY_i-\exp(\beta)\), so the maximizer solves \(\exp(\hat{\beta})=\bar{Y}\). Here \(\bar{Y}=(0+3+0+3+3+0)/6=9/6=1.5\), giving \(\hat{\beta}=\log(1.5)\approx0.4055\). The fitted mean is therefore \(\exp(\hat{\beta})=1.5\) counts per observation.

In this sample, the sample mean and fitted mean equal the true mean of 1.5. Under independent sampling from this distribution, the sample mean converges to 1.5, and the intercept estimate converges to \(\log(1.5)\approx0.4055\). The example shows how the criterion can target the mean despite a non-Poisson distribution; it does not imply that every finite-sample estimate equals the population parameter.

What the Result Does—and Does Not—Say

Consistency concerns the estimator’s probability limit as the sample size increases. It does not establish that a finite-sample estimate is exact, that the outcome distribution is Poisson, or that inference based on the Poisson distribution’s variance formula is valid. When the conditional variance differs from the Poisson variance, standard errors must account for that difference; consistency of the point estimate does not resolve the inference problem.

The result also depends on the conditional mean being correctly specified. A different outcome distribution is allowed, but a misspecified mean function is not covered by this theorem: in that case, the population criterion may be maximized at a parameter that does not reproduce the true conditional mean. Identification matters too. If different parameter values give the same linear predictor almost surely, the data cannot distinguish them using this criterion.

Common Mistakes

Equating QMLE with a Poisson distributional assumption. The Poisson criterion can be used when the conditional distribution is not Poisson, provided the conditional mean is correctly specified and the required conditions hold.
Treating a zero expected score as a consistency proof. A zero expected score does not establish a unique population maximum, uniform convergence, or control of sample maximizers. Check identification and the consistency conditions as well.
Assuming a unique population maximum is always sufficient. On a noncompact parameter space, the criterion may approach its maximum far from the target. A separated maximum, with appropriate uniform convergence and maximizer conditions, is needed. Compactness and continuity with a unique maximum provide a standard sufficient framework.
Claiming that mean correctness validates every Poisson-based calculation. Correctly specifying the mean supports consistency of the QMLE under the stated conditions; it does not make a non-Poisson variance equal to the mean.

Key takeaway: Poisson QMLE consistently estimates a correctly specified exponential conditional mean when the criterion identifies the target, converges uniformly to its population version, and its maximizer is kept near a separated population maximum. The outcome distribution itself need not be Poisson.

Check Your Understanding

Answer each question using the conditional-mean model and consistency conditions described above.

Which part of the Poisson QMLE consistency argument uses the conditional mean, and which part requires identification?
For a candidate parameter, define \(d=X'(\beta-\beta_0)\). Why is the expected criterion difference nonpositive?
In the intercept-only example, why is the fitted mean 1.5 not proof that the population mean is 1.5?
What role does a separated maximum play in the consistency argument?
Does consistency of the Poisson QMLE establish that the conditional variance equals the conditional mean? Explain.