Inference Under Misspecified Moments

Why Misspecification Changes Inference

GMM is often introduced through population moment restrictions that hold exactly at a parameter value: the expected moment vector is zero. In applications, the chosen moments may not all be exactly compatible with the data-generating process. The weighted GMM estimator can still converge, but it converges to a parameter that minimizes the population criterion rather than to a value that makes every moment zero. That parameter is called the pseudo-true parameter.

Inference around this minimizer requires more than substituting a covariance estimate into the usual correctly specified formula. When the population moments are nonzero, random changes in the sample derivatives affect the criterion’s gradient. The curvature of the population criterion also includes a term involving the second derivatives of the moments. Both features matter for a valid sandwich covariance.

For the derivations below, suppose the observations are independent and identically distributed, the moments are twice continuously differentiable near \(\theta^*\), and the required expectations and variances are finite. Assume the sample criterion has a consistent minimizer near \(\theta^*\), the population minimizer is locally unique, and the criterion Hessian there is nonsingular and positive definite. For time-series data, the same structure applies with a long-run covariance and suitable stationarity and central limit conditions.

Write \(G_i(\theta)=\partial g_i(\theta)/\partial\theta'\), a \(q \times p\) matrix, and \(G(\theta)=E[G_i(\theta)]\). The sample criterion is \(Q_n(\theta)=\bar g_n(\theta)'W\bar g_n(\theta)\), where \(\bar g_n(\theta)=n^{-1}\sum_{i=1}^n g_i(\theta)\). We use one-half of this criterion when taking derivatives; this changes neither its minimizer nor the resulting sandwich covariance.

The Population Gradient and Curvature

Differentiating \(Q(\theta)/2\) gives the population first-order condition

This is an orthogonality condition: the weighted population moment vector is orthogonal to the columns of the moment Jacobian. It does not require the weighted moments, or the moments themselves, to be zero.

The Hessian of \(Q(\theta)/2\) at \(\theta^*\), denoted \(A\), is

where \(G=G(\theta^*)\), \(m=m(\theta^*)\), and \(g_{ij}\) is the \(j\)th component of \(g_i\). The first term is the familiar Jacobian-weight-Jacobian term. The second term vanishes under correct specification because then \(m=0\); under misspecification it generally does not.

The sample first-order condition is \(G_n(\hat\theta)'W\bar g_n(\hat\theta)=0\), where \(G_n(\theta)=n^{-1}\sum_i G_i(\theta)\). Even at \(\theta^*\), its random fluctuation has two sources. The sample moments fluctuate around \(m\), and the sample Jacobian fluctuates around \(G\). Define the \(p\)-dimensional influence score

The first term is the familiar contribution from moment variation. The second captures derivative variation multiplied by the nonzero weighted population moments. Its expectation is zero. A first-order expansion of the sample condition yields

If \(B=E[\psi_i\psi_i']\) is nonsingular or positive semidefinite, the iid asymptotic covariance of \(\sqrt n(\hat\theta-\theta^*)\) is \(A^{-1}BA^{-1}\). Since \(A\) is a symmetric Hessian, this is also \(A^{-1}BA^{-T}\). For dependent observations, replace \(B\) with the long-run covariance of \(\psi_i\).

Worked Examples

Estimating the Covariance

A practical estimator replaces population quantities in the score and Hessian with sample analogues evaluated at \(\hat\theta\). For fixed \(W\), define \(\hat m=\bar g_n(\hat\theta)\), \(\hat G=G_n(\hat\theta)\), and \(\hat A\) as the sample Hessian of \(Q_n(\theta)/2\) at \(\hat\theta\):

For each observation, form the estimated score

With iid observations, set \(\hat B=n^{-1}\sum_i\hat\psi_i\hat\psi_i'\). The estimated covariance matrix of \(\hat\theta\), rather than of \(\sqrt n(\hat\theta-\theta^*)\), is

For stationary dependent data, estimate the long-run covariance of the estimated scores instead. For example, a HAC estimator combines sample score autocovariances across lags, with a kernel and bandwidth chosen to satisfy the relevant consistency conditions. Long-run covariance estimation concerns the score \(\psi_i\), not just the original moment vector \(g_i\).

Estimated Weights, Interpretation, and a Common Pitfall

The formulas above treat \(W\) as fixed, or as fixed for the first-order expansion. If \(W\) is estimated and its sampling variation is of order \(n^{-1/2}\), it can contribute to the score under misspecification. In particular, the population gradient \(G'Wm=0\) does not generally imply \(G'(W_n-W)m=0\). A full first-order calculation must then add the influence of the estimated weight to \(\psi_i\), using the joint influence representation for \(W_n\) and the moments. Ignoring this term is justified only when weight estimation has no first-order effect, or when the analysis explicitly conditions on the estimated weight under an appropriate argument.

The resulting interval and tests describe uncertainty about the pseudo-true parameter under the chosen moments and weighting matrix. They do not establish that the moments are correctly specified, nor do they automatically provide inference for a structural parameter that would have been identified under exact moment restrictions. Moment validity and parameter interpretation remain substantive questions.

• Common mistake: “The estimator is consistent, so I can use \(G'WG\) and the covariance of \(G'Wg_i\).” This drops both the nonzero-moment term in the Hessian and the derivative-variation term in the score.
• Correction: Differentiate the actual weighted criterion at its pseudo-true minimizer. Use its full Hessian and the variance or long-run covariance of the resulting score.
• Another pitfall: Treating a confidence interval for the pseudo-true parameter as evidence that all population moments equal zero. The interval concerns the minimizer of the chosen criterion; it is not a specification test.