GMM Under Misspecified Moment Conditions

Why Misspecified Moments Change the GMM Target

GMM chooses parameter values by making a vector of sample moments small under a chosen weighting scheme. Under correct specification, there is a parameter at which every population moment is zero. When no such parameter exists, GMM cannot satisfy all the population restrictions simultaneously. It instead seeks a parameter that provides the best fit according to the GMM criterion. Because “best fit” depends on how the moments are weighted, the target can depend on the weighting matrix.

This distinction matters for interpretation. Under misspecification, the GMM target is generally not a structural parameter that satisfies the proposed population restrictions. It is a parameter that optimizes a particular criterion over a specified parameter space. The minimization must be global, and it must attain its infimum before a pseudo-true parameter is defined.

Population Criterion and Pseudo-True Parameters

Let $\theta\in\Theta\subseteq\mathbb{R}^p$ be the parameter, let $g_i(\theta)\in\mathbb{R}^q$ be the vector of moment functions for observation $i$, and define the population moment vector by $m(\theta)=E[g_i(\theta)]$. Assume the expectations exist for the parameter values under consideration. In sample GMM, the sample moments are weighted by a matrix; for the population target considered here, take a fixed, symmetric positive-definite matrix $W\in\mathbb{R}^{q\times q}$.

Definition: The population GMM criterion for a fixed positive-definite weight matrix $W$ is $Q_W(\theta)=m(\theta)'Wm(\theta)$. The moments are misspecified on $\Theta$ if there is no $\theta\in\Theta$ for which $m(\theta)=0$. If the criterion attains its infimum on $\Theta$, its set of global minimizers is the pseudo-true parameter set: $$ \Theta_W^*=\operatorname*{arg\,min}_{\theta\in\Theta} Q_W(\theta). $$ If this set contains exactly one element, that element is the pseudo-true parameter, denoted $\theta_W^*$.

Positive definiteness ensures $Q_W(\theta)\geq 0$, with equality exactly when $m(\theta)=0$. Thus, under misspecification, the criterion is strictly positive at every $\theta\in\Theta$. But this fact does not by itself guarantee that a global minimum exists. If a minimizer does exist, it may not be unique; in that case the target is the set $\Theta_W^*$, not a single parameter.

A useful sufficient condition for existence is that $\Theta$ is compact and $Q_W$ is continuous on $\Theta$. The extreme value theorem then guarantees that the criterion attains its minimum. Uniqueness requires an additional condition, such as strict convexity of $Q_W$ over a convex parameter space. These are sufficient conditions, not necessary ones: on an unbounded parameter space, a minimum can still exist, for example when the criterion is coercive and continuous.

Worked Example: A Scalar Target Changes With the Weight

Suppose $\Theta=\mathbb{R}$ and the two population moments are $m(\theta)=(\theta,\theta-2)'$. The first moment is zero at $\theta=0$ and the second at $\theta=2$, so no parameter makes both zero. For a diagonal weight matrix $W=\operatorname{diag}(w_1,w_2)$, where $w_1,w_2>0$, the population criterion is $$ Q_W(\theta)=w_1\theta^2+w_2(\theta-2)^2. $$

Differentiating gives $Q_W'(\theta)=2w_1\theta+2w_2(\theta-2)$. The second derivative is $2(w_1+w_2)>0$, so the unique stationary point is the unique global minimum. Solving $Q_W'(\theta)=0$ gives $$ \theta_W^*=\frac{2w_2}{w_1+w_2}. $$

With $W=I$, the target is $\theta_I^*=1$, with moments $m(1)=(1,-1)'$ and criterion $Q_I(1)=1^2+(-1)^2=2$. With $W=\operatorname{diag}(4,1)$, the target is $\theta_W^*=2/(4+1)=0.4$. Its moments are $m(0.4)=(0.4,-1.6)'$, and its criterion is $4(0.4)^2+(-1.6)^2=0.64+2.56=3.20$. Each target is the unique best fit for its own criterion; the change in target reflects the changed relative penalty on the first moment. Neither target makes both population moments zero.

Linear Moments: A Weighted Projection

For linear population moments, the target has a closed form. Write $m(\theta)=a-B\theta$, where $a\in\mathbb{R}^q$ and $B\in\mathbb{R}^{q\times p}$. Suppose $W$ is symmetric positive definite and $B$ has full column rank. These assumptions ensure that $B'WB$ is invertible. The criterion is the weighted squared distance between $a$ and $B\theta$.

Result: For $m(\theta)=a-B\theta$, symmetric positive-definite $W$, and full-column-rank $B$, the unique global minimizer over $\mathbb{R}^p$ is $$ \theta_W^*=(B'WB)^{-1}B'Wa. $$ At this minimizer, the residual moment vector satisfies $B'Wm(\theta_W^*)=0$. This is a weighted orthogonality condition; by itself, a first-order condition is not a general substitute for establishing a global minimum.

To justify the result, expand the criterion as $Q_W(\theta)=(a-B\theta)'W(a-B\theta)$. Its gradient is $-2B'W(a-B\theta)$, so setting the gradient to zero yields $B'WB\theta=B'Wa$. For any nonzero $v\in\mathbb{R}^p$, full column rank gives $Bv\ne 0$, and positive definiteness gives $$ v'B'WBv=(Bv)'W(Bv)>0. $$ Thus $B'WB$ is positive definite and invertible, giving the stated solution. The Hessian of $Q_W$ is $2B'WB$, which is positive definite, so the criterion is strictly convex and the stationary point is its unique global minimum. Substituting the minimizing equation into the gradient condition also gives $B'Wm(\theta_W^*)=0$.

Worked Example: Two Parameters and Three Incompatible Moments

Let $\theta=(\theta_1,\theta_2)'\in\mathbb{R}^2$, $W=I$, and $$ m(\theta)= \begin{pmatrix} \theta_1-1\\ \theta_2-2\\ \theta_1+\theta_2-5 \end{pmatrix}. $$ The first two moments would both be zero at $(1,2)'$, but then the third is $-2$. In fact, the first two being zero fixes that parameter, so there is no parameter at which all three moments vanish. The criterion is continuous and strictly convex because its three linear moment gradients span the two-dimensional parameter space; equivalently, the matrix $B$ in $m=a-B\theta$ has full column rank. Its unique minimizer exists.

Here $$ Q_I(\theta)=(\theta_1-1)^2+(\theta_2-2)^2+(\theta_1+\theta_2-5)^2. $$ Setting each partial derivative equal to zero gives $$ (\theta_1-1)+(\theta_1+\theta_2-5)=0,\qquad (\theta_2-2)+(\theta_1+\theta_2-5)=0, $$ or $2\theta_1+\theta_2=6$ and $\theta_1+2\theta_2=7$. Solving yields $\theta_I^*=(5/3,8/3)'$. The residual moments are $(2/3,2/3,-2/3)'$, and the minimum criterion is $$ Q_I(\theta_I^*)=(2/3)^2+(2/3)^2+(-2/3)^2=4/3. $$ The nonzero residuals confirm that this is a best-fitting parameter, not an exact solution to the moment restrictions.

Nonlinear Moments and Global Minimization

For nonlinear moments, the same definition applies, but a first-order condition usually identifies only candidate points. A candidate must still be compared with other candidates and with boundary values if $\Theta$ has a boundary. Existence and uniqueness should be checked using the parameter space and the shape of the full criterion, rather than inferred from a stationary point alone.

Worked Example: A Unique Nonlinear Pseudo-True Parameter

Let $\Theta=\mathbb{R}$, $m(\theta)=(\theta-1,\theta^2+1)'$, and $W=\operatorname{diag}(5,2)$. The second moment is always positive, so the moments cannot both be zero. The criterion is $$ Q_W(\theta)=5(\theta-1)^2+2(\theta^2+1)^2. $$ It is continuous and tends to infinity as $|\theta|$ tends to infinity, so it attains a global minimum.

Differentiating gives $Q_W'(\theta)=10(\theta-1)+8\theta(\theta^2+1)=2(4\theta^3+9\theta-5)$. The derivative of the expression in parentheses is $12\theta^2+9>0$, so it is strictly increasing. It equals zero at $\theta=0.5$; hence this is the only stationary point, and the criterion decreases before it and increases after it. Therefore $\theta_W^*=0.5$ is the unique global minimizer. At this value $m(0.5)=(-0.5,1.25)'$, and $$ Q_W(0.5)=5(-0.5)^2+2(1.25)^2=1.25+3.125=4.375. $$ The criterion is positive because the moments remain incompatible.

Why Existence and the Weight Matrix Matter

An infimum is the greatest lower bound of a criterion’s values; it need not be attained at any parameter. For example, let $\Theta=\mathbb{R}$, $m(\theta)=(e^\theta,e^\theta+1)'$, and $W=I$. Then $$ Q_I(\theta)=e^{2\theta}+(e^\theta+1)^2=2e^{2\theta}+2e^\theta+1. $$ For every finite $\theta$, $e^\theta>0$, so $Q_I(\theta)>1$. Yet $Q_I(\theta)$ tends to $1$ as $\theta$ tends to $-\infty$. Its infimum is $1$, but no $\theta\in\mathbb{R}$ attains it. Thus this criterion has no pseudo-true parameter, despite being continuous and having a finite infimum.

The weight matrix is part of the target’s definition under misspecification, not merely a device for computing uncertainty. In the scalar example, changing $W$ changed which moment discrepancy received more penalty and moved the unique target. In the linear case, this dependence is explicit in $(B'WB)^{-1}B'Wa$. By contrast, if the moments are correctly specified and there is a unique parameter satisfying $m(\theta)=0$, every positive-definite $W$ gives criterion zero at that same parameter. The target’s sensitivity to $W$ is therefore a feature of misspecification, not a contradiction.

When sample moments converge to their population counterparts and the sample weight converges to a fixed $W$, the population minimizer is the natural candidate limit of the GMM estimator. A formal consistency argument requires further regularity conditions; the key interpretive point here is that under misspecification, changing the limiting weight can change the population target itself.

Common Mistakes

Assuming that a minimizer must exist. A criterion can have a finite infimum that it never attains. Check existence conditions, such as continuity on a compact parameter space, or establish attainment by another argument.
Calling a first-order solution the target without checking it. A stationary point need not be a global minimum, and multiple global minima can occur. Define the target by global minimization and establish uniqueness before naming a single pseudo-true parameter.
Assuming the target is independent of the weight matrix. Under misspecification, different weights generally define different best fits. State the weight matrix when describing the target.
Interpreting the pseudo-true parameter as making the moments valid. It minimizes the weighted population discrepancy; under misspecification its moments need not be zero.

Key takeaway: Under misspecification, the pseudo-true GMM target is the set of global minimizers of $m(\theta)'Wm(\theta)$, if that minimum is attained. Existence, uniqueness, and the choice of $W$ all matter: the target can be absent, nonunique, or weight-dependent.

Check Your Understanding

Try these questions using the definitions and arguments above.

For $m(\theta)=(\theta-1,\theta-3)'$ and $W=I$, find the unique pseudo-true parameter and its residual moments. Why can no parameter make both moments zero?
Why does continuity of $Q_W$ alone not guarantee that a pseudo-true parameter exists on an unbounded parameter space?
For linear moments $m(\theta)=a-B\theta$, what condition on $B$ and $W$ makes $B'WB$ invertible, and why?
Under what condition will changing a positive-definite weight matrix leave the target unchanged?
Explain why a first-order condition is not enough, by itself, to establish a unique pseudo-true parameter.