Two-Step Feasible GMM

Why Feasible GMM Needs Two Steps

Efficient GMM uses the inverse covariance matrix of the moment conditions as its weight. That matrix is generally a population quantity and therefore unknown. Two-step feasible GMM replaces it with an estimate: first obtain a consistent preliminary estimate, then estimate the moment covariance at that value, and finally use the estimated inverse covariance to calculate a refined estimate.

The word feasible signals that the procedure can be carried out using sample data. The second step is not simply a re-expression of the first: when there are more moment conditions than parameters, changing the weight generally changes the estimate. Valid inference must also account for the variability of the moments, rather than treating the estimated weight as if it made the moments independent or error-free.

For clarity, the main formulas below first use independent, identically distributed observations and correctly specified moments. Define \(S=E[g_i(\theta_0)g_i(\theta_0)']\), assumed finite and positive definite. Under dependence, \(S\) is replaced by the long-run covariance of the moment sequence, estimated using an appropriate method for dependent data. The preliminary estimator must be consistent for \(\theta_0\); its weight \(W_0\) must be positive definite and fixed or converge to a positive-definite limit. Identification and smoothness conditions must ensure that the population moments locally identify the parameter. In particular, the \(q \times p\) Jacobian \(G=E[\partial g_i(\theta_0)/\partial\theta']\) must have full column rank.

The Two-Step Construction

The preliminary estimate \(\hat\theta_1\) minimizes the GMM criterion using \(W_0\). A common choice is the identity matrix, though another positive-definite weight can be used. Compute the sample covariance of the moment contributions evaluated at this estimate:

For independent data, this centered covariance is a consistent estimator of \(S\) under the stated conditions. The second-step weight is \(\hat W_2=\hat S_1^{-1}\), provided the estimated covariance is nonsingular. The two-step estimate minimizes the criterion using this weight:

The estimated weight is calculated from the preliminary estimate and then held fixed during this minimization. This defines the two-step procedure: it does not continuously recompute the covariance as a function of each candidate value of \(\theta\). For dependent observations, the corresponding long-run covariance estimate replaces the sample covariance above; the covariance-estimation method must match the dependence structure.

Second-Step Weight and Estimate

In the same hypothetical sample, suppose the centered sample covariance of the moment contributions evaluated at the preliminary estimate is \(\hat S_1=\begin{pmatrix}4&1\\1&2\end{pmatrix}\). This matrix is symmetric and positive definite: its leading diagonal entry is positive and its determinant is \(4(2)-1(1)=7>0\). It can therefore be inverted to form the second-step weight. Suppose the sample averages have the form given in the preceding example throughout the minimization.

The preceding calculation also gives a useful closed-form expression when sample average moments are linear, \(\bar g_n(\theta)=a-B\theta\), where \(B\) is \(q \times p\) and has full column rank. For any fixed positive-definite \(W\), differentiating the criterion and setting its gradient to zero yields \((B'WB)\hat\theta=B'Wa\). If \(B'WB\) is invertible, the solution is \(\hat\theta=(B'WB)^{-1}B'Wa\). Nonlinear moment models instead require numerical minimization, with the estimated weight held fixed in this step.

Why the Efficient Weight Is \(S^{-1}\)

Under correct specification and standard regularity conditions, the asymptotic covariance of a GMM estimator using a fixed weight \(W\) is \((G'WG)^{-1}G'WSWG(G'WG)^{-1}\). Among positive-definite weights, \(W=S^{-1}\) yields the smallest asymptotic covariance in the positive-semidefinite ordering. Two-step feasible GMM estimates this weight from the data rather than assuming that \(S\) is known.

A short justification explains why using an estimated weight does not add a first-order term to this limiting distribution. The first-order condition for the second-step estimate is approximately \(G'\hat W_2\bar g_n(\hat\theta_2)=0\). Expand the sample average moments around \(\theta_0\): \[ \sqrt n\,\bar g_n(\hat\theta_2) =\sqrt n\,\bar g_n(\theta_0)+G\sqrt n(\hat\theta_2-\theta_0)+o_p(1). \] Because \(\hat W_2\) converges to \(S^{-1}\), substituting this expansion into the first-order condition gives \[ \sqrt n(\hat\theta_2-\theta_0) =-(G'S^{-1}G)^{-1}G'S^{-1}\sqrt n\,\bar g_n(\theta_0)+o_p(1). \] The central limit theorem gives \(\sqrt n\,\bar g_n(\theta_0)\) a limiting covariance \(S\). Taking the covariance of the displayed linear expression gives \((G'S^{-1}G)^{-1}G'S^{-1}SS^{-1}G(G'S^{-1}G)^{-1}=(G'S^{-1}G)^{-1}\), as claimed. The argument uses consistency of the estimated weight and correct specification of the moments.

Standard Errors and Confidence Intervals

In applications, the asymptotic covariance must be estimated. A robust sandwich estimate uses the second-step weight and estimates of both the Jacobian and the moment covariance. Let \(\hat G\) estimate \(G\), and let \(\hat S_2\) estimate the moment covariance evaluated at the final estimate. Then the estimated covariance of \(\hat\theta_2\), rather than of its \(\sqrt n\)-scaled version, is

For independent observations, \(\hat S_2\) can be the centered covariance of moment contributions at \(\hat\theta_2\). With dependent observations, use a consistent estimate of their long-run covariance. In a correctly specified two-step procedure, \(\hat W_2\) and \(\hat S_2^{-1}\) converge to the same population matrix. Keeping the sandwich form is still useful: it makes clear what variability is being estimated and provides a robust variance calculation rather than silently imposing an exact covariance value in finite samples.

Common Pitfalls and Practical Checks

A common mistake is to use the preliminary covariance itself as the final standard error, or to report the identity-weight first-step estimate as if it were the efficient two-step estimate. The correction is to distinguish the roles: the preliminary estimate is used to estimate the moment covariance; its inverse weights the second-step criterion; and inference uses the sandwich variance with the final Jacobian and an appropriate moment covariance estimate.

Other checks matter in practice:

• Check invertibility. If \(\hat S_1\) is singular or nearly singular, the inverse weight may not be computable or may be unstable. This can arise when moment conditions are redundant or sample information is limited; it should not be concealed by treating the inverse as automatically well behaved.
• Match the covariance estimator to the data. The ordinary sample covariance addresses independent observations. Serial or other dependence requires a long-run covariance estimator appropriate to that dependence.
• Use the same weight convention consistently. This tutorial defines the criterion as \(n\bar g_n'W\bar g_n\); multiplying the entire criterion by a positive constant does not change its minimizer, but covariance and standard-error formulas must use a consistent scaling convention.
• Do not interpret moment fit as a causal conclusion. GMM uses the stated moment restrictions to estimate parameters. A causal interpretation requires substantive identification assumptions, not merely a small sample average of the moments.

Two-step feasible GMM is useful when the efficient moment weight is unknown but can be estimated consistently. Its central logic is straightforward: begin with a consistent estimate, learn how the moments vary, then use their estimated covariance to reweight the moment conditions. The resulting standard errors reflect both the parameter sensitivity of the moments and their covariance.