One-Step Difference-GMM Estimation

From Difference-GMM Moments to an Estimate

Difference GMM turns valid moment restrictions into an estimator by choosing parameter values that make the sample moments small. The weighting matrix specifies how discrepancies across those moments are combined. In one-step estimation, that matrix is fixed in advance for the estimation step; the estimate is obtained by minimizing the resulting quadratic criterion.

For a dynamic panel, differencing removes the unit-specific effect from an equation such as \(y_{it}=\rho y_{i,t-1}+x_{it}'\beta+\eta_i+\varepsilon_{it}\). Write \(\theta=(\rho,\beta')'\) for the parameter vector. For each unit \(i\), stack the usable differenced outcomes in \(\Delta y_i\), the corresponding differenced regressors in \(X_i\), and the instruments specified for those differenced equations in \(Z_i\). The resulting equation is \(\Delta y_i=X_i\theta+\Delta\varepsilon_i\). The instruments and the conditions that justify them are developed in the earlier material on Difference-GMM instruments; here, the task is to use their moment restrictions to estimate \(\theta\).

Definition: The unit-level moment contribution at parameter value \(\theta\) is \(m_i(\theta)=Z_i'(\Delta y_i-X_i\theta)\). The sample moment vector is \(g_N(\theta)=N^{-1}\sum_{i=1}^{N}m_i(\theta)\). If \(Z_i\) has \(q\) columns and \(\theta\) has \(k\) components, then \(g_N(\theta)\) has \(q\) components. Validity requires \(E[m_i(\theta_0)]=0\) at the true parameter \(\theta_0\).

Define \(h_N=N^{-1}\sum_i Z_i'\Delta y_i\) and \(G_N=N^{-1}\sum_i Z_i'X_i\). Then \(g_N(\theta)=h_N-G_N\theta\). The dimensions of \(G_N\) are \(q\) by \(k\). A one-step difference-GMM estimator uses a specified \(q\) by \(q\) weighting matrix \(A_N\) and minimizes the weighted squared sample moments.

\widehat{\theta}_1 = \underset{\theta}{\operatorname{arg\,min}}\; Q_N(\theta), \qquad Q_N(\theta)=g_N(\theta)'A_Ng_N(\theta).

The standard admissibility condition is that \(A_N\) is symmetric positive definite: \(A_N'=A_N\), and \(v'A_Nv>0\) for every nonzero \(q\)-vector \(v\). This ensures that the criterion measures moment discrepancies with a positive quadratic form. The matrix is specified for the one-step procedure; it might, for example, be the identity matrix or a prespecified matrix based on a maintained error-variance structure. Its choice is part of the estimator, not an incidental computational detail.

Solving the Sample Minimization

Because the moment vector is linear in \(\theta\), the minimizer follows from differentiating a quadratic function. Expanding the criterion gives \((h_N-G_N\theta)'A_N(h_N-G_N\theta)\). Symmetry of \(A_N\) yields the first-order condition below.

\frac{\partial Q_N(\theta)}{\partial\theta} =-2G_N'A_N(h_N-G_N\theta)=0, \qquad G_N'A_NG_N\widehat{\theta}_1=G_N'A_Nh_N.

If \(G_N'A_NG_N\) is nonsingular, this condition has the unique solution shown here. Since \(A_N\) is positive definite, nonsingularity holds when \(G_N\) has full column rank. This is the sample rank condition: the instrumented regressors must contain enough independent variation to identify all \(k\) parameters.

\widehat{\theta}_1 = (G_N'A_NG_N)^{-1}G_N'A_Nh_N.

The first-order condition gives the minimizer, rather than merely a stationary point: the Hessian is \(2G_N'A_NG_N\), which is positive definite under the stated rank condition. In an overidentified model, \(q>k\), the \(q\) sample moments generally cannot all be set to zero, so the weighting matrix determines how their discrepancies trade off. If \(q=k\) and \(G_N\) is nonsingular, the estimator instead solves \(h_N-G_N\widehat{\theta}_1=0\). Any admissible symmetric positive-definite weighting matrix then gives the same coefficient.

Worked Example: An Overidentified One-Parameter Estimate

Consider four hypothetical unit-level contributions to a differenced-panel moment calculation, summarized by one scalar regressor and two instruments. The averages are \(G_N=(1,\;1.5)'\) and \(h_N=(2,\;3.75)'\), and use the prespecified weighting matrix \(A_N=\operatorname{diag}(1,\;0.25)\). There are two moments for one coefficient, so the model is overidentified.

The estimate is the weighted least-squares solution for the moments:

\widehat{\theta}_1 = \frac{G_N'A_Nh_N}{G_N'A_NG_N} = \frac{(1)(1)(2)+(1.5)(0.25)(3.75)} {(1)(1)(1)+(1.5)(0.25)(1.5)} = \frac{3.40625}{1.5625} =2.18.

At this value the sample moment vector is \(g_N(\widehat{\theta}_1)=(2-2.18,\;3.75-1.5(2.18))'=(-0.18,\;0.48)'\). Neither moment is zero, as is typical in an overidentified sample. The criterion is \(Q_N(\widehat{\theta}_1)=(-0.18)^2+0.25(0.48)^2=0.09\). The estimate is the value minimizing this weighted discrepancy, not a value that makes each sample moment vanish.

Conventional Asymptotic Covariance

For inference, the conventional large-sample covariance is a sandwich that reflects both the chosen weighting matrix and the variability of the unit-level moments. The relevant asymptotic setting has many independent panel units and a fixed number of periods. Assume the moment restrictions are valid, the moments have finite second moments, the rank condition holds in the limit, and \(A_N\) converges to a fixed symmetric positive-definite matrix \(A\). Under these conditions, the linear moment structure gives the following covariance formula.

Theorem: Let \(G=\operatorname{plim}G_N\), \(A=\operatorname{plim}A_N\), and \(S=E[m_i(\theta_0)m_i(\theta_0)']\). If \(G'AG\) is nonsingular and the independent-unit central limit conditions hold, then the asymptotic covariance of \(\widehat{\theta}_1\) is

\operatorname{Avar}(\widehat{\theta}_1) = \frac{1}{N} (G'AG)^{-1}G'ASAG(G'AG)^{-1}.

A conventional sample estimate replaces \(G\) and \(A\) by their sample counterparts and estimates \(S\) using the unit-level moment contributions evaluated at the estimate:

\widehat{S} = \frac{1}{N}\sum_{i=1}^{N}m_i(\widehat{\theta}_1)m_i(\widehat{\theta}_1)', \qquad \widehat{\operatorname{Avar}}(\widehat{\theta}_1) = \frac{1}{N}(G_N'A_NG_N)^{-1}G_N'A_N\widehat{S}A_NG_N (G_N'A_NG_N)^{-1}.

The \(1/N\) factor is part of the covariance of the estimator, not of the moment covariance \(S\). This formula allows the moments to have a general covariance matrix; it does not require the specified one-step weight to equal the inverse of that covariance. Its large-sample justification relies on independence across units (or an appropriate alternative dependence structure accounted for in \(S\)). It is not a finite-sample guarantee, especially when \(N\) is small.

Worked Example: Computing a Scalar Sandwich Standard Error

Suppose a hypothetical exactly identified calculation has \(N=4\), \(G_N=2\), \(h_N=6\), and \(A_N=3\). Its four fitted unit-level moment contributions are \(-2,\;1,\;0,\;1\). These are illustrative summary quantities, not data from an actual study.

The coefficient is \(\widehat{\theta}_1=h_N/G_N=6/2=3\), so the displayed contributions are evaluated at the fitted value. Their second-moment estimate is

\widehat{S} = \frac{(-2)^2+1^2+0^2+1^2}{4} = \frac{6}{4} =1.5.

For one parameter, the sandwich formula reduces to

\widehat{\operatorname{Avar}}(\widehat{\theta}_1) = \frac{1}{4} \left(\frac{1}{2\cdot3\cdot2}\right) (2\cdot3\cdot1.5\cdot3\cdot2) \left(\frac{1}{2\cdot3\cdot2}\right) = \frac{1}{4}\frac{1.5}{2^2} =0.09375.

The corresponding conventional standard error is \(\sqrt{0.09375}=0.3062\), in the units of the estimated coefficient. The computation illustrates the asymptotic formula; with only four units, it should not be read as assurance that a normal approximation is accurate.

Why the Weight Matters

With more moments than parameters, changing \(A_N\) can change the estimate because it changes the relative influence of moment discrepancies. In the previous overidentified calculation, keep \(G_N\) and \(h_N\) fixed but use \(A_N=\operatorname{diag}(1,\;4)\) instead. This is also symmetric positive definite.

Worked Example: A Different Prespecified Weight

For the same hypothetical moment averages, the estimate with the new weighting matrix is

\widehat{\theta}_1 = \frac{(1)(1)(2)+(1.5)(4)(3.75)} {(1)(1)(1)+(1.5)(4)(1.5)} = \frac{24.5}{10} =2.45.

The fitted moments are \(2-2.45=-0.45\) and \(3.75-1.5(2.45)=0.075\). The criterion is \(0.45^2+4(0.075)^2=0.225\). The second moment receives more weight than it did in the first calculation, and the estimate shifts accordingly. Neither result can be declared preferable solely from these arithmetic calculations; the weighting choice must be justified by the estimation procedure and its assumptions.

Exact Identification and a Common Pitfall

When the number of moments equals the number of parameters, the moment equations can be solved directly if \(G_N\) is nonsingular. The resulting estimate \(G_N^{-1}h_N\) does not depend on the admissible weighting matrix. Positive definiteness remains important: it guarantees a valid minimization criterion, including in the exactly identified case.

Common pitfall: It is incorrect to say that any nonsingular weighting matrix gives the same coefficient in an exactly identified model. A nonsingular matrix need not be symmetric positive definite. For example, with \(q=k=1\), \(G_N=1\), \(h_N=0\), and the nonsingular weight \(A_N=-1\), the criterion is \(Q_N(\theta)=-\theta^2\). It has no minimum, even though the zero-moment equation has solution \(\theta=0\). The invariance claim applies to admissible symmetric positive-definite weights, which make the zero-moment solution a minimum.

A second practical distinction is between the sample moments and their unit-level contributions. The covariance estimate uses the individual vectors \(m_i(\widehat{\theta}_1)\); the aggregate \(g_N(\widehat{\theta}_1)\) alone does not reveal how moments vary across units. Retaining the unit-level contributions is therefore essential for calculating the conventional sandwich covariance.

Check Your Understanding

Use the definitions and calculations above to answer the following questions.

What are the dimensions of \(G_N\), \(A_N\), and \(g_N(\theta)\) when there are \(q\) moments and \(k\) parameters?
Why does positive definiteness of \(A_N\), together with full column rank of \(G_N\), make the first-order-condition solution a unique minimum?
In an overidentified model, why can changing the weighting matrix change the coefficient estimate?
What information from the sample is needed to estimate \(S\) for the conventional sandwich covariance?
In an exactly identified model, under what conditions is the estimate invariant to the weighting matrix, and why is nonsingularity alone insufficient?