Optimal instruments
Optimal instruments are a way to make estimators more efficient in conditional moment models using the generalized method of moments (GMM). From one model you can write many moment conditions; optimal instruments pick a best finite set of these conditions to minimize the estimator’s variance.
In a simple nonlinear regression with a scalar y, a predictor vector x, and parameter θ, the error u = y − x′θ satisfies E[u | x] = 0. By multiplying this with different functions z(x), you get many moment conditions E[u z(x)] = 0. The goal is to choose z(x) so the GMM estimator has the smallest possible asymptotic variance.
If data are iid, the estimator’s variance depends on σ²(x) = E[u² | x]. The optimal instruments z*(x) are the ones that lead to the smallest variance, and the resulting estimator is a generalized least squares form. However, z*(x) uses σ²(x), which is unknown, so the exact optimal instruments can’t be used directly. Researchers have developed ways to characterize and estimate them, including nonparametric methods.
Historically, researchers showed how many instruments are best in certain models and described the optimal form of the instruments. In linear conditional moment models with iid data, the optimal GMM estimator is the generalized least squares estimator.
Bottom line: using optimal instruments can yield a more precise (lower-variance) estimator than using arbitrary instruments.
This page was last edited on 2 February 2026, at 18:35 (CET).