The GMM is an estimation technique that is helpful where the distribution of the data may be unknown and therefore alternative estimation techniques such as maximum likelihood are not applicable. For a given model, a certain number of moment conditions would be specified, which are functions of the model parameters and the data, such that their expectation is zero at the parameters' true values. The GMM estimator of a linear instrumental variable (IV) model has been applied to individual-level data in Economics since the 1980s.

In the context of IV estimation, one moment condition is usually specified per IV. The GMM method then minimizes a certain norm of the sample averages of the moment conditions. For identification, at least as many IVs as there are parameters are required to be estimated. For binary outcomes, GMM estimation of structural mean models can be used to estimate parameters such as the causal risk ratio in MR studies using individual-level data.

## References

- Clarke PS, Palmer T, Windmeijer F. Estimating Structural Mean Models with Multiple Instrumental Variables Using the Generalised Method of Moments. ArXiv 2015; 30: 96-117.
- Hansen LP. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica 1982; 50: 1029-1054.

## Other terms in 'One-sample MR methods':

- MR with a time-to-event outcome
- Non-parametric methods with bounds of causal effect
- Polygenic risk score approach
- Structural Mean Models (SMMs)
- Two-stage least squares (TSLS)
- Two-stage least squares (TSLS) with binary outcomes
- Two-stage predictor substitution estimators
- Two-stage residual inclusion estimators
- Within-family MR