Generalized method of moments fitting of structural mean models

In this talk we describe how to fit structural mean models (SMMs), as proposed by Robins, using instrumental variables in the generalized method of moments (GMM) framework using Stata's gmm command. The GMM approach is flexible because it can fit overidentified models in which there are more instruments than endogenous variables. It also allows assessment of the joint validity of the instruments using Hansen's J test through Stata's estat overid gmm postestimation command. In the case of the logistic SMM, the approach also allows different first-stage association models. We show the relationship between the multiplicative SMM and the multiplicative GMM estimator implemented in the ivpois command of Nichols (2007). For the multiplicative SMM, we show—analogously to Imbens and Angrist (1994) for the linear case--that the estimate is a weighted average of local estimates using the instruments separately. To demonstrate the models, we use a Mendelian randomization example, in which genotypes found to be robustly associated with risk factors from genome-wide association studies are used as instrumental variables, thereby investigating the effect of being overweight on the risk of hypertension in the Copenhagen General Population Study.