Structural Equation Model Averaging: Methodology and Application

The instrumental variable (IV) methods are attractive since they can lead to a consistent answer to the main question in causal modeling, that is, the estimation of average causal effect of an exposure on the outcome in the presence of unmeasured confounding. However, it is now acknowledged in the literature that using weak IVs might not suit the inference goal satisfactorily. We consider the problem of estimating causal effects in an observational study in this article, allowing some IVs to be weak. In many modern learning jobs, we may face a large number of instruments and their quality could range from poor to strong. To incorporate them in a 2-stage least squares estimation procedure, we consider a model averaging technique. The proposed methods only involve a few layers of least squares estimation with closed-form solutions and thus is easy to implement in practice. Theoretical properties are carefully established, including the consistency and asymptotic normality of the estimated causal parameter. Numerical studies are carried out to assess the performance in low- and high-dimensional settings and comparisons are made between our proposed method and a wide range of existing alternative methods. A real data example on home price is analyzed to illustrate our methodology.