  Instrumental variable approaches have gained popularity for estimating causal
effects in the presence of unmeasured confounding. However, the availability of
instrumental variables in the primary population is often challenged due to
stringent and untestable assumptions. This paper presents a novel method to
identify and estimate causal effects in the primary population by utilizing
instrumental variables from the auxiliary population, incorporating a
structural equation model, even in scenarios with nonlinear treatment effects.
Our approach involves using two datasets: one from the primary population with
joint observations of treatment and outcome, and another from the auxiliary
population providing information about the instrument and treatment. Our
strategy differs from most existing methods by not depending on the
simultaneous measurements of instrument and outcome. The central idea for
identifying causal effects is to establish a valid substitute through the
auxiliary population, addressing unmeasured confounding. This is achieved by
developing a control function and projecting it onto the function space spanned
by the treatment variable. We then propose a three-step estimator for
estimating causal effects and derive its asymptotic results. We illustrate the
proposed estimator through simulation studies, and the results demonstrate
favorable performance. We also conduct a real data analysis to evaluate the
causal effect between vitamin D status and BMI.
