A Note on Modelling Bidirectional Feedback Loops in Mendelian Randomization Studies

Structural equation models (SEMs) involving feedback loops may offer advantages over standard instrumental variables estimators in terms of modelling causal effects in the presence of bidirectional relationships. In the following note, we show that in the case of a single "exposure" and "outcome" variable, modelling relationships using a SEM with a simple bidirectional linear feedback loop offers no advantage over traditional instrumental variables estimators in terms of consistency (i.e. both approaches yield consistent estimates of the causal effect, provided that causal estimates are obtained in both directions). In the case of finite samples, traditional IV estimators and SEM exhibited similar power across many of the conditions we examined, although which method performed best depended on the residual correlation between variables and the strength of the instruments. In particular, the power of SEM was insensitive to the residual correlation between variables, whereas the power of the Wald estimator/2SLS improved (deteriorated) relative to SEM as the magnitude of the residual correlation increased (decreased) assuming a positive causal effect of the exposure on the outcome. The power of SEM improved relative to the Wald estimator/2SLS as the instruments explained more residual variance in the "outcome" variable.