Instrumental Variables, Errors in Variables, and Simultaneous Equations Models: Applicability and Limitations of Direct Monte Carlo

A Direct Monte Carlo (DMC) approach is introduced for posterior simulation in theInstrumental Variables (IV) model with one possibly endogenous regressor, multipleinstruments and Gaussian errors under a flat prior. This DMC method can also beapplied in an IV model (with one or multiple instruments) under an informativeprior for the endogenous regressor's effect. This DMC approach can not be appliedto more complex IV models or Simultaneous Equations Models with multiple endogenous regressors. An Approximate DMC (ADMC) approach is introduced thatmakes use of the proposed Hybrid Mixture Sampling (HMS) method, which facilitates Metropolis-Hastings (MH) or Importance Sampling from a proper marginalposterior density with highly non-elliptical shapes that tend to infinity for a pointof singularity. After one has simulated from the irregularly shaped marginal distri-bution using the HMS method, one easily samples the other parameters from theirconditional Student-t and Inverse-Wishart posteriors. An example illustrates theclose approximation and high MH acceptance rate. While using a simple candidatedistribution such as the Student-t may lead to an infinite variance of ImportanceSampling weights. The choice between the IV model and a simple linear model un-der the restriction of exogeneity may be based on predictive likelihoods, for whichthe efficient simulation of all model parameters may be quite useful. In future workthe ADMC approach may be extended to more extensive IV models such as IV withnon-Gaussian errors, panel IV, or probit/logit IV.