  Linear regression with measurement error in the covariates is a heavily
studied topic, however, the statistics/econometrics literature is almost silent
to estimating a multi-equation model with measurement error. This paper
considers a seemingly unrelated regression model with measurement error in the
covariates and introduces two novel estimation methods: a pure Bayesian
algorithm (based on Markov chain Monte Carlo techniques) and its mean field
variational Bayes (MFVB) approximation. The MFVB method has the added advantage
of being computationally fast and can handle big data. An issue pertinent to
measurement error models is parameter identification, and this is resolved by
employing a prior distribution on the measurement error variance. The methods
are shown to perform well in multiple simulation studies, where we analyze the
impact on posterior estimates arising due to different values of reliability
ratio or variance of the true unobserved quantity used in the data generating
process. The paper further implements the proposed algorithms in an application
drawn from the health literature and shows that modeling measurement error in
the data can improve model fitting.
