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 for 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.

中文翻译：

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看似无关的具有测量误差的回归：通过马尔可夫链蒙特卡罗估计法和平均场变贝叶斯近似法

协变量中带有测量误差的线性回归是一个受到广泛研究的话题，但是，统计/计量经济学文献几乎没有提及估计带有测量误差的多方程模型。本文考虑了一个在协变量中具有测量误差的看似无关的回归模型，并介绍了两种新颖的估计方法：纯贝叶斯算法（基于Markov链蒙特卡洛技术）及其平均场变贝叶斯（MFVB）近似。MFVB方法的另一个优点是计算速度快并且可以处理大数据。与测量误差模型相关的一个问题是参数识别，这可以通过对测量误差方差进行先验分布来解决。这些方法在多种模拟研究中均显示出良好的效果，在这里我们分析了可靠性比率或数据生成过程中使用的真实未观测量方差的不同值对后验估计的影响。本文进一步从健康文献中得出的应用中实现了所提出的算法，并表明对数据中的测量误差建模可以改善模型拟合。