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We all know that a simple linear regression of x on Y does not give the reciprocal of the simple linear regression of Y on x. However, I recently realised that when you use instrumental variables this “anomaly” does not occur. The forward IV estimate is cov(y,z)/cov(x,z) and the reverse IV estimate of the reciprocal slope is cov(x,z)/cov(y,z).
At least this all works out for scalar variables. Turns out it does not for multiple IV variables and the difference forms a specification test by Hahn and Hausman (2002).
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