OLS Estimation when Two Noisy Measures of a Regressor are Available: Instruments, Tests and Pitfalls

It is well known that if the regression coefficient of y on w (beta say) has a constant probability limit but we only have two noisy measures of w - x and z - then we may obtain consistent estimates of beta as long as a) the measurement errors are classical and b) the measurement errors are uncorrelated. We propose a simple test of a) and a test for b) as part of a composite null. To effect the latter we instrument x with z and functions of z and vice versa to obtain two sets of overidentifying restrictions tested via a standard J test of instrument validity. If no test in this sequence rejects we then combine the orthogonality conditions to obtain a single efficient estimate of beta. We discuss the likely prior validity of the various instruments and the pitfalls in using the test procedure. Unlike standard overidentification tests which diverge in heterogeneous response settings even when each instrument is valid, our tests only diverge when one or more instrument is invalid. We apply the test sequence and estimation procedure to analyse i) the cyclical component of wages and ii) the effect of state level unemployment on burglaries in the US. Correcting for measurement error raises the estimates of beta in both applications.