Information about the net effect of the positive correlation arising from simultaneity and the negative correlation arising from measurement error is provided by the difference between the instrumental variables estimate of в and the ordinary least squares estimate. The last row of Table 2 reports the latter. Because the OLS estimate is substantially smaller than the IV estimate, measurement error is the more important of the two influences.

Under the assumption that there is no true simultaneity problem, that is, e is uncorrelated with S, we can calculate the standard deviation of true social infrastructure, as, from the difference between the IV and OLS estimates. A standard result in the econometrics of measurement error is that OLS is biased toward zero by a multiplicative factor equal to the ratio of the variance of the true value of the right-hand variable to the variance of the measured value. Thus,

That is, we can estimate the standard deviation of true social infrastructure relative to the standard deviation of measured social infrastructure as the square root of the ratio of the OLS and IV estimates. With our estimates, the ratio of the standard deviations is 0.800.

If the correlation of S and e is positive, so true simultaneity is a problem, additional information is required to pin down as. The positive correlation from endogeneity permits a larger negative correlation from measurement error and therefore a larger standard deviation of that measurement error.