On the Stock-Yogo Tables

A standard test for weak instruments compares the first-stage F-statistic to a table of critical values obtained by Stock and Yogo (2005) using simulations. We derive a closed-form solution for the expectation that determines these critical values. Inspection of this new result provides insights not available from simulation, and will allow software implementations to be generalized and improved. Of independent interest, our analysis makes contributions to the theory of confluent hypergeometric functions and the theory of ratios of quadratic forms in normal variables. A by-product of our developments is an expression for the distribution function of the non-central chi-squared distribution that we have not been able to find elsewhere in the literature. Finally, we explore the calculation of p-values for the first-stage F-statistic weak instruments test.