The Distribution, When the Residuals Are Small, of Statistics Testing Overidentifying Restrictions

In the estimation of simultaneous equation econometric models, overidentifying restrictions improve estimates of the remaining parameters. Natural test statistics for the hypothesis that an equation is overidentified have been developed by Anderson and Rubin and by Basmann. If the residuals are jointly normal, serially uncorrelated, and small, both the above overidentification test statistics have the Snedecor F distribution asymptotically as the variance of the residuals get small. This gives analytic confirmation of Monte Carlo results of Basmann. The results given apply to linear models in which predetermined variables can be exogenous or lagged endogenous.