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.
Kadane, Joseph B., "The Distribution, When the Residuals Are Small, of Statistics Testing Overidentifying Restrictions" (1968). Cowles Foundation Discussion Papers. 484.