A General Approximation to the Distribution of Instrumental Variables Estimates - Sargan, J. D.; Mikhail, W. M. (Econometrica, 1971)
This paper develops approximations of the Gram-Charlier type to the cumulative distribution function of the instrumental variables estimator on classical assumptions. In the special case where there are only two endogenous variables in the estimated equation, exact values of the cumulative distribution function are computed by numerical integration and compared with the approximations. Although the error in the approximation depends critically on the parameters of the stochastic model, the approximation is good for the special case even for small sample size over a wide range of values of the parameters. THIS PAPER was originally conceived as a study of the finite sample distribution of two stage least squares estimates. Since it was found that the distribution of a more general class of instrumental variables estimates can be discussed in the same way with a trifling complication of the algebra, the paper was modified to cover these estimates. The basic approach is somewhat similar to that of Nagar [15], since it involves expanding the formulae for the estimator as a series of terms of 0(1), O(T-+), O(T- 1), O(T- 1+), etc., and from this a similar expansion is found for the cumulative probability of the form