THIS PAPER IS INTENDED to stress the distinction between the conditions for lack of identification in models linear with respect to the variables but nonlinear in the parameters in the sense originally defined by Fisher [2], and the less numerous set of conditions required for first order lack of identification. The latter set of conditions involve only the first derivatives of the coefficients as functions of the parameters. It is argued that if the model suffers from first order lack of identification, it will generally be the case that the usual estimators are consistent, although not asymptotically normally distributed. In a leading special case the asymptotic distribution is discussed, and the simulation of a simple model illustrates the extent to which this asymptotic distribution approximates the actual finite sample distribution.

[This paper is concerned with giving general conditions for the validity of a approximation to a distribution function by means of the weighted sum of a set of Chi-squared distribution functions. The conditions are very similar to those of [3]. The approximation is illustrated by an example of its use.]

[This paper discusses the constraints on a dynamic equation represented by the possibility of factoring out an autoregressive error specification from a general lag structure. A suitable Wald test is defined and applied to a practical case.]

[A specialization of the Edgeworth type formulae due to Chambers [4] to approximate the marginal distribution of an econometric estimator is presented, and its application to improvement of the use of asymptotic limits in significance testing is discussed. Appendices discuss the validity of Nagar approximations to estimator moments, the exact distribution of the instrumental variable estimator, the Edgeworth approximations for 3SLS and FIML estimators, and the use of Monte Carlo procedures for assessing the appropriate probability to attach to a given significance test.]

This paper obtains Edgeworth or Gram-Charlier expansions for the t ratio of instrumental variable and k-class estimators, and uses them to give approximations to the confidence intervals obtained from these t ratios. These confidence intervals for large sample size are more accurate than the usual asymptotic confidence interval. Charlier expansions is applied to the t ratio of 2SLS and non-stochastic k-class estimators. Previous general theorems in this field, with the exception of those given by Chambers [2], such as those in [3] have assumed that the statistic has moments of appropriate orders. The theorem proved here assumes only that it can be expressed as a function of other variables with moments of all orders with appropriate properties in some neighborhood of the origin. It can be applied to a wide range of

[This paper establishes certain conditions for the validity of Nagar's approximations to the mean and second moments of econometric estimations, where the estimators are rational functions of the OLS estimator of the reduced form coefficients, and of the corresponding estimate of the equation error variance matrix. The criteria depend upon the existence, and asymptotic orders of magnitude, of the moments of the estimators.]