Properties of t ratios associated with the LIML, TSLS, and OLS estimators in a structural form estimation are studied. The existence of moments of these t ratios including the LIML form is proved first. Second, Monte Carlo simulations are performed to find out real sizes of the t test and the likelihood ratio test. Third, asymptotic expansions of the distributions of t ratios are derived under the null hypothesis to find out deviations of real sizes from nominal sizes theoretically. The asymptotic power functions are also derived

[In the estimation of structural coefficients it is well-known that both two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators are consistent and asymptotically efficient, and that the exact mean of the LIML estimator does not exist. Then the TSLS estimator, which is computationally simpler, has appeared a proper choice to empirical researchers. In this article asymptotic properties of the k-class and related estimators are sorted out according to the ratio between the total number of exogenous variables and the number of observations. It is found that the TSLS distribution deviates far from its traditional asymptotic distribution; the LIML distribution stays stable about its traditional asymptotic distribution. The LIML estimator now seems more attractive than the TSLS estimator except for the fact that its exact moments do not exist. A modified estimator is proposed which is asymptotically better than the LIML estimator and whose exact moments exist.]

[This article first illuminates possible discrepancies between normal and logistic models in the bivariate dichotomous case of qualitative response analyses. Then a Cox-type test statistic for separate families of hypotheses is proposed for the purpose of comparing the two models. An asymptotic distribution of the new test statistic is derived, and also the consistency of the test is shown. Before applying the Cox-type tests to actual data, Berkson's minimum chi-square estimators of the two models are explained in Section 3. In Section 4, logistic, normal, and linear models are worked out for a set of economic data, and the Cox-type tests are applied to compare them.]

[We apply the third-order efficient method of estimation to the estimation problem of a system of structural equations in econometrics. The maximum likelihood estimator (hereafter m.l.e.) of structural equations is proved to give uniformly higher probability of concentration about true values than any regular best asymptotically normal estimator, if its asymptotic bias is properly adjusted. For instance, the full-information or limited-information m.l.e give asymptotically uniformly higher probability of concentration than the three-stage or two-stage least-squares estimators, given that these estimators are adjusted to have the same biases. The same result holds for he subsystem m.l.e. We prove the asymptotic completeness of Fuller's modified estimator. Asymptotic expansions of the distributions of the full-information m.l., subsystem m.l., and limited-information m.l. estimators are derived to terms of order O(T extasciicircum-1). Our general theorem is also applied to the multi-equation seemingly unrelated regression (SUR) model.]

[We often see that the F test is applied to testing significance of a subset of coefficients even in a structural equation. This is obviously a doubtful method because the sum of squared errors is not distributed as χ ^2 in a simultaneous equation system. It is known, however, that the likelihood ratio test is asymptotically distributed as χ ^2 with proper degrees of freedom. We analyze the asymptotic properties of these two kinds of test statistics. We find the likelihood ratio method associated with limited information maximum likelihood estimation is reliable in practice.]