Reexamination of the Limited-Information Maximum Likelihood Estimation

Abstract

With the revision of the Limited-Information Maximum Likelihood (Henceforth, LIML) estimation proposed by Fuller (1977) as a start, the LIML has been reappraised by Fujikoshi et al. (1982) and Morimune (1983), mainly because the modified LIML estimators exhibit fairly well behaved small sample properties. However, the first step of the LIML estimation procedure, a minimisation of a ratio of quadratic forms, has been solved a formula based upon the assumption that the denominator is positive definite, while the positive definiteness is apt to collapse when the sample size is small relative to the number of all predetermined variables included in the model under construction, even if the order condition of the identifiability is fulfilled. On the other hand, we have the canonical corregression proposed by Professor Paelinck (1985; pp.242–47) as an estimation method completely free of the restriction stated above. Therefore, it is of some value to develop a computational procedure which makes the LIML estimation executable without the positive definiteness so far presumed and to synthesise the LIML and the canonical corregression. In addition, it is needless to say that the proposed procedure includes the usual one as a special case and keeps the well behavedness of the LIML estimators aforementioned almost intact.