Exact Distributions of the Limited Information Maximum Likelihood Estimator and the Two-Stage Least Squares Estimator
The exact finite sample distributions of the two estimators differ. We shall discuss briefly the main results about these distributions as well as their approximations. The discussion is very brief because there are several excellent survey articles on this topic by Mariano (1982), Anderson (1982), Taylor (1982), and Phillips (1983).
In the early years (say, until the 1960s) most of the results were obtained by Monte Carlo studies; a summary of these results has been given by Johnston (1972). The conclusions of these Monte Carlo studies concerning the choice between LIML and 2SLS were inconclusive, although they gave a slight edge to 2SLS in terms of the mean squared error or similar moment criteria. However, most of these studies did not realize that the moment criteria may not be appropriate in view of the later established fact that LIML does not have a moment of any order and 2SLS has moments of the order up to and including the degree of overidentifiability (Kw — N, in our notation).
The exact distributions of the two estimators and their approximations have since been obtained, mostly for a simple two-equation model.6 Anderson (1982) summarized these results and showed that 2SLS exhibits a greater median-bias than LIML, especially when the degree of simultaneity and the degree of overidentifiability are large, and that the convergence to normality of 2SLS is slower than that of LIML.
Another recent result favors LIML over 2SLS; Fuller (1977) proposed the modified LIML that is obtained by substituting X~c/(T— K2), where c is any constant, for A in (7.3.3). He showed that it has finite moments and dominates 2SLS in terms of the mean squared error to 0( T~2). Fuller’s estimator can be interpreted as another example of the second-order efficiency of the bias-corrected MLE (see Section 4.2.4).
However, we must treat these results with caution. First, both Anderson’s and Fuller’s results were obtained under the assumption of normality, whereas the asymptotic distribution can be obtained without normality. Second, Anderson’s results were obtained only for a simple model; and, moreover, it is difficult to verify in practice exactly how large the degrees of simultaneity and overidentifiability should be for LIML to dominate 2SLS. Third, we should also compare the performance of estimators under misspecified models. Monte Carlo studies indicate that the simpler the estimator the more robust it tends to be against misspecification. (See Taylor, 1983, for a critical appraisal of the finite sample results.)