APPROXIMATE DISTRIBUTIONS OF THE LIKELIHOOD RATIO STATISTIC IN A STRUCTURAL EQUATION WITH MANY INSTRUMENTS

This paper studies properties of the likelihood ratio (LR) tests associated with the limited information maximum likelihood (LIML) estimators in a structural form estimation when the number of instrumental variables is large. Two types of asymp-totic theories are developed to approximate the distribution of the likelihood ratio (LR) statistic under the null hypothesis H0: β = β0: a (large sample) asymptotic expansion and a large-Kn asymptotic theory. Size comparisons of two modified LR tests based on these two asymptotics are made with Moreira’s conditional likelihood ratio (CLR) test and the large K t-test. Key words and phrases: Asymptotic expansion, large-Kn asymptotics, many instru-ments. 1