Simulation-Based Finite-Sample Inference in Simultaneous Equations

Abstract

In simultaneous equation (SE) contexts, nuisance parameter, weak instruments and identification problems severely complicate exact and asymptotic tests (except for very specific hypotheses). In this paper, we propose exact likelihood based tests for possibly nonlinear hypotheses on the coefficients of SE systems. We discuss a number of bounds tests and Monte Carlo simulation based tests. The latter involves maximizing a randomized p-value function over the relevant nuisance parameter space which is done numerically by using a simulated annealing algorithm. We consider limited and full information models. We extend, to non-Gaussian contexts, the bound given in Dufour (Econometrica, 1997) on the null distribution of the LR criterion, associated with possibly non-linear- hypotheses on the coefficients of one Gaussian structural equation. We also propose a tighter bound which will hold: (i) for the limited information (LI) Gaussian hypothesis considered in Dufour (1997) and for more general, possibly cross-equation restrictions in a non-Gaussian multi-equation SE system. For the specific hypothesis which sets the value of the full vector of endogenous variables coefficients in a limited information framework, we extend the Anderson-Rubin test to the non-Gaussian framework. We also show that Wang and Zivot's (Econometrica, 1998) asymptotic bounds-test may be seen as an asymptotic version of the bound we propose here. In addition, we introduce a multi-equation Anderson-Rubin-type test. Illustrative Monte Carlo experiments show that: (i) bootstrapping standard instrumental variable (IV) based criteria fails to achieve size control, especially (but not exclusively) under near non-identification conditions, and (ii) the tests based on IV estimates do not appear to be boundedly pivotal and so no size-correction may be feasible. By contrast, likelihood ratio based tests work well in the experiments performedSimultaneous Equation, Weak Instruments, Monte Carlo test, Identification