*Journal of Econometrics* 11 (1979) 275-302

**STATISTICAL INFERENCE FOR A SYSTEM OF SIMULTANEOUS, NON-LINEAR, IMPLICIT EQUATIONS IN
THE CONTEXT OF INSTRUMENTAL VARIABLE ESTIMATION**

**A. Ronald GALLANT**

North Carolina State University, Raleigh, NC 27607, USA

**Dale W. JORGENSON**

Harvard University, Cambridge, MA 02138, USA

Received February 1979

Statistical inference for a system of simultaneous, non-linear, implicit equations is discussed. The discussion considers inference as an adjunct to two- and three-stage least squares estimation rather than in a general setting. For both of these cases the non-null asymptotic distribution of a test statistic based on the optimization criterion and a test based on the asymptotic distribution of the estimator is found; a total of four. It is argued that the tests based on the optimization criterion are to be preferred in applications. The methods are illustrated by application to hypotheses implied by the theory of demand using a translog expenditure system and data on personal consumption expenditures for durables, non-durables, and energy for the period 1947-1971.