econometric

Journal

of Econometrics

41 (1989) 205-235.

North-Holland

TESTING INEQUALITY CONSTRAINTS IN LINEAR
ECONOMETRIC MODELS
Frank A. WOLAK*
Stanford

Received

lJniversi[v,

February

Stunford,

CA 94305, tiSA

1986, final version received July 1988

This paper develops three asymptotically equivalent tests for examining the validity of imposing linear inequality restrictions on the parameters of linear econometric models. First we consider the model .v = X/3 + e. where r is N(O,8), and the hypothesis test H: R/l 1 r versus K: p E R”. Later we generalize this testing framework to the linear simultaneous equations model. We show that the
Joint asymptotic distribution of these test statistics and the test statistics from the hypothesis test
H: RP = I versus K: R/3 2 r is a weighted sum of two sets of independent
X’-distributions.
We also derive a useful duality relation between the multivariate inequality constraints test developed here and the multivariate one-sided hypothesis test. In small samples, these three test statistics satisfy inequalities similar to those derived by Berndt and Savin (1977) for the case of equality constraints. The paper also contains an illustrative application of this testing technique.

1. Introduction
Estimation
under inequality restrictions has a long history in regression analysis and its application has become more widespread with the increase in sophistication of computer software. Judge and Takayama (1966) introduced least squares regression under inequality restrictions and suggested its formulation as a quadratic programming problem. Liew (1976) discussed the largesample properties of the estimator as well as presented results of a simulation study of the small-sample properties of the inequality constrained least squares
(ICLS) estimator. The increased use of this estimation technique suggests the need for a hypothesis testing procedure to examine its validity.
An

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