Testing inequality constraints in linear econometric models

Abstract

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.

Empirical Implications of Alternative Models of Firm Dynamics

“…). It is true, however, that the first implication, that is that hazard rates are nonincreasing in size at a given age, both persists and is consistent with the data from every empirical study we are aware of [Churchill, 1955;Wedervang, 1965;Evans 1987a and1987b;Samuelson, 1988 and1989]. However, most other models that allow for mortality, including…”

“…and that the variance in growth rates (again usually conditional on size) is nonincreasing in age {These implications are discussed in Jovanovic, 1982;Evans, 1987a and1987b;andSamuelson, 1988 and1989).…”

Testing for changes in the SES-mortality gradient when the distribution of education changes too

“…The null distribution of this test statistic is a mixture of chi-squares, one for each possible number of violations, where the mixture weights are the probability of that number of violations. As suggested in Wolak (1989), we compute these weights via simulation using the estimated variance matrix for the vector of mortality changes among all groups. (Additional detail is provided in the appendix.…”

“…Thus, under the null, the test statistic is a mixture of K chi-squared random variables with degrees of freedom varying from 0 to K. The mixture weight on the χ 2 with exactly n degrees of freedom is given by the probability of observing exactly n violations if the true differences(β k t −β k t ) are all exactly zero (i.e., on the boundary of violations are difficult to obtain in closed form and depend on the variance-covariance matrix of the estimated coefficients. We compute these weights via simulation as suggested in Wolak (1989). Accordingly we simulate draws from a K-dimensional multivariate normal with mean zero and variance matrix equal to the estimated variance matrix for the vector of differences,…”

“…5 While multiple two-sided hypothesis tests are straightforward to implement using standard methods and software packages, multiple one-sided hypothesis tests of the type we are considering here are less straightforward. In Section 2.2 we adapt results from Kodde and Palm (1986) and Wolak (1989) to our context and thereby derive the required test statistics and critical values.…”