Wald Test Statistics Research Articles

Kuhn-Tucker, likelihood ratio and Wald tests for nonlinear models with inequality constraints on the parameters

This paper considers the problem of testing statistical hypothesis in nonlinear regression models with inequality constraints on the parameters. First, the Kuhn-Tucker test procedure is defined. Next, it is shown that the distribution of the Kuhn-Tucker, the likelihood ratio and the Wald test statistics converges to the same mixture of chi-square distributions under the null hypothesis. To illustrate these results two examples are considered: (1) the problem of testing that individual effects are missing in an error component model, and (2) the problem of testing equilibrium for a model of markets in disequilibrium.

Conflict Among Testing Procedures in a Linear Regression Model with Autoregressive Disturbances

Silvey [10]. For a model with nonstochastic regressors we show that a systematic inequality relation exists among the test statistics; namely, the value of the Wald statistic is greater than or equal to that of the LR statistic which, in turn, is greater than or equal to that of the LM statistic. When the null hypothesis is true, we find that the Wald, LR, and LM test statistics have identical limiting chi-square distributions. Since for a large sample test the three procedures employ the same critical region, the inequality relation among the test statistics implies that there exists a significance level such that the tests will produce conflicting inferences. These results are parallel to those obtained by Berndt and Savin [2] in the context of a multivariate regression model with independent disturbance vectors. We also consider the Wald and LR tests for a model with a lagged dependent variable. In this case the Wald statistic is not the same as in the nonstochastic regressor case with the result that the inequality between the Wald and LR test statistics no longer holds. We conclude the paper with an empirical example which illustrates the relation among the test statistics.

Assessing proportionality assumption in the adjacent category logistic regression model

Ordinal logistic regression models are classified as either proportional odds models, continuation ratio models or adjacent category models. The common model assumption of these models is that the log odds do not depend on the outcome category. This assumption is also known as the or parallel logits assumption. Non-proportional and partial proportional models are proposed for the proportional odds and continuation ratio model. The non-proportional and the partial proportional versions of the adjacent category model are also feasible. Prior to fitting any of the ordinal logistic regression models, it is important to check whether the assumption of proportionality is satisfied by each independent variable. In the proportional odds model, the proportional odds assumption is checked by Brant's Wald test statistic, and the standard Wald test statistic can be used in the continuation ratio model. However there is no valid approach to test whether the proportionality assumption is satisfied by each independent variable in the adjacent category model. The aim of the study is to determine the variables in the adjacent category model that violate the proportionality assumption. For this purpose, a Wald test is proposed for testing the proportionality assumption in the adjacent category model. The validity of the proposed test is examined under H-0 with a Monte Carlo simulation study. Moreover, the proposed method is compared with the likelihood ratio test in terms of type I error rate and power under different scenarios.