The Effect of Nuisance Parameters on the Power of LM Tests in Logit and Probit Models

In econometrics, most null hypotheses are composite, dividing the parameters into parameters of interest and nuisance parameters. The domain of the nuisance parameters can influence the size-corrected critical value and hence the power of a test. We show that the domain of the nuisance parameters determines which version of the LM test to use in logit and probit models. In these models there are two commonly used ways to construct the LM test: it can be based on the Hessian matrix or the outer product (OP) matrix of the score vectors. For the OP based LM test, the domain of the nuisance parameters strongly influences the size-corrected critical value whereas the same is not true for the Hessian based LM test. A theoretical explanation is developed using large nuisance parameter asymptotics. For empirically relevant domains, the experimental evidence shows that the Hessian based LM test has better finite sample power than the OP based LM test.