The Power of Hessian and Outer Product Based Wald and LM Tests.

Wald and Lagrange Multiplier (LM) tests can be based on three commonly used estimators of the information matrix : the expectation of the Hessian matric, the Hessian matrix without the expectation operator or the outer product (OP) matrix of the score vectors. Although the Wald and LM tests are asymptotically equivalent, they typically have different powers in finite samples. We prove that Hessian based tests are superior to OP based tests in the normal linear regression model.