New Directions in Information Matrix Testing: Eigenspectrum Tests

Model specification tests are essential tools for evaluating the appropriateness of probability models for estimation and inference. White (Econometrica, 50: 1–25, 1982) proposed that model misspecification could be detected by testing the null hypothesis that the Fisher information matrix (IM) Equality holds by comparing linear functions of the Hessian to outer product gradient (OPG) inverse covariance matrix estimators. Unfortunately, a number of researchers have reported difficulties in obtaining reliable inferences using White’s (Econometrica, 50: 1–25, 1982) original information matrix test (IMT). In this chapter, we extend White (Econometrica, 50: 1–25, 1982) to present a new generalized information matrix test (GIMT) theory and develop a new Adjusted Classical GIMT and five new Eigenspectrum GIMTs that compare nonlinear functions of the Hessian and OPG covariance matrix estimators. We then evaluate the level and power of these new GIMTs using simulation studies on realistic epidemiological data and find that they exhibit appealing performance on sample sizes typically encountered in practice. Our results suggest that these new GIMTs are important tools for detecting and assessing model misspecification, and thus will have broad applications for model-based decision making in the social, behavioral, engineering, financial, medical, and public health sciences.