Chapter 13
WALD. LIKELIHOOD RATIO, AND LAGRANGE
MULTiPLIER TESTS IN ECONOMETRICS
ROBERT F. ENGLE*
University of California
Contents
1. Introduction
2. Definitions and intuitions
3. A general formulation of Wald, Likelihood Ratio, and Lagrange
Multiplier tests
4. Two simple examples
5. The linear hypothesis in generalized least squares models
5.1. The problem
5.2. The test statistics
5.3. The inequality
5.4. A numerical example
5.5. Instrumental variables
6. Asymptotic equivalence and optimality of the test statistics
7. The Lagrange Multiplier test as a diagnostic
8. Lagrange Multiplier tests for non-spherical disturbances
8.1. Testing for heteroscedasticity
8.2. Serial correlation
9. Testing the specification of the mean in several complex models
9.1. Testing for non-linearities
9.2. Testing for common factor dynamics
9.3. Testing for exogeneity
9.4. Discrete choice and truncated distributions
10. Alternative testing procedures
11. Non-standard situations
12. Conclusion
References