Applying estimated score tests in econometrics

The estimated score test is a hypothesis testing procedure that can improve on the standard score, or Lagrange multiplier (LM) test. The score for a parameter of interest will usually contain nuisance parameters. Essentially, the test replaces nuisance parameters by estimates and then takes the difference between this estimated score and its expectation as the critical region for a test. If the expectation is zero the test coincides with the standard score, or LM, test, but if the expectation is non-zero the small sample properties of the tests differ. In some cases even asymptotic properties differ. This paper examines the scope for the application of estimated score tests in econometrics and illustrates with examples. Comparisons with the standard tests emphasize differences between the tests in terms of the true, as distinct from nominal, sizes of tests.