M Tests with a New Normalization Matrix

This paper proposes a new family of M tests, building on the work of Kuan and Lee (2006) and Kiefer, Vogelsang and Bunzel (2000). The new test replaces the asymptotic covariance matrix in the conventional M test with an alternative normalization matrix, constructed using moment functions estimated from (K + 1) recursive subsamples. It is simple to implement, automatically accounts for the e¤ect of parameter estimation uncertainty, and allows for condi- tional heteroskedasticity and serial correlation of general forms. It converges to the central F distribution under the fixed-K asymptotics, and to the Chi-square distribution if K is allowed to approach in?nity. We illustrate its applicability using three simulation examples. They are: (1) specification testing for conditional heteroskedastic models, (2) nonnested testing with serially correlated errors, and (3) testing for serial correlation with unknown heteroskedasticity. The test exhibits good size properties and its power can be substantially higher than the test of Kuan and Lee (2006). Overall, the results suggest that, by integrating "self-normalization" and "fixed-bandwidth asymptotics" into the M-testing framework, we obtain an analytically simple yet widely applicable approach to misspeci?cation testing.