<italic>M</italic> Tests with a New Normalization Matrix

This paper proposes a new family of <italic>M</italic> tests building on the work of Kuan and Lee (2006) and Kiefer et al. (2000). The idea is to replace the asymptotic covariance matrix in conventional <italic>M</italic> tests with an alternative normalization matrix, constructed using moment functions estimated from (<italic>K</italic> + 1) recursive subsamples. The new tests are simple to implement. They automatically account for the effect of parameter estimation and allow for conditional heteroskedasticity and serial correlation of general forms. They converge to central <italic>F</italic> distributions under the fixed-<italic>K</italic> asymptotics and to chi-square distributions if <italic>K</italic> is allowed to approach infinity. We illustrate their applications using three simulation examples: (1) specification testing for conditional heteroskedastic models, (2) non-nested testing with serially correlated errors, and (3) testing for serial correlation with unknown heteroskedasticity. The results show that the new tests exhibit good size properties with power often comparable to the conventional <italic>M</italic> tests while being substantially higher than that of Kuan and Lee (2006).