M Tests with a New Normalization Matrix

Abstract

This paper proposes a new family of M 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 M tests with an alternative normalization matrix, constructed using moment functions estimated from (K + 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 F distributions under the fixed-K asymptotics and to chi-square distributions if K 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 M tests while being substantially higher than that of Kuan and Lee (2006).

Keywords:

• Hypothesis testing
• Misspecification
• Model diagnostics
• Self-normalization

JEL Classification:

• C12
• C52

ACKNOWLEDGMENT

We thank the editor Esfandiar Maasoumi and an anonymous referee for very helpful comments and suggestions.

Notes

¹Notation: ∇_θ^⊤ 𝔼m _t (θ _o ) and ∇_θ^⊤ 𝔼s _t (θ _o ) are the partial derivatives of 𝔼m _t (θ) and 𝔼s _t (θ) evaluated at θ _o .

p is the number of restrictions being tested. κ denotes the maximum power difference.

ℳ_3T : the proposed test. ℳ_2T : the test of KL. ℳ_1T : the conventional test. T is the sample size and h is the horizon. All results are at 5% nominal level.

²When computing the tests, the likelihood function is maximized using the constrained maximum likelihood (CML) routine for the Gauss environment, under the restriction that the parameters in the variance equation are non-negative.

³ℳ_2T is constructed by setting the starting subsample size equal to 20.

ℳ_3T : the proposed test. ℳ_2T : the test of KL. ℳ_1T : the conventional test. T is the sample size and h is the horizon. All results are at 5% nominal level.

⁴The readers may refer to Choi and Kiefer (2008) for a detailed description about the source of the data set and the construction of the variables.

⁵As an alternative, we tried to model the regressors r _t and z _t as a VAR(1) and simulated them along with ϵ _t . The results showed no difference.

ℳ_3T : the proposed test. ℳ_2T : the test of KL. ℳ_1T : the conventional test. All results are at 5% nominal level.

{x _t } and {u _t } are i.i.d. N(0, 1) and are independent of each other.

ℳ_3T : the proposed test. ℳ_2T : the test of KL. ℳ_1T : the conventional test. T is the sample size and q is the number of restrictions being tested. All results are at 5% nominal level.

ℳ_3T : the proposed test. ℳ_2T : the test of KL. ℳ_1T : the conventional test. 𝒲ℒ: Wooldridge's test. T is the sample size and q is the number of restrictions being tested. All results are at 5% nominal level.