MTests 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, Vogelsang and Bunzel (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 e¤ect of parameter estimation and allow for conditional heteroskedasticity and serial correlation of general forms. They converge to central F distributions under the …xed-K asymptotics and to Chi-square distributions if K is allowed to approach in…nity. We illustrate their applications using three simulation examples: (1) speci…cation testing for conditional heteroskedastic models, (2) nonnested 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).

The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series

“…Kuan and Lee (2006) apply the approach to a class of specification tests, the so-called M tests, which are based on the moment conditions involving unknown parameters Chen and Qu (2015). propose a procedure for improving the power of the M test, by dividing the original sample into subsamples before applying the self-normalization procedure.7 The fixed-b asymptotic has been further studied byBunzel et al (2001), Vogelsang (2002, 2005),Sun et al (2008),Kim and Sun (2011) andSun and Kim (2012) among others.…”

Fixed-smoothing asymptotics for time series

“…Given the higher order expansions presented in Section 2, it seems natural to investigate if bootstrapping can help to improve the first-order approximation. Though the higher order corrected critical values can also be obtained by direct estimation of the leading error term, it involves estimation of the eigencomponents of the kernel function and a choice of truncation number for the leading error term ℵ T (x; ∞) [see (5)] besides estimating the second-order properties of the time series. Therefore it is rather inconvenient to implement this analytical approach because numerical or analytical calculation of eigencomponents can be quite involved, the truncation number and the bandwidth parameter used in estimating second-order properties are both user-chosen numbers, and it seems difficult to come up with good rules about their (optimal) choice in the current context.…”

“…The t-statistic-based approach was extended by Bester et al [3] to the inference of spatial and panel data with group structure. In the context of misspecification testing, Chen and Qu [5] proposed a modified M test of Kuan and Lee [15] which involves dividing the full sample into several recursive subsamples and constructing a normalization matrix based on them. In the statistical literature, Shao [27] developed the self-normalized approach to inference for time series data that uses an inconsistent LRV estimator based on recursive subsample estimates.…”