Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

TLDR

The paper investigates the properties of the quasi‑maximum likelihood estimator in dynamic mean–variance models under nonnormality and derives simple formulas for asymptotic standard errors. The authors employ quasi‑maximum likelihood estimation with auxiliary regressions to compute robust LM tests and obtain closed‑form asymptotic standard errors for the mean–variance model. The QMLE is consistent and asymptotically normal, and the proposed robust inference procedures—requiring only first derivatives—perform well in simulations and empirical stock‑return volatility data, outperforming standard Wald and IM tests with negligible bias.

Abstract

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.