Robust Testing for ARCH by Wavelet Shrinkage

Abstract

Understanding the nature of volatility is helpful in risk management. For example, the computation of the value at risk (VaR) relies critically on a good volatility measure. Given the popularity of the GARCH-type models in modeling financial volatility, we consider, in this paper, a robust testing for GARCH effect in the general context of a possibly mis-specified conditional mean, assuming that correct inference regarding the conditional variance is of primary interest to the researcher. It is well known in the literature that, when the conditional mean equation is mis-specified, the popular LM test for GARCH effect often leads to over-rejection of the null hypothesis of no GARCH, even when no such effect exists in the true data generating process. In this paper, we propose an alternative robust test of the GARCH effect by estimating the unknown conditional mean equation based on the wavelet shrinkage algorithm. An important feature of the wavelet shrinkage is its ability to remove noise while preserving non-smooth features, such as large spikes in the signal. Our experiments show that the wavelet shrinkage method indeed can significantly remove distortions introduced by nonlinearity without sacrificing the power of the test. To demonstrate the empirical relevance of the proposed test, we apply our robust test to study daily returns of the SP500 index from 11/20/1985 to 12/7/1989. Our result shows that the wavelet shrinkage method proposed here not only successfully picks up the structural change caused by the October 1987 stock market crash, it also prevent researchers from falsely rejecting the null of no GARCH effect in such a situation. This demonstrates the practical relevance of the robust test proposed in this research.