One-Sided Testing for ARCH Effect Using Wavelets

There has been an increasing interest in hypothesis testing with inequality restrictions. An important example in time series econometrics is hypotheses on autoregressive conditional heteroskedasticity (ARCH). We propose a one-sided test for ARCH using the wavelet method, a new analytic tool developed in the last decade or so. The test is based on a wavelet spectral density estimator at frequency zero of the square of estimated residuals from a regression model. The square of an ARCH\ process is positively correlated at all lags, resulting in a spectral mode at frequency zero. In particular, it has a spectral peak at frequency zero when there exists persistent ARCH, or when ARCH effect is small at each lag but carries over a long distributional lag. Because wavelets can effectively capture spectral peaks, we expect that the wavelet test is more powerful than the kernel counterpart when there exists persistent ARCH or when ARCH effect has a long distributional lag. This is confirmed in a simulation study, which also compares a number of important one-sided and two-sided ARCH tests.