One-Sided Testing for ARCH E¤ect 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 e¤ect is small at each lag but carries over a long distributional lag. Because wavelets can e¤ectively capture spectral peaks, we expect that the wavelet test is more powerful than the kernel counterpart when there exists persistent ARCH or when ARCH e¤ect has a long distributional lag. This is con...rmed in a simulation study, which also compares a number of important one-sided and two-sided ARCH tests.