On diagnostic checking in ARMA models with conditionally heteroscedastic martingale difference using wavelet methods

Wavelet-based tests for lack-of-fit in semi-strong autoregressive moving average models with conditional heteroscedastic martingale difference innovations are investigated. The chi-square distributions of the Box-Pierce-Ljung methods are not necessarily adequate in this context and adjustments appear necessary. Seasonal irregularities in the spectral density of the innovations can affect the power of the classical tests, providing motivations for studying wavelets. Using the Franklin wavelet, the asymptotic distributions of the empirical wavelet coefficients are derived, and the asymptotic chi-square distributions of the wavelet-based tests are established. Monte Carlo simulations are conducted to study the performance of the methodology under the null and alternative hypotheses, including seasonal alternatives.