Approximating volatilities by asymmetric power GARCH functions

ARCH/GARCH representations of financial series usually attempt to model the serial correlation structure of squared returns. Although it is undoubtedly true that squared returns are correlated, there is increasing empirical evidence of stronger correlation in the absolute returns than in squared returns. Rather than assuming an explicit form for volatility, we adopt an approximation approach; we approximate the th power of volatility by an asymmetric GARCH function with the power index chosen so that the approximation is optimum. Asymptotic normality is established for both the quasi-maximum likelihood estimator (qMLE) and the least absolute deviations estimator (LADE) in our approximation setting. A consequence of our approach is a relaxation of the usual stationarity condition for GARCH models. In an application to real financial datasets, the estimated values for are found to be close to one, consistent with the stylized fact that the strongest autocorrelation is found in the absolute returns. A simulation study illustrates that the qMLE is inefficient for models with heavy-tailed errors, whereas the LADE is more robust.