  We attempt to unveil the fine structure of volatility feedback effects in the
context of general quadratic autoregressive (QARCH) models, which assume that
today's volatility can be expressed as a general quadratic form of the past
daily returns. The standard ARCH or GARCH framework is recovered when the
quadratic kernel is diagonal. The calibration of these models on US stock
returns reveals several unexpected features. The off-diagonal (non ARCH)
coefficients of the quadratic kernel are found to be highly significant both
In-Sample and Out-of-Sample, but all these coefficients turn out to be one
order of magnitude smaller than the diagonal elements. This confirms that daily
returns play a special role in the volatility feedback mechanism, as postulated
by ARCH models. The feedback kernel exhibits a surprisingly complex structure,
incompatible with models proposed so far in the literature. Its spectral
properties suggest the existence of volatility-neutral patterns of past
returns. The diagonal part of the quadratic kernel is found to decay as a
power-law of the lag, in line with the long-memory of volatility. Finally,
QARCH models suggest some violations of Time Reversal Symmetry in financial
time series, which are indeed observed empirically, although of much smaller
amplitude than predicted. We speculate that a faithful volatility model should
include both ARCH feedback effects and a stochastic component.
