  We make an extensive empirical study of the market impact of large orders
(metaorders) executed in the U.S. equity market between 2007 and 2009. We show
that the square root market impact formula, which is widely used in the
industry and supported by previous published research, provides a good fit only
across about two orders of magnitude in order size. A logarithmic functional
form fits the data better, providing a good fit across almost five orders of
magnitude. We introduce the concept of an "impact surface" to model the impact
as a function of both the duration and the participation rate of the metaorder,
finding again a logarithmic dependence. We show that during the execution the
price trajectory deviates from the market impact, a clear indication of
non-VWAP executions. Surprisingly, we find that sometimes the price starts
reverting well before the end of the execution. Finally we show that, although
on average the impact relaxes to approximately 2/3 of the peak impact, the
precise asymptotic value of the price depends on the participation rate and on
the duration of the metaorder. We present evidence that this might be due to a
herding phenomenon among metaorders.
