be a worthwhile endeavor, we do not discuss Rice’s formula in detail. Inter-
ested readers can find more specifics on this in the references at the end of
the chapter.
As mentioned earlier, our focus remains on the estimation of the zero
crossing rate directly from the given data sample. Obviously, the most direct
approach would be to count the number of zero crossings of the residual se-
ries and calculate the zero-crossing rate by dividing the number of crossings
by the total time. However, given that we are doing this with a sample size
of 1, the result is likely to be strongly biased to the residual series at hand.
The dilemma facing us is that we have only one realization of the residual se-
ries. The general idea of averaging across many sample residual series to rid
ourselves of the bias is not possible.
To resolve this, we resort to a resampling technique popularly known as
thebootstrap. In the bootstrap, however, we look to estimate the distribu-
tion of the time between two crossings; that is, the reciprocal of the cross-
ing rate. Note that the time between zero crossings is directly related to the
time that we expect to hold the paired position. Thus, the test for tradabil-
ity also leads us to an estimate of the holding period, which can be used as a
benchmark for time-based stops. The time between two crossings relates di-
rectly to the trading horizon and is therefore of direct relevance for trading.
114 STATISTICAL ARBITRAGE PAIRS
FIGURE 7.1 Density Plot of ArcSin Distribution.02040 60 80 100
Time Steps0.0100.0150.0200.0250.030Density