Final_1.pdf

then the number of times we can expect the time series to cross its equilib-
rium value in unit time. Thus, the zero-crossing frequency provides us with
a quantitative characterization for the mean reversion property.
Notice that if the zero-crossing rate is very high, then the time to revert
to mean is short, implying that the time we need to hold the paired position
is small. The signal-to-noise ratio is bound to be good, and we could be
more comfortable with the idea that the pair is tradable. Thus, a high zero-
crossing rate for the residual series is a preferred trait and directly appeals to
our requirements.
A high zero-crossing rate is also indicative of a stationary series. To
strengthen this conviction, we make an observation in contrast by consider-
ing the example of Brownian motion a nonstationary series. Even though the
distribution of Brownian motion is symmetric about the mean, the zero-
crossing event is very infrequent. The theoretical explanation for the phe-
nomenon is well captured by the famous arcsine law for Brownian motion
discovered by P. Levy. The law provides us with information on the last pas-
sage time, or the last time that the Brownian motion visited zero. More pre-
cisely, let us consider a Brownian motion starting at zero, or time t= 0 and
stopped at time T. If gis the last time when zero is visited, then the proba-
bility distribution satisfies

(7.7)

The density function corresponding to this is given as

(7.8)

A quick plot of the density graph in Figure 7.1 shows that we expect the zero
crossing to have occurred with high probability only at the extremes of the
time interval. Thus, we would expect very few zero crossings for Brownian
motion a nonstationary time series.
For a stationary ARMA time series with known parameters, the theo-
retical zero-crossing rate may be calculated using the formula developed by
Rice. The Rice formula is the sum of a series of zero-crossing probabilities
at various time steps. The probability at each lag is calculated using the auto-
correlation function. Thus, if the residual series is stationary and the ARMA
parameters are known, we can apply Rice’s formula to obtain the zero-
crossing rate. Note that this requires us to estimate the ARMA parameters
with the assumption that the series is actually stationary. This is something
we wish to avoid. We favor a model free approach and will focus on that to
obtain an estimate of the zero-crossing rate. Therefore, even though it would

Pu

uT u

()=

()−

11

π

Pg u

u

T

()< =

2

π

arcsin

Testing for Tradability 113