maximum liquidity price is the volume weighted average price, commonly
termed as the VWAP price. We could therefore construct our time series
with VWAP prices.
By choosing the VWAP price we are arguably within reason to assume
that the error distributions of the VWAP price from the maximum liquidity
price is about the same regardless of the magnitude of the price range. We
can therefore resort to the simple ordinary least squares version in our re-
gression analysis. Of course, the manner in which the calculation of standard
error forming the basis for the tstatistics tests would still need alteration. In
our scenario of cointegration or tradability testing, the emphasis on the tsta-
tistic is rather low, and therefore we contend that this is something we can
live with. Additionally, note that using the VWAP price has the tendency to
temper extreme values and therefore has the added benefit of minimizing
the effect of outliers on the regressions. In conclusion, the time series con-
structed with the VWAP price is better suited to understand equilibrium re-
lationships and should be the preferred approach.
However, this does not do away with the need to decide which of the
two price series we should use for the independent variable. We will avoid
being repetitive and just say that the same idea as was adopted in the multi-
factor model case may be applied here, also.
Testing Residual for Tradability
Subsequent to estimating the equilibrium relationship we need to construct
the residual time series. Although we advocated using VWAP prices to esti-
mate the equilibrium relationship, we recommend constructing the residual
time series by applying the equilibrium relationship to the time series of
stocks constructed using the close-close method. If the two series are indeed
cointegrated, constructing such a time series provides us with a fairly good
picture of the oscillations about the equilibrium value.
We begin by reviewing the ideal situation. In an ideal situation for trad-
ability, the two stocks would be cointegrated, and the residual series would
be stationary. It is therefore desirable that the properties of the residual se-
ries exhibit the characteristics of stationary series. One of the properties of
stationary series, the property of mean reversion, is of particular importance
to us. This is relevant because pairs trading is a bet that the residual series
will revert to its mean or equilibrium value. In other words, deviations from
the mean are quickly corrected by the series moving back toward the mean.
It would therefore be nice if we could quantify the degree of mean reversion
of a given time series.
It turns out that highly mean-reverting series are also characterized by a
high frequency of zero-crossings. A zero-crossingis defined as the transition
of the time series across its long-run mean. The frequency of zero-crossing is
112 STATISTICAL ARBITRAGE PAIRS