(^1) There is a one-step test for cointegration originally proposed by Johannsen, which
I do not discuss here. However, references for it are in the appendix.
the two stock series (not to be confused with the correlation between the
first differences or returns of the series). A strong correlation is usually evi-
denced by a good fit in the regression of one time series against the other.
Additionally, a good fit is characterized by a strong tstatistic and r-squared
measure. Therefore, deducing from the preceding information, we could
look for a good value for the tstatistic and the r-squared measure from the
regression of the two time series, infer a strong correlation between the two
series, and declare the existence of a common trend. This sounds logical,
does it not? However, it is not true. Let us see why that is.
Upon careful examination of the preceding argument, it can be seen that
it is actually incomplete. The argument would be complete if we could assert
that the good fit upon regression or strong correlation property is a trait ex-
clusive to cointegrated systems. Only if that were true could we conclude
that evidence of the property implies cointegration. Unfortunately for us, the
good fit on regression property is not exclusive to cointegrated systems. As
a matter of fact and rather surprisingly, completely independent random
walks when regressed against each other can also result in a high r-squared
measure. This rather counterintuitive phenomenon was reported in the find-
ings of a simulation experiment conducted by Granger and Newbold, who
aptly coined the phrase “spurious regression” to describe it. Thus, as evi-
denced by spurious regression, the strong correlation property is not exclu-
sive to cointegrated systems and therefore cannot be used as a definitive test
for cointegration.
We now turn to another property of cointegrated systems. This is the
existence of a linear combination of two time series that is stationary and
mean reverting. That property is in fact a defining property of cointegration
systems. We could therefore design a cointegration test based on the verifi-
cation of this property. Such a test for cointegration was first prescribed by
Engle and Granger. The rationale behind the test proceeds somewhat like
this: If two time series are cointegrated, then a simple regression of one time
series against the other should produce a good estimate of the linear rela-
tionship. Also, the spread series resulting from the linear relationship, that
is, the residual series from the regression, must be stationary. Therefore, to
test for cointegration all we need to do is estimate the linear relationship be-
tween the two series given using simple regression and test for stationarity of
the residuals. If the residuals form a stationary time series, then we have a
cointegrated pair. Thus, cointegration testing is a two-step process:^1
- Determination of the linear relationship.
- Stationarity testing on the residuals.
Testing for Tradability 105