196 The Basics of financial economeTrics
equilibrium in cointegration analysis means that if variables are apart, they
show a greater likelihood to move closer together than further apart.
More formally, consider two time series xt and yt. Assume that both
series are nonstationary and integrated order one. (Integrated order one
means that if we difference the variable one time, the resultant series is sta-
tionary.) These series are cointegrated if
zt = xt − aytzt is stationary for some value of a.
In the multivariate case, the definition is similar but vector notation
must be used. Let A and Y be vectors (a 1 , a 2 ,... , an) and (y 1 t, y 2 t ,... ,
ynt)′. Then the variables in Y are cointegrated if each of the y 1 t ,... , ynt are
nonstationary and, Z = AY, Z is stationary. A represents a cointegrating
vector.
Finding cointegration between two variables represents a special
case. We should not expect most nonstationary variables to be cointe-
grated. If two variables lack cointegration, then they do not share a long-
run relationship or a common stochastic trend because they can move
arbitrarily far away from each other. In terms of the present value model
of stock prices, suppose stock prices and dividends lack cointegration,
then stock prices could rise arbitrarily far above the level of their divi-
dends. Using a U.S. stock index and dividend data from 1887 through
2003, Gurkaynak illustrated that whenever stock prices are not cointe-
grated with dividends, stock prices rose far above the level justified by
the level of dividends.^6 This would be consistent with a stock market
bubble. Even if it is not a bubble, it is still inconsistent with the efficient
market theory.
In terms of stock market linkages, if the stock price indices of different
countries lack cointegration, then stock prices can wander arbitrarily far
apart from each other. This possibility lends support to proponents who argue
that investors would benefit from international portfolio diversification.
Testing for Cointegration
There are two popular methods of testing for cointegration: the Engle-
Granger tests and the Johansen-Juselius tests. We illustrate both in the
remainder of this chapter.
(^6) Refet Gurkaynak, “Econometric Tests of Asset Price Bubbles: Taking Stock,”
Journal of Economic Surveys 22 (2008): 166−186.