Cointegration 193
relationship is reinforced with a very high R^2 value. However estimating the
same regression over a different time period, 1990−2007, reveals
Japanese Stock index=2905.67−0.29 U.S. Stock Index
t-statistic (30.54) (2.80) R^2 = 0.04This regression equation suggests there is a strong negative relationship between
the two stock market indices. Although the t-statistic on the slope coefficient
(2.80) is large, the low R^2 value suggests that the relationship is very weak.
The reason behind these contradictory results is the presence of stochas-
tic trends in both series. During the first time span, these stochastic trends
were aligned, but not during the second time period. Since different economic
forces influence the stochastic trends and these forces change over time, dur-
ing some periods they will line up and in some periods they will not. In
summary, when the variables have stochastic trends, the OLS technique may
provide misleading results. This is the spurious regression problem.
Recall that the OLS method requires that the observations are inde-
pendent and identically distributed, and because the monthly values of
the Japanese stock index (as well as those of the U.S. stock index) are not
independent and identically distributed, the use of OLS regression for such
monthly series is meaningless.
Another problem is that when the variables contain a stochastic trend,
the t-values of the regressors no longer follow a normal distribution, even
for large samples. Standard hypothesis tests are no longer valid for these
nonnormal distributions.
At first, researchers attempted to deal with these problems by removing
the trend through differencing these variables. That is, they focused on the
change in these variables, Xt − Xt− 1 , rather than the level of these variables,
Xt. Although this technique was successful for univariate Box-Jenkins analy-
sis, there are two problems with this approach in a multivariate scenario.
First, we can only make statements about the changes in the variables rather
than the level of the variables. This will be particularly troubling if our
major interest is the level of the variable. Second, if the variables are subject
to a stochastic trend, then focusing on the changes in the variables will lead
to a specification error in our regressions.
The cointegration technique allows researchers to investigate variables
that share the same stochastic trend and at the same time avoid the spurious
regression problem. Cointegration analysis uses regression analysis to study
the long-run linkages among economic variables and allows us to consider
the short-run adjustments to deviations from the long-run equilibrium.
The use of cointegration in finance has grown significantly. Surveying
this vast literature would take us beyond the scope of this chapter. To narrow