Cointegration 195
The simplest example of a nonstationary variable is a random walk. A
variable is a random walk if
Xt = LXt + etwhere et is a random error term with mean 0 and standard deviation σ. L is
defined as a lag operator, so that LXt is Xt− 1.^5
It can be shown that the standard deviation σ(Xt) = tσ, where t is time.
Since the standard deviation depends on time, a random walk is nonsta-
tionary.
Nonstationary time series often contain a unit root. The unit root
reflects the coefficient of the Xt− 1 term in an autoregressive relationship of
order one. In higher-order autoregressive models, the condition of nonsta-
tionarity is more complex. Consider the p-order autoregressive model
(1 − a 1 L^1 −... − apLp)Xt = et + a 0 (10.1)
where ai terms are coefficients
Li is the lag operator
If the sum of polynomial coefficients equals 1, then the Xt series are non-
stationary.
If all the variables under consideration are stationary, then there is
no spurious regression problem and the standard OLS estimation method
can be used. If some of the variables are stationary, and some are nonsta-
tionary, then no economically significant relationships exist. Since non-
stationary variables contain a stochastic trend, they will not exhibit any
relationship with the stationary variables that lack this trend. The spurious
regression problem occurs only when all the variables in the system are
nonstationary.
If the variables share a common stochastic trend, we may overcome
the spurious regression problem. In this case, cointegration analysis may be
used to uncover the long-term relationship and the short-term dynamics.
Two or more nonstationary variables are cointegrated if there exists a linear
combination of the variables that is stationary. This suggests cointegrated
variables share long-run links. They may deviate in the short run but are
likely to get back to some sort of equilibrium in the long run. It is impor-
tant to note that, here, the term “equilibrium” is not the same as used in
economics. To economists, equilibrium means the desired amount equals
the actual amount and there is no inherent tendency to change. In contrast,
(^5) Similarly L (^2) Xt = L(LXt) = Xt− 2 and more generally, LpXt = Xt−p, for p = 0,1,2,...