ECONOMICS 266, Spring, 1997
Bent E. Sørensen
March 1, 2005
The survey by Campbell and Perron (1991) is a very good supplement to this chapter - for fur-
ther study read Watson’s survey for the handbook of econometrics Vol. IV, and for multivariate
models use Johansen’s (1995) book.
Cointegration theory is definitely the innovation in theoretical econometrics that has cre-
ated the most interest among economists in the last decade. The definition in the simple case
of 2 time series xt and yt, that are both integrated of order one (this is abbreviated I(1), and
means that the process contains a unit root), is the following:
xt and yt are said to be cointegrated if there exists a parameter α such that
ut = yt − αxt
is a stationary process.
This turns out to be a pathbreaking way of looking at time series. Why? because it seems that
lots of lots of economic series behaves that way and because this is often predicted by theory.
The first thing to notice is of course that economic series behave like I(1) processes, i.e. they
seem to “drift all over the place”; but the second thing to notice is that they seem to drift in
such a way that the they do not drift away from each other. If you formulate this statistically
you come up with the cointegration model.
The famous paper by Davidson, Hendry, Srba and Yeo (1978), argued heuristically for models
that imposed the “long run” condition that the series modeled should not be allowed to drift
arbitrarily far from each other.
The reason unit roots and cointegration is so important is the following. Consider the re-
yt = α0 + α1xt + ut .
A: Assume that xt is a random walk and that yt is an independent random walk (so that
xt is independent of ys for all s). Then the true value of α1 is of course 0, but the limiting
distribution of α̂1 is such that α̂1 converges to a function of Brownian motions. This is called
a spurious regression, and was first noted by Monte Carlo studies by Granger and Newbold
(1974) and Phillips (1986) analyzed t