A paper to illustrate the concept of cointegration, a concept with a wide use in econometrics.

Consider a random walk \(x_t\), which the step is guided by a stationary white-noise

\[x_t - x_{t-1} = w_t\]

Usually, a random walk is modeled as the footstep of a drunk. In the paper, it is assumed that a drunk is walking with a unleashed dog, which both of them follow a random but independent trail

\[x_t - x_{t-1} = w_t \\ y_t - y_{t-1} = u_t\]

Such random walks are non-stationary as \(\lim_{t\to\infty} x_t\) is hard to determine. Now consider the case that there is a correction between the drunk and the dog such that \(\lvert x_t - y_t\rvert\), while random, shall not be too large. Then the walks \(x_t\) and \(y_t\) are said to be cointegrated.

If a non-stationary series would become stationary if it is differentiated for \(n\) times, then the series is said to be “integrated of order \(n\)”. Hence a stationary series is by definition integrated of order zero. The series above are integrated of order one, since

\[x'_t = w_t\]

and the white-noise term \(w_t\) has zero mean.

For a set of series, if there is a linear combination of them with non-zero weights that, the linear combination is integrated of order \(n\), then the set of series is said to be “cointegrated of order \(n\)”. Obviously, a set of series which each of them is integrated of order \(n\) will be cointegrated of order less than \(n\).

Consider the random walks above, it is better to be modelled as

\[x_t - x_{t-1} = w_t - c(x_{t-1} - y_{t-1}) \\ y_t - y_{t-1} = u_t + d(x_{t-1} - y_{t-1})\]

Where the second term on the right is the error-correcting term. If the error-correcting term is non-stationary, the series are non-stationary too (consider that a stationary series must be integrated of order zero). But if \(c\) and \(d\) above are positive constants, however small are they, \(x_t-y_t\) will stablise around zero.

Bibliographic data

   author = "Michael P. Murray",
   title = "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction",
   journal = "The American Statistician",
   volume = "48",
   number = "1",
   pages = "37--39",
   month = "Feb",
   year = "1994",