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

@article{
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",
}