A Drunk and Her Dog: An Illustration of Cointegration and Error Correction
Journal article by Michael P. Murray; The American Statistician, Vol. 48, 1994
Journal Article Excerpt
A drunk and her dog: an illustration of cointegration and error correction. by Michael P. Murray 1. INTRODUCTION Teachers of statistics have long used the drunkard's walk to introduce nonstationary processes. Here I adapt the drunkard's walk to clarify the more recently introduced notion of cointegration (Granger 1981) and to concretize the link between cointegration and error-correction models (Engle and Granger 1987). The mathematics of cointegration and error correction are sophisticated, but the concepts themselves are simple enough to allow their introduction at elementary levels as a straightforward extension of the drunkard's walk. The link between cointegration and error correction arises naturally from the humorous tale of the drunk and her dog. 2. THE TALE OF THE DRUNK AND HER DOG The drunk is not the only creature whose behavior follows a random walk. Puppies, too, wander aimlessly when unleashed. Each new scent that crosses the puppy's nose dictates a direction for the pup's next step, with the last scent forgotten as soon as the new one arrives. Thus, the meanderings, [x.sub.t] and [y.sub.t], of both drunks and dogs along the real line can be modeled by the random walk: (1) [x.sub.t] - [x.sub.t-1] = [u.sub.t] and (2) [y.sub.t] - [y.sub.t-1] = [w.sub.t], where [u.sub.t] and [w.sub.t] are stationary white-noise steps that the woman and dog take each period. One key trait of random walks is that the most recently observed value of the variable is the best forecaster of future values. If I come out of a bar with a friend who asks me, "Where is that puppy we saw out here earlier?", I am likely to answer. "Well, he was right over there when I went in." We might have the same exchange about a drunk we saw earlier as well. A second key trait of random walks is that the longer we have been in the bar, the more likely it is that the puppy or the drunk has wandered far from where we last saw them. If my friend and I had been in the bar a long while, I'd say about either the dog or the drunk, "But heaven only knows where they've got to by now." This growing variance in location characterizes the "nonstationarity" of random walks. But what if the dog belongs to the drunk? The drunk sets out from the bar, about to wander aimlessly in random-walk fashion. But periodically she intones "Oliver, where are you?", and Oliver interrupts his aimless wandering to bark. He hears her; she hears him. He thinks. "Oh, I can't let her get too far off; she'll lock me out." She thinks, "Oh, I can't let him get too far off; he'll wake me up in the middle of the night with his barking." Each assesses how far away the other is and moves to partially close that gap. Now neither drunk nor dog follows a random walk; each has added what we formally call an error-correction mechanism to her or his steps. But if one were to follow either the drunk or her dog, one would still find them wandering seemingly aimlessly in the night; as time goes on, the chance that either will have wandered far from the bar grows. The paths of the drunk and the dog are still nonstationary. Significantly, despite the nonstationarity of the paths. one might still say, "If you find her, the dog is unlikely to be very far away." If this is, right, then the distance between the two paths is stationary, and the walks of the woman and her dog are said to be cointegrated of order zero. To understand the phrase cointegrated of order zero, we should first define integrated series. Nonstationary series that become stationary when differenced n times are called integrated of order n. For a set of series to be cointegrated, each member of the set must be integrated of the same order, n; thus the term cointegration. A set of series, all integrated of order n, are said to be cointegrated if and only if some linear combination of the series--with nonzero weights only--is integrated of order less than n. Such a linear combination is called a cointegrating relationship (see Engle and Granger [1987] for more details).Notice that cointegration is a probablistic concept. The dog is not on a leash, which would enforce a fixed distance between the drunk and the dog. The distance between the drunk and the dog is instead a random variable, but a stationary one despite the nonstationarity ... |
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