Let’s start our journey with the form of causality most popular with those engaged in time series forecasting. It is named after its developer Clive W.J. Granger, winner of the Nobel prize in economics (along with Robert Engle) in 2003. In his own words:

“The basic ‘Granger Causality’ definition is quite simple. Suppose that we have three terms,

*X*and

_{t}, Y_{t},*W*and that we first attempt to forecast

_{t},*X*+1 using past terms of

_{t}*X*and

_{t}*W*. We then try to forecast

_{t}*X*+1 using past terms of

_{t}*X*and

_{t}, Y_{t},*W*. If the second forecast is found to be more successful, according to standard cost functions, then the past of

_{t}*Y*appears to contain information helping in forecasting

*X*+1 that is not in past

_{t}*X*or

_{t}*W*. … Thus,

_{t}*Y*would ‘Granger cause’

_{t}*X*+1 if (a)

_{t}*Y*occurs before

_{t}*X*+1; and (b) it contains information useful in forecasting

_{t}*X*+1 that is not found in a group of other appropriate variables.

_{t}"Naturally, the larger

*W*is, and the more carefully its contents are selected, the more stringent a criterion

_{t}*Y*is passing. Eventually,

_{t}*Y*might seem to contain unique information about

_{t}*X*+1 that is not found in other variables which is why the 'causality' label is perhaps appropriate."

_{t}Granger’s concept, as applied to time series, essentially says that although the current value of a time series can often be predicted from its own past values, the introduction of a second time series can improve predictive accuracy. This second time series, however, must be related to the first in a particular way. Otherwise, pairs of non-stationary time series can be highly correlated but not causally related. For instance, bread prices in Britain and sea levels in Venice both rise over time and hence are correlated, but they are clearly not causally connected. Enter the concept of cointegration.

In his Nobel lecture Granger explained: “if a pair of series [is] cointegrated then at least one of them must cause the other.” What does it mean for two series to be cointegrated? Here are three ways of picturing cointegration. First, Granger’s own. He compares a time series to a roughly stretched out string of pearls. Suppose, he says, that there were two similar strings of pearls, both laid out (or thrown) on the same table. “Each would represent smooth series but would follow different shapes and have no relationship. The distances between the two sets of pearls would also give a smooth series if you plotted it. However, if the pearls were set in small but strong magnets, it is possible that there would be an attraction between the two chains, and that they would have similar, but not identical, smooth shapes. In that case, the distance between the two sets of pearls would give a stationary series and this would give an example of cointegration.”

Here’s another example of cointegration offered by Thomas Karier in his book

*Intellectual Capital: Forty Years of the Nobel Prize in Economics*(Cambridge University Press, 2010). (I can only assume he doesn’t own an untrained basset hound.) “Suppose a person and a dog are free to wander in any direction and we track their movements. If the person and the dog are unrelated, then there may not be any apparent relationship between the two paths. However, if the dog belongs to the person, then their paths should coincide more frequently and Granger would say that the two paths are cointegrated.” (pp. 270-71)

And finally, from Kevin D. Hoover comes what many would consider the best image for participants in the markets: the randomly-walking drunk and his faithful, sober friend who follows him to make sure he does not hurt himself. “Because he is following the drunk, the friend, viewed in isolation, also appears to follow a random walk, yet his path is not aimless; it is largely predictable, conditional on knowing where the drunk is.”

Pairs trading is often based on cointegration because theoretically this approach guarantees mean reversion in the long run although, as we know, not profits. (Those interested in pursuing this line of thinking might want to start with the Trading with Matlab blog post "Pairs Trading—Cointegration Testing." But, like the untrained basset hound, I’m wandering.)

The question is whether Granger causality is the best we can come up with. What are its flaws? That will be the subject of the next post in this series.

>the randomly-walking drunk

ReplyDeleteThis is very poor analogy independently of untrained bassset hound ownership. The drunk's walk is never random, just unstable.