I was re-reading Michael Murray’s explanation of cointegration:
and marvelling at the calculus.
Of course it’s not any subtraction. It’s subtracting a function from a shifted version of itself. Still doesn’t sound like a universal revolution.
(But of course the observation that the lagged first-difference will be zero around an extremum (turning point), along with symbolic formulæ the (infinitesimal) first-differences of a function, made a decent splash.)
Jeff Ryan wrote some R functions that make it easy to first-difference financial time series.
Here’s how to do the first differences of Goldman Sachs’ share price:
require(quantmod) getSymbols("GS") gs <- Ad(GS) plot( gs - lag(gs) )
Look how much more structured the result is! Now all of the numbers are within a fairly narrow band. With
length(gs) I found 1570 observations. Here are 1570 random normals
plot(rnorm(1570, sd=10), type="l") for comparison:
Not perfectly similar, but very close!
Looking at the first differences compared to a Gaussian brings out what’s different between public equity markets and a random walk. What sticks out to me is the vol leaping up aperiodically in the $GS time series.
I think I got even a little closer with drawing the stdev’s from a Poisson process
plot(rnorm(1570, sd=rpois(1570, lambda=5)), type="l")
but I’ll end there with the graphical futzing.
What’s really amazing to me is how much difference a subtraction makes.